Differential and Integral Inequalities: Functional Partial, Abstract and Complex Differential Equations v. 2: Theory and Applications: Functional

=

p/2,

m(tl) = p ,

and

m(t)

> p/2,

t E [t., tJ.

7.4. PERTURBED

63

SYSTEMS

We see from these relations that D+m(t,) >, 0.

(7.4.3)

On the other hand, it follows, from (ii), that IIYt(t0 ,do)llo


for

to

< t < tl

*

Let 4 = yt,(to,do), and, because of uniqueness, Ytz+h(to> do)

Thus,

D+"(t2) = lim SUP h-l[V(t, h-O+

= lim SUP h-l[V(t, h 4 +

+ Vt2 +k

= Ytz+h(t27

h 3 0.

41,

+ h, Ytz+h(to,do>> + h, Yt,+h(h

V ( t , Yt,(to > do))]

-

-

I d > )

!

V(t2

+ h,

,4 4 - V t 2 +)I,

Xt,+h(f,

9

Xt,+h@,

,4))

+

where ~ ( t,4) , is the solution of (7.1.1) with an initial function at t = t, . Using the Lipschitzian character of V , the assumption (7.4.2), and the preceding relation, we obtain (7.4.4)

D+m(t2) G K.111d 110 - ""(t2).

Since q > 0 is sufficiently small, there exists a y > 0 such that Kq < a - y , and, hence, the fact that 11 (I, V ( t ,d), together with the inequality (7.4.4), implies that

<

+ K.11

< &!)[-a < -Ym(tz)

Df"(t2)

< 0,

since m ( t 2 )= p / 2 > 0. This contradicts (7.4.3) and proves that m(t) < p , t 3 to . Consequently, by ii, it follows that

II YdtO > do)llo <

P2

whenever 11 c $ ~ (lo < p/2K. Thus, setting before, we arrive at the inequality

D+v(4Y d t o

9

do))

3 to

t

4

9

and arguing as

= y l ( t o ,+o)

< -Y w, Yt(fo do)), >

and therefore, by Theorem 1.4.1,

w, Y&O

>

do))

<w

o

9

do) exP[- Y ( t

-

toll,

t

2 to *

64

7

CHAPTER

Because of ii, from this inequality results a further inequality

proving the stated result.

THEOREM 7.4.2. Assume that the trivial solution of (7.1.1) is generalized exponentially asymptotically stable and thatf(t, 4) is linear in 4. Suppose that g E C [ J x R, , R,] and g(t, 0) = 0, g ( t , u) is nondecreasing in u for each t E J , and

/I R ( t , +)I1

< At, I1 d Ilo),

t

E

4E c, -

f,

(7.4.5)

If y(to , +o) be the solution of (7.4.1) such that uo = K(t,)j/#JO /IO

I1YdtO > 4 O ) l l O G

r ( t , to

>

UO),

t

3 to

, then (7.4.6)

9

where r ( t , t o , uo) is the maximal solution of 24’

existing for t ug

< p.

=

-p’(t).

+ K(t)g(t,

U),

3 to and satisfying

u(t,) = U o

r(t, t o ,uO)< p, t

,

(7.4.7)

to , whenever

Proof. Lety(to ,#Jo) be any solution of (7.4.1) such that /I & ( ( o Setting #J = yt(t, , #Jo), we have Yt+h(tO >

d o ) = Yt+*(t,

d),

h

< p/K(to).

2 0,

because of uniqueness of solutions. Suppose now that ~ ~ + #J), ~ (h t2, 0, is the solution of (7.1.1) through ( t ,4). If 11 yl(to, $0)110 < p, t 3 to , we should have

Thus, since #J

= yl(t,

,#JO), it follows that

7.4.

65

PERTURBED SYSTEMS

By Theorem 1.4.1, it now results that

V t ,Y t ( t 0 4 0 ) ) < r ( t , t o uo), t b t o choosing uo = K(to)/I$, I(o , r(t, to , uo) being the maximal solution of 7

7

9

(7.4.7). Furthermore, from (ii) in the proof of Theorem 7.4.1,

II Yt(t0 > 4o)llo

e v, Yt(t0

9

$0)).

Also, by assumption, r(t, t o ,uo) < p, whenever uo < p. Our choices II do I10 < p / K ( t o ) and uo = W t o ) II $0 I10 imply that a0 < P. Hence,

I/ Yt(t0

9

4o)llo

< P,

t

2 to .

Thus, the estimate (7.4.6) holds.

THEOREM 7.4.3.

Under the assumptions of Theorem 7.4.2, the stability properties of the trivial solution of the scalar differential equation (7.4.7) imply the corresponding stability properties of the trivial solution of the perturbed system (7.4.1). T h e proof of this theorem is immediate from the relation (7.4.6). However, the following special cases are of importance. COROLLARY 7.4.1. T h e function g ( t , u ) = h ( t ) u, where X E C [ J ,R,], is admissible in Theorem 7.4.3, provided there exists a continuous function q(t) > 0, t E J , such that exp

[p(ro)- p ( t >

+

Jt to

W) ~

s ds] )

< q(to),

t

> to.

(7.4.8)

Proof. It is enough to show that, under the assumptions of the corollary, the trivial solution u = 0 of (7.4.7) is equistabIe. For, the general solution u(t, t o , uo)of u' = -p'(t)u

+ K(t)X(t)u,

is given by u(t, t o , uo) = uo exp

[p(to)- p ( t )

u(to) = uo ,

+ Jt W W ds] to

9

t

2 to

9

and, hence, equistability follows from (7.4.8). COROLLARY 7.4.2. T h e functions g(t, a ) = h(t)u, h E C [ J , R,], p ( t ) = cut, 01 > 0, and K ( t ) = K > 0 are admissible in Theorem 7.4.3, provided (7.4.9) lim sup ( t - to)-l ~ ( s )dsl < c u j ~ t-w

[

jt

to

66

CHAPTER

Proof.

7

I n this case, the general solution u(t, to , uo) of (7.4.7) is given by

and, therefore, the condition (7.4.9) shows that the trivial solution of (7.4.7) is uniformly asymptotically stable. Hence, by Theorem 7.4.3, the trivial solution of (7.4.1) is also uniformly asymptotically stable. T h e foregoing results can be extended to the case when f ( t , $ ) is nonlinear, on the basis of Theorem 7.2.2. Furthermore, as in the case of ordinary differential equations, we can show that, if the trivial solution of the unperturbed system is uniformly asymptotically stable, then it has certain stability properties under different classes of perturbations. For example, the concepts of total stability may be formulated parallel to the definitions of Sect. 3.8, and corresponding results may be proved. Likewise, boundedness, Lagrange stability, integral stability, and partial stability can be discussed. We shall omit such results as exercises to the reader. I n the following sections, we shall only concentrate on extreme and perfect stability criteria and existence of almost periodic solutions.

7.5. Extreme stability Associated with the system (7.1.1), let us consider the product system x’ == f ( t ,

4

(7.5.1)

Y’= f ( 4 Yt).

7

DEFINITION 7.5.1. T h e system (7.1.1) is said to be extremely unqormly stable if, for every E > 0, to E J , there exists a S(E) > 0 such that implies

I/ do - $0

I10

<S(4

I1 XdtO do) - Y4tO > $o)llo < 7

€9

t 3

t0

;

extremely quasi-uniform asymptotically stable if, for every and to E J , there exists a T ( E7) , > 0 such that

implies

I/ 4 0 II X 4 t O , 4 0 1 - YdtO

9

- $0

#o)llo

/lo

<

€9

< ?1 t 3 to

E

> 0, 7) > 0,

+ w,7).

If the preceding two concepts hold simultaneously for $o , +o E 5 P , we shall say that the system (7.1.1) is extremely unqormly completely stable.

7.5.

67

EXTREME STABILITY

T h e following theorem provides necessary conditions for the system

(7.1 -1) to be extremely uniformly completely stable.

THEOREM 7.5.1. Assume that (i) f~ C [ J x %%,R"],and, for every llf(4

01

> 0, if II 4 1l0 <

< L(4114

4) - f ( 4 #)I1

-

*

110

<

II $ 110

01,

01,

;

(ii) the solutions of (7.1 . l ) are uniformly bounded; (iii) the system (7.1.1) is extremely uniformly completely stable. Then, there exists a functional V(t,4, $) satisfying the following conditions:

(1") V E C[J x

I V t ,dl ,$1)

x gfl,R+],and V(t,4, $) satisfies

%n -

V t ,42

< W?"I 4 1

42)l

9

- $2 110

+ II

$1 - *z 1101,

for di, lJli E C , , (i = 1,2), t 3 0, where M ( q ) is a positive continuous function; (2") there exist functions a, b E X such that

4d(3") D'V(4

4, + ) ( 7 . 5 . 1 )

=

*

< w #>4 < 414

110)

+ h,

lim sup h - l [ V h-Of

Xt+h(4

-

+

110);

41,Y t + h ( 4 *)

W d ,41

-

< -V(4 4, $1. Let 7 be an arbitrary nonnegative number. Consider the case C,, . Then, corresponding to each E > 0, to E J , there exists a T ( E9) , > 0 such that, if t >, to T ( E.I,), then

Proof.

do,$o

E

+

II %(to do) 9

-

We assume that, if E > 1, T(E,q) ness of solutions, it follows that

II %(to for all t

7

d0)llO

Yt(4l =


t

< E.

*o)llo

T(1,q). By the uniform bounded-

II Yt@O

?

~0)Ilo

< Y(71)

2 to . Furthermore, if 11 4 \lo < y(q), 11 $ /lo < y(v), Ilf(4 4)- f ( 4 9911 < L(Y(?))ll d - 9 110

we have

We can assume that T (Eq), , y(y), and L ( y ( 7 ) ) are continuous. Moreover, if we consider a positive function A(€,7) for E > 0 and 7 3 0, such that

4% 71) = exP[{w4?))

+ 11 %, ?)I + 2 Y ( d exp[%

d

1

9

(7.54

68

CHAPTER

7

there exist two continuous functions g ( c ) and M ( 7 ) such that for E > 0, g(0) = 0, M ( 7 ) > 0, and

d.1

A(%7) G

g(E)

Wd-

>0

(7.5.3)

We now define V J t ,4,$) as follows:

vdt, 4,$1 = g ( W 0SUP Gk(ll X t + o ( 4 4)- Yt+& 30 for k

=

(7.5.4)

$)llo)eo,

1 , 2, 3,..., where u Gk(U)

u 3 K-1,

- k-1,

0

10,


(7.5.5)

K-1.

I t is easy to see that

by (7.5.3). Moreover, fort& , PoE %%' , there exists a p

I1 X t ( h 4,) 9

-

Ydt,

>

c0)Ilo

< B(Ild0

- $0

Ilo),

EX

t >to

such that I

because the system (7.1.1) is extremely uniformly stable. We thus have

c, $1 < g ( W Gk(B(II c exp[W-l, < P(ll c $ M(lld - $

V7C(t,

- $110))

-

110)

+,

II d

~

$ Il0)l

(7.5.9)

110).

We shall next show that Vk(t, $) satisfies a Lipschitz condition with respect to 4 and $. For any 4, , $, E C, ,

7.5.

EXTREME STABILITY

69

By (7.5.10) and the continuity of x t ( t 0 , c $ ~ in ) t , it follows that V k ( t ,4, $) is continuous in ( t , 4,$). Let us now consider D+V,(t, 4,$) with respect to the product system (7.5.1). Since

T h e desired Lyapunov functional may now be defined by

I n view of (7.5.8), this V ( t ,4,$) can be defined for all t E J , 4,$ E P. From (7.5.7) and (7.5.91, it is easy to see that there exist functions a, b E 3-satisfying (2"). Furthermore, because of the inequality

we obtain, for 4,41 , #, t,bl E C, ,

which proves that V ( t ,4,$) satisfies the Lipschitz condition as described in (1'). Finally, we shall show that (3")is satisfied. By the definition,

70

CHAPTER

7

if h is small enough,

<

D+r(t, 9,

$)(7.5.1)

N (2k))-1

li

k=l

-

vk(t,

9,

and therefore, by (7.5.1 I),

As N is arbitrary, this implies

T h e proof is complete. Sufficient conditions for the extreme complete stability of the system (7.1.1) are given by the following result.

THEOREM 7.5.2.

Assume that

(i) j ' C~[ J x q n , R n ] ,and, for every +o E Vn,the solutions x ( t 0 ,$o) exist in the future; (ii) V E C [ J x V n x Vn,R,],and, for t 2 to , D'Vt,

XdtO

t

(iii) g E C [ J x R,

do), YdtO 7

40))

< g(t, w ,%(to

, R], and g ( t , 0) = 0;

7

do), yt(t0 ,$0)));

7.5.

71

EXTREME STABILITY

(iv) there exist functions a, b E X on the interval [0, co) such that, for t 0, +, E Vn,

+

w, 4, 1cI) < 411d

b(lld - 9 110) G

-

*

110).

Then, if the trivial solution of (6.1.2) is uniformly completely stable, the system (7.1.1) is extremely uniformly completely stable.

Proof. Assume that the trivial solution of (6.1.2) is uniformly completely stable. This means that (S,*) and (S,*) hold at the same time. If E > 0, to E J are given, there exists a 6 = 6 ( ~ > ) 0 such that uo 6 implies

<

u(t, t o , uo)

<

w,

3 to.

t

Choose 6, = ~ ' ( 8 ) and uo = V(t,, + o , +,). yields, because of conditions (i) and (ii),

q t , %(to > do), Ydt ,

9

$0))

< r(t, t o

?

Then, Theorem 1.4.1 t b to

uo),

9

where r(t, to , uo) is the maximal solution of (6.1.2). We therefore obtain b(ll %(to

do)

?

YdtO *o)llo)

< b(4,

Yt@O ? 1cIo)llo

< E?

9

~

t

3 to

9

which implies that

I1 %(to

?

40)

~

3 to

t

3

<

provided 11 +o - +o \lo 6,. This proves extreme uniform stability of the system (7.1.1). a. Let now 01 > 0, E > 0, and to E J be given. Suppose I/ 4, - +o 1l 0 Let 01, = a(a). Since (&*) holds, given 01, > 0, b ( ~ > ) 0, and to E J , there exists a positive number T = T(to, 01, E) such that, if uo a , ,

<

u(t, t o , uo)

< bk),

t

t to

<

+ T.

As previously, it results that

4

%(to

do)

9

~

Y4tO > 1cIo)llo)

and this shows that, whenever

I1 %(to

7

90) -

< b(+

/I +, - +, 1l 0

YdtO ,+o)llo

< E?

t 3 to

<

01,

t

+ T,

we have

3 to + T.

I t therefore follows that the system (7.1.1) is extremely uniformly completely stable. COROLLARY 7.5.1. in Theorem 7.5.2.

T h e function g(t, u) = -01u,

01

> 0,

is admissible

12

7

CHAPTER

7.6. Almost periodic systems We shall continue to consider the functional differential system (7.I. 1). For the purpose of this section, however, we take f €

C[(-Co,

03)

x

c,, R"].

All the results that follow are extensions of the results of Sect. 3.18 to functional differential systems.

DEFINITION 7.6.1. A functional f E C[(-co, CO) x C, , Rn]is said to be almost periodic in t unqormly with respect to q5 E S for any compact set S C C, if, given any q > 0, it is possible to find an Z(q) such that, in any interval of length [(q),there is a T such that the inequality

Ilf(t + 4 ) -f(4 4111 < 77 is satisfied for t E (- GO, co), q5 E S. 7 7

We shall first prove a uniqueness result.

THEOREM 7.6.1.

Assume that

(i) f~ C[(-co, co) x C,, , Rn] and f ( t , q5) is almost periodic in t uniformly with respect to E S, S being any compact set in C, ;

+

(ii) I/ E C [ J x C, x C, , R,], V(t, q5, q5) Lipschitz condition in q5, $ for a constant M b(l/ 4

-

# 110)

< q t , 4, $1,

= 0, V ( t ,q5, $) satisfies = M ( p ) > 0, and

a

b E .f;

(iii) g E C [ J x R, , R],g(t, 0) = 0, and, for t 3 0,

n+V t ,4 , 4)< g(4 V ( t ,4, #I), where D+V(t, +, 9) is defined with respect to the product system x' = f ( t , X t ) , y' = f ( t , Y t ) ; (iv) the maximal solution of (6.1.2), through the point (T,,, 0), T,, 0, is identically zero. Then, there exists a unique solution of the almost periodic system (7.1 . l ) , to the right of t,, E (- co, a).

Proof. Sincef(t, +) is continuous, there exists at least one solution for a given to E (- co, co) and a q5,, E C, . Suppose that, for some to E (- co,co) t C, , there exist two solutions ~ ( t, , and and y(t,, ,+,) of (7.1.1). Then, at some t, > t o , we should have

II %,(to

* 40)

-

Yt,(t,

>

4o)lIo

=

€9

(7.6.1)

7.6.

where we may assume E < p. For to constant p1 < p such that

II -%(to

9

73

ALMOST PERIODIC SYSTEMS

d0)llo

< P1

9

< t < t, , there exists a positive I/ Yt@o

7

4o)llo

< P1

*

These solutions are uniformly continuous functions and bounded by p1 on the interval to - T t t , , and, hence, there exists a compact set S C C, such that

< <

%(to

6

9

do) E s,

Yt(4J,do) E s

for

t E [ t o , tll.

By Lemma 1.3.1, given b ( ~ ) /and 2 a compact set 6(~> ) 0 such that

, TI, there is a

[ T ~

=

r ( t , 7 0 , 0,s)

<w / 2 ,

t

E [To

(7.6.2)

, TI,

where r(t, T~ , 0, 6) is the maximal solution of U' = g(t, U)

+ 6,

.(To)

=

0.

Let 8 be a 6/2M translation number for f ( t , 4) such that to + 8 3 0, that is, (7.6.3) Ilf(t 8,d) -f(4 4111 < V M , t E (-00, a),

+

provided

belongs to a compact set S C C, . We consider the function m(t) = v t

Then,

+

+ 8, %(to

+

f

do), Y&o > do)),

t

E

[ t o 7 tll.

where x * ( t 8, xt), y*(t 8, y t ) are the solutions of (7.1.1) such that 4++o(t 8, x t ) = X f = X d t , > 4 0 ) and YTfdt 8, rt) = Y t = Yt(& , 4 0 ) , respectively. We thus have

+

+

74

CHAPTER

because of (7.6.3). Defining inequality m(t)

T,,

=

< r(t + 6,

to

70

7

+ 8, we get, by Theorem 1.4.1, the

, 0, a),

t E [ t o , tll, (7.6.4)

Assumption (ii) and (7.6.1), on the other hand, lead to 4tl)

2 w,

contradicting (7.6.4). It therefore follows that there is a unique solution for the system (7.1.1) to the right of to . COROLLARY 7.6.1. 7.6.1.

T h e function g(t, u) = 0 is admissible in Theorem

DEFINITION7.6.2. If, for any p that l l f ( t , #)]I < M(p), whenever bounded.

> 0, there exists an M ( p ) > 0 such # E C, , we shall say that f ( t , 4) is

The next theorem gives the sufficient conditions for perfect stability criteria of the trivial solution of (7.1.1).

THEOREM 7.6.2. L

Suppose that

(i) V E C [ J x C, , R,],V ( t ,#) is Lipschitzian in > 0, and, for ( t ,#) E J x C, ,

# for a constant

= L(p)

b(ll9 110)

< q t , 41,

(ii) g E C [ J x R,, R],g(t,0)

b E .f;

(7.6.5)

= 0, and, for ( t ,4) E J x C, ,

D+V(t>9)(7.1.1) G A4 V ( t ,4));

(iii) f E C [ ( - CO, CO) x C, , R"],f ( t , 0) = 0, f ( t , #) is bounded, and 4)is almost periodic in t uniformly with respect to 4 E S, S being any compact set in C, .

f(t,

Then, the null solution of (7.1.1) is (I") perfectly equistable if the trivial solution of (6.1.2) is strongly equistable; (2") perfectly uniform stable if the trivial solution of (6.1.2) is strongly uniform stable;

7.6.

75

ALMOST PERIODIC SYSTEMS

(3") perfectly equi-asymptotically stable if the trivial solution of (6.1.2) is strongly equi-asymptotically stable; (4") perfectly u n i f o r d y asymptotically stable if the trivial solution of (6.1.2) is strongly uniformly asymptotically stable. We shall prove only the statement corresponding to (4"). Let us suppose that the trivial solution of (6.1.2) is strongly uniformly asymptotically stable. Let 0 i E < p and to E (- 03, 03) be given. Then, given b ( ~> ) 0, T,,E J , and any compact interval K = [T,,, t*], there exist an 7 = q ( ~> ) 0 and a 6 = S(T,, E ) > 0 such that Proof.

u(t, 70 uo I 7)

< b(t-),

t s[To

7

t*l,

(7.6.6)

whenever u,,< 6, where u(t, T,,,u o , q) is any solution of (3.18.6). Choose LS, = 6 and u,, = L 11 +o I/, , L being the Lipschitz constant for V(t,$). Consider a solution x(t0 ,#,,) of (7.1.1) such that to E (- 03, a), I/ +o I/, < 6, . Suppose that, at some t , we have

/I X d t , > 9o)llo

=

6.

Then, there exist t , and t, such that to < t , II x&o , +o)llo = and that

< t , , (1 x t , ( t , , +,,)\lo

=

a,,

€9

81

< II %(to 9o)llo < 6, 7

tl

< t < t2

*

Clearly, there exists a compact set S C C, such that xt(t, , +,) E S for t t, . Let 0 be an q/L translation number off ( t , +) for E S such that to 0 3 0, that is, to

< < +

Consider the function

Then we have

+

76

CHAPTER

where xl*,,(t

x *( t

+ 8, x l ( t , , &))

is a

7

(7.1.1) such

solution of

+ 0, x t ( t , , do)) = xt(t, , do).It, therefore, follows that

Since x , ( t o , +,J E S for to D+m(t)

that

< t < t, , we obtain, using

< g(t + 8, m ( t ) ) + 7,

t

E

[to,

bl,

and hence, by Theorem 1.4.1,

< r(t + 8, *, 4,

m(t>

letting T~ : t, of (3.18.6). At t

+ 0, =

h(6) s; q t ,

70,

t

9

E

+

(7.6.8)

[ t o , 421,

where r(t 0, T ~ a,, , 7) is the maximal solution t, , we are led to an absurdity

+ 8, %,(to

7

40))

< r ( t , + 8,

70

, *o , 7 )

< 44,

in view of relations (7.6.5), (7.6.6), and (7.6.8). Thus, the perfect uniform stability of the trivial solution of (7.1.1) is proved. By assumption, given b ( c ) > 0, T , >, 0, there exist positive numbers S o , 7 = T ( E ) and T = T ( E )such that u(t, 7 0 , * o , ~ )< b(c),

3 70

t

+ T,

(7.6.9)

whenever uo 6 6 , . Choose u, = L (1 4, )lo and L8, = 8, , and let 6,* min[S,, So], where 8, = 6(p). Suppose now that x(t, ,4,)is any solution of (7.1.1) such that 11 4,1, So*, t, E (-a, a). Since [ / x l ( t , ,+o) l o ,< p for all t 2 to a n d f ( t , 4)is bounded and consequently ~ ' ( t , +,)(t) , is bounded by some constant, there exists a compact set S such that xt(t,, E S for all t 3 t, . As before, let 6 be an v/l-translation number of f ( t ,+) so that (7.6.7) is satisfied. Considering the function

<

:

m(t) = F(t

+ 8, ",(to ,$J~))

for

t

3 to ,

it is easy to obtain, as previously, m(t)

-= 1-(t + 8,

70,

*o

, 7),

t

2 to *

This, together with (7.6.5) and (7.6.9), yields

411%(to ,4o)llo)

< m ( t ) G r(t + 8,

70

, *o

,v) < b ( 4 ,

7.6.

for t

77

ALMOST PERIODIC SYSTEMS

+ 6 3 + T , and, consequently, T~

<

whenever 11 +o So*. This completely establishes (4"). COROLLARY 7.6.2. T h e function g(t, u) = -mu, N > 0, is admissible in Theorem 7.6.2 to yield perfect uniform asymptotic stability of the trivial solution of (7.1.1). As remarked in Sect. 3.18, if the functional f ( t ,+) is not almost periodic and f E C [ J x C, ,R"],then, from the strong stability properties of the trivial solution of (6.1.2), we may deduce strong stability properties of the trivial solution of (7.1. l), on the basis of Theorem 7.6.2. Finally, the following theorem assures the existence of an almost periodic solution.

THEOREM 7.6.3.

Suppose that

(i) V E C [ J x C, x C, , R,],V ( t ,+, +) is Lipschitzian in for a constant L = L ( p ) > 0, and, for t E J , +, E C, ,

+

b(II 4 - 1cI IIo)

< v(t,4,$) < .(I1 4

-

(ii) g E C [ J x R, , R ] ,and, for t E J , +, t,h

D+q4 4,$1

a , b E S;

$ llo), E

+ and + (7.6.10)

C, ,

< g(t, U t ,4,$1);

(iii) f E C [ ( - co, 00) x C, , R n ] , f ( t +) , is bounded, almost periodic in t uniformly with respect to E S, S being any compact subset in C, , and f ( t ,+) is smooth enough to ensure the existence and uniqueness of solutions of (7.1.1); (iv) for any b(c) > 0, CY > 0, and 5, E I , there exist positive numbers 7 = q ( c ) , T = T ( E CY) , such that, if uo CY and 5 3 5, T,

+

<

4 5 , 50

9

+

(7.6.11)

uo ,.I)< b(c),

where u( 5 , to, uo , 7)is any solution of u' = g(5, ).

+ q,

450)

=%,

5, 2 0;

(7.6.1 2)

(v) there exists a solution x(to, +,J of (7.1.1) such that

II %(to

,40)llo

< B < P,

t

2 to >

to E (-a, 00).

Then, (7.1.1) admits a bounded almost periodic solution, with a bound B.

78

CHAPTER

7

Proof. T h e proof runs, naturally, parallel to the proof of Theorem 3.18.5. Hence, we shall only indicate necessary changes. Let %(to, + o ) be the solution of (7.1.1) such that 11 xt(t,, , 40)ilo B. Since f ( t , 4) is assumed to be bounded, we have, consequently, that (1 %'(to,40)(t)\lis bounded for a constant B, , for t 3 t o . Let S be the compact subset of C, consisting of functions that are bounded by B and are Lipschitzian for a constant B, . Lct { T ~ be } any sequence such that rk -+ co as k -+ a3 and

<

uniformly for t E (- co, co), 4E S. Let /3 be any number, and let U be any compact subset of [p, co). Let 0 < E < p, and choose 01 = a(2B). Then, let 7 and T be the numbers defined in assumption (iv), for this choice. Let k, = k,(P) be the smallest value of R such that

Choose an integer n,

=

llf(t

+

no(€,p) Tkl >

2 k, so large that, for k2 3 k, 2 no ,

4) - f ( t

+

T/c2>

4111 < 7 / 3 4

(7.6.13)

for all t E (- GO, co), q5 E S. Let 8 be an y/3Ltranslation number for f ( t , 6) such that t, 8 >, 0, that is,

+

a), 4 E S. for t E (-a, Consider the function, for t

where t ,

=

where x*(t

t

+

3 to,

and x t = xt(to, +o). Then,

T~~ - T ~ ,

+ 8, xt),y*(t + 0, xt,) are the solutions of (7.1.1) such that

7.6.

79

ALMOST PERIODIC SYSTEMS

+

+ 6, x t ) = x t ,y&(t 6, xt,) = x t , , respectively. Thus, in view of the Lipschitzian character of V ( t ,4, $) and assumption (ii), we get

x&(t

o+m(t) < g ( t + e, fl(t>>+

+ /I

lim sup h4’

< g(t + 0, 4 t ) ) + L[ll %’(to

+ II

X’(t0

9

+ 6,

< g(t + 8, d t > + L[llf(t, Tkz

+

T7cl

Xt,)(t

-f(4

+ e,

Xt)l10

+ 6, %)(t + 6)ll

+ e)lll

+ e,xt)ll + Ilf(4

-f(t

XtJ

?

do)(t>- x*’(t

7

+o)(G) - Y*’(t

+ llf(t + - f ( t + 6, %,)Il .

X t + h - X?+-ts+h(t

+ 6, ~t,>iioI

- Y?+e+h(t

Xtl+h

h-’[l l

XtJ

Xt,)

Since t rkl 2 to + T , for t E U , we obtain, using the relations (7.6.13) and (7.6.14), D+m(t

+ < g(t + + 0, m(t + Tkl

.k,>

TkJ)

+ 17,

which implies, by Theorem 1.4.1, if uo = m(to), m(t

+

< r(t +

Tkl)

Tkl

+ 6, + 6, uo, 4, to

where r(4, t o ,u,, ,7)is the maximal solution of (7.6.12). By assumption (iv), it follows that r(6, t o ,

But,

for

all

UO,

t E U, t

5 = t + Tk, + 6, to= to m(t

17)

+

< &)

> to + + T .

+ 0, we get +

t 3 t o + T.

if

T ~ ,

Tkl)

Consequently, for all t E U , k,

II Xt+.rkl

Hence,

T

< b(€),

t

E

identifying

u.

> k, 3 no , we have, in view of (7.6.10), - X t + T k Z /lo

< E,

which, in turn, leads to the inequality

II 4%> do)(t

+

T k J ~X(t0

9

+o>(t

+ %z)lI <

t

€9

u.

This proves the existence of a function w ( t ) defined on [/3, co) and bounded by B. Since is arbitrary, w(t) is defined for t E (-00, a), and we have x(t0 ,do)(t

+

7Rl)

-~

-

( t )0

uniformly on all compact subsets of (-

00,

as

a).

4

-

m,

80

CHAPTER

7

Following closely the rest of the proof of Theorem 3.18.5, we can show that w ( t ) satisfies (7.1 . l ) and is almost periodic. This completes the proof. COROLLARY 7.6.3. If, in addition to the hypothesis of Theorem 7.6.3, the trivial solution of (6. I .2) is strongly uniformly asymptotically stable, then the system (7.1.1) admits an almost periodic solution that is perfectly uniformly asymptotically stable. I n particular, g(t, u ) = --NU, 01 > 0, is admissible.

