NONLINEAR DELAY EQUATIONS WITH NONAUTONOMOUS PAST
Genni Fragnelli Dipartimento di Ingegneria dell’Informazione Universit` a di Siena Via Roma 56, 53100 Siena, Italy
Dimitri Mugnai Dipartimento di Matematica e Informatica Universit` a di Perugia Via Vanvitelli 1, 06123 Perugia, Italy
Abstract. Inspired by a biological model on genetic repression proposed by P. Jacob and J. Monod, we introduce a new class of delay equations with nonautonomous past and nonlinear delay operator. With the aid of some new techniques from functional analysis we prove that these equations, which cover the biological model, are well–posed.
1. Introduction and motivations In 1961, Francois Jacob and Jacques Monod presented a visionary gene control model, for which they received the Nobel Prize in Physiology or Medicine in 1965. In their model the gene is transcribed into a specific RNA species, the messenger RNA (mRNA). Nowadays the mathematical model they introduced to study genetic repression in eucharyotic cells (which, as opposed to bacteria, have well-defined cell nuclei), see, e.g., [19] or [21], is well known. In 2006 the Nobel Prize in Physiology or Medicine was awarded to Andrew Z. Fire and Craig C. Mello who discovered a new mechanism for gene regulation. In the same year the Nobel Prize in Chemistry was awarded to Roger D. Kornberg for his fundamental studies concerning the transfer of information stored in the genes to those parts of the cells that produce proteins. However, already four decades ago Goodwin suggested that time delays caused by the processes of transcription and translation as well as spatial diffusion of reactants could play a role in the behavior of the system ([20]). Later studies on these models included either time delays (see, e.g., [3], [23] or [33]) or spatial diffusion (see, e.g. [25]). The fundamental models which include time delays and spatial diffusion are 1991 Mathematics Subject Classification. Primary: 34G20, 47A10; Secondary: 47D06, 47H20, 47N60. Key words and phrases. nonlinear delay equations, evolution family, local semigroup, genetic repression. The research of the second author is supported by the MIUR National Project Metodi Variazionali ed Equazioni Differenziali Nonlineari. 1
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proposed in [6], [24] and [36], where the following system of equations is considered: (1) du1 (t) = h(v1 (t − r1 )) − b1 u1 (t) + a1 (u2 (t, 0) − u1 (t)), t ≥ 0, dt dv1 (t) = −b2 v1 (t) + a2 (v2 (t, 0) − v1 (t)), t ≥ 0, dt 2 ∂u (t, x) ∂ u (t, x) 2 2 = D1 − b1 u2 (t, x), t ≥ 0, x ∈ (0, 1], 2 ∂t ∂x 2 ∂v2 (t, x) = D2 ∂ v2 (t, x) − b2 v2 (t, x) + c0 u2 (t − r2 , x), t ≥ 0, x ∈ (0, 1], ∂t ∂x2 with boundary conditions ∂u2 (t, 0) = −β1 (u2 (t, 0) − u1 (t)), t ≥ 0, ∂x ∂v2 (t, 0) (2) = −β1∗ (v2 (t, 0) − v1 (t)), t ≥ 0, ∂x ∂u2 (t, 1) = ∂v2 (t, 1) = 0, t ≥ 0, ∂x ∂x and initial conditions u1 (s) = f1 (s), u1 (0) = u1,0 , v (s) = g (s), v (0) = v , 1 1 1 1,0 (3) u2 (s, x) = f2 (s, x), u2 (0, x) = u2,0 (x), v2 (s, x) = g2 (s, x), v2 (0, x) = v2,0 (x), for x ∈ (0, 1], f1 , g1 : [−r1 , 0] → R and f2 , g2 : [−r2 , 0] → L1 [0, 1]. The functions fi , gi for i = 1, 2 describe the prehistory of the system and they have to satisfy the following compatibility conditions f1 (0) = u1,0 , g1 (0) = v1,0 , f2 (0, ·) = u2,0 (·), g2 (0, ·) = v2,0 (·). In this model, the interval (0, 1] corresponds to the cytoplasm Ω \ ω, since the nucleus ω is localized at 0. The constants bi are the kinetic rates of decay, ai denote the rates of transfer between ω and Ω \ ω and they are directly proportional to the concentration gradient. The constants Di are the diffusivity coefficients and the constant c0 is the production rate for the repressor. The