7.7. Notes T h e results of Sect. 7.1 are adapted from the work of Driver [3]. See also Halanay [22] and Krasovskii [5]. Theorem 7.1.4 is new. Theorems 7.2.1 and 7.2.2 are taken from Hale [l]. See also Yoshizawa [3]. Theorems 7.2.3 and 7.2.4 are new. Theorems 7.2.5 and 7.2.6 are based on Halanay [221T h e results on autonomous systems in Sect. 7.3 are taken from the work of Hale [S], which may also be referred to for a number of illustrative examples. For the results on perturbed systems of Sect. 7.4, see Corduneanu [2], Halanay [22], and Hale [l]. Theorem 7.5.1 is due to Yoshizawa [I], whereas Theorem 7.5.2 is new. Section 7.6 contains the work of Lakshmikantham and Leela [3]. See also Hale [6] and Yoshizawa [2, 31. For closely related results, see Driver [3], Halanay [22], Hale [5], J. Kato [l], Krasovskii [5], Lakshmikantham and Leela [2], Liberman [I], Miller [l], Razumikhin [2, 61, Reklishkii [l-51, Seifert [I], Sugiyama [S], and Yoshizawa [3].

Chapter 8

8.0. Introduction I n what follows, we wish to treat the solutions of the functional differential system (7.1.1) as elements of euclidean space for all future time except at the initial moment. Our main tool, in this chapter, is therefore a Lyapunov function instead of a functional. T h e derivative of a Lyapunov function with respect to the functional differential system will be a functional, which may be estimated either by means of a function or a functional. While estimating the derivative of the Lyapunov function in terms of a function, a basic question is to select a minimal class of functions for which this can be done. Thus, by using the theory of ordinary differential inequalities and choosing the minimal sets of functions suitably, several results are obtained. If, on the other hand, the estimation of the derivative of the Lyapunov function by means of a functional is considered, the selection of a minimal set of functions is unnecessary. Nevertheless, this technique crucially depends on the notion of maximal solution for functional differential equations and the theory of functional differential inequalities. This method also offers a unified approach, analogous to the use of general comparison principle in ordinary differential equations. Moreover, it is important to note that the knowledge of solutions is not demanded in either case.

8.1. Basic comparison theorems

+

Let V E C[[-T, co) x S o , R+], and let E C,, . We define D+V(t,+(O), +), D-V(t, #(O), +) with respect to the functional differential system (7.1.1) as follows:

+ h, 4(0) + w, 4)) lim inf W V ( t + h, 4(0) + hf(t, 4))

D+V(t,4(0),4) = lim SUP h - V ( t

D-V(t, +(0),4) =

~

h-O+

-

h-0-

81

V ( 44(0))1, V ( t ,d(0))l.

(8.1.1)

82

8

CHAPTER

We need, subsequently, the following subsets of qn,defined by Q, =

[+E c, : I

Qo = [+ E

and Q,

=

v,lo = w, +(ON,

t E J1,

(8.1.2)

c, : v(t + s, W)) < W V ,+(O))),

[d E c, : I VtA, lo =

where A(t) > 0 is continuous on

+

w d(O)A(t)),t

t E 11,

(8.1.3)

J1,

(8.1.4)

E

a),

[-7,

(i) I Vt lo = sup,-<,Go V(t s,4(s)); (ii) L(u) is continuous on R , , nondecreasing in u, and L(u) u > 0; and (iii) I VtAt ( 0 = sup V(t s, +(s))A(t s).

+

-T<s
+

> u, for (8.1.5)

We now state a few fundamental comparison results.

THEOREM 8.1.1.

Let V E C[[-T, co) x So , R,] and V(t,3) be locally Lipschitzian in x. Assume that the functional D-V(t, +(O), +), defined by (8.1. I), verifies the inequality

o-v(t,+(O), 4) < g(t, l’(t, +(O))),

t

> to

9

4E Q,,

(8.1.6)

where g E C[J x R , , R,], and r ( t , to , uo) is the maximal solution of the 0. Let scalar differential equation (6.1.2), existing to the right of to %(to,+o) be any solution of (7.1.1) defined in the future, satisfying vto

SUP

--7<s
Then,

v,%(to

9

90)(t))

7

+0(4>

< r(t, t o

>

< uo uo),

(8.1.7)

t 2 to

-

(8.1.8)

Proof. Let %(to, +o) be any solution of (7.1.1) with an initial function E C, at t = to . Define the function

49 For

E

=

V(t,

4 t O 1 #O)(t)).

> 0 sufficiently small, consider the differential equation u‘

= g(t, u)

+

E,

u p o ) = uo 2 0,

(8.1.9)

whose solutions u(t, E) = u(t, to , uo , E ) exist as far as ~ ( t o, , uo) exists,

8.1.

83

BASIC COMPARISON THEOREMS

to the right of t o . Since lim u(t, e )

= r(t, to ,uo),

c-0

the truth of the desired inequality (8.1.8) is immediate, if we can establish that t 2 to. m(t) < u(t, E ) , Supposing that this is not true and proceeding as in the proof of Theorem 6.3.3, we can see that there exists a t, > to such that

<

< <

(i) m ( t ) u(t, E ) , t o t t, ; (ii) m(tl) = ~ ( t, E, ) , t = t , . From (i) and (ii), we get the inequality D-m(t1) 3 u p , , .)

= g(t1 , U ( t ,

4 ) + 6-

(8.1.10)

Sinceg(t, u) + E is positive, the solutions u(t, E ) are monotonic increasing in t , and therefore, by (i) and (ii),

I mtl Setting that

#

= xt,(t0 ,$,)

10 = m(t1) =

4 4 4. 9

and noting that #(O)

= x(t,

,~ ) ~ ) ( t , it ) , follows

This means that q5 E L?,, and, consequently, using the Lipschitzian character of V(t,x) in x and the relation (8.1.6), we obtain, after simple computation, the inequality D-m(t1)

< g(t1

>

4tl)).

Th i s is incompatible with (8.1.10), on account of (ii). It therefore follows that (8.1.8) is true, and the proof is complete. T h e following corollary is a useful tool in itself in certain situations.

COROLLARY 8.1.1. Let V E C [ [ - T , 00) x S , , R,] and locally Lipschitzian in x. Assume that, for t > to ,q5 E .R, , D-V(t, 4(0),4)

V(t, x) be

< 0.

Let x(t, , be any solution of (7.1.1) such that % ( t o +,)(t) , E So for t E [to , t,] C J. Then,

84

CHAPTER

8

Proceeding as in Theorem 8.1.1 with g = 0, we arrive at the inequality Proof.

w,4 t o

?

G

+o)(tN

w z

9

4 t o do>(tz)>, 7

where t, E (to , t l ) . Since V ( t , , ~ ( t,&)(t,)) , > 0, the assumptions L(u) imply that

< < t, . T h e rest of the proof is

which shows that x l ( t 0 , +,) E Qo , to t similar to the proof of Theorem 8.1.1.

T h e next comparison theorem gives a better estimate.

THEOREM 8.1.2. Let the assumptions of Theorem 8.1.1 hold except that the inequality (8.1.6) is replaced by

+

+ C(ll d(0)lI) < g(t, V t ,+(O)>>,

D+V(t,d(O), 4)

(8.1.11)

for t 3 t o , E C, , where the function C E X . Assume further that g ( t , u ) is monotone nondecreasing in 2c for each t . T h e n (8.1.7) implies

+

Set == x,(t, ,+o) so that +(O) = x(to ,+,,)(t). We then obtain, using the condition (8.1.1 I), the inequality D+m(t1)


7

4tl)).

Here, we have used the monotonicity of g(t, u ) in u and the fact that

q t ,4 t o

9

+o)(t))

G4th

while applying the assumption (8.1.11). With these changes, it is easy to prove the stated result, following the arguments in the proof of Theorem 8.1.1. T h e following variant of Theorem 8.1.1 is more useful in certain situations.

8.1.

BASIC COMPARISON THEOREMS

85

THEOREM 8.1.3. Assume that the hypotheses of Theorem 8.1.1 hold except that the inequality (8.1.6) is replaced by

for t > t o ,4 E SZ, Then,

, where A(t) > 0 is continuous on [-T, a).

and therefore, in view of the assumption (8.1.13), it follows that D-L(t, +(O),C)

e g(t, L(t, C(O))),

for t > t o , 4 E Q, , where Q, , in this case, is to be defined with L(t, x) replacing Y(t,x) in (8.1.2). It is clear that L(t, x) is locally Lipschitzian in x, and, thus, all the assumptions of Theorem 8.1.1 are satisfied, with L(t, x) in place of V ( t ,x). T h e conclusion is now immediate from Theorem 8.1.1. On the basis of the comparison theorem for functional differential inequalities developed in Sect. 6.10, we are now in a position to prove the following result, which plays an equally vital role in studying the behavior of solutions of functional differential systems.

THEOREM 8.1.4. Y E C[-T, a)x S , , R,], and V(t,x) is locally Lipschitzian in x. Assume that, for t E 1,4 E C, , D+V(t,(b(O),

where

Y t = Y(t

+ s, 4(s)),

4)< g(4 w,+(ON, -T

< s < 0,

g

E

Vt),

(8.1.15)

C [ J x R, x % + , R],

86

CHAPTER

8

g(t, u , a) is nondecreasing in u for each ( t , u ) , and r ( t o ,o0)is the maximal

solution of the functional differential equation

(8.1.16)

u‘ = g(t, u, U t )

with an initial function uo E %?+, at t = t o , existing for t >, t o . If x(t0 ,40) is any solution of (7.1.1) defined in the future such that Vt0 = J”t0

+ s, 4o(s)) <

(70

(8. I .17)

>

then we have

Proof.

Let x(t0 ,q50) be any solution of (7.1.1) such that

Set 4 = xt(t,, $,), which implies that +(O) m ( t ) = C'(t, X ( t ,

9

=

~ ( t,+,)(t). , Define

+o)(t)),

so that

v(t + s, d(s)). Since (8.1.17) holds, we have mto < uo . Moreover, for small h > 0, "1

=

because of the fact that V ( t ,x) satisfies a Lipschitz condition in x. This, together with (8.1.15), yields the inequality (8.1.19)

<

Then, it follows that D+v(t) 0, in view of (8.1.19). By Lemma 1.2.1, v(t) is nondecreasing, and therefore D_v(t) 0, which implies that D-4t)

< g(t, m ( t ) ,mt),

<

t >to.

T h e desired result (8.1.18) now follows from Theorem 6.10.4.

8.2.

87

STABILITY CRITERIA

8.2. Stability criteria We shall, in what follows, give sufficient conditions for various stability notions in terms of Lyapunov functions. This will be accomplished in two different ways. In one approach, the theory of ordinary differential inequalities will be used, as before, whereas in the other, the theory of functional differential inequalities play a major role. Since, for the purposes of this chapter, it is convenient to interpret the solutions of (7.1.1) as elements of euclidean space, the definitions of stability and boundedness have to be modified accordingly. For example, the definition 7.1.1 would appear in the following form.

DEFINITION 8.2.1. The trivial solution of (7.1.1) is said to be (S,) equistable if, for each E > 0, to E J , there exists a positive function

6

=

8 ( t o , c) that is continuous in to for each

E,

such that, whenever

I14 0 /lo < 6,

we have

I/ 4 t O do)(t)ll < 1

€7

t

2to.

With this understanding, we can prove the following results.

THEOREM 8.2.1. Let there exist functions V ( t ,x) and g ( t , u) enjoying the following properties : (i) V E C [ - T , GO) x S, , R,], V ( t ,x) is positive Lipschitzian in x, and V ( t ,).

< 4, II x ll),

definite,

( 4 4 E J x S" ,

where a E C[J x [0, p), R,], and a E 3'for each t E J ; (ii) g E C [ J x R, , R,],g ( t , 0) = 0, and, for t > to , D-Vt,

d(O>,4

locally (8.2.1)

E

SZ, ,

< g ( 4 q t , d(0"

Then the trivial solution of (7.1.1) is (1") equistable if the trivial solution of (6.1.2) is equistable; (2") uniform stable if the trivial solution of (6.1.2) is uniform stable and, in addition, V(t,x) is decrescent.

Proof.

Let x(t, , &) be any solution of (7.1.1). Choose

88

CHAPTER

< u,, , by

so that V(to,+,,) yields the estimate

8

(8.2.1). An application of Theorem 8.1.1

< r(t, t u , 4,

L'(t, 4 t o + o ) ( t ) ) 7

t 3 tu ,

(8.2.2)

where r ( t , t , , uo) is the maximal solution of (6.1.2). Also, because of the positive definiteness of V ( t ,x), we have

411x 11)

< t'(4

( 4 4E

x),

J x

s,,

bEx-.

(8.2.3)

Let 0 < E < p and to E ] be given. Assume that the null solution of (6.1.2) is equistable. Then, given b ( ~ > ) 0, to E J , there exists a 8 = 8(t,, E ) > 0 satisfying 4 4 t o , uo) < &),

<

provided u,, 8. Moreover, there exists a 6,

It +u

110

< 8,

(8.2.4)

t 2 to,

and

4tO

>

=

a1(t,,,e) such that

<

II d o ),1

(8.2.5)

hold together, because of the assumption on a(t, u).I t now follows, from

It d o I / d 81 that 9

b(ll4,

9

do)(t)ll)

< v, 4,+u)(t)) < r(t, to , a,,) I

<

and, consequently,

<

/I X ( t , ,do)(t)il <

w,

t 3 to,

t >, t o f

€7

whenever 11 +u I/, 6, . Thus, (1") is proved. If V ( t , x ) is decrescent, there exists a function a E X satisfying

w,4 < .(I1

x 1%

(4 4 E J x

s,

*

Hence, if we assume that the trivial solution of (6.1.2) is uniform stable, it is easy to see, from the foregoing proof, that 6, does not depend o n t o , proving (2"). T h e proof is complete.

COROLLARY 8.2.1. 8.2.1.

T h e function g ( t , u )

= 0 is admissible in Theorem

THEOREM 8.2.2. Assume that there exist functions V(t,x), g ( t , u), and A ( t )satisfying the following conditions: (i) A(t) > 0 is continuous on

[-T,

a),and A(t)-+ a as t -+a;

8.2.

89

STABILITY CRITERIA

(ii) V E C [ [ - T , co) x S o , R,], V(t,x) is positive definite, locally Lipschitzian in x, and verifies (8.2.1); (iii) g E C [ J x R, , R + ] , g ( t ,0) = 0, and, for t > t o , E SZ, ,

4)D - W

4)+ v(4 $(ON D - W

+(O),

+

< g(4 U t ,+(ON W).

Then, the trivial solution of (7.1.1) is equi-asymptotically stable if the trivial solution of (6.1.2) is equistable. Proof. If x(to,+o) is any solution of (7.1.1) such that 4 t o )4

0

9

I1$0

110)

=

uo 9

we have, by Theorem 8.1.3,

4 4 q t , 4 t O ,+o>(t)) < y ( t , t o

7

uoh

t

2 to

(8.2.6)

Let 0 < E < p and to E J be given. Let 01 = min-7Gi<mA(t).By assumption on A(t),it is clear that 01 > 0. Set q = 01b(~).Then, proceeding as in the proof of Theorem 8.2.1 with this q instead of b ( ~ )it, is easy to prove that the trivial solution of (7.1.1) is equistable. To prove equi-asymptotic stability, let q* = ab(p). Let S l ( t o , p ) be such that 11 #Io [lo 6, implies 11 %(to,#Io)(t)ll p, t 3 to . This is possible by equistability. Designate 8,(t0) = 8,(t, ,p), and suppose that (1 +o 8, . Since A(t)-+ co as t -+ 00, there exists a positive number T = T ( t o ,E ) such that

5

<

<

A(t)4)> ab(p),

t >, to

+ T.

(8.2.7)

We then have, using (8.2.3), (8.2.6), and the fact that u ( t , t o , uo) < q* if uo

<

S(t0

9

PI,

4 4 b(llX(t0

>

+o)(t)ll)

w,

%(to > +o)(t)> G G r(t, to uo) < T * = ab(p), t 3 t o . 9

If t >, to + T , it follows, from the foregoing inequality and (8.2.7), that provided 11 +o

[lo

< 8,.

This concludes the proof of the theorem.

COROLLARY 8.2.2. The functions g ( t , u ) are admissible in Theorem 8.2.2.

= 0 and A(t) = eat,

01

> 0,

90

8

CHAPTER

THEOREM 8.2.3. Assume that there exists a function V(t,x) satisfying the following conditions: (i) V t C[[-T, CO) x S, , R,], V ( t ,x) is positive definite, decrescent, and locally Lipschitzian in x; (ii) for t > t o , E Q, ,

+

u-If, m,4)< -C(ll d@)ll),

c: E .x.

Then, the trivial solution of (7.1 . l ) is uniformly asymptotically stable.

Proof. Since V is positive definite and decrescent, there exist functions a, b E .f satisfying

4 11) < V ( t ,4 < 4 x ll), Let 0

s :’

E

< p, t,

E

(4 ).

J be given. Choose 6

=

E

J x

s, .

(8.2.8)

> 0 such that

6(e)

a(S) < b(€).

(8.2.9)

<

6, then (1 x(t, ,&,)(t)/l< E , t 3 to . Suppose We claim that, if /I 4,,1 that this is not true. Then, there exists a solution ~ ( t,c, $ ~ of ) (7.1.1) with I/ 4o 1, 6 such that

<

II “ ( t o > 4 o ) ( b l l and so that

/I X ( t ,

?

4o)(t)ll

I,’(tz

?

<

t

€9

4 t , 4o)(tz>> J

=

6

E

[to >

4, (8.2.10)

3

because of (8.2.8). Furthermore, this means that x(t, , +,,)(t)E S, t E [t,,, t.1, Hence, the choice u,,= a(ll4, ),1 and the condition

,

give the estimate

If,44l

9

$o)(tN

< 41

$0

lid,

t

E

[to 9

f21,

(8.2.11)

because of Corollary 8.1.1. Now the relations (8.2.10), (8.2.1 l), and (8.2.9) lead t o the contradiction

This proves that the trivial solution of (7.1 . I ) is uniformly stable.

8.2.

91

STABILITY CRITERIA

T o prove uniform asymptotic stability, we have yet to show that the null solution of (7.1.1) is quasi-uniform asymptotically stable. For this , be any solution of (7.1.1) such that 11 +o (lo a,, purpose, let ~ ( t,+o) where 6, = 6(p). It then follows from uniform stability that

<

X(t0

Let now 0

< E < 6,

,do)(t) E so, t 2 t o *

be given. Clearly, we have b(€) < 4 8 , ) .

I n view of the assumptions on L(u), which occurs in the definition of Qo , it is possible to find a ,8 = P(E) > 0 such that

+ /3

L(u) > u

if

b(6)

< u < a(6,).

Moreover, there exists a positive integer N inequality b(€)

If, for some t

+ N/3 >

=

(8.2.12)

N ( E ) satisfying the (8.2.13)

480).

t o , we have

w4to

do>(t))2 b ( 4 ,

it follows that there exists a 6, = a,(€) > 0 such that 11 ~ ( t, cjo)(t)lI , >, 6, because of (8.2.8). This, in turn, implies that C(lI X ( t 0 do)(t)ll) ?

2 C@,) = 8 2 -

(8.2.I 4)

Obviously, 6, depends on E. With the positive integer N chosen previously, let us construct N numbers t,i = t k ( t o ,E), k = 0, 1, 2,..., N , such that 9

€)

=

tk+l(tO

7

9

t k ( t O , €)

€)

+ @is,) +

,

+1

T*

It then turns out that

+ and, consequently, letting T ( E )= N[(,8/8,)+ tk(tO

3

+

€1=

'[(/3/'2)

T],

tN(t0,

4 = t o + T(+

we have

Now, to prove quasi-uniform asymptotic stability, we have to show that

II X ( t ,

P

90>(t)II

<

€7

t

2 to

+T(4

92

CHAPTER

whenever 11 $,

8

< 6, . It is therefore sufficient to show that

/lo

q t , ,'(to 4")(4) < h ( E ) 9

i -( N - k ) 8,

2 t,

t

(8.2.15)

1

for h 0, 1, 2,..., N . For k 0, (8.2.15) follows from the first part of the proof and (8.2.13). We wish to prove the desired inequality (8.2.15) -

by induction. Suppose that, for some k

and, for some t , >, t,.,.,

-

< N , we have

(pis,),

It then follows that

and, consequently, we derive from (8.2.12) the inequality L(V(t* "( to 4o)(t*))) 9

for t,

7

< [ < t , . This implies that

> h(6) + ( N 4 B > "(to $ o ) ( E ) ) ,

w,

-

t

L ( V ( t , ,4(0)))'> V(t* t- s, $(S))>

-7

< s < 0,

where 4 = xl,(t,, , +J, so that +(O) = %(to, $J(t,). Hence, 4 E Qo . It therefore follows from condition (ii) and the relation (8.2.14) that

n- V ( t ,4(0),4)<

< 0.

-82

This shows that, if ever LXt, "( to

,4o)(t)) < b(6)

+(N

-

1)

-

B 7

for some t >, t x t l (pis,), then the same inequality will hold henceforth. Now, supposing that ~

q t , "(to, 40)(t)) 2 b ( E ) for tl,+, - (pis,) 1 'ft*. Xfto

t

+ (N

-

k

-

1) B

< t < t*, we deduce, by arguing as before, do)(t*)) < b(6) -t ( N - R, P

< q t * , 4 t O ,$o)(t*))

- 82(t* -

82(t*

- tk+l

+

- GC+l),

@/'2))

8.2.

93

STABILITY CRITERIA

<

which proves that t* t,,,, . I n other words, we have verified the truth of (8.2.15) for all k. T h e proof of the theorem is therefore complete. T h e next theorem may be useful in some situations.

THEOREM 8.2.4. Assume that there exist functions V ( t ,x) and g(t, u) satisfying the following conditions:

(i) V E C[[-T, GO) x S o ,R,], V(t,x) is positive definite, decrescent, and locally Lipschitzian in x; (ii) g E C [ J x R, , R,], g(t, 0 ) = 0, g(t, u) is monotone nondecreasing in u for each t , and, for t t o ,(b E C , ,

> D+ V(4 4(0)7 a, + C(Il 4(0)il) < g(4

d(O))),

where C E Z. Then, the trivial solution of (7.1 .l) is uniformly asymptotically stable if the trivial solution of (6.1.2) is uniform stable.

Proof.

By Theorem 8.1.2, we have

v4 4 t o 4o)(t)) 9

<

As this implies,

~

st

C(llX(t0

+ r(4 t o , to

UO),

< r(4 t o , U o ) ,

V t , 4 t o +o)(t)) 9

7

4o(s)ll) ds

t

2 to *

t 3 to ,

on the basis of Theorem 8.2.1, it is clear that the trivial solution is uniform-stable. Let 8, = S,(p), where S,(p) is the number obtained for uniform stability. Similarly, let S,(E) be the number corresponding to E, and suppose that 11 x(t0 ,(bo)(t)J( >, S,(E) for t E [to, to -t TI, where T is chosen to satisfy the inequality (8.2.16) Since r(t, to , uo) follows that

< b(p), t

b(llX(t0

9

2 to ,

4o)(to

whenever uo = u(ll $o

+ T)I/ < - C ( U E ) )

Ilo)

< Q),

it

T -1- b(P).

<

This implies, because of (8.2.8), that 0 < b(S,(c)) 0. This contradiction proves that there exists a t , E [to , to T ] such that

+

II 44l do)(t1)ll < S1(+ 7

94

CHAPTER

8

I t therefore follows, from the decrescent nature of V ( t ,x), that, in any case,

II 4 t " , (bo)(t)ll < whenever

< 8,.

I/ +o 11"

t

€7

3 to

+ 1'9

T h e proof is therefore complete.

COROLLARY 8.2.3. T h e function g(t, u) = 0 is admissible in Theorem 8.2.4. We shall now consider Eq. (7.1.8) and illustrate the practicality of using Lyapunov functions instead of functionals. Take L(t, x) = A(t) V ( t ,x) = eatx2, a: > 0. Then, the set QA is defined by

which implies#Jz(s) (7.1.9), we see D - L ( ~+(o>, , +)

< +2(0)e-as,

= eat [ -

q

- T

2(" ~

~

< s < 0. Hence, using the relation

+ b) ~ ( 0 )

+(s) ds

2ab$(0)

-

2b2$(0)

-7

When

#J

E QA

/"

I$(S

~

T) ds]

.

-7

, this reduces to, assuming b 3 0,

=

[.2(a + b ) + (4ub + 4b2 eaT)(@+

N

-

"1

q t , $(O)).

We wish to apply Theorem 8.2.2 with g ( t , u ) e 0. This means that a, 6 , OL, and T must satisfy the condition

Clearly, by Theorem 8.2.2, the trivial solution of (7.1.8) is uniformly asymptotically (exponentially) stable, provided (8.2.17) is verified. Observing that e" - 1 3 x,x > 0, and ex >, 1 , it follows, from (8.2.17), that 01

+ 2b(U + b ) < 2 ( U f b). 7

8.2.

Now, choosing condition

01

=

+ b), 0 < y < 1,

2y(a

< br < 1

0

95

STABILITY CRITERIA

we readily deduce the

< 1,

-y

which is the known condition for asymptotic stability. Since OL is arbitrary, letting 01 -+0 and noting that lim

m

earl2 -

4

1

__ -

01

7

-

2’

the condition (8.2.17) yields 0
< 1,

which is the condition obtained before for uniform stability. If we wish to obtain the stability region in a different way, we have to compute D L ( t , 4(0),4) using (7.1.8) directly, instead of (7.1.9) as before. We then get the inequality D-L(t,

W), 4) < [a

-

2(a

-

I b I ear)]L(t, d(0)).

T o apply Theorem 8.2.2 with g = 0, we must have 0

< 0112 < a - I b I eur,

which implies that the region for uniform asymptotic stability is 1 b 1 < uerar. If 01 -+ 0, this condition reduces to 1 b I a, which is the region for uniform stability for Eq. (7.1.8). Notice that the latter condition I b I < a may also be obtained by taking V ( t ,x) = x2, computing D-V(t, +(O), 4) for 4 E Q, , and using Theorem 8.1.1 with g = 0. Finally, take V ( t ,x) = x2,L(u) = u/q, 0 < q < 1, and

<

C(U) =

241

- q 1 / 2 ) 242.

Then, one can verify the conditions of Theorem 8.2.3 by a calculation similar to the foregoing. The trivial solution of (7.1.8) is uniformly asymptotically stable if I b I < The following theorem shows that the study of the stability properties of the functional differential system can be reduced to that of a single functional differential equation. Naturally, this technique depends heavily on Theorem 8.1.4, as anticipated. Analogous to the stability definitions with respect to the scalar differential equation (6.1.2), if we say that the trivial solution of (8.1.16) is equistable, we mean the following.

96

CHAPTER

8

DEFINITION 8.2.2. T h e trivial solution of (8.1.16) is said to be equistable if, for each E .: 0, to E J , there exists a 6 = 6 ( t o , E), which is continuous in t, for each E , such that

to , o o ) ( t ) <

<

t >, t o

€9

provided 1 uo lo 6, where u ( t o , uo) is any solution of (8.1.16) with an initial function uo E 9,at t = t o . Other definitions may be understood similarly.

THEOREM 8.2.5. Assume that there exist a function V ( t ,x) and a functional g(t, u, u) satisfying the following conditions: (i) Y E C[[-T,

CO)

4 x 11)

x S, , R,], V(t,x) is locally Lipschitzian in x,and

< Ut, 4 < 4 x I\),

(4 2) E J x

s,,

where a, b E .X; (ii) g E C[J x R , x %+ , R],g(t, 0 , O ) E 0, g ( t , u, a) is nondecreasing in o for each ( t , u), and, for t E J , 4 E C, ,

W),$1 < g(4 v(4 W)), Vt).

D'CIt,

Then, the trivial solution of (7.1.1) satisfies one of the stability notions, if the trivial solution of (8.1.16) obeys the corresponding one of the stability definitions. Suppose that the trivial solution of (8.1.16) is equistable. Let ) 0, to E J , there exists a p and to E J. Then, given b ( ~ > 6 ( t o , e ) > 0 such that 1 uo lo 6 implies

Proof.

0 <E 6

=

< :

<

4 t O , oo)(t) < b(c),

t 3 to *

Let ~ ( t,+o) , be any solution of (7.1.1) existing for t 3 to such that V(to+ s, do(s)) u0(s).We then have, by Theorem 8.1.4, the inequality

<

L7(t,

to

9

+o)(t))

G

r(tn

, oo)(t),

t

2 to ,

where r ( t o , a,) is the maximal solution of the functional differential equation (8.1.16). Choose 6, > 0 such that 6, = a-'(6), and let jl +o 1, 6, . I t turns out that

<

41 to ,+n)(t)ll) G

V t , 4 t n +o>(t>)

< r(tn

on)(t)

< b(€),

t

>, to

9

8.3.

97

PERTURBED SYSTEMS

which implies that

I1 4 t o > +o)(t)ll < E ,

<

t

2 to >

provided that 11 +o 1l0 6 , , showing that the trivial solution of (7.1.1) is equistable. T h e proof corresponding to other notions may be given by modifying the arguments suitably. T h e theorem is therefore proved. COROLLARY 8.2.4. T h e functional g ( t , u, U) = 0 is admissible in Theorem 8.2.4 to yield uniform stability of the trivial solution of (7.1.1).

8.3. Perturbed systems Let us consider the perturbed system (7.4.1) corresponding to the unperturbed system (7.1.1). We then have

THEOREM 8.3.1. Suppose that (i) V E C[[-T, co) x S o , R,], V ( t ,x) is positive definite and satisfies > 0; (ii) for t > to ,4 E 9, ,

a Lipschitz condition in x for a constant L = L ( p )

D - V , 4(0),+)(,.IA

e 0;

(iii) w E C[J x R, , R,],w ( t , 0) = 0, and, for t

> t o ,4 E 9, ,

I/ R(t,4111 e w(t9 V(4 d(0))). Then, the trivial solution of (7.4.1) is equistable (uniform stable) if the trivial solution of (6.1.2) with g(t, a) = Lw(t, a) is equistable (uniform stable). Proof. Let t > to and and (iii), we have D-V(t, 9(0),4)(,.4.1)

4 E Q, . Then,

in view of assumptions (i), (ii),

< limk-0-inf lz-l[V(t + h, 4(0) + h { f ( t , 4 ) + q t , 4)) -

v(t + d(0) + h f ( 4 4111

+ lim inf W V ( t + h, +(O) + V ( t ,4)) < L II R(t,9)ll + D-C'(t, +(O), h9

-

h 4 -

4h7.1.1)

< Lw(t, q t , W)))= g(tt F ( t , #(ON.

T h e desired result is now a consequence of Theorem 8.2.1.

I.-(t, d(0))l

98

CHAPTER

8

THEOREM 8.3.2. Suppose that (i) V E C[[-T, 00) x S, , R,], V(t,x) is positive definite and satisfies a Lipschitz condition in x for a constant L = L ( p ) > 0; A(t)400 as t 00, and, (ii) A ( t ) > 0 is continuous on [ - T , a), fort>t,,+EQ,, ---f

+ V t ,4(0))D - 4 t ) < 0; C [ J x R, , R,],w(t, 0) 0, and, for t > t o ,+ A f t )D-V(t, 4(0),6)

(iii) w

E

E

A ( 4 I/ R(t,6111

< 4 4 JV,

E

Q, ,

A@)).

Then, the trivial solution of (7.4.1) is equi-asymptotically stable if the null solution of (6.1.2) with g ( t , u ) = Lw(t, u ) is equistable.

Proof.

If t

> to , + E QA , it follows that

which implies the inequality a t ) D-V(4 +(Oh

d4(7.4.1)

+ v(t,(b(0))D - 4 ) < Lw(4 w, $((I)) A@)).

Consequently, the conclusion follows by Theorem 8.2.2.

THEOREM 8.3.3.

Suppose that

(i) V E C[[-T, 00) x S, , R,], V ( t ,x) is positive definite and satisfies a Lipschitz condition in x for a constant L = L(p) > 0; (ii) for t >, to , E C, , and C E X ,

+

D'V(t3

4h.l.l)

< -C(ll4(0)11);

(iii) w E C [ J x R, , R,.], w(t, 0 ) = 0, w(t, u) is nondecreasing in u for each t E J , and, for t 3 t o , E C,, ,

+

/I R(t, 4111 < w(t, V ( t ,4(0))). Then, the uniform stability of the trivial solution of (6.1.2) with g ( t , u ) = Lw(t, u ) assures the uniform asymptotic stability of the trivial solution of (7.4.1).

Proof,

Let t

3 to and + E C, . Then, as previously,

D+V(t,+('),

$)(7.4.1)

d(o)li) + Lw(t, V(t,+(O))),

and, therefore, the uniform asymptotic stability of the trivial solution of (7.4.1) follows by Theorem 8.2.4.

8.3.

THEOREM 8.3.4.

99

PERTURBED SYSTEMS

Assume that

(i) V EC[--7, CO) x S o ,R,], V(t,x) satisfies a Lipschitz condition in x for a constant L = L ( p ) > 0, and b(ll x II)

in

< V t , 4,

(4 4 E J x

s,,

b E .x;

(ii) g, E C [ J xR, x V, , R],gl(t, 0, 0) 3 0, gl(t, u, u) is nondecreasing u for each ( t , u), and, for (t,4) E J x C, , D+V(t,#('),

gl(t, v(t,#(O)), ;)6'

#)(7.1.1)

(iii) g, E C[J x V+ , R,],gz(t,0) = 0, gz(t,u) is nondecreasing in u for each t E J , and II R(4 #)I1 < gz(4 II # 11)Then the stability properties of the trivial solution of (8.1.16) with g(4 u, 4 = gl(4 u, 4

+ Lg&, b - W

imply the corresponding stability properties of the trivial solution of (7.4.1). Proof.

Let t E J and #J D'v(t, #(O),

E

#)(7.4.1)

Co . Then, D'v(t, #(O), #)(7.1.1) < gl(C v(4#(O)), V t ) < gl(4 v(4#(O)), V t ) = g(4 W)), Vt),

w,

+ 11 R ( f ,#>I1 + Lgz(4 II 4II) + Lgz(4 b-YVt))

because of assumptions (i), (ii), and (iii). Now, Theorem 8.2.5 can be applied to yield the stated results.

COROLLARY 8.3.1.

T h e functions gl(4 u, 0) =

--01u

+

PT

sup -T<S
44,

> 0, p > 0, and g2(t, 0) = y sup-,GsGo u(s), y being sufficiently small, are admissible in Theorem 8.3.4, provided b(u) = u and 0 < T < (01 - y ) / P , to guarantee that the trivial solution of (7.4.1) is exponentially asymptotically stable. 01

Proof. Under the assumptions, it is easy to see, as in Theorem 6.10.7, that u' = -01u (PT y ) sup-,GsGo ut(s) admits a solution r ( t o ,uo) that tends to zero exponentially as t + CO, and, therefore, the conclusion follows from Theorem 8.3.4.

+

+

100

CHAPTER

8

8.4. An estimate of time lag We wish to estimate the time lag r in order that the solutions of an ordinary differential system (8.4.1)

x’ = f ( t , x)

and a functional differential system Y’

=

(8.4.2)

F(4 Y t )

may have the same behavior, namely, exponential decay. Since Eq. (8.4.2) may also be written as x’

where

=f

( 4 x)

R(4 x, 4

+ R ( t , x, 4,

= F(4

(8.4.3)

4 - f ( k 4,

it is sufficient to consider the perturbed system (8.4.3).

THEOREM 8.4.1. Suppose that (i) I/ E C [ S , , R,], V ( x ) is positive definite and satisfies a Lipschitz condition in x for a constant L = L(p) > 0 ; (ii) f E C[/ x S , , R”], and, for (t, x) E J x S,, ,

< -aL’(x),

D+V(X)(&4.1)

(iii) R

E

01

> 0;

C [ / x S, x C, , R”],and for 4 E C, ,

/I R ( f >+(Oh +)I1

< NT

SUP

--7<s
%w).

Then, if 0 < r < a/LN, every solution ~ ( t, +o) , of (8.4.3) with +o E C, at t = t o , which is defined in the future, tends to zero exponentially as t 4 a.

Proof. Let t E J and 4 E C, . Then, using assumptions (i), (ii), and (iii), we have

8.5.

101

EVENTUAL STABILITY

It is clear that g(t, u, a) satisfies the conditions of Theorem 6.10.4, and, consequently, J+(t,

7

<4 t O

4o)(t))

9

t

oo)(t),

2 to

Y

is any solution of (8.4.3) with the initial function E C,, where x(t0 , at t = to , defined in the future, and r ( t , , uo) is the maximal solution of (8.1.16) with uo E V+ at t = to such that V(+,(s))< uo . All that remains to be shown is that r(to , uo) exists in the future and tends to zero exponentially as t -+ CO, if 0 < T < a/LN. I n fact, as we have seen in the proof of Theorem 6.10.7, the equation u'

-mu + L N T

= g(t, u , u , ) =

sup

u,(s)

-T<S
admits solutions ~ ( t,,uo) that tend to zero as t + co exponentially, if 0 < T < a/LN. T h e theorem is therefore established. COROLLARY 8.4.1.

Assume that

(i) f E C [ J x S,,, P

II x (ii) R

E

I , and, for all sufficiently small h > 0,

+ hf(4 .)I1 < II x ll(1

-

ah),

01

> 0,

( 4 x) E J x so;

C [ J x S,, x C, , R"], and, for ( t , + )E J x C, ,

II R(4 d(O), 4111 G NT --r<s
8.5. Eventual stability We shall extend the results on eventual stability discussed in Sect. 3.14 to the functional differential system (7.1.1). Following the definition 3.14.1, we can formulate eventual stability concepts in the present case. For example, the set = 0 is said to be [with respect to the system (7.1.1)] ( E l ) eventually uniform@ stable if, for every E > 0, there exists a 6 = S ( E ) > 0 and T~ = T ~ ( E )> 0 such that

+

+, <

I/ X ( t 0 ,4o)(t)ll < E ,

t

2 t o 2 70

f

provided 11 6. Analogously, we shall denote by ( E l * ) to (E4*)the corresponding notions of the set u = 0 with respect to the functional differential equation (8.1.16).

102

CHAPTER

THEOREM 8.5.1.

8

Let the following conditions hold:

(i) V E C[[--7,

GO)

x S, , R,], V ( t ,X) is locally Lipschitzian in X, and

411x 11)

< V t , "> < 4x 11)

for 0 < OL < 11 x 11 < p and t >, O(ct), where a, b E X and O(u) is continuous and monotonic decreasing in u for 0 < u < p ; (ii) g E C [ J x R+ x V+ , R ] , and the set a = 0 is eventually uniformly stable with respect to (8.1.16); (iii) f E C [ J x C,, R"], and

for every

+

D+V(t,d@), 4) E

C, such that 0 < a

Then, the set (7.1.1).

+

=

e g(t, L T 4 %(0)),Vd

< I/ +(O)lI < p and t >, O(a).

0 is eventually uniformly stable relative to the system

Proof. Let 0 < E < p. By assumption, given b ( ~ > ) 0, there exists a T ~ ( E> ) 0 such that 6, = a,(<) > 0 and a T ,

<

1 q,lo

4 t o , uo)(t)< 44,

t

2 t o 2 71

(8.5.1)

9

8,. Let us set 6 = ~ ~ ( 6 ,and ) T , ( E ) = 0[8(~)]. Let max[T1(E), T ~ ( E ) ] . We claim that (El)is satisfied with this choice of S ( E ) and T ~ ( E ) .Suppose that this is not true. Then, there would exist a solution x(t, , +,) of (7.1 . I ) and two numbers t, , t, such that t, > t, > to 2 T o ( € ) , whenever =-

T"(E)

=

II "(to and 8

=

> sbO)(tl)ll

< ll "(to

7

II 443

8,

4o(t)ll

< El

3

t

%o)(tz>lI E (tl

7

=

tz).

<

E,

+

Moreover, 11 xt(t0 , $o) l o E , t E [to , t l ] . Setting = x,(t, , +o) for t t (tl , t,), we see that S < jl +(O)lI < E . We choose I u0 10 = a(ll $1Il,), where +1 = xl,(t,, +"). Then, condition (iii) and Theorem 8.1.4 yield the inequality I '(t>Y ( t l +l)(t))

<4tl

?

t

oo)(t),

E [tl

> t21,

(8.5.2)

where y(tl , is any solution of (7.1 . I ) through (tl , +1), and Y ( t , , a,) is the maximal solution of (8.1.16) through ( t l , a,,). I t turns out that (8.5.2) is also true for ~ ( t, +,J , on the interval t, t t, . Hence, we get

< <

b(E)

-< r,-(t, , "(to ,% " ) ( t Z ) )

Y(t,

3

uo)(tz)

< b(E),

8.5.

103

EVENTUAL STABILITY

on account of the fact that t, > t, > to >, T ~ ( E )and the uniformity of the relation (8.5.1) with respect to t o . This contradiction shows that ( E l ) is valid, and the theorem is proved. COROLLARY 8.5.1. The uniform stability of the trivial solution u = 0 of (8.1.16) is admissible in Theorem 8.5.1 in place of the eventual uniform stability of the set u = 0. I n particular, g ( t , u, a) = 0 is admissible.

THEOREM 8.5.2. Assume that (i) V EC[[-T, 00) x S, , R,], V(t,x) is Lipschitzian in x for a constant L = L(p) > 0, and

&ll x II)

e v(44 < 4 x ll),

for 0 < 01 < 11 x 11 < p and t >, O(a), where a,b E X and O(u) is continuous and monotonic decreasing in u for 0 < u < p ; (ii) f E C [ J x C, , R"],and Df

w,W),4)e 0,

for every 4 E C, such that 0 < 01 < 11 +(O)lI < p and t 2 O(cx); (iii) R E C[J x C,, R"], and, for every 4 E C,*, p* < p and t >, 0,

Then the set 4 = 0 is eventually uniformly stable with respect to the perturbed system (8.5.3) x' = f ( 4 X t ) R(t, 4.

+

Proof. Let 0 T ~ ( E )such that

< E < p* 248)

be given. Choose the numbers 6

< b(e)

11 R(t, $)I[. Define h(t) = max,,+,,oGDt to find a T ~ ( E ) > 0 such that rrca

and

T~(E= )

0(8(e)),

=

a(€) and (8.5.4)

Since h(t) is integrable, it is possible (8.5.5)

provided to >, T ~ ( c ) where , L is the Lipschitz constant for V ( t , x). Let = m a x [ ~ ~ (T~(E)I. ~),

TO(€)

104

CHAPTER

8

Suppose that there exists a solution x(t0 , of the perturbed system (8.5.3) and two numbers t , , t, such that t , > t , >, to >, T,,(E),

At t where = xl,(to, (8.5.5) and the fact that I/

< V(tz ,

X(t,

7

t,

=

< 6,

, we therefore obtain, in view of

Co)(tz)>

< 4 8 ) + 4) = 24%

which is incompatible with (8.5.4). This shows that ( E l ) holds, and the theorem is established.

THEOREM 8.5.3. Let assumption (i) of Theorem 8.5.1 hold. Suppose further that f E C [ J x C, , R"] and

+

D+

$(O),

4 ) < -C(ll4(0)ll),

for every E C, such that 0 < 01 < I/ +(O)lI < p and t >, e(u) and C E %. Then, the set = 0 is eventually uniformly asymptotically stable.

+

Proof. T h e eventual uniform stability of the set C$ = 0 follows by Corollary 8.5.1. Let 0 < E < p be given. Choose 6, = 6(p), T~ = ~ ( p ) , and T ( E )= T ( E ) {a(p)/C[G(e)]). Assume that to >, T~ and 11 I(o 6,. It is sufficient to show that there is a t , E [to T ( E ) , to T ( E )such ] that

+

+

+

II X ( t 0 4 O ) ( ~ l ) I l < Y E ) , Y

in order to complete the proof. Suppose, if possible, that

< /I "(tn

8(~)

4n)(t)ll <

PF

t E [to

+

4e)t

to

+ T(E)I.

<

8.6,

Letting #I

=

105

ASYMPTOTIC BEHAVIOR

xl(t0 ,#Io), we see that

G )G 114(0)11 < P and 1) $ )lo


Hence, it follows that

It then turns out that

which is an absurdity. This means that there exists a ti E

[to

+

~ ( c ) , tn

+ T(6)I

such that 11 ~ ( t,+,,)(t,)ll , < a(€), and therefore, in any case,

/I 4%do)(t)ll < 3

provided 11 +o (I,

€9

t 2 to

+T(4

to

2 44,

< 6, . T h e proof is complete.

8.6. Asymptotic behavior I n this section, some of the results of the Sect. 3.15 will be extended to the functional differential systems.

THEOREM 8.6.1. Assume that (i) V E C[[-T, a)x R", R,], V(t,0) = 0, V ( t ,x) satisfies a Lipschitz condition in x for a continuous function K ( t ) 0, and

>

411x 11) G

w 4,

where b E X is such that b(u) -+ co as u + 00;

106

8

CHAPTER

(ii) g E C [ J >: R, x %+' , R ] ,g ( t , u, a) is nondecreasing in u for each ( t , u),and, for t E J , 4,4 E Vn,

n+ If,+(O) =

--

4 $1

$(Oh

-

lirn sup h-YC'(t h-Of

+ h, 4(0)

Then, every solution ~ ( t,, t + co.

Proof.

Let t

E

-

$(O)

+ h { f ( t ,4)- f ( 4 $)I

of the system (7.1.1) tends to zero as

en.Then,

J and y5 E

+ h, 4(0) + W ( t ,4)) - v(t,d(0))l < K(t)llf(t, 0)ll + lim sup h-l[V(t + A, 4(0)

I)+ I -(t,4(0), 4 ) -= lim sup h-l[V(t h-04

+ h { f ( t ,+) - f ( C

h 4 f

0)))

~

V t , 4(0))l

< K(t)llf(t, 0)Ii + g(t, U t , W)),l~'t>,

in view of assumption (ii). Let ~ ( t, $ , ,) be any solution of (7.1.1) such that V(t, S , $ ~ ( S ) ) u,(s). It then follows, by Theorem 6.9.4, that

+

<

v(f,"(to

9

+o)(t>)

< r(to

3

oo)(t),

t b to >

(8.6.2)

where r ( t , , u0) is the maximal solution of the functional differential equation (8.6.1). T h e stated result is now a direct consequence of the hypotheses (i) and (iii). T h e next theorem is very useful in applications, since it does not demand V ( t ,x) to be positive definite.

THEOREM 8.6.2. Assume that (i) f E C[J x C,, R"], and

llf(f, 4111 < M ,

tE

1,

I1 4 I10

< P*


8.6.

107

ASYMPTOTIC BEHAVIOR

(ii) V E C [ [ - T , 03) x S, , R,], V ( t ,x) is locally Lipschitzian in x,and

o+w,W),4)< -C[d(O)l, for t E J and 4E 52, where C(x) is positive definite with respect to a closed set 52 in S, ; (iii) all the solutions x(to ,$,J of (7.1.1) are bounded for t >, to . Then, every solution of (7.1,l) approaches the set 52 as t + CO.

Proof. Let x ( t 0 , 40) be any solution of (7.1.1). By assumption (iii), it is bounded, and, hence, there exists a compact set Q in S, , such that t 2 to to ,do)(t) E Q , Moreover, it also follows that 11 x , ( t o , +o)llo < p* < p , t 3 t o , and

therefore, by assumption (i), we have lI.f(t9

%(to

9

d0))ll

< M-

Suppose that this solution does not approach 52 as t + CO. Then, for some E > 0, there exists a sequence {trc},t, + co as k + CO, such that 4 t n > ddtd E

S(Q3

€1" n 8,

where S(Q, E)" is the complement of the set S(52, e ) = [x : d(x,Q) < €1. We may assume that t, is sufficiently large so that, on the intervals t, t t, ( E / ~ Mwe ) , have

< < +

4 t o 2 do)(t) E S(Q,

4v nQ-

(8.6.3)

These intervals may be supposed to be disjoint, by taking a subsequence of (t,), if necessary. By Theorem 8.1.2 and assumption (ii), we get V

9

4 t O do)(t))

<

SUP --7<s
Uto>do(4

-

st

to

C[x(to9 CO)(S)I ds,

(8.6.4)

for t >, to . Since C(x) is positive definite with respect to 52, the relation (8.6.3) shows that there exists a 6 = 6 ( ~ / 2 )> 0 such that C[x(tO

~ d O ) ( ~ ) l2

It therefore turns out that

tk

< <

t7c

f (E/2M).

(8.6.5)

I08

CHAPTER

8

on account of (8.6.4) and (8.6.5). T h e foregoing inequality leads to an absurdity as tZ 03, since, by assumption, V ( t ,x) >, 0. As a con! as t -+ 00, and the sequence, any solution x(t, , $o) tends to the set 2 theorem is proved. Making use of two Lyapunov functions, we can extend Theorem 4.2.1 to functional differential system (7.1.1). ---f

THEOREM 8.6.3.

Let the following assumptions hold:

(i) f~ C [ J x C,, , R n ] , f ( t ,0) = 0, andf(t, 4)is bounded on J x C,, ; (ii) V , E C[[-T, CO) x S o ,R,], Vl(t,x) is positive definite, decrescent, locally Lipschitzian in x, and, for t E J , (b E C, ,

< 44(0)) < 0,

D+Li(t,4(0i 4)

where ~ ( x is) continuous for x E So ; (iii) V , E C[[-T, co) x S o ,R,], Vz(t,x) is bounded on J x S, and is locally Lipschitzian in x. Furthermore, given any number a, 0 < 01 < p, there exist positive numbers [ = ,$(a)> 0, 7 = ~ ( a> ) 0, 7 < 01 such that

d(O), 4)> 4 that < II $(O)ll < p

D’L72(t,

for every t where

3 0, $ E C,, such E

OL

= [x €

so: a(.)

=

and d(+(O),E )

< 7,

01,

and d(x,E ) is the distance between the point x and the set E. Then, the trivial solution of (7.1.1) is uniformly asymptotically stable.

Pmof. As the proof requires appropriate changes in the proof of Theorem 4.2.1, we shall indicate only the modifications. Let 0 < E < p and to E J. Since Vl(t,x) is positive definite and decrescent, there exist functions a, b E X , satisfying b(lI x 11)

Let u s choose S

< l’d4 x) < 4

ll),

( 4 ). fzJ x

so.

6 ( ~> ) 0 such that

:

b ( € ) > a(6).

Then, by Corollary 8.2.4, the uniform stability of the trivial solution of (7.1.1) results. Let us designate 6, = S(p). Assume that ]I +o ]lo < S o . T o prove the theorem, it is sufficient to show that there exists a T = T ( E )such that, , < S ( E ) . As in the proof of for some t , E [t, , to ?’],jl ~ ( t,+o)(t,)jl Theorem 4.2.1, this will be achieved in a number of steps:

+

8.6.

(1") If d[x(t,), x(t,)]

109

ASYMPTOTIC BEHAVIOR

> r > 0, t, > t, , then Y

< Mnl/2(t,

t,),

~

<

where x(t) = x(t, ,+,)(t) and Ilf(t, +)\I M , ( t ,+) E J x C, . There is no change in the proof of this statement. (2") By assumption (iii), given 6 = a(€), 0 < 6(c) < p, there exist two positive numbers ( = ((E), 7 = q ( ~ )7, < 6, such that

for every t E J , sider the set U

+

=

D+V,(t, 4(0), 4 ) > 5 E

< 11 +(O)lI < p,

C, such that 6

[x E S, : 6

< 11 x 11 < p

and

d(+(O),E ) < 7. Con-

d(x, E ) < q],

and suppose that S U ~ V,(t, ~ ~x) ~= L. ~ Assume ~ < ~ that, at t = t, , we have x(tl) = x(t, , +,)(t,) E U. Then, for t > t, , it follows, by letting m ( t ) = V2(t,x(t)),that D+m(t) 2 D+V,(t,

4)> E,

because of condition (iii) and the fact that V2(t,x) satisfies a Lipschitz condition locally in x. I n obtaining the foregoing inequality, we have set 4 = xt(t,, 4,) so that +(O) = x(t) and used the inequality 11 4 1, = 11 xt(t, ,$,)1 0 < p, which is a consequence of uniform stability. Since m(t)

~

m(tl) =

It

D+m(s) ds,

tl

as long as x(t) remains in U , we should have

> E(t

-

tl).

Th i s inequality can be realized simultaneously with m ( t ) t

< t,

+( W E ) .

< + ( 2 L l f )with the

Hence, it follows that there exists a t, , t , < t, t, property that x(t2)is on the boundary of the set U. (3") Consider the sequence {tic} such that t , = to

+ k(2L/5)

< L only if

(k = 0, 1, 2 ,..,,).

I LO

CHAPTER

8

Defining n(t) = Vl(t,x(t)) and using assumption (ii), we obtain D+n(t)

+

< D+li(t, W),4)< 0,

where = xt(to , do), as before. Let A, = h7/2Mr~l/~,where h = inf[l(w(x)i, 6 < I/ x /I < p, and d ( x , E ) >, 71/21. Suppose that x(t) satisfies, for t, t t,,, , the inequality 6 < 11 x(t)ll < p. Then, arguing as in the proof of Theorem 4.2.1, with obvious changes, we can show that

< <

f71(Gc+2 > 4 t k f 2 ) )

< Vdtk

> 4tk)) -

A,

*

We now choose an integer K* such that h,K* > a(S,) and let T = T ( E )= 4K*L/l. Assuming that, for to t to T , we have

< < +

/I 4t)lI 2

w

7

we arrive at the inequality, as in Theorem 4.2.1, I yi(to

+

7’9

X(tn

+ T ) ) < Vi(to, 40) ~

< a@,)

-

K*h,

K*hi

< 0,

which is absurd, since Vl(t, x) is positive definite. It therefore turns out TI such that (1 x(t*)ll < 6, and this proves that there exists a t* E [to , t, the uniform asymptotic stability of the trivial solution of (7.1 .I).

+

8.7. Notes T h e comparison theorems 8.1.1 and 8.1.3 are due to Lakshmikantham [ I , 61. See also Driver [3]. Theorem 8.1.4 is new. Theorems 8.2.1 and 8.2.2 are adapted from the work of Lakshmikantham [l, 61, whereas Theorem 8.2.3 is based on the result of Driver [3]. See also Krasovskii [2, 51. Theorems 8.2.4 and 8.2.5 are new. T h e examples in Sect. 8.2 are taken from Lakshmikantham [6] and Driver [3]. All the results of Sect. 8.3 are based on the work of Lakshmikantham [6], whereas Theorem 8.3.4 is new. Section 8.4 contains new results. See also Halanay [22] for particular cases. T h e results of Sects. 8.5 and 8.6 are new. For many similar results for delay-differential equations, see Oguztiireli [I]. For related work, see Driver [3], El’sgol’ts [4], Krasovskii [l-51, Lakshmikantham [I], Oguztoreli [l], and Razumikhin [2, 61. For the use of vector Lyapunov functions in studying the conditional stability criteria of invariant sets, see Lakshmikantham and Leela [2].

PARTIAL DIFFERENTIAL EQUATIONS

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Chapter 9

9 .O. Introduction T hi s chapter is devoted to the study of partial differential inequalities of first order. We consider some basic theorems on partial differential inequalities, discuss a variety of comparison results, and obtain a priori bounds of solutions of partial differential equations of first order in terms of solutions of ordinary differential equations as well as solutions of auxiliary partial differential equations. We also treat the uniqueness problem, error estimation of approximate solutions, and simple stability criteria. We make use of Lyapunov-like functions to derive sufficient conditions for stability behavior. For systems of partial differential inequalities of more general type, we merely indicate certain analogous results.

9.1. Partial differential inequalities of first order We shall use the well-known notation am at

mi=-,

m, =

am

ax

-

wp a2m

mxg=

whenever convenient. It is, however, necessary to caution the reader not to confuse the symbol m l with the one used while considering functional differential systems. Let 01, ,l3 E C [ J ,R ] , and suppose that a ( t ) < P(t), t E J. Assume that a’(t), P ’ ( t ) exist and are continuous on J . For to E J , we define the following sets: 113

114

CHAPTER

9

E

=

[ ( t ,x): to

< t < co,.(t) < x < P ( t ) ] ,

E, aE,

=

[ ( t ,2 ) : t ,

< t < co,.(t) < x < P ( t ) ] ,

=

[ ( t ,x): t

aE,

== [ ( t ,x):

aE,

=

t o , &(to) < x < &to)], < t < 03,x = &(t)],
=

to

[ ( t , %): t ,

Let us first prove a lemma that is very useful in the subsequent discussion.

LEMMA 9.1.1.

Assume that

(i) m E C [ E ,R ] , and m(t, x) possesses partial derivatives (not necessarily continuous) on E and total derivative on aE, u aE, ; (ii) if ( t l , xl)E E, , " ( t 1 , xl) = m,(t, , xl) = 0, then m,(tl , xl) < 0; (iii) if ( t , , xl)E a E 2 , m(t, , xl) = 0, and m,(t, , x,) 0, then

<

mdt, , XI)

(iv) if ( t , , xl)

E

9

< 0;

Xl)

aE, , m(t, , xl) = 0, and m , ( t l , x,) 3 0, then m d t , x1) 7

(v) m(t, x)

+ 4 t l ) %(t,

+

P'(t1)

mdt, , x1)

< 0;

< 0 on aE, .

Under these assumptions, m(t,x)

<0

on

(9.1.1)

E.

Proof. Suppose that the inequality (9.1.1) is false. Then, by assumption (v) and the continuity of m(t, x) on E, it follows that there exists a point ( t l , x,) E E , t , > t, , such that m ( t , , x,) = 0

Evidently, m(t, , x) situations arises:

and

m(t, x)

< 0,

t

< t, ,

( t , x) E E .

< 0 for (tl , x) E E , and, hence, one of the following <

, XI) E aJ% and m,(t,, XI) 0; (b) ( t l , x1) E and m,(t,, XI) 3 0; (c) ( t l , xl) 6 aE, u aE, and m,(t, , x,) = 0. (a) (tl

(9.1.2)

Consider the case (a). Because of condition (iii), we have

9.1.

115

PARTIAL DIFFERENTIAL INEQUALITIES OF FIRST ORDER

On the other hand, it turns out, by the relation (9.1.2), that

which is absurd in view of (9.1.3). Similarly, in the case (b), we obtain

because of condition (iv). Again, we get a contradiction to (9.1.4), since the relation (9.1.2) also yields

I n the last case (c), the contradiction results from assumption (ii). This proves the truth of the inequality (9.1.1). REMARK9.1.1. Notice that it is necessary for the proof only that the derivatives mt , m, or the total derivative exist at such points (t, x) for which m(t, x) = 0. We shall now prove some basic results in partial differential inequalities of the first order.

THEOREM 9.1.1. L e t f E C [ E x R x R, R ] , u, ‘u E C [ E ,R ] , and suppose that the following conditions hold: (ao) u, ‘u posses derivatives u L,‘ut ,u, ,u‘ , on E (not necessarily continuous), and the total derivative on aE2 u aE, ;

(ad

<

f(t, x, u, %), v t > f(t, x, v, v,) Ut

on

E;

(a2)f ( t , x,z , p ) - f ( t , x,z , q) 3 - a ’ ( t ) ( p - q), p (a31 f ( t , x,.,p) - f ( t , x,z , 4 ) -P’(t)(p - q), P (a4) u(t, x) < v(t, x) on aE, .

<

Then, u(t,x)


on

3 q, on aE2 ; 3 9, on aE3 ;

E.