nonlinear function h appearing in (1) is a decreasing function and represents the production of mRNA (messenger ribo nuclein acid). It is of the form 1 , 1 + kxρ where k is a kinetic constant and ρ is the Hill coefficient. The delay r1 > 0 is the transcription time, i.e., the time necessary to the transcription reaction, and r2 > 0 is the translation time. The constants β1 and β1∗ are the constants of Fick’s law (see, e.g., [1, Chapter VI]). We underline the fact that all biological constants are positive. Concerning the Hill coefficient, generally it results ρ > 1 if more than one molecule of type v1 is needed to repress a molecule of type u1 and ρ ≤ 1 if every molecule of type v1 interacts only with one molecule of type u1 . According to this model, the eucharyotic cell Ω consists of two compartments where the most important chemical reactions take place. Such compartments are enclosed within the cell wall ∂Ω, unpermeable to the mRNA and to the repressor, and separated by the permeable nuclear membrane. The first compartment ω is the nucleus where mRNA is produced. The second compartment, denoted by Ω \ ω,
(4)
h(x) =
NONLINEAR DELAY EQUATIONS WITH NONAUTONOMOUS PAST
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is the cytoplasm in which the ribosomes are randomly dispersed. The process of translation and the production of the repressor take place here. We denote by ui and vi the concentrations of mRNA and of the repressor, respectively, in ω if i = 1 and in Ω \ ω if i = 2. These two species interact to control each other’s production. In the nucleus ω, mRNA is transcribed from the gene at a rate depending on the concentration of the repressor v1 . The mRNA leaves ω and enters the cytoplasm Ω \ ω where it diffuses and reacts with ribosomes. Through the delayed process of translation, a sequence of enzymes is produced, which in turn produce a repressor v2 . Such a repressor goes back to ω, where it inhibits the production of u1 . But, according to this model, the repressor in the cytoplasm at time t and position x depends on the mRNA that was at time t − r2 at the same position x. This assumption, however, is unrealistic. For this reason in [16] the author presented a system of modified equations, which take into account the diffusion in the past of the mRNA contained in the cytoplasm. To include such a phenomenon in the previous model, the author supposes, for simplicity, that this migration is given by a diffusion of the form et∆D , where ∆D := d2 dx2 is the Laplacian with Dirichlet boundary conditions. To be more precise, she considers the Laplacian ∆D with domain (5)
D(∆D ) := {f ∈ W 2,1 [0, 1] : f (0) = f (1) = 0}
on the Banach space L1 [0, 1]. Then the evolution family U :=(U (t, s))−1≤t≤s≤0 solving the corresponding Cauchy problem (see [17, Example 6.1]) is (6)
U (t, s) := T (s − t), −1 ≤ t ≤ s ≤ 0,
where (T (t))t≥0 = (et∆D )t≥0 is the heat semigroup on L1 [0, 1]. Here r1 = r2 = 1. Thus, assuming that the mRNA in the cytoplasm is subject to a diffusion in the past of the form et∆D , the term u2 (t − r2 , x) must be modified. Let u e2 (t − r2 , x) be the modification of u2 (t − r2 , x) governed by (U (t, s))−1≤t≤s≤0 , i.e. ½ U (−r2 , 0)u2 (t − r2 , x), 0 ≤ t − r2 , u e2 (t − r2 , x) := U (−r , t − r )f (t − r , x), 0 ≥ t − r2 , 2 2 2 2 ½ (7) T (r2 )u2 (t − r2 , x), 0 ≤ t − r2 , = T (t)f2 (t − r2 , x), 0 ≥ t − r2 . Then, system (1) becomes (8) du1 (t) = h(v1 (t − r1 )) − b1 u1 (t) + a1 (u2 (t, 0) − u1 (t)), dt dv (t) 1 = −b2 v1 (t) + a2 (v2 (t, 0) − v1 (t)), dt ∂ 2 u2 (t, x) ∂u2 (t, x) = D1 − b1 u2 (t, x), ∂t ∂x2 2 ∂v2 (t, x) = D2 ∂ v2 (t, x) − b2 v2 (t, x) + c0 u ˜2 (t − r2 , x), ∂t ∂x2
t ≥ 0, t ≥ 0, t ≥ 0, x ∈ (0, 1], t ≥ 0, x ∈ (0, 1].
In order to study the well-posedness and the stability of (8) with boundary condition (2) and initial conditions (3), in [13] the author considers the simplified and linearized system around the steady-state solutions of (8) (see [24, Section 5]).
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She rewrites (8) as a delay equation with nonautonomous past of the form ˙ = Bu(t) + Φe ut , t ≥ 0, u(t) (N DE) u(0) = x ∈ X, u e0 = f ∈ Lp (R− , X), p ≥ 1, where X is a Banach space, (B, D(B)) is a closed, densely defined operator on X, the delay operator Φ : D(Φ) → X is a bounded, linear operator, and the modified history function u et : R− → X is given by ( e (τ, t + τ )f (t + τ ) for t + τ ≤ 0, U (M HF ) u ˜t (τ ) := e (τ, t + τ )u(t + τ ) for t + τ > 0, U e (t, s))t≤s on X (see, e.g., [9]). Therefore for some backward evolution family (U problem (8) describes the behaviour of systems where the history function is modified as time goes by. Recently, in the framework of linear delay equations with nonautonomous past, G. Fragnelli and G. Nickel used a semigroup approach to discuss well-posedness and qualitative properties of equations of the form (N DE) (see [13] and [17]). In particular, they showed that solving (N DE) is equivalent to solving the abstract Cauchy problem ˙ t ≥ 0, U(t) = CU(t), Ã ! (ACP ) x U(0) = f on the product space E := X × Lp (R− , X), where C is defined by the operator matrix µ ¶ B Φ C := 0 G on the domain ½µ ¶ ¾ x D(C) := ∈ D(B) × D(G) : f (0) = x , f for a suitable operator (G, D(G)). Using perturbation theory for C0 –semigroups they proved the generator property of C and then obtained results on the asymptotic behavior of (ACP ), and hence of (N DE). Such partial functional differential equations with ”nonautonomous past” were introduced by S. Brendle and R. Nagel in [5], where the existence of mild solutions was shown by constructing an appropriate semigroup on a space of continuous functions. Under appropriate conditions, classical solutions were found in [15]. e (t, s))t≤s ≡ Id, one For the general theory of (N DE) with semigroups when (U can see, e.g., [4], [12]. The aim of this paper is to study a nonlinear version of (N DE) by a local semigroup approach. To be more precise, we consider the nonlinear problem ˙ = Bu(t) + Φ(˜ ut ), t ∈ [0, Tmax (x, f )), u(t) (N N DE) u(0) = x ∈ X, u ˜0 = f ∈ Lp (−T, 0; X), p ≥ 1, on some Banach space X where T > 0 is fixed, possibly ∞, (B, D(B)) and u et are defined as before, and the delay operator Φ is nonlinear.