Proof. It will be shown that the function m(t, x) defined by m(t, x) = u(t, x)

-

n(t, x)

(9.1.5)

116

9

CHAPTER

verifies the hypotheses of Lemma 9.1.1. Clearly, assumptions (i) and (v) are satisfied in view of (ao)and ( a 4 )of the theorem. Furthermore, m(t, x) fulfills condition (iv) of Lemma 9.1 . l . For, let ( t l , xl)E aE, ,m(t, ,xl)= 0, and m,(t, , xl) 3 0. Then, we have

4f, 4

“(t, , x1)

?

%(tl , x1)

and

2 %(tl , 4.

Consequently, by the condition (a,), we derive f(tl x1 1 4 t l

<

>

-f(h

uz(t1 , x1))

.%)Y

-P’(fJu,(t,

x1) -

9

%it, 7

>

x1 9 4 t l > 1 XI), t i%

%)I.

>

4)

I t now results, using the inequalities (a,), that 4 f l

?

x1) -

vdt,

9

XI)

<

-P’(t1>[uz(t,

, x1)

- %(tl

,%)I,

which implies that %(tl

?

x1)

i

P’(t1) %(tl

> Xl)

< 0.

This verifies condition (iv) of Lemma 9.1.1. T h e remaining assumptions may be similarly checked, and the conclusion (9.1.5) is immediate by the application of Lemma 9.1.1. T h e proof is complete. I t is clear, from Theorem 9.1.1, that one of the inequalities (al)must be strict for the validity of the claim (9.1.5). T h i s can, however, be dispensed with, if the function f satisfies a uniqueness condition. T o that effect, we present the following.

THEOREM 9.1.2. Let the assumptions of Theorem 9.1.1 hold, except that the differential inequalities (al) are replaced by

Suppose further that f(t, S , ui

>

P) p f ( t , X,

uz

>

P)

< g ( t , ui 4,

~1

> ~2

9

(9.1.6)

where g E C [ J x R, ;R ] , and the maximal solution r(t, to , 0 ) of y’

= g(t,y),

At”) = 0

(9. I .7)

is identically zero. Then, u(t, x)

< v(t,x)

on

i3E,

(9.1.8)

9.1.

117

PARTIAL DIFFERENTIAL INEQUALITIES OF FIRST ORDER

implies u(t, x)

< v(t, x)

Proof. Consider the solutions y(t, differential equation Y’

for all sufficiently small

+

= &,Y) E

on

=

€)

Y(Q

E,

> 0, which

u(t, x,

= y ( t , t,

E)

=

exist for t

u(t, x)

-

(9.1.9)

E.

, 0,

E)

of the ordinary (9.1.10)

E,

2 t, . Define (9.1.1 1)

y ( t , €).

Then, because of the relation

Y V ?4 > g(t, Y ( 4 E ) ) ,

we obtain x,

€1 = u,(4 x)

> 0, it follows,

Y’(t, .)

44 4,U , ( t , 4)- g(t, Y ( 4

< f(t,

Since y(t, e )

-

x>

€))-

(9.1.12)

from (9.1.1l), that

u(t, x,€1 < 44 x),

and hence, by (9.1.6), there results the inequality f ( t , x, u ( t , X ) > % ( t ,

4)-f ( 4 X? 44 x, .),

in view of the fact that u,(t, x, E) yields a&, x,

€1
= u,(t,

%44 x,

6))

d g(4 r(t,€>>,

x). This, together with (9.1.12),

x, u(t, x,

€1, %(C

x,

€1).

Moreover, we have also the inequality u(t, x,

6)

< v(t, x)

on

aE,.

Hence, an application of Theorem 9.1.1 shows that u(t, x,

E)

< v(t, x)

on

E.

Since, by assumption, lim,,,y(t, E) = 0, the stated result (9.1.9) is an immediate consequence of (9.1.11). Th is completes the proof. Consider the s’pecial case f ( t , x, u, u,)

= F ( t , x,

u ) - A(t, x) u,

.

Let a ( t ) , P ( t ) be any functions satisfying the ordinary differential inequalities

118

CHAPTER

9

such that a ( t ) < P(t). Suppose that we define the set E by means of these functions a ( t ) , /3(t). Then, we have the following corollaries. COROLLARY 9.1.1. Let F E C [ E x R, R ] ,X E C [ J x R, R ] ,u, E C[E,R], and the assumptions (ao), ( a l ) , and ( a 4 ) of Theorem 9.1.1 hold with f = F ( t , x, u) - X ( t , x) u, . Then, (9.1.5) is valid. COROLLARY 9.1.2.

Suppose that

F ( t , -r, %) - F ( 4 x, u2)

< g(4 u1

- uz),

u1

> u2,

where g(t, u ) satisfies the same assumptions of Theorem 9.1.2. Assume further that the hypotheses of Corollary 9.1.1 hold except that the inequalities (al) are replaced by (-al*) of Theorem 9.1.2. Then, (9.1.8) implies (9.1.9). For the proof of the foregoing corollaries, it is sufficient to show that the conditions (a,), ( a 3 ) of Theorem 9.1.1 are fulfilled. Verification of this fact is trivial in view of the choice of functions a(t) and /3(t). Suppose, on the other hand, we assume that If(4

-r,

u, P) - f ( 4 x, u, 411


-

q I.

(9.1.13)

+

Choose n(t) a - L(t - to), P ( t ) = --a L(t - to), and y < aL. Let to t < to y . Then, we may define the set E, as before, by means of these functions and the time interval. When (9.1.13) is verified, it can be easily checked that f satisfies the conditions (a2),(a3)of Theorem 9.1.1. From this observation stem the following corollaries.

<

+

:

COROLLARY 9.1.3. Let the hypotheses of Theorem 9.1.1 hold except the conditions ( a 2 ) and (a3). If f verifies the inequality (9.1.13), the conclusion (9.1.5) is valid. COROLLARY 9.1.4. Let the hypotheses of Theorem 9.1.2 hold except the conditions ( a 2 ) and (a3). I f f verifies the condition (9.1.13), then (9.1 .S) implies (9.1.9). 9.2. Comparison theorems

I n dealing with the applications of ordinary differential inequalities to partial differential equations, we have to estimate the solutions of such equations, which are functions of several variables, by functions of one variable. While doing so, the following lemmas prove to be useful.

9.2.

119

COMPARISON THEOREMS

LEMMA 9.2.1. Let G be a bounded open set G C R". Let u E C [ f x G, R], and let M ( t ) = max,.c u(t, x). If M(t,) = u(t, , x,) and au(t, , xo)/at exists, for (to, x,,)E (0, co) x G, then

< au(t0, x,>/at.

D-m(t,)

Proof.

(9.2.1)

Choose a sequence {t,}, so that tk < to , t, + to as k + co, and (9.2.2)

For k sufficiently large, we have (9.2.3)

On the other hand, by the definition of M ( t ) and the fact that M(t,,) = u(t, , x,,), for k sufficiently large, it follows that

which, on account of the relations (9.2.2) and (9.2.3), yields (9.2.1).

LEMMA 9.2.2. Let G be a bounded open set such that G C R n and C[J x G, R].Let

uE

w ( t ) = max I u(t, x)I, XtG

M ( t ) = max u(t, x), XEG

and N ( t ) = maZ[-u(t, x)]. X€G

Let ( t o ,xo)E (0, m) x G. Then,

(1") w(t,) > 0 implies either w(t,) = M(to)or w(to) = N(t,,); (2") eu(to) > 0 and w(t,) = M(t,) implies D-w(t,) < D-M(t,,); (3") w(to) 0 and w(t,,) = N(t,,) implies D-w(t,) D-N(t,,).

<

>0

Proof. Suppose that w(t,) the inequality U(to

, x)

< I u(to

9

.)I

and w(t,,)

=

u(t, , x,,). It follows from

< w(to) == ~ ( t,,xo),

x E G,

I20

9

CHAPTER

, If, on the other hand, we suppose that that M(to)= ~ ( t , xo). w(to) = -u(t0, xo) > 0, the inequality -u(t,

7

x)

< I u(t0

3

)I.

<

to)

=

to, xg),

x

E G,

shows that N ( t o )= - u ( t 0 , x,,). Assume now that w(to)= M(to),and choose a sequence {tk} such that t, < to , t, -+to as K + co, and

Since we have M(t,)

< +k)

and

M(t0) = w(Q,

there results the inequality

<W

-4 t O )

4 t k )

t,

-

k ) - M(t0)

1 ,

I

tk -

and, consequently, it follows that D-w(t,)

< D-M(t,).

T h e proof for the case (3") is analogous. We shall prove the following simple comparison theorem.

THEOREM 9.2.1. Suppose that continuous on Q, where

uE

C[Q,R],&/at, auldx exist and are

a = [ ( t , x): to < t < to + a, I x I < /3 and /3

> Ma.

< M I W a x I + g(t, I u I)

+

where g E C [ [ t o to , a ) x R, , R,] r ( t , t, ,yo)is the maximal solution of Y'

Y(t0) = Yo

=dt,Y),

9

)I.


for

I 4 4 41 < r(t, t o Yo) 3

on

Q,

(9.2.4)

and g ( t , 0) = 0. Assume that

< t < to + a. Then, I+,

implies that

M ( t - to)],

Let

1Wat I

existing on to

-

>0

Ix I on

(9.2.5)


(9.2.6)

Q-

(9.2.7)

9.2. Proof.

COMPARISON THEOREMS

121

Consider the function m(t) =

max

1x1<&M(t-t,)

1 u(t, x)I,

and assume that r ( 4 to ,Yo)

<4 t ) .

This means that t > to and m(t) > 0. Moreover, m ( t ) = u(t, xo). T h e point ( t , xo) may be an interior point of 9 or x,, = & [ p - M ( t - to)]. Suppose that (t,xo)E int Q. Then au(t, x,,)/ax = 0, and, as a consequence, by (9.2.4) and Lemmas 9.2. and 9.2.2, we get

Suppose now that xo = fi - M ( t - to).Then, there are two possibilities: (a) 4 t ) = u(t, B (b) m(t) = -u(t,

-

M ( t - to)), M ( t - to)).

-

Consider the case (a). If x is sufficiently close to xo = p - M ( t - to), u(t, x) ,< u ( t o ,x,,), and, therefore, we get

B

-

w

ax

-

to))

>, 0.

As a result, we obtain the inequality (9.2.8)

at ( t , x,,). Furthermore,

U(S,

fi - M ( s - to))< m(s),s < t, and

Hence, using (9.2.8), we deduce that

< I-Mat I ax aU

aU

122

9

CHAPTER

Suppose that the possibility (b) holds. Notice that -u(t, /3

Hence, - u ( t , x)

< -u(t,

-

M(t

p

-

-

to)) = max 1 u(t, .)I.

M ( t - to)),and, thus,

Since

we get (9.2.9)

at

(2,

xo). On the other hand, --u(s,

p

-

M(s - to)) < m(s),

s

< t,

and -u(t, /3

-

M(t

-

to))= m(t),

which implies that D-m(t)

<

au

--

at

+ M ax -

(9.2.10)

at ( t , xo). By the relations (9.2.9) and (9.2.10) and the fact au

au

-+M-< at

ax

I at I au

-

au ax

+M-

results the inequality D - 4

< At, m(t)).

T h e proof for the case xo = - [ p - M ( t - to)] is similar. Thus, we have shown that the inequality r ( t , to , y o ) < m(t) implies that D-m(t)

< g(t, m(t>).

<

This yields, by Theorem 1.4.1, that m ( t ) r(t, to ,yo), contradicting our supposition. Hence (9.2.7) is true, and the theorem is proved.

We shall next prove a more general result whose proof is similar to the one just presented.

9.2.

123

COMPARISON THEOREMS

THEOREM 9.2.2. Letf E C [ E x R, x R, , R,], u E C [ E ,R],u possess partial derivatives on E and total derivative on aE2 u aE, and on E, (9.2.11)

Assume further that aE, , (9.2.12) aE, , (9.2.13) (9.2.14)

(9.2.15)

where r(t, t o , yo) is the maximal solution of (9.2.5), existing for t

> to .

Proof. Let us define m(t) = maxm(t)GzGo(t) I u(t, .)I and suppose that r(t, t o ,y o ) < m(t). This implies, as before, t > t o , m ( t ) > 0 for t > t o , and there is an xo for each t such that m(t) = I u(t, xo)l. Suppose that the point ( t , xo) E E, . Then, au(t, xo)/ax = 0, and, as a result, D-m(t)

<

1 g 1 Gf(4

xo

9

I u(t, X0)L I 4, X0)l)

>

X0)L

0)

XoN

&l

= At, m ( t ) ) ,

by (9.2.11), (9.2.14), and Lemmas 9.2.1 and 9.2.2. Suppose that ( t , xo) E aE, , t > t o . Then, there are two possibilities: (a) (b)

4 4 = u(t, B(t)), Nt)= - 4 t , &I).

Let us consider the case (a). We have u(t, 4

d 4 4 xo),

124

CHAPTER

if x is sufficiently close to x,

= P(t),

9

and, therefore,

As a result, it follows that

Also, for

s

< t, u(s, P(s))

< m(s)

I u(t, B(t))l

and

=

m(t).

Hence, we deduce, using (9.2.12), (9.2.13), and (9.2.14), that D_m(t)

< D-

I u(t, B ( W

If (b) holds, we notice that -4t,

and hence au(t, “,,)/ax

4

< -44 P(%

< 0. Consequently,

On the other hand, -u(s, P(s))

< wz(s),

s

< t,

and

-u(t, ~ ( i ) =.m(t). )

This implies, as before, using (9.2.1 1), (9.2.12), and (9.2.14), that D-m(l)

<

au

- - - f3’(t) -

at

au ax

:1 I

<

aU

- - P‘(t) ax

9.2.

125

COMPARISON THEOREMS

It therefore follows that, in any case, the assumption r(t, to , y o ) < m ( t ) implies the differential inequality D-m(t) ,< g(t, 49).

T h e proof for the case ( t ,x,,) E aE, , t > to , is similar. By Theorem 1.4.1, we have m ( t ) r(t, t,, ,yo), which contradicts the assumption m(t) > r(t, to , y o ) , Hence, the conclusion (9.2.15) is valid, and the proof is complete.

<

Theorem 9.2.2 may be proved by reducing it to Lemma 9.1.1 as follows.

Secondproof of Theorem 9.2.2. Let y ( t , c) = y(t, to ,yo , c) be a solution of (9.2.16) Y' = g(t,y) c, y(t0) = yo E,

+

for sufficiently small

E

+

> 0. It is then

enough to show that

< Y ( t ,4

1 4 4 .)I

on

E,

since lime+,, y(t, e ) = r(t, to ,yo). Let us first show that


u(t, x)

on

c)

E.

Set m(4 4 = 44 -4 - Y ( t , 4

so that

m(t,x)

<0

on

Suppose that (tl , xl)E aE, , m(tl , xl) implies 4 t l , .1) = Y(t1 , 4

aE,.

= 0,

and

>0

and m,(tl , xl) >, 0, which %(tl,

x1)

3 0.

Hence, using (9.2.11), (9.2.13), and (9.2.14), we arrive at the inequality mdt1

?

Xl) = U d t l p.1)

G f(tl > x1

YYtl

~

9

4tl

,<

-S'(t1)

=

-S'(t1) %(tl

mdt1 9 x1)

> 21)) -

.1>?

, x1)

%,ti

which leads to

9

> c)

, x1)

g(t1 > YO1 >

c

~

~

+ S'(t1)

€7

>

x1)

< 0.

6))

~

6

126

CHAPTER

9

It turns out that we have so far verified conditions (i), (v), and (iv) of Lemma 9.1.1. T h e remaining assumptions of Lemma 9.1.1 may easily be checked. This implies, by Lemma 9.1.1, that u(t, x)

< y(t,E )

on

(9.2.17)

E.

+

Defining n(t, x) = - [ y ( t , E) u(t, x)] and proceeding in a similar way, we can show that n(t, x) satisfies the hypotheses of Lemma 9.1.1. Consequently, we obtain - u ( f , x)

< y(t,E )

on

E.

(9.2.18)

T h e relations (9.2.17) and (9.2.18) together imply that

I 44 .)I < Y ( t , .)

on

E,

and this proves the theorem, as observed earlier. Finally, we state the following useful comparison theorem.

THEOREM 9.2.3. Let m E C[E,R + ] , ~CE[ E x R, x R, R ] , satisfying the following assumptions: (i) m(t, x) possesses continuous partial derivatives ut , u, on E and total derivative on aE, u aE, ; (ii) for ( t ,x) E E, (9.2.19)

(iii) the relations (9.2.12), (9.2.13), and (9.2.14) hold. Then, the inequality m(t, x) 4 44

< y o on aEl implies

< r(4 t o , yo)

on

E,

where r ( t , to ,y o ) is the maximal solution of (9.2.5). Proof.

Consider the function v(4 x)

=

m ( 4 x)

-

Y ( 4 4,

where y(t, E) is any solution of (9.2.16) for small E > 0. We wish to show that v ( t , x) satisfies the conditions of Lemma 9.1.1. Clearly, assumptions (i) and (v) of Lemma 9.1.1 hold. Suppose that ( t l , xl>E aE, , n(tl , xl)= 0, and nz(t, , xl) 0, which assures that

9.3.

127

UPPER BOUNDS

m,(t, , xl) >, 0. Hence it follows, in view of the assumptions (9.2.13), (9.2.14), and (9.2.19), that

744 , x1)

Y’(4 4 x1 > 4 4 > X 1 h %it1 < -B’(Q %it, ,x1) - 6 = -P’(h> 4 1 , x1) - E,

=4

1

9

Xl) -

7

>

- g(t1 >

x*))

Y(tl

%

€1) -

from which we infer the inequality 4 t l 9 x1>

+ B’(tl>%(tl

9

x1)

< 0.

This verifies condition (iv) of Lemma 9.1.1. Since it is easy to verify the other conditions of Lemma 9.1.1,the proof of the theorem is complete by the application of Lemma 9.1.1.

COROLLARY 9.2.1. Under the assumptions of Theorem 9.2.3, if m(t, x) 0 on aE, and if the maximal solution r(t, t o , 0) is assumed to 0 on E. be the trivial solution of (9.2.5), then m(t, x)

<

<

9.3. Upper bounds

I n this section, we shall obtain upper bounds of solutions of partial differential equations of first order: Ut =

(9.3.1)

f(4 x, U, %).

DEFINITION9.3.1. Any function u(t, x) is said to be a solution of (9.3.1) if the following conditions hold: (i) u E C[E,R], u(t, x) possesses continuous partial derivatives on E, and total derivative on aE, u aE, ; (ii) u(t, , x) = +(x), where $(x) is continuous on ..(to) x P(t,,); (iii) u(t, x) satisfies the equation (9.3.1) for ( t , x) E Eo .

< <

The following theorem gives an estimate on the growth of solutions of Eq. (9.3.1).

128

CHAPTER

Assume that f(4

9

< g(t, u),

x, u, 0 )

(9.3.4)

2 0,

u

where g E C[J x R, , R] and g(t, 0) = 0. Suppose that r(t) = r(t, to , y o ) is the maximal solution of (9.2.5) existing for t >, t o . If u(t, x) is any solution of (9.3.1) such that

< Yo

l4(x)I we have

on

< r(t)

I u(t, x)i

>

(9.3.5)

E.

(9.3.6)

aEi

on

Proof. Let u(t, x) be any solution of (9.3.1) satisfying (9.3.5), and let y ( t , e ) = y ( t , t o ,y o , e ) be a solution of (9.2.16) for small E > 0. Since lim,,,y(t, c) = r ( t ) , it turns out that it is enough to show that 1 4 4 41 < Y ( 4 4

on

(9.3.7)

E,

to prove (9.3.6). In fact, we prove -u(t, x)

< y(t, 6)

and

u(t, x)

< y ( t , c)

on

E,

so that (9.3.7) holds. First of all, consider the function m ( t , x)

=

u(t, x) - y ( t ,

€).

It is clear that conditions (i) and (v) of Lemma 9.1.1 are satisfied. Let us suppose that (tl , xl) E

aE, ,

m(tl , xl)

=

0,

and

m,(tl

, xl)

0.

(9.3.8)

Hence, it follows, on account of (9.3.3) and (9.3.4), that m*(t,, x1)

x1)

= Udtl =f(t1

Y

x1

-

Y’(h > .)

44

!

x1>, %(tl ? x1))

<

pP‘(t1)

%it, x1) - €

=

-P’(Q

m,(t,

- g(t1 >

Y(tl

>

4-E

9

9

.1>

- 6.

Consequently, we have

4 4

> XI)

+ P’(t1)

%(tl

> XI)

<

0 7

verifying condition (iv) of Lemma 9.1.1. T h e verification of the remaining assumptions is similar. Thus, by Lemma 9.1.1, u(t, x)

< y ( t , C)

on

E.

9.3.

129

UPPER BOUNDS

We consider next the function

4,4 = -[Y(4 4

+

UP, XI19

and, proceeding as before, it can be shown that m(t, x) satisfies the hypotheses of Lemma 9.1.1, so that

< y(t, E )

-u(t, x)

on

E.

T h e proof is therefore complete, as observed earlier.

COROLLARY 9.3.1. T h e conclusion of Theorem 9.3.1 remains true even if we replace (9.3.4) by

If@,

< g@,

0)l

x,

I I).

THEOREM 9.3.2. Suppose that, instead of (9.3.4), we have the weaker assumption

f(4

3,

u, 0 )

< g(t7

4

(9.3.9)

7

for u E Q, where SZ is defined by Q

= [u: r ( t )

< u < r ( t ) + E o , t >, to]

for some e0 > 0. Then, the assertion of Theorem 9.3.1 is valid.

Proof. Let [ t o , TI be any given compact interval. Then, by Lemma 1.3.1, the maximal solution r(t, e) = r(t, t o ,y o , E ) of (9.2.16) exists on [ t o , TI, for all sufficiently small E > 0 and lim r(t, E ) = r ( t ) , €4

uniformly on [ t o , TI. As a result, there exists an E* > 0 such that r(t, E ) < r(t) c0 for t E [ t o , TI. Furthermore, we have, by Theorem 1.2.1,

+

which implies the inequality r ( t ) < r ( t , .)

< r(t)

+

t

€0,

E

(9.3.10)

[ t o , TI.

T o prove (9.3.6), it is sufficient to show that

I aft, )I.

< r(t, c )

on

E.

We proceed to show, as in Theorem 9.3.1, that u(t, x)

< r(t, c )

and

-u(t,

x)

< r(t, c )

on

E.

130

CHAPTER

9

Following the proof of Theorem 9.3.1, we wish to verify the assumptions of Lemma 9.1.1. I n verifying assumption (iv) for m(t, x) = u(t, x)

-

r(t, 6 ) ,

we arrive at the step u(tl

,

=

~ ( t,,c)

uz(t, , xl)

and

2 0,

analogous to (9.3.8). This, together with (9.3.10), leads to the fact u = ~ ( t, ,xl) E Q, and, hence, we use the weaker assumption (9.3.9) to show that mdt,

4 -tP'(t1) %At, , x1) < 0,

7

as before. With this change in the argument, the rest of the proof is similar. Instead of the maximal solution of (9.2.5), if we know a positive solution of the differential inequality Y'

R(CY),

Y(t0) = Yo

> 0,

(9.3.1 1)

it is easy to prove the following result.

THEOREM 9.3.3. Let f~ C [ E x R x R, R], and let the boundary conditions (9.3.2) and (9.3.3) hold. Assume that y ( t ) > 0 is a solution of (9.3.1 1). Suppose that f ( t 7

x, u, 0)

< g(t, u),

(9.3.12)

for u E Q", where L?*

2 to].

[u: u = y ( t ) ,t

Then, (9.3.5) implies

I u(t, x)l < y ( t ) Proof.

on

E.

All that is necessary to show is that u(t, x) iy ( t )

and

-u(t, x)


on

While verifying the assumptions of Lemma 9.1.1, with

E.

9.3.

131

UPPER BOUNDS

This implies that u = u(t, , xl)E Q*, and, hence, we can use (9.3.12) to verify the hypotheses of Lemma 9.1.1. With this observation, the rest of the proof runs similar to the proof of Theorem 9.3.1. T h e next theorem allows us to estimate the difference of any solution

of (9.3.1) and any solution of

ZIt

THEOREM 9.3.4. Letf, F

E

(9.3.13)

= F ( t , x, ZI, ZIJ.

C [ E x R x R, R], and

whereg E C [E x R, , R,] andg(t, 0) = 0. Assume that r(t) = r ( t , to , y o ) is the maximal solution of (9.2.5) existing for t 3 t o . Suppose that

and

Let u(t, x), v(t, x) be any two solutions of (9.3.1) and (9.3.13), respectively, such that u ( t o ,x) = +(x), v ( t o ,x) = $(x), on aE, , and

Under these assumptions,

I u ( f ,x) - v(t, x)I

< r(t)

on

E.

T h e proof is very much the same as the proof of Theorem 9.3.1.

COROLLARY 9.3.2. T h e function F ( t , x, u, u,) = f ( t , x,u, u,) is admissible in Theorem 9.3.4 to yield an estimate for the difference of any two solutions of (9.3.1). I n fact, instead of (9.3.14),

is sufficient. We shall next consider a priori bounds for solutions of (9.3.1) in terms of solutions of another partial differential equation. Sometimes, this has an advantage of obtaining sharper bounds.

132

CIIAPTER

THEOREM 9.3.5.

Let f E C [ B x R

9

x R, R], and

I f(t>2 , 1 4 %)I cz G(t,x, I I, I u, I), where G E C [ B x R, x R+, R , ] .Suppose that G(t, v, z,P)

G(t, v,

3 -u’(t)(p

2, q)

-

p 3 q,

q),

on

aE,,

(9.3.15)

and

where g E C [ J x R , , R ] , and the maximal solution r(t, t o , 0) of (9.1.7) is identically zero. Let u ( t , x) be a solution of (9.3.1) and x ( t , x) 3 0 be a solution of Zt =

(9.3.18)

G(t,x,Z, %),

such that zs(t, x) >, 0 and

< z(t, x)

1 u(t, x)l

on

aE,

Under these assumptions, we have j u ( t , )I.

Pmof.

<<

x ( t , x)

on

E.

Let u(t, x) be any solution of (9.3.1). We shall show that

I u ( t , x)I

--I z ( t , x)

+ y(t,e)

on

E,

where y ( t , E ) y ( t , to , 0, c) is any solution of (9.1.10), for sufficiently small E > 0. First of all, define the function ~

nz(t, x) = u(t, x)

-

z(t, x) ~- y ( t , c ) .

Evidently, m(t, x) 0 on aE, . Suppose that ( t l , xl) and m,(tl , xl) 3 0. We then derive that and

“(4 , %)

~

%(tl

Z(t,

,4

E

, x1) = Y(tl , 4 > 0

> z,(h , 4.

aE, , m ( t , , xl)

= 0,

9.3.

I33

UPPER BOUNDS

It therefore follows that

verifying condition (iv) of Lemma 9.1.1. T h e other assumptions of Lemma 9.1.1 may be checked similarly. Thus, by Lemma 9.1.1, we have

O n the other hand, defining

it is easy to show that n(t, x) satisfies the hypotheses of Lemma 9.1.1, and, consequently, -u(t, x)

< z(t,x) + y(t,

on

C)

E.

It therefore turns out that Iu(4 I).

since lim,,,y(t,

e)

<

4

on

E,

= 0. T h e proof is complete.

T h e following theorem may be proved, using the arguments similar to the preceding one with necessary changes.

THEOREM 9.3.6. Let f,F

E

C [ E x R x R, R ] , and

lf(4 x,u, us) - q t , x,a,%)I

< G(t,x, I u

a

I, I us

- us

I),

where G E C [ E x R, x R, , R,] satisfies the conditions (9.3.15), (9.3.16), and (9.3.17) of Theorem 9.3.5. Suppose that the maximal solution r ( t , t o , 0) of (9.1.7) is identically zero. Let u(t, x), v(t, x) be any

134

CHAPTER

9

solutions of (9.3.1) and (9.3.13), respectively. Then, if z(t, x) >, 0 is the solution of (9.3.18) such that

1 u(t, .Y) we have

-

1 u(t, x)

v(t, x)I

-

on

S z(t, X)

< z(t,x)

v(t, x)i

aE, ,

on

E.

9.4. Approximate solutions and uniqueness We shall consider the partial differential inequality

< 6.

I ut -f(4 x,u, u,)l

(9.4.1)

DEFINITION 9.4.1. A function u(t, x,6) is aid to be a &approximate solution of (9.3.1) if (i) u E C [ E ,K],and u(t, x, 6) possesses continuous partial derivatives LJ aE, ; /3(to); (ii) u(to, x,6) = (b(x), where +(x) is continuous on .i(to) x (iii) u ( t , x,6) satisfies the inequality (9.4.1) on E, .

on E, and total derivative on aE,

< <

T h e following theorem estimates the error between a solution and a &approximate solution of (9.3.1).

THEOREM 9.4.1.

LetfE C [ E

f ( 4 x,u , P ) where g E C[J

-f(t,

x R x R, R ] , and satisfy

x, 'u, P )

< g(4 u

-

(9.4.2)

> u,

n),

x R, , R] and g(t, 0) = 0. Assume that r(t, 6 )

=

r ( t , t o , 0,s)

is the maximal solution of Y'

existing for t

-I- 6 ,

Y(b)

=

0,

3 to , and the inequalities

f(f,x,z , P) - f ( f , f ( f , .y,

= R(4Y)

x, z , 9 )

z , P ) -f(t, x,2, 4)

2 -.'(t>(P

-< -P'(t)(P

-

-

91,

P >, 9,

on

aE,, (9.4.3)

4),

P 2 4,

on

aE,,

(9.4.4)

hold. I,et u(t, x), u(t, x,6) be a solution and a &approximate solution of (9.3.1). Then,

I u ( t , x)

-

u(t, x, S)l

< r(t, S)

on

B.

(9.4.5)

9.4. Pyoof.

APPROXIMATE SOLUTIONS AND UNIQUENESS

135

Consider the function m(t, x)

where y ( t , e )

=

4 4).

6,

c)

= y ( t , to , 0,

-

44 x, 8) - y ( 4 e ) ,

is a solution of

Y’ = g(4y)

+8 +

Y(td

6,

= e,

for small E > 0. Suppose that, for (tl ,xl) E aE, , m ( t , , xl) = 0 and , xl) 3 0. This shows that

m,(t,

> 4 t l , x1 , 8 )

4 t l 7 x1)

and

%it,

I

2 %it1

4

9

x1 , 8).

Thus, using (9.4.2) and (9.4.4), we have mt(t1

9

44

x1) = f(tl x1 7

-f(tl

f

7

+f@l 1

x1 9

P

Xl),

udt1 9 x1>>

U ( t l7 ’1

4% x1 9

UZ(tl

J

7

?

- g(t1

< -B’(t1)

,A t 1

9

%(tl

so that %(f,

I

-f@1 x1 4)- E - 6 8)

x1

Xl>

9

9

x1)

-

.1>)

61, %it, , .1>>

- f ( G , x1 > 4 t , > x1 , s>,%(tl -

9

, x1 , 8)) 9

4 t l I x1 7 8 ) , 4 1 , x1 9 8))l

€9

+ B’(tl>

,x1)

< 0.

This proves condition (iv) of Lemma 9.1.1. It is easy to show that the other assumptions of Lemma 9.1.1 also hold. Hence, by Lemma 9.1.1, ~ ( tx),

-

u(t, x,

6)

< y (t , e )

on

E.

on

E.

Proceeding similarly, we can show that ~ ( tX,, 8 )

-

u(t, x)

< y ( t ,E )

T h e estimate (9.4.5) results immediately, noting that limy(t, 6) €4

T h e proof is complete.

=

~ ( tto, 0,8).

136

9

CHAPTER

COROLLARY 9.4.1. If the function g ( t , u ) = Ku, K takes the form

1 u(t, x)

-

u(t, x,

s)i < (S/K)[exp K ( t - to) - 11

> 0,

then (9.4.5)

on

E.

We next state a uniqueness theorem of Perron type whose proof is an immediate consequence of Theorem 9.4.1 or Corollary 9.3.2.

THEOREM 9.4.2. Let f E C [ E x R x R,R] and the condition (9.4.2) hold. Assume further that the boundary conditions (9.4.3) and (9.4.4) are satisfied. If y ( t ) = 0 is the maximal solution of Y'

= g(4

Y),

Y(to) = 0,

for t >, t o , then the partial differential equation (9.3.1) admits atmost one solution.

THEOREM 9.4.3. Under the assumptions of Theorem 9.4.2, given E > 0, there exists a 8 ( ~ > ) 0 such that I d(x)

~~

$I).(

<6

on

aE,

I u(t, x)

implies

-

v(t, x)l

<E

on

E,

where u(t, x), v(t, x) are any two solutions of (9.3.1) satisfying u(t, , x) = 4 ( x ) , " ( t o ,x) = #(x), respectively, provided the solutions y ( t , t o ,yo) of (9.2.5) exist for t 3 to and are continuous with respect to the initial values (to,yo).

P ~ o o f . By Corollary 9.3.2, we have, choosing Yo

= max

%

l4(4

~

Icr(x)I,

the inequality

I 44 4 - v ( t ,41 < r ( t , t o , Yo)

on

E.

By assumption, given E > 0, there exists 8 = 8 ( ~ > ) 0 such that, if y o S, then ~ ( tt,, , y o ) < E , t 3 to , where r ( t , to , y o ) is the maximal solution of (9.2.5). T h e stated result follows at once.

9.5. Systems of partial differential inequalities of first order Many of the results obtained in the earlier sections of this chapter can be extended to more general systems. I n this section, we shall indicate some basic theorems only.

9.5.

137

SYSTEMS OF INEQUALITIES

Let 01, /3 E C [ J ,R n ] such that a ( t ) < /3(t),t E J. Assume that a’(t), ,G’(t) exist and are continuous on J . For t o e J , we may now define the sets E, E,, , aEl , aE, , and aE,, following Sect. 9.1. First of all, we shall extend the Lemma 9.1.1.

LEMMA 9.5.1.

Assume that

(i) m E C[E,RN] and m(t, x) possesses partial derivatives on E and total derivative on aE, u aE, ; (ii) for some index j , 1 \ ( j N , if (tl , xl)E E,, , mj(tl , xl) = 0, mi(t, , xl) 0, i # j , and m,J(tl , xl) = 0, then m,i(tl , xl) < 0; (iii) for some index j , 1 < j N , if (tl , xl) E aE, , d ( t , , xl) = 0, mi(tl , x,) 0, i # j , and m,j(t, , x,) 0, then

<

<

<

<

<

< < N , if

(iv) for some index j , 1 j mi(tl , xl) 0, i # j , and m,j(tl

<

( t , , xl)

, xl) 3 0, then

c n

mAt,

(v) m(t, x)

> 21)

i-

k=l

BkYtl) mk,(t,

E

aE, , mj(tl , xl) = 0,

, XI) < 0 ;

< 0 on aE, .

Then, we have m(t, x)

<0

on

E.

(9.5.1)

Proof. Suppose that the inequality (9.5.1) is false. Then, the set 2 given by Z

u N

=

[ ( t ,X)

E

E : mi(t, X)

i=l

01

is nonempty. Take the projection of Z on t-axis and denote by t, its greatest lower bound. It follows from assumption (v) that t, > t o . T h e set 2 is closed, and, hence, we conclude that there is a point ( t , , xl)E E and an index j , 1 j N , such that

< <

m y t , , xl) = 0, d ( t , x)

< 0,

t

< t, ,

and, for i # j , m y t , , x)

< 0.

( t ,x) E E,

(9.5.2)

138

CHAPTER

9

Now, one of the following situations arises:

in view of condition (iii). O n the other hand, it follows by (9.5.2) that d

m3(t1 , 4 t l ) )

2 0,

which is incompatible with (9.5.3). Following now the proof of Lemma 9.1.1 and that just given, it is easy to obtain a contradiction for the cases (b) and (c) proving that the inequality (9.5.1) is true. An extension of Theorem 9.1.1 to systems of inequalities is the following.

THEOREM 9.5.1.

the fol'

(i) z1 on aE, (ii)

Let f E C [ E x RN x Rn, R N ] ,u, v ing assumptions: ,assess

+E,;

E

C[E,R N ]satisfy

continuous partial derivatives on E and total derivatives

o n E , i = 1 , 2,..., N ; (iii) for vach i,

UtZ

G f " 4 x,U , U5z),

a?

>f " t ,

x,TI, U,Z)

9.5.

139

SYSTEMS OF INEQUALITIES

(v) u(t, x) < v(t, x) on aEl ; (vi) f ( t , x, u, p ) is quasi-monotone in u for each fixed ( t , x, p ) . These conditions imply that u(t, x)

Proof.

< v(t, x)

on

(9.5.4)

E.

Consider the function m(t, x)

=

u(t, x)

-

v(t,x).

Evidently, m ( t , x) verifies assumptions (i) and (v) of Lemma 9.5.1, on account of assumptions (i) and (v). Suppose that, for some index j , 1 <j N , ( t l , xl)E aE, , we have mj(t, , xl) = 0,mi(tl , xl) 0, i # j , and m,J(tl , xl) 3 0. This implies

5

<

(9.5.5)

Since f ( t , x,u, p ) is quasi-monotone in u, we have, because of relations (9.5.5). the ineaualitv

which implies that

140

CHAPTER

9

This verifies assumption (iv) of Lemma 9.5.1. T h e checking of the remaining conditions is similar, and the stated result follows, by Lemma 9.5.1. Making use of a similar argument, we can prove the following theorem, which generalizes Theorem 9.1.2.

TIIEOREM 9.5.2. Let the assumptions of Theorem 9.5.1 hold, except that condition (ii) is replaced by ulz

vtz

f"t,

x,u, uz",

:f z ( ( t ,x , v , azz)

on

(i L- I , 2 ,..., N ) .

E

Assume further that, for each i, f"t,

x , a, p )

x,u, p ) - f " 4

< g(4

- 4,

uz > az,

where g E C [ J x R, , R ] , and the maximal solution r(t, t o , 0) of (9.1.7) is identically zero. Then, whenever

u(t, .x)

< v(t,x)

on

E

u(t, x)

< v ( t ,x)

on

aE,.

We shall next prove a comparison theorem analogous to Theorem 9.2.3.

THEOREM 9.5.3. Let m E C [ E ,R+N], f that the following conditiQns hold:

E

C [ E x RN x Rn, RN] such

(i) m ( t , x) possesses continuous partial derivatives on E and total derivative on aE, LJ aE, ; I , 2,..., N , (ii) for ( t , x) E B and i (iii)

m t ( t , x )
f ( 4 -t", u, 0)

< !At, 4,

(9.5.6) (9.5.7)

where g E C[J x R+N,R N ] ,g ( t , 0 ) = 0, and g ( t , u ) is quasi-monotone in u for each t E (iv) for each i,

1;

9.5. Then, m(t, x)

141

SYSTEMS OF INEQUALITIES

< y o , on aEl , implies m(t, 4

< r ( t , t o , Yo)

on

B,

where r ( t , to ,yo)is the maximal solution of the differential system existing for t

Y’

3 to .

Proof. Let y(t, E ) differential system

= s(t,Y),

Y(to)

= y ( t , t o ,yo , E )

Y’

= g(t,y)

for sufficiently small vector

E

+

= Yo

Y

be any solution of the ordinary Y(td = yo

6,

+

6,

> 0. Since it is known

lim Y(4 .) t-0

=

that

r(t, t o 7 Yo),

it is sufficient to show that m ( t , x)

< y(t,

on

E)

(9.5.10)

B.

It is easy to verify that v(t,x) = m(t, x) - y ( t , E ) satisfies the hypotheses of Lemma 9.5.1. Clearly, assumptions (i) and (v) hold. Moreover, for an index j , 1 < j N , (tl , xl) E aE, , suppose that d ( t , , xl) = 0, v i ( t 1 ,xl) 0, i f j , and vE:3(tl, xl) 3 0. Then,

<

<

and

, 4 > 0,

mj(t1 , X l ) = Yqtl

mE(t, , x1)

m,J(t1 , x1)

< y“t1 ,

i

E),

#i

t 0.

Thus, using the relations (9.5.6), (9.5.7), (9.5.8), and the quasi-monotone character of g ( t , u), we get m,3(t,, x1)

-


Y

dY’P(t1

, €1

x1 m(t1 Y x1>, mz’(t1 , x1)) 9

-P(t1

, Y(tl 4) - 8 7

, x1 , m(t1 , Xl), mzj(t1 , x1)) - g3(tl , mYt1 , x1), ..., y3(tl , 6),..., mN(tl , xl)) c3 , 0) +f’(tl , x1 , m(t1 , Xl>,O)


-

7

7

-

i?(tl

c

7

9

m(t1 x1)) 9

-

8

n

<

-

=

-

k=l

Pkl(t1)

?

21)

- E3

n

1

k=l

Pk’(t1) 4 # 1

, x1)

-

2.

142

CHAPTER

9

proving that hypothesis (iv) of Lemma 9.5.1 is valid. Arguing similarly, it is easy to verify the assumptions of Lemma 9.5.1, and consequently (9.5.10) is true. This proves the theorem. Following the proofs just presented, the growth estimates of solutions and approximate solutions and a uniqueness result for the systems of the type (i = 1, 2 )...,N ) , u: f y t , x,u, u,i) 1

where f E C [ E x RN x R”, RN],may be proved. We leave it as an exercise to the reader. Finally, we shall give below a theorem on differential inequalities for overdetermined systems of inequalities. For such systems, the time variable t also may be a vector, such that t,O t, < GO, y = 1, 2,..., k. T h e sets E , E, , aE, , aE, , and aE, may be understood with this change. Then, we have

<

THEOREM 9.5.4. Let f E C [ E x RN x Rn, RN], u, v E C[E,RN] satisfy the following conditions: (i) u(t, x), v(t, x) possess continuous partial derivatives on E and total derivative on aE, v aE, ; (ii)

4,< fy((4x, u, u;),

’fy2(4 x,‘u, ‘uri), on E , i = 1, 2,..., N , y (iii) for each i and y,

1, 2,..., k ;

(iv) the functionf(t, x, u, p ) satisfies the quasi-monotone property in

u for each ( t , x,

(v) ~ ( tx),

p);

v(t, x) on aE, .

9.5.

143

SYSTEMS OF INEQUALITIES

Under these assumptions, u(t,x) < v(t,x)

Proof.

on

E.

Introduce Mayer’s transformation t,

where A, >, 0 ( y

=

=

t,O

iA,$,

1, 2,..., k). Define the functions

c(s, x,A ) E(S,

+ As, x), v(to+ As, x).

= .(to

x,A)

=

By assumption (v), it follows that Zi(0, x,A)

< qo, x, A),

and the functions u“, v” are defined on

where s is a single variable. Moreover,

c c

~,%,(tO

+

A,v,,(to

+

k

cs =

ES =

v=l

k

,=l

AS,

x),

AS,

x).

We define also the functions for each i,

so that the inequalities (ii) reduce to

cs < P ( S , x,c, C2), fjs

> Fi(s, x,fi,

on

E.

Evidently, F(s, x, z , p ) satisfies quasi-monotone property in z for each ( t , x, p ) . Furthermore, the inequalities (iii) reduce to

-w,x, z , 4 ) 3 --k 1 n

Fi(S, x,x,P)

and

.j’(t)[Pj

-

%I,

p 3 q,

on

aE,

Pj’(t)[Pj

-

4j1,

P 2 4,

on

aE,.

j=1

c n

Fi(S, x,.,P) -FFi(S,x,z, 4 ) $ --R

j=1

144

9

CHAPTER

I t then follows that the functions u", 5, and F satisfy the assumptions of Theorem 9.5.1, and, consequently,

< 5(s, x, A)

C(s, x, A)

on

E,

and, in particular, for s = 1 , 1(1,x, A)

< $(I,

x, A).

Let, now, ( t , x) be an arbitrary point, and let A

=

(t

-

t,)

=

(tl

-

tl,,

..., t ,

- t,O)

be scuh that A, >, 0. Then, on E, we have u(t, x ) = C(1, x , t

-

to) < 5(l, x,t

~

to) =

v(t, x),

which is exactly the relation we have to prove. This completes the proof. One could formulate and prove analogous results for systems of the type u;, = Fji(t, x,u, u;),

on the strength of Theorem 9.5.4. We shall not attempt such a formulation of the results.

9.6. Lyapunov-like function Let us consider a first-order partial differential system of the form U t = f(t,

x, u, uzi),

(9.6.1)

where f E C [ E x RN x R", R N ] .We wish to estimate the growth of solutions of (9.6.1) by means of a Lyapunov-like function. T o this end, we have

THEOREM 9.6. I .

Assume that

(i) V c C [ J x RN,R , ] , V(t,a ) possesses continuous partial derivatives with respect to t and the components of u, and at

zL)

i ~.f(t, x , u, uZi) a2l

< G(t,x , l,'(t, u), V,(t, u)),

where G E C [ E x R-, x R, R] and V J t , u) = (aV/aa). (&/ax);

9.6.

145

LYAPUNOV-LIKE FUNCTION

(ii) G(t,x,x , p ) - G(t,x,z , 0) 3 -a’(t)p, p 3 0, on aE, ; (iii) G(t, x,x , p ) - G(t, x,z, 0) -,B’(t)p, p 3 0, on aE, ; (iv) G(t,x,x, 0 ) d g(t, x), z > 0, where g E C [ J x R, , R]; (v) the maximal solution r ( t , to , yo) of (9.2.5) exists for t 3 to .

<

Then, any solution u(t, x) of (9.6.1) satisfying

wo

?

4 ( 4 < yo

on

(9.6.2)

aE,

allows the estimate V ( t , 4 4 4)d

t o , Yo)

on

E.

(9.6.3)

Proof. Let u(t, x) be any solution of (9.6.1) such that (9.6.2) holds. Consider the function m(t, x)

=

V(t,u(t, x).

By assump.tion (i), we have

and

< G(4 x,m(t, 4,m,(t, XI), on

m ( t 0 , x)
aE, *

I t is evident that the hypotheses of Theorem 9.2.3 are fulfilled, and, as a result, on E, V ( t >44 XI) = m ( t , 4 < r ( t , to ,yo) where r(t, to ,yo) is the maximal solution of (9.2.5). T h e proof is therefore complete.

THEOREM 9.6.2.

Assume that

(i) the assumption (i) of Theorem 9.6.1 is satisfied; (ii) G(t,x,z, p ) - G(t,x,x, Q) 3 - ~ ’ ( t ) ( p- Q), p 3 q, on aE2 ; (iii) G(t,x,z , p ) - G(t,x,z , p) -P’(t)(p - Q), p 3 4, on a - 4 ; (iv) G(t, x, z1,p ) - G(t, x,z2,P) g(t, z1 - 4,XI 3 Z Z ; (v) the maximal solution r ( t , t o ,0) of (9.1.7) is identically zero.

<

<

Then, if z(t, x) >, 0 is the solution of (9.3.18) such that x(t,, x) = #(x) 3 0 on aE, and V t , > $(XI) < ?4) on >

146

CHAPTER

we have V ( t ,u(t, x))

9

< x ( t , x)

on

E.

Proof. If u(t, x) is any solution of (9.6.1), we obtain, as in Theorem 9.6.1, the inequality am(t’ at

< G(t, X, m(t, x), m,(t, x)).

Since z(t, x) is the solution of (9.3.18), we have

As the hypotheses of Theorem 9.1.2 are satisfied, the conclusion follows immediately, and the theorem is proved. Let us now assume the existence of solutions of (9.6.1). Suppose also that the system (9.6.1) has the trivial solution u = 0. We may then formulate the definition of stability of the trivial solution of (9.6.1). DEFINITION 9.6.1. T h e trivial solution u = 0 of (9.6.1) is said to be stable if, for every E > 0 and to E J , there exists a S > 0 such that 11 +(x)ii < 6 on aE, implies

/I u (4 .)I1 < 6

on

E,

where u(t, x) is any solution of (9.6.1) with u(t, , x) = +(x) on aE, . T h e trivial solution u = 0 of (9.6.1) is said to be asymptotically stable if it is stable and, for every E > 0, to E J , there exist positive numbers 6, and T such that 11 $(x)li < 6, on 8E, implies

On the strength of Theorem 9.6.1, it is easy to state the sufficient conditions for the stability behavior of the trivial solution of (9.6.1). THEOREM 9.6.3. further that

Let the assumptions of Theorem 9.6.1 hold. Suppose b(l/ 2.1 11)

<:

q t ,4

< 41u Il),

(9.6.4)

where a, b E X . Then, the stability or asymptotic stability of the trivial solution of the ordinary differential equation (9.2.5) implies the stability or asymptotic stability of the trivial solution of the system (9.6.1).

9.6.

147

LYAPUNOV-LIKE FUNCTION

Proof. Suppose that the trivial solution of (9.2.5) is stable. Let E > 0 and to E J . Then, given b ( ~> ) 0 and to E J , there exists a 6 > 0 such that y o 6 implies

<

Y(t3 to ,Yo)

< 6,

t

2 to

(9.6.5)

7

where y(t, to ,yo ) is any solution of (9.2.5). By Theorem 9.6.1,

w,44 4)< r(t, to

I

Yo)

on

(9.6.6)

E,

for any solution u(t, x) of (9.6.1), r(t, to ,y o ) being the maximal solution of (9.2.5). Choose a positive number 6, such that a(6,) = 6, and assume that (1 d(x)ll 6, . This implies that

<

V t o>4(4

< 4 $(.)I

< 4%)= 8.

Choose yo = supzsaEIV(to,C#J(x)). It then follows, by the relations (9.6.4), (9.6.5), and (9.6.6), that

4114 4 a)<

w ,4 4 4) < r(4 to ,Yo) < &)

on

E,

which leads to a further inequality

I1 44 )1.I

<E

on

E,

<

provided II+(x)ll 6,. This proves the stability of the trivial solution of (9.6.1). Now, suppose the trivial solution of (9.2.5) is asymptotically stable. Let E > 0 and to E J- Then, given b ( ~ > ) 0 and toE J , there exist two positive numbers 6, and T such that yo 6, implies

<

to ,Yo)

< b(E),

t

+ T.

2 to

As before, we choose y o = supzsaE, V(to, +(x)). Furthermore, let ~ ( 8 , )= 8, and assume that 11 C#J(x)lI . These considerations show that, as previously,

< so

M u ( 4 41) < V(t>44 4)

< r(4 t o

9

Yo) < 44,

+ T and m ( t ) < x < ,B(t).From this follows the inequality t >, t o + T , II u(t, 1)I. < m ( t ) < x < P(t), provided 11 +(x)II < so. It is easy to see that this assures the asymptotic for t 3 to

E,

stability of the trivial solution of (9.6.11, in view of the foregoing proof. T h e theorem is completely proved.

148

CHAPTER

9

Theorem 9.6.2 may also be used to discuss stability properties of the trivial solution of (9.6.1). For this purpose, let us assume that Eq. (9.3.18) possesses the trivial solution and that all the solutions x(t, x) with ~ ( t, ,x) = $(x) 3 0 are nonnegative on E. Then, we can define stability notions with respect to the trivial solution of (9.3.18), noting that all the solutions are nonnegative. DEFINITION 9.6.2. The trivial solution of (9.3.18) is stable if, for every c > 0 and to E J , there exists a 6 > 0 such that x(t, x) < t on E, provided $(x) 6 on aE, . The definition for asymptotic stability may be similarly formed.

<

THEOREM 9.6.4. Let the assumptions of Theorem 9.6.2 hold, and let V ( t ,ZL) satisfy the inequality (9.6.4). Then, the stability or asymptotic stability of the partial differential equation (9.3.18) implies the stability or asymptotic stability of the trivial solution of the partial differential system (9.6.1). 9.7. Notes The results of Sect. 9.1 are due to Plis [6]. Lemmas 9.2.1 and 9.2.2 and Theorem 9.2.1 are adapted from Szarski [8]. Theorems 9.2.2 and 9.2.3 are new. T h e contents of Sects. 9.3 and 9.4 are modeled on the basis of the work of Plis [6] and are new. For the result of the type given in Theorem 9.5.4, see Szarski [S]. T h e other results of Sect. 9.5 are new. Section 9.6 contains new results. For further related work, see Plis [l-51 and Szarski [I-3, 6, 81.

Chapter 10

10.0. Introduction I n this chapter, we shall investigate partial differential equations of parabolic type. First of all, we shall concentrate in obtaining certain results concerning parabolic differential inequalities in bounded domains and comparison theorems connected with such inequalities. We consider different kinds of initial boundary-value problems, obtain bounds and error estimates, and prove uniqueness of solutions. Stability criteria of the steady-state solutions is discussed. Many of the results have been extended to systems of parabolic differential equations and inequalities in bounded domains. Introducing the concept of Lyapunov functions, we give sufficient conditions for stability and boundedness of various types. Criteria for conditional stability and boundedness are discussed in terms of several Lyapunov functions. Regarding the parabolic differential equations in unbounded domains, we have basic results concerning parabolic differential inequalities and uniqueness of solutions. Finally, we treat the exterior boundary-value problem. We have given uniqueness criteria only.

10.1. Parabolic differential inequalities in bounded domains Let H be a region of ( t ,x) space in Rn+l satisfying the following conditions: (i) H is open, contained in the zone to < t < co, to 3 0, and the intersection of H , the closure of H , with any zone to t T is bounded; (ii) for any t , E [ t o ,co), the projection Ptl on Rn of the intersection of R with the plane t = t , is nonempty; and for every sequence {tic} such that (iii) for every ( t l , xl) E

< <

149

150

CHAPTER

tk t [ t o ,GO), t,<- f t , as k xk t P t kand xli ---t x1 as k

-+ co,

---f

10

there is a sequence {xk} satisfying

60.

We shall denote by aH that part of the boundary of H which is contained in the zone to < t < co. It is easy to see that the topological product H = ( t o ,a)x D, where D C Rn is an open, bounded region, satisfies the preceding conditions. Also, the pyramid defined by the inequalities to

< t < t" + T ,

1 xi

-:x

1

< a, -L(t

- to),

i

=

1 , 2,..., n,

where L >, 0, ai > 0, and T = min(ui/L), verifies the requirements of H . On the other hand, H = H , u H 2 , where

< t < 1 , 0 < x < 21, 1 ,< t < 2 , 0 < x < I],

Hl

=

[(t,x): 0

Hz

=

[ ( t ,x):

does not satisfy condition (iii) at the point (1,

g).

Let us prove some basic lemmas, which we shall use frequently.

LEMMA 10.1.1. Assume that (i) m E C [ H , R ] , the partial derivatives m , , m, , mzz exist and are continuous in H ; (ii) m(t, x) 0 on P f 0u a H ; (iii) if ( t l , xl) E Ptl , m ( t , , xl) = 0, m,(t, , xl) = 0, and the quadratic form C,

for arbitrary vector A, then m,(t, , xl)

< 0.

These assumptions imply that m ( t , x) ,0

Bmf.

on

(10.1.1)

H.

Suppose that the inequality (10.1.1) is not true. Then, the set

z = [(t, ).

€

H : m(t, ). >, 01

is nonempty. Let 2, be the projection of Z on t axis and t , We thcn have m(t, x) 0 on IZ n [ t o , t l ] .

=

inf Z , .

10.1.

PARABOLIC DIFFERENTIAL INEQUALITIES

151

We assert that m(tl , x) has a maximum equal to zero for some x1 E Ptl . If this is false, on account of assumption (ii), we must have m(t, x) < 0 for all-x E Ptl . This contradicts the definition of t, . Hence, there is an x1 E int Ptl such that m(t, , xl) = 0. It therefore follows that %(tl

3

4 2 0.

(10.1.2)

Since m(t, x) attains its interior maximum at ( t l , xl), m,(t, , xl) = 0 and the quadratic form

we obtain

for an arbitrary vector A. We then deduce by condition (iii) that mt(t, , x,) < 0, which is incompatible with (10.1.2). Thus, the set 2 is empty, and the inequality (10.1.1) is true. Let m ( t , x) be continuous on aH. We shall denote by aH, that part of the boundary aH on which a(t, x) > 0, that is, aH,

=

[ ( t , X)

E

aH: a(t, X) > 01.

(10.1.3)

Let T be a direction defined at each point of aH, in a continuous manner. We shall say that a direction T at ( t l , x,) E aH, points into the interior of H , if there exists a finite ray with (tl , xl) as its origin, all of whose is interior points lie in H . For any continuous function u(t, x), ~ a directional defined as follows. If T points into the interior of H , a u / a is derivative, that is, (10.1.4)

where (tl , x,) lies on the ray issuing from ( t , x) in the direction of T, the distance from (t, , xl) to ( t , x) being 47. We shall always assume that the direction T points into the interior of H , and we shall take aul3-r to be defined in the sense of (10.1.4). If, in addition, the direction T is orthogonal to the t axis, then it is denoted by T ~ and , the corresponding directional derivative will be given by

3 4 4 X) a70

=

lim u(t, 4

Ar,,+O

-

u(t, x1)

AT^

,

where (t,xl)lies on the ray issuing from ( t , x) in the direction of T ~ the , We . shall use both types of distance between ( t , xl) and ( t , x) being h O directional derivative in the sequel.

152

CHAPTER

10

We are now in a position to prove a variant of Lemma 10.1.1.

LEMMA 10.1.2. Let the hypotheses of Lemma 10.1.1 remain the same except that condition (ii) is replaced by (iia) m ( t , x) < 0 on P l 0 and aH - aHa ; (iib) if ( t l , x,) E aH, and m ( t , , x,) = 0, then a m ( t , , xl)/a7 where, for each ( t , x) E aH, , am/aT is assumed to exist.

< 0,

Then, the conclusion (10.1.1) is valid. Proof. T h e proof is similar to the proof of Lemma 10.1.1, the only difference being that (tl , xl) cannot lie on aH, , in view of condition (iib). For, if ( t , , x,) E aH, , since m ( t , , xl) = 0, we have, by condition (iib), (10.1.5)

On the other hand, from the fact that m(t, x) deduce that

< 0 on A n [ t o ,t l ] , we

contradicting (10.1.5). Let us now consider the partial differential operator given by qu1

= Ut

-f(C x,u, u, , u,,),

where f E C [ H x R x Rn x Rn2,R ] , u, u,, = (a2u/ax12, azu/ax, ax, ,..., a2u/axn2).

=

( 10.1.6)

(au/ax,,..., au/axn), and

DEFINITION 10.1 .l. T h e function f ( t , x, u, P, R ) is said to be eZZiptic at a point (tl , xl) if, for any u,P i , Qik , Ri, (i, k = 1, 2,..., n), the quadratic form

for arbitrary vector A, implies

If the foregoing property holds for every ( t , x) E H , thenf(t, x, u, P, R ) is said to be elliptic in H. Moreover, we shall say that the differential operator T is parabolic iff ( t , x,u, P, R ) is elliptic.

10.1.

PARABOLIC DIFFERENTIAL INEQUALITIES

153

A fundamental result in parabolic differential inequalities may now be proved.

THEOREM 10.1.1. Assume that (i) u, z, E C [ H ,R],the partial derivatives u l , u, , u,, , v 1 , v, , vz, exist and are continuous in H ; (ii) f~ C[H x R x Rn x Rnz,R ] ,the differential operator T is parabolic, and T [ u ]< T[v] on H; (iii) u(t, x)

< v(t,x)

on Pt0u aH.

Under these assumptions, we have u(t, x)

< v(t, x)

on

(10.1.7)

H.

Proof. We consider the function m(t, x)

=

u(t, x)

-

v(t, x).

Clearly, m(t, x) satisfies assumptions (i), (ii) of Lemma 10.1.1. We shall show that condition (iii) of Lemma 10.1.1 is also true. Suppose that (tl , xl) E Ptl , m ( t , , xl) = 0, m,(t, , xl) = 0, and the quadratic form

5

mzizk(tl >

hXk

i,k=l

<

for arbitrary vector A. This implies that (10.1.8)

and

Consequently, using the ellipticity of the function f,we obtain At,

> x1 >

4tI


7

7

4, %(tl

x1 > 4 t l

7

x1>, U,,(tl

,Xl),

>

U d t l > Xl),

x1))

%dtl

, x1)).