NONLINEAR DELAY EQUATIONS WITH NONAUTONOMOUS PAST
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The main result concerning the well–posedness in the sense of Hadamard for problem (N N DE) is described in Theorem 4.2. As an application of such a result, we will prove that problem (8) is well–posed when h has a form similar to (4) with ρ ≤ 1 (see Section 5 and Theorem 5.5). Stability properties for solutions of (N N DE) are still under investigation. The paper is organized as follows: • in Section 2 we list the tools that will be used in the rest of the paper; • in Section 3 we rewrite the nonlinear delay equation with nonautonomous past (N N DE) as a nonlinear abstract Cauchy problem (N ACP ); • in Section 4 we prove that the nonlinear abstract Cauchy problem (N ACP ) is well–posed; • in Section 5 we apply the results proved in the previous sections to the biological model (8). 2. Preliminaries 2.1. Well-posedness of Nonautonomous Cauchy Problems. In this subsection we adapt the concept of well-posedness of the nonautonomous Cauchy problem (see, e.g., [27]) to our situation, i.e., we replace R with [−T, 0] and consider the problem ( u(t) ˙ = −A(t)u(t), −T ≤ t ≤ s ≤ 0, (N CP ) u(s) = x ∈ X, on a Banach space X, where (A(t), D(A(t)))t∈[−T,0] is a given family of (unbounded) linear operators. Definition 2.1. For a family (A(t), D(A(t)))t∈[−T,0] of linear operators on the Banach space X, the nonautonomous Cauchy problem (N CP ) is said well-posed with regularity subspaces (Ys )s∈[−T,0] if the following conditions hold: (i) (Existence) For all s ∈ [−T, 0] the subspace Ys := {x ∈ X : there exists a classical solution for (N CP )} ⊂ D(A(s)) is dense in X. (ii) (Uniqueness) For every x ∈ Ys the solution us (·, x) of (N CP ) is unique. (iii) (Continuous dependence) The solution depends continuously on s and x, i. e., if sn → s ∈ [−T, 0], xn → x ∈ Ys with xn ∈ Ysn , then kˆ usn (t, xn ) − u ˆs (t, x)k → 0 uniformly for t in compact subsets of [−T, 0], where ( us (t, x) if s ≥ t, u ˆs (t, x) := x if s < t. If, in addition, there exist constants Mω > 0 and ω ∈ R such that kus (t, x)k ≤ Mω eω(s−t) kxk for all x ∈ Ys and t ≥ s, then (N CP ) is called well-posed with exponentially bounded solutions.
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As in [27, Proposition 2.5], we can show that for each well-posed (N CP ) there exists a unique backward evolution family (U (t, s))−T ≤t≤s≤0 solving (N CP ), i.e., the function t 7→ u(t) := U (t, s)x is a classical solution of (N CP ) for s ∈ [−T, 0] and x ∈ Ys . 2.2. Evolution Families and Semigroups on Lp (−T, 0; X). We first review the basic notations and results on backward evolution families in order to describe the modification of the history function given in (M HF ). Definition 2.2. A family (U (t, s))−T ≤t≤s≤0 of bounded linear operators on a Banach space X is called an (exponentially bounded, backward) evolution family if (i) U (t, r)U (r, s) = U (t, s), U (t, t) = Id for all −T ≤ t ≤ r ≤ s ≤ 0, (ii) the mapping (t, s) 7→ U (t, s) is strongly continuous, i.e. the map (t, s) 7→ U (t, s)x is continuous ∀ x ∈ X, (iii) kU (t, s)k ≤ Mω eω(s−t) for some Mω > 0, ω ∈ R and all −T ≤ t ≤ s ≤ 0. In this paper we will use evolution semigroup techniques, for which we refer to, e.g., [7], [11, Section VI.9], [22], [26], [32], [34]. To this purpose, we first exe (t, s))−T ≤t≤s , and from now now, tend (U (t, s))−T ≤t≤s≤0 to an evolution family (U though not explicitly stated, we always assume that t ≥ −T also in the extensions. Definition 2.3. (1) The evolution family (U (t, s))−T ≤t≤s≤0 on X is extended to e (t, s))−T ≤t≤s by setting an evolution family (U for − T ≤ t ≤ s ≤ 0, U (t, s) e U (t, s) := U (t, 0) for − T ≤ t ≤ 0 ≤ s, U (0, 0) = Id otherwise. e := Lp (I; X), p ≥ 1, we then define the (2) Setting I := [−T, ∞), on the space E e corresponding evolution semigroup (T (t))t≥0 by e −T ≤ s ≤ s + t ≤ 0, U (s, s + t)f (s + t) e e e e (T (t)f )(s) := U (s, s + t)f (s + t) = U (s, 0)fe(s + t) −T ≤ s ≤ 0 ≤ s + t, e f (s + t) otherwise, e s ∈ R, t ≥ 0. for all fe ∈ E, e We As in [28], we have that the semigroup (Te(t))t≥0 is strongly continuous on E. e e denote its generator by (G, D(G)). Note that we do not assume any differentiability e (t, s))−T ≤t≤s , and hence the precise description of the domain D(G) e is difficult for (U (see Section 2.1 below). However, in [30, Proposition 2.1] the following important e is proved. property of D(G) e of G, e the generator of (Te(t))t≥0 on E, e is a dense Lemma 2.4. The domain D(G) subset of C0 ([−T, ∞), X) := {f : [−T, ∞) → X : f is continuous and lim f (t) = 0}. t→∞
e D(G)) e is a local operator (see [11, Proposition 2.3] and [30, Theorem Since (G, 2.4]), we can restrict it to the space E := Lp (−T, 0; X), p ≥ 1, by the following definition.
NONLINEAR DELAY EQUATIONS WITH NONAUTONOMOUS PAST
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Definition 2.5. Take e D(G) := {fe|[−T,0] : fe ∈ D(G)} and define
e fe)|[−T,0] Gf := (G
for f = fe|[−T,0] ∈ D(G).
The operator G is not a generator on E. However, if we identify E with the e : f (s) = 0 ∀ s ≥ 0}, then E is invariant under (Te(t))t≥0 . As a subspace {f ∈ E consequence, we obtain the following lemma. Lemma 2.6. The semigroup (T0 (t))t≥0 induced by (Te(t))t≥0 on E is given by ( U (s, s + t)f (t + s), −T ≤ s + t ≤ 0, (9) (T0 (t)f )(s) = 0, otherwise, for any f ∈ E. Let us recall that the growth bound ω0 (T0 (·)) of the semigroup (T0 (t))t≥0 is defined as n o (10) ω0 (T0 (·)) := inf ω ∈ R : ∃ Mω ≥ 1 s.t. kT0 (t)k ≤ Mω eωt ∀ t ≥ 0 , which implies that for all ω1 > ω0 (T0 (·)) there is Mω1 ≥ 1 s.t. kT0 (t)k ≤ Mω1 eω1 t
(11)
∀ t ≥ 0.
The following lemma characterizes the generator of the semigroup (T0 (t))t≥0 (see [17] and [28]). Lemma 2.7. The generator (G0 , D(G0 )) of (T0 (t))t≥0 is given by e ∩ E : f (0) = 0}, D(G0 ) = {f ∈ D(G)
G0 f = Gf.
Moreover, (T0 (t))t≥0 is nilpotent, so that σ(G0 ) = ∅ and s(G0 ) = ω0 (T0 (·)) = −∞. As a consequence, ρ(G0 ) = R, where ρ(G0 ) is the resolvent set of (G0 , D(G0 )), i.e. © ª ρ(G0 ) := α ∈ R : s.t. (αI − G0 ) is invertible . In addition, if U := (U (t, s))−T ≤t≤s≤0 , analogously to the growth bound for semigroups, we set n o ω0 (U) := inf ω ∈ R : ∃ Mω ≥ 1 s.t. kU (t, s)k ≤ Mω eω(s−t) ∀ t ≤ s ; it is clear that ω0 (T0 (·)) = ω0 (U) (see for example [11, VI Section 9.6]). Therefore if λ > ω1 , setting R(λ, G0 ) = (λI − G0 )−1 , we have (12)
kR(λ, G0 )k ≤
Mω1 , λ − ω1
where Mω1 and ω1 are as in (11) (see, e.g. [11, Proposition II.3.8]). e D(G)), e Therefore, we end up with operators (G0 , D(G0 )) ⊂ (G, D(G)) ⊂ (G, where only the first and the third are generators. Since the domain D(G) of the generator G is not given explicitly, then it is very important to find a core of it, i.e. a dense set D in D(G), endowed with the graph norm, which is invariant under the semigroup (T0 (t))t≥0 . To this aim we recall the following result.