This, together with the inequality T[u]< T [ v ] ,yields

154

CHAPTER

10

T h e next theorem allows nonlinear boundary conditions and is, as such, more general than Theorem 10.1.1.

THEOREM 10.1.2. Let the assumptions of Theorem 10.1.1 remain the same except that the boundary condition is replaced by (iiia) u(t, x) < v(t, x) on Ptoand aH - aH, ; (iiib) for each ( t , x) E aH, , a u / a ~av/ar , exist, and .(t, x)

where Q E C[aH, x

y+

Q(t, x, u(t, x))

< 0,

R,R].

Then the inequality (10.1.7) is true.

Proof. T h e proof is similar to the proof of Theorem 10.1.1 except that we have to verify conditions (iia), (iib) of Lemma 10.1.2. T h e definition of m(t, x) and condition (iiia) imply condition (iia). If ( t l , xl) E aH, and m ( t 1 ,xl) = 0, then it is easily checked that a m ( t l , xl)/& < 0 because of condition (iiib) and the fact that a(t, , xl) > 0 whenever ( t l , xl) E aH, . This verifies condition (iib), and therefore Lemma 10.1.2 assures the desired result. COROLLARY 10.1.1.

Assume that

(i) u E C[H,R], the partial derivatives u t , u, , u, exist and are continuous in H ; (ii) f E C[H x R x R" x R"', R],f ( t , x,0, 0, 0) 3 0, the differential operator is parabolic, and T [ u ] < 0 on H ; (iii) u(t, x) < 0 on Pt0u aH. Then, u(t, x)

<<0 on H.

COROLLARY 10.1.2. T h e Corollary 10.1.1 remains true even if condition (iii) is replaced by (iii*) u(t, x) 0 on Pt0and aH - aH, ; (iv) for each ( t , x) E aH, , au/ar exists and .(t, x)

aT

+ Q(t, x, u(t, x)) < 0

on aH, , where Q E C[aH, x R,R] and Q(t, x, 0) = 0.

10.2. COMPARISON

155

THEOREMS

10.2. Comparison theorems

A comparison theorem that plays a prominent part in applications is the following result.

THEOREM 10.2.1. Suppose that (i) m E C [ g ,R,], m(t, x) possesses continuous partial derivatives m, , m, , and m,, in H ; (ii) f E C[H x R x R" x Rnz,R ] ,the differential operator T is parabolic, and T[m] 0 on H ;

<

(iii) 6 E C [ J x R, , R ] , and r ( t ) = r ( t , t o ,yo) 3 0 is the maximal solution of Y' = A t l Y ) , Y ( t d = yo 2 0, (10.2.1) existing on [ t o ,0 0 ) ; (iv) m(t, x) r(t, to ,yo) on Pt0 u aH.

<

Then, one of the assumptions

<

( 4 f (4 x,u, 090) g(t, u), > 0, (b) f ( t , x, u, 0 , O ) < g ( t , u), r(t) < u implies that m(t, x)

Proof.

< r ( t ) + e 0 , for some c0 > 0,

< r(t, t o , Yo)

on

H.

(10.2.2)

We consider the function v(t, x) = m(t, x)

where r(t, c)

=

-

r(t, E),

r(t, to , y o , c) is the maximal solution of Y'

=g(4y)

+

c,

At,)

= yo

+

6,

for sufficiently small E > 0, which exists on any compact interval [to , to 81, 8 > 0, by Theorem 1.3.1. Also,

+

lirn r(t, c) = r(t, to ,yo). OE'

Furthermore, this implies that there exists an c0 r ( t ) < r(4 6)

< r(t>

+

€0

, [ t o , to

> 0 such that

+ PI.

(10.2.3)

We shall show that, under either one of the assumptions (a) or (b), v(t, x) satisfies the hypotheses of Lemma 10.1.1. Clearly, the conditions

I56

CHAPTER

10

of Lemma 10.1.I hold. T o verify condition (iii), let (tl , xl)E Ptl , a ( t , , xl) = 0, u,(t, , xl) = 0, and the quadratic form

for arbitrary vector A. I t therefore follows that m(t1 , x1)

= = r(t1

, €1,

mx(t, , x1) = 0,

(10.2.4)

and

Since the function f is elliptic, we obtain (10.2.5) Notice that the relations (10.2.3) and (10.2.4) yield r ( h ) < m(t1 >I.,

< dtl)

+

€0

-

Also, m(t, , x,) > 0, since ~ ( t, ,6) > 0. Thus, in any case, the assumption (a) or (b) may be used. Hence,

on account of (10.2.5) and the assumption (a) or (b). By Lemma 10.1.1, we then have v(t, .x) < O on IT, which implies that m(t, ,Y)

< r ( t , to , y o )

on

H,

since lim,,,, r ( t , 6 ) .= r ( t , to , yo).T h e proof is complete. T h e boundary condition (iv) of Theorem 10.2.1 may be replaced by nonlinear boundary conditions to achieve the same conclusion. This we state as a corollary, the proof of which may be deduced by reducing to Lemma 10.1.2.

10.2.

157

COMPARISON THEOREMS

COROLLARY 10.2.1. T h e conclusion of Theorem 10.2.1 remains valid, if, in place of boundary condition (iv), we have

<

(a) m(t, x) r ( t , t o ,yo) on P i o and aH - aH, ; (b) for each ( t ,x) E aH, , amja-,,, exists and

whereQ E C[aH, x R, ,R,], Q(t,x,u ) > 0 if u

> 0, for each ( t ,x) E aH, .

COROLLARY 10.2.2. Let assumptions (i) and (ii) of Theorem 10.2.1 hold. Suppose that g E C [ J x R, , R] and that y ( t ) > 0 is a differentiable function satisfying the differential inequality Y'(9

>g(t,y(t)),

Y(t0) = Yo

< g(4 u),

u =Y(t),

> 0.

Then, the assumption f(t, x, u, 090)

implies that m(t, x)


on

t

3 to

9

H,

provided either (a) m(t, x) < y ( t ) on Pt0u aH; or (b) m(t, x) < y ( t ) on Pt0and aH - aH, and

where Q E C[aH, x R,, R] and Q(t, x,u ) ( t , x) E aH, .

>0

if u

> 0,

for each

I n certain situations, the next theorem is more suitable in applications, since it offers a better estimate. Moreover, it shows that the strict inequality T[u]< T[v]in Theorem 10.1.1 may be relaxed i f f satisfies certain additional restrictions.

THEOREM 10.2.2. Assume that (i) m, ZI E C [ n ,R,], the partial derivatives m , , m, , m,, , u i, u, , u,, exist and are continuous in H ; (ii) f E C [ H x R, x Rn x Rnz,R ] , the differential operator T is parabolic, and T [ m ] < T[v] on H;

158

10

CHAPTER

(iii) g E C [ J x R, , R ] ,g(t, 0) = 0, the maximal solution of Y’

= g(t,y),

Y(tJ

=

(10.2.6)

0

is identically zero, and f ( t , -x,2 1 P , R ) - f ( t , X, 2 2 , P, R ) 9

(iv) m(t, x)

< g(t,

21

-

zZ),

21

> ZZ

;

(10.2.7)

< v(t, x) on PLOu aH.

Under these assumptions, m(t, x )

< v(t,x)

on

Proof. Consider the solutions y ( t , C ) differential equation Y’

for sufficiently small

E

== g

> 0.

+

(t,~)

= y ( t , to , 0, E )

=

€9

(10.2.8)

H.

of the ordinary

€9

(10.2.9)

Define the function

z(t,x)

:-=

v(t, x)

+y(t,

€).

Clearly, m ( t , x) < z(t, x) on PLO u a p . Furthermore, observing that y ( t , e ) > 0 and using the relation (10.2.7), we have

qz) = % ( t ,.x) - f ( f , x , z(t, x ) , % ( t ,x), %.dt, x ) ) = v,(t,). -f(t,

x, v(t,).

+ f ( t , .2^,

+ Y ( t ?€1, % ( t ,4,%(t, x))

v(t,x ) , v r ( t , XI1

%Z(t,

- K t , x , n(t, x), %(tr x ) , v&,

3 T[vI

+

+ Y’(L

6)

x))

4)

E.

This implies, in view of the fact that T [ v ]3 T [ m ] , the inequality T[m] T [ x ] . Hence, applying Theorem 10.1. I to functions m(t, x), z ( t , x), we obtain the relation I

,

m(t, x )

< z(t, x )

on

H.

Since, by assumption, limG+,,y ( t , c) = 0, the desired result (10.2.8) follows immediately. T h e proof is complete. 10.2.3. Let the assumptions of Theorem 10.2.2 remain the same except that boundary condition (iv) is replaced by rhEOREM

(a) m(t, x)

< v(t, x) on PtOand aH

-

aH, ;

-

10.2.

I59

COMPARISON THEOREMS

(b) for each ( t , x) E aH, , am/aTO,a v / a ~ ,exist, and

.(t, x)

+ Q(t, x, v(t, x)) > 0

870

where Q E C[aH, x R, , R],and Q is increasing in u for each ( t , x).

Proof. T h e proof is similar to the proof of Theorem 10.2.2. I n the present case, we verify the assumptions of Theorem 10.1.2. Evidently, m(t,x)

< z(t,x)

on

Pi0

and

aH

-a€€, .

T h e monotonicity of the function Q in u, together with the fact that z(t, x) > v(t, x), shows that, on aH,,

> -Q@,

x,

4 4 4).

T h e application of Theorem 10.1.2 now yields the stated result. As an immediate consequence of Theorems 10.2.2 and 10.2.3, we derive weak maximum and minimum principles.

COROLLARY 10.2.3. Assume that (i) u E C [ a ,R],u(t, x) possesses continuous partial derivatives u t , u, , and u, in El; (ii) f E C[H x R x Rn x Rn2,R ] , and the differential operator T is parabolic; (iii) f ( t , x, z, 0, 0) 0 if z > 0 and T[u] 0; (iv) either

<

<

< <

(a) u(t, x) 0 on Ptou aH; or (b) u(t, x) 0 on Pto and aH - aH, au/aTo exists and a(t, xJ

370

, and for each ( t , x) E aH, ,

+ Q(t, x, u(t, x)) < 0

on

where Q E C[aH, x R , R ] and Q(t, x, z ) is increasing in z for each ( t , x). Then we have u(t, x)


on

B.

I60

CHAPTER

10

COROLLARY 10.2.4. Let assumptions (i) and (ii) of Corollary 10.2.3 hold. Assume further that (iii*) f ( t , x,z , 0, 0) 3 0 if z (iv*) either

&/&,

< 0 and

T ( u ) 3 0;

(a) u(t, x) >, 0 on Pt0u aH; or (b) u(t, x) 3 0 on Pt0 and 8H - aH, , and for each ( t , x) E aH, , exists and on

where Q

E

C[aH, x R, R] and Q(t, x,z ) is increasing in z for each ( t , x).

Then, we have u ( t , x)

O

on

H.

Finally, we shall prove a comparison theorem that will be useful in considering the stability of steady-state solutions of nonlinear diffusion equations. Let G be an open bounded region in Rn and aG be its boundary. Denote by H the topological product [0, a)x G.

THEOREM 10.2.4. Suppose that (i) u, v E C[H, R ] ,the partial derivatives u i, v t , u, , a,, u, , v,, exist and are continuous in I€; R ] , the differential operator T is para(ii) f~ C [ H x R x Rn x R7&', bolic, q / a u exists and is continuous, and T[v] T[u]on H ; , the direction of the (iii) the derivatives h ( t , x)/~T,,, &(t, x ) / ~ T ,in outward normal to the hypersurface (0, a)x aG exist, and

<

where Fl, F, are continuous functions with closed domain and bounded derivatives, such that Fl(4 < F2(u),

if u belongs to the common domain of definition of Fland F, ;

Then, we have v(t, x)

< u(t, x)

on

H.

(10.2.10)

10.2.

161

COMPARISON THEOREMS

Proof. We divide the proof into two parts. T h e first is a proof of the theorem if the condition T [ v ] T[u] is replaced by strict inequality, that is, T[v]< T[u].We consider the function

<

m(t, x) = v(t,x)

-

u(t, x),

and proceed as in the proof of Lemma 10.1.I. T h e only difference in the proof is in showing that m ( t l , xl) has a maximum equal to zero for some x1 E G. Suppose that x1 E aG. Then, by assumption, Wtl

9

31) -

av(t1 9 x1)

-

870

a70

= FMtl >

Wl

9

x1)

370

4 - FMtl

9

XI))

> 0, since u(t, , xl) = v(t,, xl). Let x* E G be a point on the normal to the hypersurface (0, co) x aG at xl, sufficiently close to x l . Then, we obtain, from the fact that m ( t , x) is continuously differentiable and by application of mean value theorem, that m ( t l , x*) > 0. Since m(t, x) attains its maximum equal to zero at (tl , xl), this is an absurdity. Hence, x1 E G. T h e rest of the proof is standard, and (10.2.10) is therefore true. We now prove the second part, that is, we shall not demand T[v]< T[u].Let us deny the conclusion (10.2.10). Then, there exists a t, > 0 and an x1 E G such that 4 t l 1 x1)

34tl

2

4-

(10.2.11)

We define a function w(t, x) by

where both

E

> 0 and n >, 2 will be specified later.

and w(0, x)

=

v(0,x)

+E n-1

We have

162

CHAPTER

10

Since af/au exists, it follows that

vt

+ ( t + I)” E

~

> f ( t , x , v, v, ,az,)

-

vt

=

T[v], (10.2.12)

provided

and

E

> 0 is sufficiently small. Let

We now choose n so large that

With this value of n, the inequality (10.2.13) holds for x E G and t E [0, t J . There is a number p > 0 such that F 2 M 4 x1))

-

FMt, x1))

>P

9

for t E (0, t,] and x1 E aG, since the left side of the foregoing inequality is positive on a closed set. Hence, for all w ( t , xl) belonging to the domain of F, , there results W,(t,

4 = %(4 x1) = F2(44 x1)) > Fl(44 4 ) > Fl(44 x1)) > F1

+P

+ (n

(ax1) + .(

EM

€

+

1y-1

l)(t + 1)”-1

1

= Fl(W(4 xJ),

if E is sufficiently small. Here M is an upper bound of 1 F,’ now choose E > 0 so that (10.2.12), (10.2.14), and min[u(O, x ) X€G

-

v(0,x)] > E

(10.2.14)

1. Let

us

(10.2.15)

10.3.

163

BOUNDS. UNDER AND OVER FUNCTIONS

hold. Note that there is a positive value of E satisfying (10.2.15), since the left side is the minimum of a continuous positive function on a closed set. Thus, it follows, from (10.2.12) and T [ v ] T[u],that

<

T[w]< T[v]< T[u] Also, w(0, x)

=

v(0,x)

on

xE

G,

t

+2 n - 1 < u(0, x),

E

[O, tl].

x E G.

Applying the first part of the proof, we obtain w ( t , x)

< u(t, x)

[O, tl] x G.

on

However, this, together with (10.2.1 l), leads to W(t1

?

x1)

< u(t, > XI)

< 4 4 , x1) E

(n - I)(tl

+

1),-1

This contradiction shows that there does not exist a t, (10.2.1 l), and, hence, the proof is complete.

> 0 satisfying

10.3. Bounds, under and over functions Consider the partial differential equation where f E C[H x R x Rn x Rn2,R]. DEFINITION 10.3.1. Given an initial $(t,x), which is defined and continuous on Pt0U aH, a solution of (10.3.1) is any function u(t, x) satisfying the following properties:

(i) u ( t , x) is defined and continuous for ( t , x) E H ; (ii) u(t, x) possesses continuous partial derivatives u l , u, , u,, in H and satisfies (10.3.1) for ( t , x) E H ; (iii) u(t, x) = +(t, x) for ( t , x) E Pl0u aH. T h e problem of finding a solution to the partial differential equation (10.3.1) is called a first initial-boundary-vale problem. Let Q E C[aH, x R, R ] ,and suppose that (a) u(t, x)

= $(t, x)

on Pi0 and aH

-

aH, ,

164

CHAPTER

10

(b) a ( t , x)[au(t,x ) / a ~ ]$- Q(t, x,u(t, x)) = +(t,x) on aH,, where it is assumed that a ~ / &exists for ( t , x) E aH, .

If, instead of boundary condition (iii) in the Definition 10.3.1, we ask (a) and (b) to be satisfied, we have a mixed initial-boundary-value problem.

REMARK 10.3.1. If Q(t, x,u ) = P ( t , x)u, where P ( t , x) > 0 on aH,, the problem is said to be a first mixed problem; and if is not restricted to be positive on aH, , it is called a second mixed problem. If a ( t , x) = 0, the boundary condition (b) is of Dirichlet type, and the first mixed problem coincides with the first Fourier problem. O n the otherhand, if ~ ( tx), = 1 and P ( t , x) .= 0, the boundary condition (b) is of Newmann type, in which case the mixed problem reduces to second Fourier problem. If a(t, x) = 1, so that aH, = aH and P(t, x) is continuous on aH, the problem is called a third initial-boundary-value problem. If, in addition, H is a cylinder and the directions T are inward conormals, it is said to be a second initial-boundary-value problem. I n what follows, we shall assume the existence of solutions for the two boundary-value problems just stated, that is, first and mixed initialboundary-value problems.

THEOREM 10.3.1. Assume that (i) f E C [ H x R x R” x Rnz,I?], f ( t , x, u, P, R ) is elliptic, and one of the assumptions

<

g(t, I U I), (a) l f ( 4 x,U , 0,O)I (b) l f ( t , x, u, 0, 0)l < g ( t , I I), r ( t ) < < r ( t ) € 0 , for some co > 0, holds; (ii) g E C[/ x R, , R,], and the maximal solution r ( t ) = r(t, to , y o ) of (10.2.1) exists for t 2 t , ; (iii) u(t, x) is a solution of the first initial-boundary-value problem, such that on Pt0 u aH. I u ( t , .x)1 < ~ ( tto, , y o )

Then,

1 Pyoof. of

u ( t , %)I

+

< r ( t , to , y o )

on

H.

By Theorem 1.3.1, the maximal solution r ( t , c ) Y’

for sufficiently small

= g(t,y) E

+

€7

y(to>= yo

+

=

r ( t , to , y o ,c )

E,

> 0, exists on any compact interval

[ t o ,to

+ y],

10.3.

165

BOUNDS, UNDER AND OVER FUNCTIONS

> 0, and limc+or ( t , c) = r ( t , to , y o ), uniformly on [ t o , to Furthermore, we have

y

r ( t ) < r(t, .)

for some eo

< r ( t ) + €0 ,

> 0. Let us first

t

E

[ t o , to

+ rl,

+ y].

(10.3.2)

consider the function

m(t, x) = u(t, x)

-

r(t, €).

It is easy to see that m(t, x) satisfies conditions (i), (ii) of Lemma 10.1.1. T h e condition (iii) is also verified. For, let ( t l , xl) E Ptl , m(tl , xl) = 0, m2(tl , xl) = 0, and, for an arbitrary vector A,

Since this implies that u(t1 , xl) = r ( t l ,

c)

> 0, uz(tl , xl)= 0, and

we have, on account of the ellipticity off,

f(h

1

x1

9

U ( t l 3 xl>,U Z ( t l


7

'117

UZZ(tl

> .1>>

,U(t1 , Xl), 0,O).

(10.3.3)

Moreover, because of (10.3.2), 4tl>

< I 4 t l ,%>I < r(t1)

+

€0.

(10.3.4)

Thus, either of the assumptions (a) or (b) of the theorem shows that

using (10.3.3) and the fact that u(t1 , xl) by Lemma 10.1.1, that u(t, x)

< r(t, E )

Let us next consider the function

4 4 2)

=

-[r(t,

=

on

r(t, , c). We therefore obtain,

8.

€1 + 44 4 1

(10.3.5)

166

CHAPTER

10

and show that it also fulfills the conditions of Lemma 10.1.1. It is only required to check condition (iii) of Lemma 10.1.1. Suppose that (tl , .I> E ptl n(t1 7 . 1 ) = 0, n,(t, , x1) = 0, and 7

for some arbitrary vector X f 0. Then, we have -4t1,

€1 > 0,

x1) = r(t1 9

-4, , .1) = 0,

and

By the ellipticity of the functionf, we get

Also, (10.3.4) holds. Hence, as before, we obtain

on account of (10.3.6) and either one of the conditions (a) or (b), noting that ~ ( t,I),., = -u(t, , x,) = r(t, , e ) . Hence, by Lemma 10.1.1, -u(t, x )

< r(t, 6)

a.

on

(10.3.7)

T h e two inequalities (10.3.5) and (10.3.7) now yield the desired estimate

I 44 .)I

e r ( 4 t o ,YO)

on

H,

and the proof is complete.

COROLLARY 10.3.1. T h e assertion of Theorem 10.3.1 remains true if u(t, x) is a solution of the mixed initial-boundary-value problem such that

(ad !

u(t,

41 < r ( 4 t o ,Yo) on P," and

-

aH, ;

10.3.

(a2) for some p

I 4 4 ).

167

BOUNDS, UNDER AND OVER FUNCTIONS

> 0,

=+ 87,

e Br(4 1, ,Yo>

Q(4 x, u(t, .))I

on

aH, ,

where(? E C[aH, x R,R],Q(t, x, 2) is increasing in x for each ( t ,x) E aH,, Q(t, x, -4 = -P(t, x,4, and P(t,t o 3 Yo) Q(4 x,44 t o ,Yo))-

<

T h e next theorem offers a better bound and is a variant of Theorem 10.2.2.

THEOREM 10.3.2. Assume that (i) f~ C [ a x R x Rn x Rnz,R ] , the differential operator T is parabolic, and (10.3.8) f(t, x, -u, -P, -R) = -f(t, x, u, P, R); (ii) g E C [ J x R, , R ] ,g(t, 0) = 0, the maximal solution r ( t , to , 0) of (10.2.6) is identically zero, and f(t, x, zi 3 P, R ) - f ( t , x,z2 9 P, R )

< g(t, z1 - 4,

21

> z2 ; (10.3-9)

(iii) w E C [ H ,R,], v(t, x) possesses continuous partial derivatives 2 0, w t uzz, and T[v] >, 0; (iv) u(t, x) is a solution of the first initial-boundary-value problem satisfying 1 u(t, x)I < v(t, x) on Pi,, u 8H. u,

9

Then, we have

< v(t, x)

1 u(t, x)l Prooj.

on

H.

(10.3.10)

Let us consider the function

44 4 = -[u(t, 4

+ v(t, x) + Y ( t , 41,

where y(t, E) = y ( t , t o , 0, E ) is any solution of (10.2.9), for sufficiently small E > 0. Let (tl , xl)E Pt1, n(tl , xl) = 0, n,(tl , xl) = 0, and the quadratic form

for some vector A. This means, noting y(t, E ) -4t1 -U&1

> x1) f

> 0, that

> "(tl , 4,

4 = "&,

t

XI)

b 0,

168

CHAPTER

10

and

Because of the ellipticity off, it results that f(t1

, x1 , - 4 f 1 > X l ) , --%(f1 4,-u,,(t1, x1)) 5f ( t l , x 1 , - 4 t 1 , 4,-%c(t, , 4,%it1 , XI)). 9

Furthermore, using (10.3.8), (10.3.9), and the preceding inequality, we obtain "Ltl

7

x1)

= -

G

4 t l

-,f(t1

> x1)


s < -e,

I

J

>

x1)

, x1

,r ( t 1

?

?

XI),

1

I.'@,

c)) -

- Y'(t1

9

%(tl

4

9

4,%til1

-%(t,

, XI), %&I

'UAtl 9

4,%At,

XI>,

At1

7

> Xl), U,,(tl

Xl), %1 ti

1

x1 > v(t1 9 x1),

*2^1 1 - 4 t 1

Kfl

"dtl

4tl

x1

1

-f(t1

-

7

Y(tl

7

> x1))

, x1)) - Y'(t1 €1 , XI)) , x1)) YYtl 4 9

-

9

4) - e

which implies that n,(t, , xl) < 0. Clearly, n(t, 2) satisfies all the assumptions of Lemma 10.1.1, and hence

< v(t, x)

-u(t, x)

+y(t,

c)

on

H.

Proceeding similarly, we can show, on the basis of Lemma 10.1.1, that u ( t , x)

< v(t, x) + y ( t , c)

on

H.

T h e preceding two inequalities, together with the fact that lim,,,y(t, c) 0, yield the estimate (10.3.10). T h e theorem is proved. ~

COROLLARY 10.3.2. Let the hypotheses of Theorem 10.3.2 remain the same except that condition (iv) is replaced by (iv*) u ( t , x) is a solution of the mixed initial-boundary-value problem satisfying (a) 1 u(t, .)I v(t, x) on P t 0 and i3H - aH, ;

<

(b)

I a ( t , x)

au(t, x) ~

87"

+ a t , x, 44 .))I

av(t x ) < " ( t , x) -A 370

10.3.

I69

BOUNDS, UNDER AND OVER FUNCTIONS

where Q E C[BH, x R, R ] , Q(t, x, x) is increasing in z for each ( t , x ) aH,, ~ andQ(t, x, -z) = -Q(t, x, 2). Then, (10.3.10) is valid. We shall now introduce the notion of under and over functions with respect to the parabolic equation (10.3.1).

DEFINITION 10.3.2. Let u E C [ H ,R ] , and let u(t, x) possess continuous partial derivatives u t , u,, u,, in H. If u(t, x) satisfies the parabolic differential inequality T[u] < 0

on

H,

together with u(t, x) = + ( t , x) for ( t , x) E Pt0u aH, we shall say that an under function with respect to the first initial-boundary-value problem. On the other hand, if

u(t, x) is

T[u] > 0

on

u(t, x) is said to be an over function.

satisfies

on

u(t, x) = +(t, x)

a(t, x)

If

H,

&/a7 and

Pt0

9 +

Q(t, x, u(t, x))

exists on aH, and u(t, x)

= +(t,x)

aH

-

aH,,

on

we shall say that u(t, x) is an under or over function with respect to the mixed initial-boundary-value problem according as T[u]< 0 or T[u]> 0, on H.

As a direct consequence of Theorems 10.1.1 and 10.1.2, we have the following.

THEOREM 10.3.3. Let f~ C [ H x R x Rn x Rnz,R] and the differential operator T be parabolic. Suppose that u(t, x) and v(t, x) are under and over functions with respect to the first initial-boundary-value problem. If x(t, x) is any solution of the same problem such that u(t, x)

< x ( t , x) < v(t, x)

then u(t, x)

< z(t, x) < v(t, x)

on

Pt0u a H , on

H.

(10.3.1 1)

T h e inequality (10.3.11) remains true, even when u(t, x) and v(t, x)

170

CHAPTER

10

are under and over functions with respect to the mixed initial-boundaryvalue problem, provided z(t, x) is any solution of the same problem and u(t, x)

< z ( t , x) < v(t,x)

on

and

Pt0

i3H - aH, ,

+ Q(t,x, u(t, x)) < a(t, x) !%a7?? + Q(t, x, z(t, x)) av(t,x) < "(4 x) ____ a7 + Q(4 x, 4 4 4).

a(t, x)

10.4. Approximate solutions and uniqueness We shall begin with the theorems that estimate the difference between a solution and an approximate solution of (10.3.1).

THEOREM 10.4.1. Assume that (i) f E C[H x R x Rn x Rnz7R],the operator T is parabolic, and .f(t, X, 2 1

,P, R ) - f ( t ,

X, ~

2

P,%R )

< g(t, z1 - zz),

~1

> zz ; (10-4-1)

(ii) n E C [ p ,R],v(t, x) possesses continuous partial derivatives u t , a,, v,, such that I 7-[74 G s(t>, (10.4.2) where 6 E C [ J ,R,]; (iii) g E C [ J x R, , R],g(t, 0)= 0, and r ( t , t o ,yo ) is the maximal solution of

y'

= g(t,y)

+ S(t),

(10.4.3)

Y ( t 3 = yo 2 0,

existing for t 2 to ; (iv) u(t, x) is any solution of the first initial-boundary-value problem such that on Pto u a f f . I u ( t , ). v(t, X)l r(4 t o ,Yo)

e

~

Then, the estimate

I u ( t , ).

-

v(t, .>I

< r ( t , to ,yo)

on

is valid.

Proof.

Define N t , '4

= u ( 4 x)

-

v(t, x)

-

y(t, E),

R

(10.4.4)

10.4. where y ( t , c )

= y ( t , to

Y’

,y o , E) is any solution of

= g(t,y)

+ S(t) +

+

YPO) = yo

6,

for sufficiently small E > 0. Suppose that (tl , xl) m,(t, , xl) = 0, and, for some nonzero vector A,

Since y(t, c )

171

APPROXIMATE SOLUTIONS AND UNIQUENESS

E

(10.4.5)

E,

Ptl , m(tl , xl)

= 0,

> 0, this implies that

and

T h e last inequality yields, because of the ellipticity off, At1

Y

x1

7

aft1 7


x1>7 %(t, >

> x1

?

4tl

>

x1),

Uzz(f1

4,%it,

9

4)

7

X l ) , VU,U,(tl

, El)).

It follows, in view of the preceding inequality, that %(tl ? x1> -

vUt(t1 9 x1)

G f(tl x1 9

-f(t1 -

7

4tl

?

7

x1)7

x1 7 v(t1

9

uz(t1

>

Xl),

vJt,

Xl),

9

Trz(t1

Xl),

7

4)

vzz(t1 > x1))

T[v].

Hence, the relations (10.4.1) and (10.4.2) show that %(tl 7 5 1 )

-

4 t l

x1)

< g(t1

9

Y(tl

.>I

9

+ S(t).

We thus have mdt1 ? x1) =

<

%(tl 7 x1) --E

-

< 0.

vdt,

9

x1)

- y’(t1 9

.)