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Lemma 2.8 ([14], Lemma 4.4). The set n D := f ∈ W 1,p (−T, 0; X); f (0) ∈ D(B), o f (s) ∈ Ys , s 7→ A(s)f (s) ∈ Lp (−T, 0; X) is a core of G. Moreover, Gf = f 0 + A(·)f
a.e.
for every f ∈ D. Here (A(·), D(A(·))) are the given unbounded operators which appear in (N CP ). 2.3. Local Semigroup. A typical phenomenon in nonlinear problems is that the solution of an abstract Cauchy problem may exist only locally. Thus the notion of local semigroup is fundamental. Here we recall some definitions, for which we refer to [8], [18]. Definition 2.9. A real Banach lattice X is said ordered if it contains a closed subset X+ satisfying (1) λf + µg ∈ X+ for any f, g ∈ X+ and for any λ, µ ≥ 0; (2) X+ ∩ −X+ = 0; (3) X+ − X+ = X. The set X+ is called a proper generating cone. If, for example, X = Lp , the space of nonnegative functions is a proper generating cone. In the same way, also spaces of the form Lp × Lq are ordered, since (f, g) = (f + , g + )−(f − , g − ), where u+ and u− denote, respectively, the positive and negative part of a whatever function u. In the following, with the writing ”f ≤ g in X” we will mean ”g − f ∈ X+ ”. Let X be a Banach space and A ⊂ X × X. It will be convenient to view A as a multi-valued function from X to X. Definition 2.10. The function A ⊂ X × X is defined as Af = A(f ) := {g : S (f, g) ∈ A}. The domain D(A) of A is {f : Af 6= ∅}. The range of L is R(A) := {Af : f ∈ D(A)}. We shall identify a single-valued function A : D(A) ⊂ X → X with its graph {(f, Af ) : f ∈ D(A)}. Thus, for example, I ”=” {(f, f ) : f ∈ X}. With the only purpose to set out some notations, we give the following definitions. Definition 2.11. A single–valued function A is called Lipschitz continuous if there is a constant L such that kAf − Agk ≤ Lkf − gk for all f, g ∈ D(A). The smallest constant L is called the Lipschitz seminorm of A and is denoted by kAkLip . Of course, if A is linear kAkLip is simply kAk. Definition 2.12. An operator A ⊂ X × X is (1) dissipative, if (I − αA)−1 is a (single–valued) function for all α > 0 and k(I − αA)−1 kLip ≤ 1 for all α > 0; (2) m–dissipative, if A is dissipative and R(I − αA) = X for some α > 0; (3) quasi m–dissipative, if A − ωI is an m–dissipative operator for some ω ∈ R. Definition 2.13. We call (V (t), Dt )t≥0 a local semigroup on a Banach space X if the following conditions hold:
NONLINEAR DELAY EQUATIONS WITH NONAUTONOMOUS PAST
(1) (2) (3) (4)
9
D St ⊆ Ds ⊆ X for every t ≥ s ≥ 0; t>0 Dt is dense in X; V (t) : Dt → X is continuous for t ≥ 0; if s, t ≥ 0, then
V (0)f = f for all f ∈ X, V (t)Ds+t ⊆ Ds , V (s + t)f = V (s)V (t)f for all f ∈ Ds+t ; S (5) limt&0 kV (t)f − f k = 0 for all f ∈ t>0 Dt . Definition 2.14. The infinitesimal generator of a local semigroup (V (t), Dt )t≥0 is defined as V (t)f − f Cf := lim , t&0 t where f ∈ D(C) and ( D(C) :=
) V (t)f − f f∈ Dt : lim exists . t&0 t t>0 [
The local semigroup (V (t), Dt )t≥0 is called positive if it is defined on an ordered Banach space X and 0 ≤ f ≤ g implies 0 ≤ V (t)f ≤ V (t)g for all t ≥ 0 and all f, g ∈ Dt ∩ X+ . Now we can define the following approximation procedure at 0. Definition 2.15. Let C : X → X be an operator on a Banach space X. We say that the operator C is approximated by globally Lipschitz operators Cν , ν ∈ N, if there exists a family of globally Lipschitz operators Cν : X → X such that Cν f = Cf for all f ∈ X satisfying kf k ≥ 1/ν. Remark 1. In [8] an analogous approximation method at infinity was introduced to show that the sum of two nonlinear operators can be a generator of a local semigroup (see Theorem 2.16 below). Of course the following results still hold true for approximations at infinity. Here we need the approximation at 0 for the biological application (see Section 5), since the function under consideration is H¨older continuous, but not Lipschitz continuous near 0. Proposition 1. Every operator C : X → X on a Banach space X approximated by globally Lipschitz S operators Cν , ν ∈ N, is the generator of a local semigroup (V (t), Dt )t≥0 with t>0 Dt = X. Proof. Let C be approximated by globally Lipschitz operators Cν . Let us consider the following sequence of abstract Cauchy problems ( u(t) ˙ = Cν u(t), (13) u(0) = f ∈ X.
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By Crandall–Liggett Theorem (see [18]), problem (13) has a unique local solution Vν (·)f and every Vν (t) : X → X is a nonlinear strongly continuous semigroup with kVν (t)kLip ≤ eων t for some ων ∈ R. By definition of Cν , Vν (·)f also solves ( u(t) ˙ = Cu(t), (14) u(0) = f ∈ X, provided that kVν (t)f k ≥ 1/ν for any t ∈ [0, Tν ), for a certain Tν > 0. By uniqueness of solutions for (13), the local semigroup is well defined by setting, analogously to what done in [8] for approximation at infinity, V (t)f := Vν (t)f for any f ∈ X and t ∈ [0, Tν ). Then V (·)f is a local solution of (14) defined in [0, Tmax (f )), where Tmax (f ) is the right end of the maximal interval of existence for the solution of problem (14). Moreover, V (t)f = Vν (t)f if kV (t)f k = kVν (t)f k ≥ 1/ν and V (t)f = lim Vν (t)f ν→∞
if
0 ≤ t ≤ Tmax (f ),
and, by the definition of local semigroup (V (t), Dt )t≥0 , we immediately get that S ¤ t>0 Dt = X. The next theorem states that if we have a sum of a quasi m-dissipative operator and a generator of a local semigroup, the Lie-Trotter product formula holds and the sum is a generator of a local semigroup; this theorem corresponds to [8, Theorem 15] for approximations at infinity, but the proof can be restated almost word by word for approximations at 0. Theorem 2.16. Let A be a quasi m–dissipative operator on an ordered Banach lattice space X and suppose that the semigroup (S(t))t≥0 generated by A is positive. Let F be a positive operator on X approximated by globally Lipschitz operators Fν , ν ∈ N, generating semigroups (Vν (t))t≥0 on X. Hence F generates a positive local semigroup (V (t), Dt )t≥0 , and suppose that such a semigroup leaves D(A) invariant. Finally, suppose that for every f ∈ D(A) ∩ X+ there exists a constant t0 (f ) > 0 such that the commutator inequality (15)
V (t)S(t)f ≤ S(t)V (t)f
holds for any t ∈ [0, t0 (f )]. Then the nonlinear Lie-Trotter product formula holds, i.e. for every f ∈ D(A) · µ ¶ µ ¶¸n · µ ¶ µ ¶¸n t t t t V S (16) U (t)f := lim S f = lim V f n→+∞ n→+∞ n n n n exists for any t ∈ [0, t0 (f )] and defines a (local) positive semigroup (U (t), Dt )t≥0 . This semigroup has generator (C, D(C)) with C = A + F and D(C) = D(A). Moreover the estimate (17)
V (t)S(t)f ≤ U (t)f ≤ S(t)V (t)f
holds true for any f ∈ D(A) ∩ X+ and any t ∈ [0, t0 (f )].