It is easy to see that m(t, x) satisfies the hypotheses of Lemma 10.1.1, and hence

u (t, x)

--

v(t, x)

< y(t, 6 )

on

H.

I n a similar way, considering the function n(t, x)

=

v(t, x )

-

u(t, x) - y(t, €)

I72

CHAPTER

10

and verifying the assumptions of Lemma 10.1.1, we can obtain v(t, x)

u(t, x)


on

ET.

T h e last two inequalities, together with the fact that lim Y ( t , €1 E

-0

r(t, to ,Yo),

assure the stated estimate (10.4.4).

THEOREM 10.4.2.

Let u(t, x) be any solution of the mixed initialboundary-value problem such that

<

(a) I u(t, x) - v(t, I). r ( t , t, , y o ) on Pt0 and aH - aH, ; (b) for each ( t ,x) E 8 H a , a(/ u - ZI I)/&, exists, and, for some j3

> 0,

where Q E C[aH, x R , , R ] , Q(t, x,z ) is increasing in x for each ( t , x), and p r ( t , t, , y o ) Q(t, x,r ( t , to , yo)),other hypotheses (i), (ii), and (iii) of Theorem 10.4.1 being the same.

<

Then (10.4.4) is true. REMARK 10.4.1. If u(t, x), v(t, x) are any two solutions of the boundaryvalue problem, we can deduce the estimate of the difference between them, as a consequence of Theorems 10.4.1 and 10.4.2. Similar remark holds for the theorem given below.

'Tmomni 10.4.3. Assume that (i) f , G E C [ H x R x RrLx Rnz,R ] , G(t,x, z , P, R) is elliptic, and, if x1 > z 2 , PI P, , f ( t 9

(ii)

F

x,2 1 1'1 > R , ) -At, ,x, 2 2 p2 > 4) 2 ,G(t,2, z1 - 22 , Pl - P, , R, - RZ); 7

9

(10.4.6)

C [ f I ,R ] , v(t, x) possesses continuous partial derivatives

ZI~,

(10.4.7)

10.4.

173

APPROXIMATE SOLUTIONS AND UNIQUENESS

whereg E C [ J x R, , R],g(t, 0) = 0, and the maximal solution r ( t , to , 0) of (10.2.6) is identically zero; (iv) z E C [ R ,R+], z, 2 0, z t , z,, exist and are continuous in H , and

2 G(t, x, z,

zt

Then,

I 44).

- v(l,

+ s(t, 4.

zzz)

< z(t, x)

on

Pi0 u aH

41 e 4 4 4

on

H,

1 u(t, x) - v(t, x)I implies

232,

(10.4.9)

where u(t, x) is any solution of the first initial-boundary-value problem.

Proof. As usual, we shall reduce the theorem to Lemma 10.1.1. We shall first consider the function

where y ( t , E ) = y ( t , to , 0, e) is any solution of (10.2.9) for small E > 0. Suppose that (tl , xl) E Ptl , m ( t , , xl) = 0, m,(t, , xl) = 0, and, for some nonzero vector A,

Since z(t, x) >, 0, y ( t , E )

> 0, the preceding 4 t l , x1)

%(tl x1) 9

-

supposition implies that

> 4 t l 4,

%it, 1x1)

9

= zz(t1,

x1)

2 0,

T h e ellipticity of G shows that

In view of this and the relations (10.4.6), (10.4.7), we have

174

10

CHAPTER

Hence, using (10.4.8) and (10.4.9), we derive 4 t l

,

< G(t, ~

<

~

?

x1 , 4 t l

G(t1 x1 7

R(tl > Y(tl

+

I

6))

< 0.

-€

+

, XI) Y(tl , E ) , %it, xl), &(tl , x1)) so1 ,)I. , X l ) , %it1 X l ) , Z , X ( t l ,x1)) - S(t1 , 4 9

Z(t1

9

7

--

This shows that m(t, x) verifies the assumptions of Lemma 10.1.1, and, therefore, on H. u(t, x) v(t, x) < z(t, x) + y ( t , E) -

Arguing similarly, we can prove w ( t , x)

-

u(t, x)

< z ( t , x) + y(t, 6)

on

H.

Since lim,,,y(t, e ) = 0, by assumption, it follows from the preceding two inequalities that

I u(t, .x)

-

v(t, .x)I

< z(t, x)

on

H,

proving the theorem. We shall next consider the uniqueness problem.

THEOREM 10.4.4. Suppose that (i) f~ C [ H x R x Rn x R’”, R ] , the operator T is parabolic, and

f(4 x, z1 , p,R ) -f(4

x,

2.21

(ii) g E C[/ x R, , R ] ,g(t, 0) of (10.2.6) is identically zero.

p, R) =

< f(t, z1 - 4,

z 1

>22 ;

0, and the maximal solution r ( t , to , 0)

Under these assumptions, there is at most one solution to either one of the initial-boundary-value problems. T h e proof is a direct consequence of Theorems 10.4.1 and 10.4.2.

10.5. Stability of steady-state solutions Let problem D represent the partial differential equation of the form ut = :

for x E [a, b], t

> 0, together

%(t, a )

= f,(u(t,

a)),

f(.x, u, u,

7

(10.5.1)

ux,),

with the boundary conditions u,(t,

b)

= f,(u(t,b)),

t

> 0,

10.5.

STABILITY OF STEADY-STATE SOLUTIONS

175

where fi ,fiare continuous functions with bounded derivatives. Let us assume that af/au exists and is continuous,f (x, u, P, R ) is nondecreasing in R for each (x, u, P ) . We use the notation u(t, x, $) to denote a solution of problem D such that

4 0 , x, 4 ) = 4 w ,

6

[a,

4,

where 4 E C [ [ a ,b], R]. DEFINITION10.5.1. Let u ( t , x, $) be a solution of problem D. We shall say that u(t, x, 4)is a steady-state solution if u(t, x, $) = $(x), t > 0.

DEFINITION 10.5.2. T h e steady-state solution u(t, x, 4) of problem is said to be stable if, given E > 0, there is a 6 > 0 such that

D

implies

DEFINITION10.5.3. A

Let

= [(x, u): x E [ a , b’J

< < #z(x)17

and

where z+hl , are arbitrary functions, twice continuously differentiable on [a, 61. Let B be the set of functions on [a, b] such that z+h E B implies

T h e steady-state solution u ( t , x, $) of problem D is said to be asymptotically stable if it is stable and lim[ max j u(t, x,+) - u(t, x, #)I]

t-m

x~[a,bl

=

0,

whenever I$ E B. T h e set A is called the domain of attraction. Sufficient conditions for a steady-state solution u(t, x, $) of problem D to be stable are given by the following theorem.

THEOREM 10.5.1. Assume that there exists a one-parameter family o(x, A), A E [A, , A,], of solutions of the equation f(X,

v, v z , vm)

=

0

(10.5.2)

176

10

CHAPTER

fulfilling the following conditions: (i) there is a A * E ( A , , A), such that v,(a, A*) =f,(v(a, A*)) v,(b, A * ) = J.(v(b, A")); (ii) zq,(x, A) > 0, x E [a, 61, A E [A, , A,];

(4a,(% 4 >f,(v(a, 4)and %(b, A)

f,(v(b, A)), A

(iv) a,(%

Then, if 4(x) lem D is stable.

Pyoof.

Let

E

=

given. Choose A, max

1 v(x, A*)

max

1 v ( x , A")

xt[a,b]

[A, , A*); E (A*, A,]. E

the steady-state solution u(t, x,4 ) of prob-

v(x, A*),

> 0 be

and

~

E

[A, , A*), Ao

E

(A*, A2] such that

v(x,A,,)\ < E

(10.5.3)

< €.

(10.5.4)

and xt[a,b]

-

W(X,

A*)(

Then, define the number 6 by 6

7-=

min[ min (v(x,A*)

-

v(x,An)),

xt[a,bl

min (v(x,An)

xt[u,bl

-

Since v,(x, A) > 0 for all x E [a, b ] , it is clear that 6 the inequality

and assume that $(x) a(x,A,)

5:

$A).(

= -

(10.5.5)

v(x,A*))].

> 0. Let $(x)

satisfy

n(x,A"). We then have, by (10.5.5),

s < #(x) < +(x)

+ < v(x, A,),

x E [a, b ] .

The fact that v(x,A) is a family of solutions of (10.5.2) and u(t, x, $) is a solution to the Droblem D imolies that

By a successive application of Theorem 10.2.4, we deduce that v(x, A") < u(t, x, I))

< v(x,A",

x E [a, b ] ,

t

2 0.

10.5.

177

STABILITY OF STEADY-STATE SOLUTIONS

This, because of the relations (10.5.3) and (10.5.4), yields u ( 4 x,

4)- E = v(x, A*)

and u(4 x, 4)

+

=

€

v(x, A*)

-E

< v ( x , A,) < u(t, x, $)

+

> v(x, An) > u(t, x, #).

€

It is evident from the preceding inequalities that

< 6.

whenever max,.[,,,l I $(x) - +(.)I

T h e proof is complete.

I n the situation in which it is difficult or impossible to find a oneparameter family u(x,A), satisfying the conditions of Theorem 10.5.1, it may still be possible to find an upper bound. This we state as a corollary.

COROLLARY 10.5.1. Suppose that there exists a solution u(x)of (10.5.2) satisfying %(a) < fl(44) and vdb) > f,(v(b))* Then, if u(t, x, $) is a solution to problem D such that #).(I

we have

< v(4,

u(4 x,4) < +),

x E [a, bl,

x E [a,bl,

t b 0.

A similar corollary may be stated establishing a lower bound.

THEOREM 10.5.2. Let the hypotheses of Theorem 10.5.1 hold. Suppose further that, for x E [a, b] and A E [A, , A,], fv(x,

v(x,

3,%(X,

A),

%,(X,

Then, if +(x) = u(x,A*), the steady-state solution u(t, x, totically stable, and the set A

=

[(x, u ) : x E [a, b]

and

(10.5.6)

A)) f 0-

v(x, A,)

4) is

< u < v(x, A,)]

asymp(10.5.7)

is a region of attraction.

Proof. T h e stability of steady-state solution u(t, x, #J) follows by Theorem 10.5.1. Let A be the set defined by (10.5.7), and let B be the set of functions such that E B implies [(x, +(x)) : x E [a, b]] C A. We

178

10

CHAPTER

shall first show that, for any E > 0 and any 4 E B, there exists a T I > 0 such that max [@, x, $) - 44 x, 411 < E . (10.5.8) x~Ca,bI t>T,

Let

E

> 0 be given, and let ha E (A*, v(x, AO)

u(x,A*)

-

A,] be such that

< E,

(10.5.9)

x E [a, b].

Without loss of generality, we may assume that f, > 0, in view of ,u2, p3 by

(10.5.6). We then define three positive numbers p l ,

v(x,

A)

~

'u(x,

< p1/2,

A*)

x E [a, b].

Define a positive number p4 by

Let H(A) be a function defined for A E [Ao ,A,], such that, for h

fZ('u(b,

4 - h ) -fz('u(b, 4)< P4/2.

< H(A), (10.5.12)

Consider the function w

=

w(x, A) = v ( x , A)

-

where 6 > 0 will be specified later. Since small 6 > 0, we have f(X,

w, w, > w,,) = f(x.

'u -

< f(X, =

0.

'u, u ', 'up

u ',

f, exists, for sufficiently

8 , u' , ,' u 3

= f(X, 71%'us > u ),',


8,

9

'um)

,'um)

- VV(& ~

8p2

'us u ',

, 'uz,)

+ 0(S2)

+ 0(S2) (10.5.13)

10.5.

We let 6,

179

STABILITY OF STEADY-STATE SOLUTIONS

> 0 be such that 6,

< rmin [v(x,A,) e[a,b]

- $(x)].

Let us now choose a positive 6 so small that (10.5.13) holds, and also (10.5.14)

( 10.5.15)

I t follows, from the inequality (10.5.13), that there exists a p5 > 0 satisfying f(X,

Let m

=

w ,w, w,,) 9

< -PF,

E

9

[A0,

41,

A*)

e-pt,

xE

[a, 4.

m(t, x) = w(x, h(t)), where h(t) = A*

+ (A,

-

p > 0, being a number to be specified shortly. Then, f ( x , m,m, , m,,) - m

t =f(x,

<

w ,w,

+ wAp(A,

+

-p5

9

~

WZZ)

A*)

e-pt

(10.5.16)

P 3 P ( A , ~- A*).

Choose p = p5/p3(h2- A * ) so that the right-hand side of (10.5.16) is equal to zero. Then the inequality f(x,

?%

m,

,%z)

- mt < f ( % u, u,

>

% I . )

U t -=

0

is verified for x E [a, b ] , t E [O, T I ] , where T I is the solution of the equation h ( T J = A*. T h e inequality (10.5.1 l), together with (10.5.14) and (10.5.15), shows that

fI(+? 4 - 6 ) -f1(+4 4) >

-(P4'2)-

(10.5.17)

From the definition of w , we deduce, in view of (10.5.10) and (10.5.17), that

f M 4 A))

4 6) >fi(+, 4) - (P4/2) > v,(a, 4

= f*(fl(a,

==

w,(a, A).

~

(10.5.18)

180

CHAPTER

10

Similarly, from (10.5.10), (10.5.12), (10.5.14), and (10.5.15), we derive $dw(b,

4) = fz(v(b, A) < v,(b, A)

9+ ( d 2 )

8) < f2(7@, wc(b, A).

-

=

Clearly, (10.5.18) and (10.5.19) hold for all X E [A*, A,]. Theorem 10.2.4, it follows that m(t, ).

2

u(t, x,

x E [ a , 61,

$17

t

E

[O,

(10.5.19)

On the basis of 7-11>

since m(0, x) > $(x), x E [a, b ] . From the definition of m and Tl , we get

> u(T1 , x, $),

v(x, AO)

x E [a,4 .

Thus, by Corollary 10.5.1, v(x, Ao)

> u(t, x, $),

x E [a, 61,

t

> T, ,

which, together with (10.5.9), gives us (10.5.8). T h e next step of the proof is to show that there exists a T, > 0 such that (10.5.20) max [u(4 x, 4)- 4 4 x, $11 < xe[a.bl t>T,

T h e proof of this consists of showing that there is a lower bound for $) which can be increased with time until it is within E of u ( t , x, 4) at some time T, . T h e proof of this fact is similar to the first part and differs only in minor details. Let T = max[Tl , TJ. Then, from (10.5.8) and (10.5.20), we obtain u(t, x,

max

xtla.61

I u(t, x, 4)

~

u(t, x,

$)I < E .

t>T

This completes the proof of the theorem. COROLLARY 10.5.2. T h e conclusion of Theorem 10.5.2 remains valid if (10.5.6) is replaced by either (i) f,, = 0 andf,&, v(x, 4, %(X, 4, % r ( X , (ii) f r = 0, f U Z = 0, and fv,,(x, 4x, A), x E [a, 4 , A E [A, A,].

4) f %(X,

0 ; or A), %,(x,

A)) f 0 for

9

We now give an example to illustrate Theorems 10.5.1 and 10.5.2. Consider the partial differential equation u.

-

(1

+

242)

u,. - uu,z,

xE

[I, 21,

10.6.

181

SYSTEMS OF PARABOLIC INEQUALITIES

subject to the boundary conditions I)

T h e equation (1

+

uz(4 2) = fz(u(4 2)).

and

= f&(4 1))

242)

u,,

-

uu,2 = 0

has a one-parameter family of solutions v ( x , A ) given by v(x, A ) Notice that v,(x, A) = x-l[l + &(x, A)]1/2 sinh-l v ( x , A).

=

sinh Ax.

Suppose that

+ u2)1/2sinh-l u, fl(u) < (1 + u2)1/2sinh-l u, ti(.) < +(l + u2)1/2sinh-l u, f2(u) > t(l + u2)1/2sinh-' u,

fl(u) > (1

u u u u

> 0, < 0, > 0, < 0.

fl(0)

=

0,

f2(0)= 0,

Then, by Theorem 10.5.1, u(t, x, 4) = 0 is stable. T o apply Theorem 10.5.2, we observe that fv(x, v, 0,

,v,,)

=

2vv,, - vz2 2A2 sinh2Ax - X2 cosh2Ax

=

A2(sinh2Ax- 1).

=

Thus, fv < 0 if sinh2 Ax < 1 or u < 1. Hence, u(t, x, 4) = 0 is asymptotically stable, by Theorem 10.5.2.

10.6. Systems of parabolic differential inequalities in bounded domains Let us consider now a partial differential system of the type where

u: = f i ( t , x, u,u,i, u;,),

u,i =

, d2,-*.,

i

=

1, 2 ,...)N ,

u&),

For convenience, we shall write the preceding system in the form

182

CHAPTER

10

where f E C [ w x R N x Rn x Rn', RN]and each function f is elliptic,

so that the system is parabolic.

We shall first state the following lemmas, which are extensions of Lemmas 10.1.1 and 10.1.2.

LEMMA 10.6.1. Suppose that (i) m E C [ H ,RN],m(t, x) possesses continuous partial derivatives m ,, m, mzz in H ; (ii) m(t, x) < 0 on Ptou aH; 7

<

(iii) for any (tl , xl)E Ptl and an indexj, 1 < j N , if mj(t, , xl) mi(t, , 31) 0, i # j , mzj(tl , xl) = 0, and the quadratic form

<

=

0,

h being an arbitrary vector, then mi(t,, xl) < 0.

Under these assumptions, we have m(t, x)

Proof.

<0

on

H.

(10.6.2)

Assume, if possible, that the set 2=

N

(J [ ( t ,x) E H : myt, x) 3 01 i=l

is nonempty. Let Z , be the projection of Z on t axis and t, = inf Z. I t follows from condition (ii) that t , > t o . Since the set 2 is closed and condition (ii) holds, we conclude that mi(tl , .Y)

<0

on

Ii n [to , t l ] ,

for i = 1, 2 ,..., N , and there is an indexj, 1 such that "'(tl , xl) = 0.

<j < N , and an x1 E int Pt1

We therfore have m,'(t, , x1) >, 0.

(10.6.3)

On the other hand, since mj(t, x) attains its maximum at (tl , xl)E int P t l , we get n2,'(t1 , XI) = 0,

10.6.

183

SYSTEMS OF PARABOLIC INEQUALITIES

and, for some nonzero vector A,

By condition (iii), it follows that contradicting (10.6.3). Hence, the inequality (10.6.2) IS ' true. Let 01 E C[aH,R N ] ,and denote by aHmithat part of the boundary aH on which d ( t , x) > 0. Corresponding to this change, we may define, as before, for each i = 1, 2,..., M,and ( t , x) E aHmi,aui/a?, aui/a.roi,T ~ .r0i being the directions. Then, we have an extension of Lemma 10.1.2 which may be proved by combining the proofs of Lemmas 10.1.2 and 10.6.1.

LEMMA 10.6.2, Let condition (ii) of Lemma 10.6.1 be replaced by (iia) mi(t, x) < 0 on Pt0and aH - aHmi; (iib) for any index j , 1 j N and (tl ,xl) mi(t, , xl) = 0, i j , then

+

< <

E

aH,i, if d ( t , , xl)

=

0,

where, for each ( t , x) E aHmi, ami/& is assumed to exist. Then, the assertion (10.6.2) is true if the other assumptions remain the same. We are now in a position to extend the previous results to systems of parabolic inequalities. We shall present only some typical theorems.

THEOREM 10.6.I. Assume that (i) u, z, E C[H ,R N ] ,the partial derivatives a t , u, , u,, , ot, v, , v,, exist and are continuous in H ; (ii) f E C[R x RN x R" x Rn2,R N ] ,the differential operator T is parabolic, the function f ( t , x, u, P, R ) is quasi-monotone nondecreasing in u for each ( t , k, P , R), and T[u] < T[v]

(iii) u(t, x) < v(t, x) on Pt0u aH. Then, we have u(t, x ) < v(t, x)

on

on

H;

H.

(10.6.4)

,

184

CHAPTER

10

P Y O O ~We . shall show that the vector function m ( t , x)

=

u(t, x)

~

v(t, x)

satisfies the hypotheses of Lemma 10.6.1. I n view of conditions (i) and (ii) of Theorem 10.6.1, it is sufficient to check assumption (iii) of Lemma 10.6.1. Suppose that ( t l , xl)E Ptl , mj(t, , xl) = 0, 1 < j N, mi(tl , xl) 0, i # j , m,j(t, , xl) = 0, and

<

<

for an arbitrary vector A. This means that (10.6.5)

and

Since f j is elliptic, we get

This, together with (10.6.5) and quasi-monotonicity of fi, implies that

10.6.

185

SYSTEMS OF PARABOLIC INEQUALITIES

THEOREM 10.6.2. Let assumptions (i) and (ii) of Theorem 10.6.1 hold. Suppose further that (iii*) ui(t, x) < vi(t, x) on Pt0and aH - aHai, i = 1, 2,..., N ; (iv) for each ( t , x) E aHai, aui/aTi, avi/ad exist, and uyt, x)

t, x aTi

+ gyt, x, u(t, x))

where Q E C[HH x RN,RN], Q(t, x, u ) is quasimonotone nondecreasing in u for each ( t , x). Then, the inequality (10.6.4) remains valid.

Proof. The proof proceeds as in Theorem 10.6.1. T o verify the assumptions of Lemma 10.6.2, we have only to check condition (iib). N and (tl ,xl)E aH,j. Suppose that &(t, , xl) = 0 and Let 1 < j mi(t, , xl) 0, i # j . This implies that

<

<

Uj(t1

Ui(tl

,x1) = vj(t1 , X d , , x1) < vi(tl , Xl),

#j.

Hence, by the quasi-monotonicity character of Q, we have P(t11 x1 , 4 t l 9 x1))

< Qvl,

XI, UP1

, x1)).

Thus, there results the inequality

q,, x1) amj(t1

7.1)

aTj

< P(tl =

I

x1 ,Q(t1 ,Xl))

-! a t 1

1

x1 , v(t1 , x1))

0,

which assures that amj/aTj(t, , xl) This completes the proof.

< 0,

since d ( t l , xl)

>0

on aH,j.

It is now easy to formulate and prove comparison theorems, componentwise bounds, and error estimates for systems of parabolic differential equations. We shall not include such results. However, the following comparison theorem will be needed later, and, hence, we shall merely state it.

THEOREM 10.6.3. Suppose that (i) m E C [ g , R+N],m(t, x) possesses continuous partial derivatives m, 1 m, , mzz in H ;

186

CHAPTER

10

(ii) f E C [ H x RN x Rn x Rn2,R N ] ,the differential operator T is parabolic, and T [ m ] 0 on H ; (iii) g E C [ J x R t N , R N ]g(t, , y ) is quasi-monotone nondecreasing in y for each t E J , r ( t , to ,yo) 2 0 is the maximal solution of the differential system

<

Y'

existing for t

20

3 to , and .y',

f(f,

(iv) m(t, 4

Y(t0) = Yo

g(t,y),

=:

< r(t, t o

9

x, 0,O)

< g(4 4,

z

2 0;

yo) on Pto u aH.

Then, m(t, x)

< r ( t , to , y o )

on

H.

10.7. Lyapunov-like functions

Let us continue to consider the partial differential system (10.6.1). We shall restrict ourselves to the first initial-boundary-value problem of such systems. I n what follows, a solution will always mean according to Definition 10.3.1, unless specified otherwise. Let V E C [ H x R N ,R,] and V = V ( t ,x, u ) possess continuous partial derivatives with respect to t and the components of x,u.Let

"'1.

L..,

p, p1 ~=1, v , v l - 1 , 2,..., n

T h e following theorem offers an estimate of solutions in terms of Lyapunov-like functions and is useful subsequently.

THEOREM 10.7.1. Suppose that (i) f E C [ H x R N x Rn x Rn2,R N ] ,G E C [ H x R, x Rn x Rn2,R], G is elliptic, and

a v av at + a , f ( t , x, u, u,Z, &) < G(4 v(4 x, u), V,, X,

V,,),

(10.7.1)

10.7.

I87

LYAPUNOV-LIKE FUNCTIONS

where V E C[R x RN,R,] and V , , V,, are given in the foregoing; (ii) g E C [ J x R, , R] and G(t,x, z, 0 , O )

< g(4 4;

(10.7.2)

(iii) the maximal solution r(t, to , y o ) of the differential equation (10.2.6) exists for t 3 to ; (iv) u(t, x) is any solution of the partial differential system (10.6.1) such that

v(t,X, w,x))

G r(t, to ,yo)

on

ptou aH.

(10.7.3)

a.

(10.7.4)

These assumptions imply

v(t,x, 4 4 4)< r(t, t o ,Yo)

on

Proof. Let u(t, x) be any solution of (10.6.1) satisfying (10.7.3). Define the function m(t, x)

Then,

=

V(t,x, u(t, x)).

4, ). < r(4 to ,Yo)

on

Pto u aff.

Moreover, because of (10.7.1), we obtain

Now, a straightforward application of Theorem 10.2.1 yields the estimate (10.7.4).

THEOREM 10.7.2.

Let the assumptions of Theorem 10.7.1 remain the same except that the relations (10.7.1) and (10.7.3) are replaced by 41)

av av [ x + 3F .f ( 4 x,

u, u,i, uL,]

+ V(4 x, 4 4)

< G(t,x, 4 t ) V t ,x, 4,4)v, , 4 t ) V,,),

(10.7.5)

where A ( t ) > 0 is continuously differentiable on J and A(t) v(t,3, +(t, x))

< r(t, t o ,yo)

Ptou aH-

on

Then, the inequality (10.7.4) takes the form A(t) V(t, x, u(t, x))

< r(t, to ,yo)

on

H.

(10.7.6)

188

Proof.

CHAPTER

10

Defining lTl(t,

x,).

=

A ( t ) q t , 2,

4,

it can be easily checked that Vl(t,x,u) preserves the properties of V ( t ,x,u ) in Theorem 10.7.1, and hence (10.7.6) follows from Theorem 10.7.1.

REMARK10.7.1. Taking A(t)= 1, we see that Theorem 10.7.2 reduces to Theorem 10.7.1. Since Theorem 10.7.1 is an important result in itself in the study of various problems of partial differential equations, we have listed it separately. We note that g ( t , z ) in (10.7.2) need not be nonnegative, and, hence, this has an advantage in obtaining sharper bounds. For example, taking V = 11 u 11' andg(t, u ) = L(t)u,whereL(t) is continuous on J , one can get an upper bound, from Theorem 10.7.1, as follows:

provided

where

max II + ( t o XtPd

I

.)I1

< Yo.

If we assume that V ( t ,x,u ) = 0 if and only if u may be used to prove a uniqueness result.

=

0, Theorem 10,7.€

THEOREM 10.7.3. Assume that (i) f E C [ H x R N x Rn x Rn2,R N ] G , E C[H x R , x Rn x Rnz,R ] , G is elliptic, and

-< G(t,X,

V ( t , x, u

-

v), Vz(t,x,u

-

v), Vzs(t,x,u - v));

,.

(ii) g E C [ J x R , R ] ,g(t, 0) = 0, and G(t,5,z,0,O)

< g ( 4 4,

> 0;

(iii) the maximal solution r(t, to , 0) of (10.2.9) is identically zero. Then, the parabolic differential system (10.6.1) admits a unique solution.

10.7. LYAPUNOV-LIKE FUNCTIONS

189

Let V I EC[H x R N ,R+N], and suppose that V ( t ,x, u ) possesses continuous partial derivatives V t , V , , V,, , V u. Then, the following extension of Theorem 10.7.1 may be proved.

THEOREM 10.7.4. Let the following assumptions hold: (i) f E C[H x RN x Rn x Rn', R N ] GE , C[H x R,N x Rn x Rn2,R N ] , G is elliptic, and

(ii) g E C [ J x R+N,R N ]g(t, , y ) is quasi-monotone in y for each t E J , and G(4 X, z,o, 0) < g ( t , 4, 3 0; (10.7.8) (iii) the maximal solution r(t, to ,yo) of the ordinary differential system Y'

= g(t,y),

Y(tJ

= Yo

30

exists for t 3 to ; (iv) u(t, x) is any solution of the system (10.7.1) such that

Under these assumptions, we have

Proof.

Consider the vector function

where u(t, x) is any solution of (10.6.1) such that (10.7.9) holds. By the assumption (10.7.7), we deduce the inequality m(t, 4

e G(4

X,

dt,

4, d ( 4 4, &it,

XI).

Since m(t, x) satisfies the assumptions of Theorem 10.6.3, it follows that 4 4 4 < 44 to Yo) 9

and the proof is complete.

on

H,

190

CHAPTER

10

10.8. Stability and boundedness Let u(t, x) be any solution of the partial differential system (10.6.1). We shall assume that the system (10.6.1) possesses the identically zero solution. Denote

I1 44 .)llPt

=

max I1 4 4 411. XCP,

DEFINITION 10.8.1. T h e trivial solution of the partial differential system (10.6.1) is said to be equistable if, for each E > 0 and to E J , there exists a positive function 8(t,, E ) , which is continuous in to for each E , such that

On the basis of this definition, it is easy to formulate the various definitions of stability and boundedness analogous to those in the earlier chapters. We shall now give sufficient conditions for stability and boundedness of solutions of the parabolic difierential system (10.6.1). 'rHEOREM

10.8.1. Assume that there exist functions V ( t ,x,u ) and

g(t, y ) satisfying the following conditions:

(i) g E C [ J x R , RI, g(t, 0 ) 0; (ii) V E C[H x RN,R,],V ( t ,x,u ) possesses continuous partial derivatives V , , I/, , V,, , V,, in H , and 9

~

WI 11)

< 1 -(t, x, ). < a(t, I1

ll),

(10.8.1)

where h t N ,a E C [ J x R , , R,], a F Y- for each t E J ; (iii) f E C[H Y R N x R" x Itnn,R N ] ,G E C [ H < R , x Rn x Rn',R] G is elliptic, and

a[- a v 27 + (iv) G(t, X,

Z,

.f(t,

0, 0 )

12, 21,

usL,? 1 1 J

< G(t,x, b.(t, .w, u ) , I T z , 17rE);

< g(t, z), 2 3 0.

10.8.

191

STABILITY AND BOUNDEDNESS

Then, the equistability of the trivial solution of the ordinary scalar differential equation y'

= g(4y),

Y(t0) = Yo

(10.8.2)

30

implies the equistability of the trivial solution of the partial differential system (10.6.1).

Proof. Let E > 0 and to E J be given. Assume that the trivial solution of (10.8.2) is equistable. Then, given b ( e ) > 0 and to E J , there exists a 6 = 8 ( t o ,E) > 0 that is continuous in to for each E such that Y ( t , to ,Yo)

<

<w,

t b to

(10.8.3)

?

provided yo 6, where y(& to , y o ) is any solution of (10.8.2). Let be any solution of (10.6.1) such that

u(t, x)

Then, by Theorem 10.7.1, we obtain

q t , x,u(t, .)I < r(t, t o ,yo)

on

(10.8.5)

H,

where r(t, to ,yo)is the maximal solution of (10.8.2). Choose yo such that a ( t o ,II $ ( t o , .)IIpto) = y o . Then, there exists a 6 , = S1(to, E) such that

II +(to

9

.)lIpt0

< 81

and

a(to 9 II d(to , .)IIPJ

<8

(10-8-6)

hold simultaneously. It is clear that 6 , is continuous in to for each We claim that, if (4 (b)

then

I1 +(to >

%P,,

lld(4 .)llm <

< 8;

€9

t

2 to

/I 44 .)IIP, < E ,

t

2

If this is not true, suppose that, for some t,

II 4 t , *

.>IIP,,

1

to.

> t o , we

have

36

Then, by condition (b), there exists an xo E int P,, such that

II +I

7

x0)Il

= E.

E.

192

CHAPTER

10

It now follows from relations (10.8.1), (10.8.3), and (10.8.5) that

< i;(tl

44

, xo "(t,, %)) <

, t o Yo) < 44,

9

9

which is a contradiction. This proves the equistability of the trivial solution of (10.6.I), and the proof is complete.

THEOREM 10.8.2. Let the assumptions of Theorem 10.8.1 hold except that the function a(t, y ) occurring in (10.8.1) is independent of t , that is, a ( t ,y ) = a(y)E X . Then, the uniform stability of the trivial solution of (10.8.2) implies the uniform stability of the trivial solution of (10.6.1). Proof. I n this case, it is enough to choose 6, = ~ ~ ( 6 Since ) . 6 is independent of t o , it is clear that 6, is independent of to . T h e rest of the proof is very much the same as that of Theorem 10.8.1.

THEOREM 10.8.3. Under the assumptions of Theorem 10.8.1, the equi-asymptotic stability of the trivial solution of (10.8.2) implies the equi-asymptotic stability of the null solution of (10.6.1). Proof. Assume that the solution y = 0 of (10.8.2) is equi-asymptotically stable. This implies that (S,*) and (S,*) hold. Hence, we need to prove only the quasi-equiasymptotic stability of the trivial solution of (10.6.1), as, by Theorem 10.8.1, the equistability is guaranteed. Let E > 0 and to E J be given. It then follows, on account of (S3*), that, given b(c) > 0 and to E J , there exist positive numbers 6, = so(to) and T = T ( t o ,E) such that

+

(10.8.7) 2 to T , provided yo < 6, . Choosing yo = a(t, , 1) $(to , -)I\ plo),as before, we can Y ( 4 t o 3 Yo) < b ( 4 ,

t

show the existence of a positive number

/I C(to -)llp'o ,< &I I

and

8,

=

8,(t,) such that

4 t o 7 I1 +(to

9

.)llPJ

< so

hold at the same time Suppose now that n ; (a) I/ +(to * ) l / P t o (b) I1 +(t, *>/laIf < ' t 3 to

+ T.

+

Let there exist a sequence { t k } , t, 3 to T and t,+ co as k + co, such that / / u(t, , *)llPtk 2 E for some solution u(t, x). Then, there exist x, t int P t k satisfying (1 u(t, , x,)[[ = E in view of condition (b). Thus, using relations (10.8.1), (10.8.5), and (10.8.7), we arrive at the contradiction b(c) < V(t, , X k , U ( t , , X k ) ) < Y ( t , , to ,Yo) < 4).

10.8.

193

STABILITY AND BOUNDEDNESS

Thus, the quasi-equi-asymptotic stability holds, and, as a result, the trivial solution of ( 10.6.1) is equi-asymptotically stable.

THEOREM 10.8.4. Under the assumptions of Theorem 10.8.2, the uniform asymptotic stability of the trivial solution of (10.8.2) implies the uniform asymptotic stability of the trivial solution of (10.6.1). Proof. Assume that the trivial solution of (10.8.2) is uniformly asymptotically stable. Then, we have (S,*) and (S,*). By Theorem 10.8.2, the uniform stability of the trivial solution of (10.6.1) follows. T o prove the quasi-uniform asymptotic stability of the trivial solution, we proceed as in Theorem 10.8.3 and choose 8, = u-l(S0), observing that So and T are both independent of t o . T h e proof is complete.

THEOREM 10.8.5. Assume that there exist functions V(t,x, u), g(t, y ) , and A(t) satisfying the following conditions: (i) A(t) > 0 is continuously differentiable for t E J , and A(t)+ co as t + co; (ii) g E C[J x R , , R],and g(t, 0) = 0; (iii) V E C[H x R N ,R,], V(t,x, u ) possesses continuous partial derivatives V , , V, , V,, , V, in H, and (10.8.1) holds; (iv) f E C [ H x R N x Rn x Rn', R N ] ,G E C[H x R, x Rn x Rnz,R], G is elliptic, and

(v)

< G(t,x,4 t ) w, x,4, A ( t ) vz , A ( t ) V z z ) ; G(t,x, z, 0 , O ) < g(t, 4,27 3 0.

Then, the equistability of the trivial solution of (10.8.2) guarantees the equi-asymptotic stability of the trivial solution of (10.6.1). Proof. Let E > 0 and to E f be given, and let u = min,,, A(t). By assumption (i), G > 0. Set 7 = o ~ ( E ) . Assume that (S,*) holds. Then, given 7 > 0, to E J , there exists a 6 = 6(to , E ) such that ~ ( ttn, yo) 9

whenever y o

t

(10.8.8)

to,

< 6. Let u ( t , x) be any solution of (10.6.1) such that A(t0) V(t0 P x,+(to

and

< 7,

>

4)G Yo

7

A ( t ) v(t,x,+ ( t , x)) G r(t, t o >Yo)

x E pto

9

on

aN.

194

CHAPTER

10

Then, by Theorem 10.7.2, it follows that

<

A(t) V ( t ,x, u(t, 4) r ( t , to ,yo)

on

(10.8.9)

H,

where r ( t , to ,y o ) is the maximal solution of (10.8.2). We choose yo so that a(to , 11 +(to , -)]I pt) A(to)= yo . Then, we can assert the existence of a 8 = 8 ( t o , E ) such that the inequalities

/Id(to , .)I/Pto < 8

and

A(t0)4

0

9

/I+ ( t o

7

.)IIP,,>

<6

hold together. With this 8, the equistability of the trivial solution of (10.6.1) is verified. For, otherwise, we arrive at the contradiction, proceeding as in the proof of Theorem 10.8.1, +I =

G K E )

,< 4,) Vtl

7

xo 3 4 t l

,xo))

< Y(t1

7

to ,Yo)

< 17.

For a fixed E = , we designate by 6, = 8,(t0) the number 8 ( t o , el). Let 0 < E < el and to E J be given. Suppose that 11 +(to,-)lip 6, . Since t0 A(t) 00 as t -+ 00, there is a number T = T ( t o ,E ) satisfying

<

---f

b(6) A ( t ) > 17,

t

2 to

+ T.

(10.8.10)

It is easy to show that, with this So and T , the quasi-equi-asymptotic stability holds. Suppose that this is not true. Then, there exists a sequence (tk},t, >, to T , and t, co as k + c o such that 11 u(t,, .)I1 3 E for Ptk 6, and II+(t, -)[IaH < E , some solution u ( t , x) satisfying ll+(to, -)/Ip t 0 t >, to + T . Also, there exist xlj E int Ptk such that 11 u(t, , xk)lI = E . T h e relations (10.8.8) and (10.8.9) yield

+

---f

<

4 t k )

b ( 4 < 17,

which is an absurdity in view of (10.8.10). It therefore follows that the trivial solution of (10.6.1) is equi-asymptotically stable, and the proof is complete. Let us consider the following example. Let L(u) denote the differential form

Jw c %(t, n

=

j,k=1

+ c bk(t, n

X)%j”r

4 u Z k

9

k=l

where aj,rc(t,x) and bk(t,x) are continuous functions on quadratic form n

(10.8.1 1)

g, and the ( 10.8.12)

10.8.

STABILITY AND BOUNDEDNESS

195

on R, h being an arbitrary vector. Let F E C [ H x RN,RN].Consider the system (10.8.13) ut = L(u) F(t, x,u).

+

Assume that F(t, x, 0)

= 0 and

f

i=l

u W ( t , x,u )

< A(t)

(ui)2, <=1

where h E C [ J ,R]. Taking V ( t ,x, u) = 11 u inequality (10.8.12), we obtain

and making use of the

it follows, from Theorem 10.7.1, that

provided

where If, in addition,

the application of Theorem 10.8.1 yields equistability of the trivial solution of (10.8.13). On the other hand, the assumption

1, h(s)

ds = -0

implies the equi-asymptotic stability of the trivial solution of (10.8. I3), by Theorem 10.8.3. Regarding the boundedness of solutions of the partial differential system (10.6.1), we have the following result.

196

10

CHAPTER

THEOREM 10.8.6. Assume that there exist functions V(t,x, u ) and

g(t, y ) fulfilling the following hypotheses:

(i) V E C[H x RN,R,], V ( t ,x, u)possesses continuous partial derivatives V , , V , , V,, , V , in H , and b(ll u II)

-

< v(t,x,4 < 44 ll u II),

(I 0.8.14)

where b E .X, b(y) 4 co as y a,a E C[J x R, , R+],and a E Zfor each t E J ; (ii) f E C[H x R N x R" x R"', RN], G EC[H x R, x Rn x Rnz,R], G is elliptic, and

(iii) G(t,x,x, 0, 0 )

< g(t, x), x 3 0, where g E C [ J x

R, , R ] .

Then, the equiboundedness of Eq. (10.8.2) implies the equiboundedness of the system (10.6.1).

<

Proof. Let a > 0 and to E J be given. Suppose that 1) $ ( t o , .)lip a. t0 Define u1 = a(to ,a). Let Eq. (10.8.2) be equibounded. Then, given a1 > 0 and t, E J , there exists a P1 = &(to , a ) such that ~ ( tt o, 3 yo)

if yo < a1 . Since b(y) + co as y that


b(P)

9

( 10.8.15)

t bto,

co, we can find a P

= B(to , a ) such

2 PI.

(10.8.16)

Let u(t, x) be any solution of (10.6.1). Then, if (10.8.4) holds, we have (10.8.5). Choose yo = a(t, , 11 +(to , -)IIp, ). Suppose that there exists a solution u(t, x) of (10.6.1) such that

(a) I / +(to > . ) / / P ( o (b) It 9(t, .)llm

<

01,

P, t 2 to

9

and 11 u ( t , , .)itp f l 3 /3 for some t , > to . This implies the existence of an xo E int Ptl satisfying I/ ~ ( t, xo)li , = 8. We are now lead to an absurdity, in view of the relations (10.8.14), (10.8.5), (10.8.15), and (10.8.16): b(P)

< v(t, xo , ~ ( t ,xo)) < r ( t , , to >

9

9

yo)

< B1 < b(P).

This proves the equiboundedness of the differential system (10.6. l), and the theorem is proved.

10.8.

STABILITY AND BOUNDEDNESS

197

On the strength of Theorem 10.8.6 and the parallel boundedness and Lagrange stability results in ordinary differential equations, the following theorem on various boundedness results and Lagrange stability may be proved. We only state the theorem, leaving the construction of the proof as an exercise.

THEOREM 10.8.7. Let assumptions (i), (ii), and (iii) of Theorem 10.8.6 hold, except that the relation (10.8.14) is strengthened to b(ll u 11)

< v(t, 4 < 41u 11)) X,

where a, b E X and b(y) + co as y -+ 03. Then, if the ordinary differential equation (10.8.2) satisfies one of the notions (B,*) to (&,*), the partial differential system verifies the corresponding one of the concepts (B,) to (BJ. Corresponding to the system (10.6.1), we may consider the perturbed system U t = f(t, X, % u:, 4%)F(t, X, 4, (10.8.17)

+

where F(t, x,u ) is a perturbation term. Then, we have the following result.

THEOREM 10.8.8. Assume that (i) V E C[H x R N ,R+],V ( t ,x, u) possesses continuous partial derivK , and atives V , , V,, V Z z ,V , in H , j ! aV(t, x,u ) / & 11

<

Wll * II)

< w,4 < 4 z l II)? X,

where a, b E X ; (ii) f E C [ H x R N x Rn x Rn', R N ] , F E C[H x R N , R N ] , G E C [ H x R, x Rn x Rnz,R], G is elliptic, and

for some 01 > 0; (iii) G(t, x,x, 0, 0 ) g(t, 0 ) 3 0 ; (iv) llF(t,x,u)ll

< g(t, z),

z

>, 0, where g E C [ J x R , , R],

< vV(t,x,u), and a 3 Kv.

Then, one of the stability notions of the trivial solution of (10.8.2) implies the corresponding one of the stability results of the trivial solution of the perturbed system (10.8.17).

I98

Proof.

CHAPTER

10

Using the respective assumptions in (i), (ii), and (iv), we find

I t is evident, from this inequality, that we can directly apply Theorems 10.8.1-10.8.4 to obtain the desired result. T h e proof is complete. Although we can prove a number of results by the techniques just used, that is, by reducing the study of partial differential system to the study of ordinary differential equations, in certain situations, this method does not yield all the information about the given system. For instance, consider again the example (10.8.11). Suppose we now assume that the ui,k(t, x) hihk is positive definite instead of positive quadratic form semidefinite, as demanded in (10.8.12). This stronger hypothesis has no effect. I n other words, we do not get more information because of this assumption. T o be more specific, suppose F = 0 so that g = 0. Then, we can conclude by Theorem 10.8.1 that the trivial solution of (10.8.1 1) is stable. This conclusion remains the same even when the preceding quadratic form is assumed to be positive definite. I n such situations, the following theorem is more fruitful.

xF,k=l

THEOREM 10.8.9. Assume that (i) V E C[H x R N ,R,], V ( t ,x, u ) possesses continuous partial derivatives V , , V , , V,, , Vt,in N,and b(ll u II)

< v(t,x, u ) e 44 II u Ill,

where b E Z, a E C [ ] x R, , R,], and u E X for each t E f; (ii) f E C[H x RN x Rn x Rn2,RN], G E C[f7 x R, x Rn x Rn2,R ] , G is elliptic, G(t,x, 0, 0, 0) = 0, and

10.8.

where g E C[J x I?, r(t, t o , 0) of

199

STABILITY AND BOUNDEDNESS

,R ] , g(t, 0) E 0, and the maximal soiution Y’

= &Y),

Y(t3

0

=

is identically zero. Then, the equistability of the trivial solution of zt =

(10.8.18)

G(t,x, z, z, %), 9

implies the equistability of the trivial solution of (10.6.1). Proof. Let u(t, x) be any solution of (10.6.1) such that

and

V(to,x, +(to

,.I)

,< z(t0

9

4

v(t,x, +(4 x)) < z(t, x)

on

Pto

on

aff,

where x ( t , x) >, 0 is the solution of (10.8.18). Define m(t, x)

V ( t ,x, u(t, x)).

=

Then, we get

< G(t,x, 4 4 4, m d t , 4,mm&, 4). If we write T[v]= v t - G(t, x,v, v,, vZJ, then it is clear that T[m]< T [ z ] . Furthermore, m(t, x) < z(t, x) on PL0u aH. All the assumptions of Theorem 10.2.2 being verified, we deduce that V ( t ,x, u(t, x))

< z(t, x)

on

R.

Let E > 0 and to E J be given. Assume that the trivial solution(l0.8.18) is equistable. Then, given b ( ~ > ) 0 and to E J , there exists a 6 = S(to , E) such that

<

(9 max,,pt, X ( t 0 > x) 8, (ii) maxZEaHx(t, x) < E , t b to , implies

max z(t, x) XSP,

< E,

t 3 to

Let maxZSp ~ ( t ,, x) = .(to, 11 # ( t o , -)[!pto),and let 6, = &(to, E) be the 50 same number chosen according to the inequalities (10.8.6) in the proof of Theorem 10.8.1. Suppose that I I + ( t o , ,< 6, and II+(t, - ) / I aH < E , to

200

CHAPTER

10

t 3 t o . Assume that there exists a solution u(t, x) of (10.6.1) such that, for some t , > t o , I/ u(t, , -)I/ 3 E . It then follows that there is an xo E int Pil satisfying / j u(t, , xo)lI = E . From this, we deduce the inequality VE)

< L’(t1 , xo , 4 t , , xo)) < Z(t1 , xo) < b ( E ) .

This contradiction proves the equistability of the trivial solution of (10.6.1). On the basis of this theorem, we can formulate other stability results in this setup. We notice, however, that we now have the problem of knowing the stability behavior of partial differential equation (10.8.18). I n the cases where the function G(t,x,x, x, , x,,) is simple enough to know the behavior of its solutions by other methods, this technique is useful.

10.9. Conditional stability and boundedness I n this section, we shall consider the partial differential system of the type (10.9.1) U f = f(4 x, % u, , u,,), where U, =

(F

7-7

au2 __

au,

--

ax,

7

ax,

,*--,

au, __

ax, ’-*’

au, ax, ,-*., -)ax,

au, __

and

azu,

azu,

a2u,

ax, ax, ’...’ ax,z

’*..’

a2u, ax,2

**..’

w). a2uN

I t will be assumed that the first initial-boundary-value problem with respect to (10.9.1) admits the trivial solution and that all solutions exist on H . I n the sequel, a solution of (10.9.1) will always mean a solution of the first initial-boundary-value problem. Let k < N and M(N--k)denote a manifold of ( N - K ) dimensions containing the origin. Let S(a) and represent the sets, as before,

s(a)

S ( a ) = [ u : (1 u (1

< a]

and

S(a) = [u : 1) u (1

< a],

respectively. Parallel to the conditional stability and conditional boundedness definitions (C,) to (C16)of Sect. 4.4, we can formulate the definitions of conditional stability and boundedness of the trivial solution of (10.9.1.). Corresponding to (C,), we have

10.9.

CONDITIONAL STABILITY AND BOUNDEDNESS

20 1

DEFINITION 10.9.1. T h e trivial solution of the partial differential system (10.9.1) is said to be conditionally equistable if, for each E > 0 and to E J , there exists a positive function S ( t , , E ) , which is continuous in t n for each E, such that, if (i) ). M(N--k) (ii) $(t,4 c S ( E ) , ( t , 4 E aH, 9

9

then u(t, x)

“lo

c S(E),

>

( t , x) E H.

Sufficient conditions for the conditional stability of the trivial solution of (10.9.1) are given by the following result.

THEOREM 10.9.1. Assume that (i) g E C [ J x R+N,R N ] , g ( t , 0 ) = 0, and g ( t , y ) is quasi-monotone nondecreasing in y for each t E J ; (ii) V EC[H x R N ,R + N ] , V(t,x, u ) possesses continuous partial derivatives V t , V , , V,, , V , in H , and b(lI u 11)

where b E X , a (iii)

E

N

< 1 vik x,4 < 4 4 II u Ill, i=l

C [J x R+ , R+], and a E X for each t E J ;

f E C[H x RN x RNnx RNn2,R N ] ,G E C[H x R,N x RN x RN2,R N ] , G(t,2,

2, z:,

ZLJ

is elliptic, and

av

-

at

av + au .f(t, x,

u, u,

, uzz) < G(t,X, v(t,X, u ) , I/aci, VL);

(iv) G(t,x,x, 090) < g(t, 4, z 3 0; (v) Vi(t,x, u ) = 0 (i = 1, 2 ,..., k), k < N , if u E , where M(N--k) is an ( N - k)-dimensional manifold containing the origin. Then, if the trivial solution of the ordinary differential system y‘

= g(t,y),

= yo

>, 0,

to

2 0,

(10.9.2)

is conditionally equistable (in the sense of Definition 4.4.2), the trivial solution of the partial differential system (10.9.1) is conditionally equistable.

202

Proof.

10

CHAPTER

For any

> 0,

E

if

11 u 11

=

we have from assumption (ii) that

E,

(10.9.3) Suppose that the trivial solution of (10.9.2) is conditionally equistable. ) 0 and to E J , there exists a 6 = 6(to , c) > 0 such that Then, given b ( ~ >

c Yz(4 to ,Yo) < 44, N

t

2 to

(10.9.4)

9

2=1

provided v

C

y20

<6

and

( i = 1 , 2 )...,k),

0

uzo =

(10.9.5)

2=1

where y ( t , to , y o ) is any solution of (10.9.2). Suppose that u(t, x) is any solution of (10.9.1). It follows by Theorem 10.6.4 that N

N

V 2 ( 4

44 4)< 1 Y , ( t ,

.2^,

to

9

( t , 4 E R,

Yo),

2=1

2-1

(10.9.6)

whenever N

C VZ(4l

N

9

x, d t o

4) < 1Yro >

9

and

c N

V 2 ( 4

x , 4(t, .))

2=1

E

pto>

2=1

2=1

<

N

rz(t, t o ,yo),

(4 ).

E

2=1

aff,

where r ( t , to ,y o )is the maximal solution of (10.9.2). Choose y o such that N zi=lyiO /I $(to * ) / i P , o ) and ,). M(N-k) pto so that yio = Vi(to, x,$ ( t o ,x)) = 0 (i = 1, 2,..., k), by assumption (v). I n view of the properties of a(t, Y), there exists a 6, = 6,(to , e ) > 0 such that 7

Ild.(ro

9

9

7

.)/IPt0

< 61

and

4to

>

II d(to

7

.)llPJ

9

<8

at the same time. From the choice of yio and condition (ii), it follows n M(N-k), we have the inequality that, whenever + ( t o , x) C s(6,) (10.9.6). Suppose now that there exists a solution u(t, x) of (10.9.1) which satisfies

x, s(sl) M(N-I,) > (a) (b) +(4 4 c S(C--),( 4 x) E aH, 9

pto

9

10.9.

CONDITIONAL STABILITY AND BOUNDEDNESS

203

and has the property that u(t, x) $ S(E) for some t, > to and x E Ptl. Because of relation (b), there exists an xo E int Ptl such that 11 u(t, , x,,)lI = E. Hence, by (10.9.3), (10.9.4), and (10.9.6), we are led to the following absurdity:

< c Vi(t1 , xo N

b(c)

i=l

< i=l c Ti(t1 , to ,Yo) < b(4. N

9

, .a>>

U( t 1

Consequently, the trivial solution of (10.9.1) is conditionally equistable, and the proof is complete. On the strength of Theorem 10.9.1 and the parallel theorems on conditional stability and boundedness (in Sect. 4.4), we have the following

THEOREM 10.9.2. Assume that the hypotheses of Theorem 10.9.1 hold, and suppose that a(t, r ) = a(r), a E X . Then, one of the notions (C,*) to (C&) relative to the ordinary differential system (10.9.2) implies the corresponding one of the conditional stability concepts (C,) to (C16). T h e following example, in addition to demonstrating the conditional stability, serves to show that the system (10.9.1) need not be parabolic. Consider the system

au, at

au, at

where F, F, F,

+ cos t ) u , + (1

=

(1

=

(1 - ePt)ul

=

(cos t

~

cos t)u,

+ (cos t

-

l)u, ;

+ (1 + e+)u, + (ct- l)u, ;

&)ul

~

+ (ect - cos t)u, + (e+ + cos t)u, .

204

CHAPTER

10

Assume that the quadratic forms C;j=, aiih,hj , Crj=1bi& , and C:,i=, cijhihi are all nonnegative for arbitrary vector A. Choosing the vector Lyapunov function V = ( V l , V , , V,) such that

we observe that the functions b(r) and a(t, r ) reduce to b(r) = [(u12

+ u: + u1;3)1/2]2

Furthermore, the function G

and =

n(t, r )

=

5[(u,2

+ u: + u32)1/2]2.

(G, , G, , G,) takes the form

T h e differential system (10.9.2) can be reduced to

We find that g = (g, ,g, ,8,) fulfills the monotonic requirements. Choose k = 1. T h e n the solution ~ ( tto, ,yo) of (10.9.2) is given by rl(4 t o Yo)

= YlO

r2(t, to ,yo)

T

7

exp[4(t

-

yzoexp[sin t

to)],

-

sin to],

y3(t, to ,yo) = y30exp[4eKt0 - 4.81.

Here we have A!l(N-,i) = M , , the set of points u such that u2 - u3)2 = 0. I t is clear that the condition (C,") holds, which,

(ul

+

10.10 UNBOUNDED DOMAINS

205

in its turn, implies, from Theorem 10.9.1, the conditional equistability of the trivial solution of the system considered previously.

10,lO. Parabolic differential inequalities in unbounded domains Let D be a region in Rnfl of (t, x) space, satisfying the following conditions: (i) D is open and contained in the zone to < t < m; (ii) for any t, E [ t o , m], the intersection Stl of D with the plane t = t, is nonempty and unbounded; (iii) for any t , , Stl is identical with the intersection of the plane t = t, with the closure of that part of D which is contained in the zone to t < t, .

<

We denote by aD that part of the boundary of D which is contained in the zone to < t < GO. For any 4 E C[D,R ] , if the inequality

DEFINITION 10.10.1.

+(4 4

< M e x p w I/ x IlZ)

holds, where M , K are positive constants, we shall say that 4 E E2(M,K ) . Similarly, 4 E E2(M,K ) implies that

If 4 E E,(M, K ) 4 E &(M, K ) .

and

vq4 x) 3 -M

e x p w II x

4 E &(M, K )

simultaneously, we shall say that

l ”.

DEFINITION 10.10.2. A function - f C~[ D x R x Rn x Rnz,R] is said to satisfy an L? condition if there exist positive constants Li (i = 0, 1, 2, 3, 4) such that If(4 x, u, p , R) - f ( 4 x, u, p,R)I

+ (4II x /I + L,) c I p , n

,=I

THEOREM 10.10. I .

-

< Lo

p, I

c I R,, n

z,k=l

-

R,, I

+ ( L , /I x (I2 + L4) I

u

-

21 I.

Assume that

(i) u, v E C[D,R ] , the partial derivatives u t , u , , u,, , v l , v,, a,, exist and are continuous in D ;

206

CHAPTER

10

(ii) f E C[D x R x Rn x Rnz,R],f satisfies an 9 condition, the differential operator T is parabolic, and T[u]

< T[u]

on

D;

(iii) u E &(M, K O )and v E -? KO); $(&I, (iv) u(t, x) v ( t , X) on St0w ao.

<

Then, everywhere in D,we have (10.10.1)

4 4 x) 6 v(t,.)-

Proof. Let D , denote an open bounded domain separated from D by the cylindrical surface I', , with the equation I/ x I( = R. We shall denote by S 2 , aD, the parts of the surfaces S l 0 ,aD, respectively, lying inside and on Let C, represent the part of contained in D. Furthermore, let us designate by Dh,DRh, aDh, aD,h, CRhthe parts of the sets D, D,, aD, aD, , C, , respectively, contained in the strip to t <. to h . We shall prove the theorem for a domain Dh, where h will be chosen conveniently. Consider the function

r,.

r,

+

<

m = m(t, x)

Define u

=

=

u(t, x) - u(t, x).

UH, v = GH,where

is the growth damping factor. Here p = p ( K ) , y = y ( K ) are positive constants, which will be chosen later, for K > K O .Setting z we have m

=

-

v(t, x),

xH defined in D", where h


=

z(t, x)

C ( t , x)

It then follows that

< f ( t , x, u, uxH + u H = ,R ( l ) ) -f(t, X, iZxH + U H x , R"') ZL,

+ f ( t , x, u, iizH -f(t,

X, FO,

VxH

+ u H x , R(2))

+CHz,

R'3'),

(10.10.2)

10.10

207

UNBOUNDED DOMAINS

where

R'3' = [v,*,,H

+ v,,H,, + G'.,H,* + K * , j l .

Let {RU}be an increasing sequence such that R, -+co as fixed 01, consider the domain D i m .Denote A

01

-+ co. For

a

sup - z(t, x).

=

(t,x)sDh Re

-

+

+

+

Then, there exists a (tl , xl)E D i a = D t x St aDta C t a such that A = ~ ( t, ,xl). We shall show that the constants p, v, and h may be , 0. chosen such that ( t l , xl) E D t a S t a implies the inequality ~ ( t, xl) Indeed, supposing the contrary, we have, at ( t l , xl),

+

az/at (tl

7

x1)

<

3 0,

%(tl

, x1) = 0,

and

for an arbitrary vector A. This means that

Since f is elliptic, it follows that f(t1

+

, x1 , U(t1 ,x1), "a UH,,
w + uH,

9

R'9.

Hence, using the 9 condition and the relation (10.10.2), we get

-

"1

at

=

~ F H (say).

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CHAPTER

10

From the definition of H , we deduce that 2KLon

+

{l

Setting L

=

PK II x 1 '

- p(t -

to)}'

-

4

1

-

p(t

-

to)

.

max[Li, L,], and, taking into consideration the inequalities

I xi I

I/ x /I < I/ x 11'

< I/ x 11,

+ 1,

1 - ~ ( -t 1,)

< 1,

we have FH

<

H

(1

--

P(t

-

to>Y

[4K2L,n2/ j x (1'

+ 2KLn + L3 I1 x 11' + L4

+ 2KL0n + 4KLn (1 x \I2

- pK

/I x 1 '

-

rU

- ~ ( -t to)}'].

Let us now choose Kp

=

+

4K2Lon2 4KLn

+ L, + A,

and h

=

(I

- P)/p,

A

> 0,