NONLINEAR DELAY EQUATIONS WITH NONAUTONOMOUS PAST
11
3. Nonlinear Delay Equations with Nonautonomous Past as Abstract Nonlinear Cauchy Problems In this section we want to rewrite the nonlinear delay equation with nonautonomous past ˙ = Bu(t) + Φ(e ut ), t ∈ [0, Tmax (x, f )), u(t) (N N DE) u(0) = x ∈ X, u e0 = f ∈ Lp (−T, 0; X), p ≥ 1, where the nonlinear delay operator Φ acts on a modified history function u et (see below), as a nonlinear abstract Cauchy problem ˙ t ∈ [0, Tmax (x, f )), U(t) = CU(t), Ã ! (N ACP ) x , U(0) = f for a suitable (C, D(C)) on the product space E := X × Lp (−T, 0; X). To this aim we now fix the notations and assumptions to be used in the rest of this paper. General Assumptions: (1) The operator (B, D(B)) is the generator of a strongly continuous semigroup (S(t))t≥0 on a ordered Banach lattice space X. (2) The nonlinear delay operator Φ : D(Φ) → X is Lipschitz continuous in every subdomain {z ∈ D(Φ) : kzk ≥ R}, R > 0. (3) The evolution family (U (t, s))−T ≤t≤s≤0 solves the backward nonautonomous Cauchy problem associated to the given family (A(t), D(A(t)))t∈[−T,0] on regularity subspaces Yt (see Definition 2.1). Remark 2. Let ω0 (S(·)) be the growth bound of the semigroup (S(t))t≥0 . Then, for all ω2 > ω0 (S(·)) there is a constant Mω2 ≥ 1 such that (18)
kS(t)k ≤ Mω2 eω2 t ,
for all t ≥ 0.
Moreover if α > ω2 , then α ∈ ρ(B), where ρ(B) is the resolvent set of (B, D(B)), and by (12) Mω2 (19) kR(α, B)k ≤ , α − ω2 where Mω2 and ω2 are as in (18) (see, e.g., [11, Proposition II.3.8]). Definition 3.1. The modified history function u et : [−T, 0] → X in (N DE) is defined as ( e (τ, t + τ )u(t + τ ) for t + τ ≥ 0, U u et (τ ) : = e (τ, t + τ )f (t + τ ) for t + τ ≤ 0, U ( U (τ, 0)u(t + τ ) for − T ≤ τ ≤ 0 ≤ t + τ, = U (τ, t + τ )f (t + τ ) for − T ≤ τ ≤ t + τ ≤ 0, e (t, s))−T ≤t≤s is the extension (as in Definition 2.3) of where the evolution family (U (U (t, s))−T ≤t≤s≤0 . Definition 3.2. A function u : [−T, Tmax (x, f ))) → X is said a classical solution of (N N DE) if
12
GENNI FRAGNELLI AND DIMITRI MUGNAI
(1) u ∈ C([−T, Tmax (x, f )), X) ∩ C 1 ([0, Tmax (x, f )), X), (2) u(t) ∈ D(B), u ˜t ∈ D(Φ), t ∈ [0, Tmax (x, f )), (3) u satisfies (N N DE) for all t ∈ [0, Tmax (x, f )). We say that (N N DE) is well-posed if ¡ ¢ (1) for every fx in a dense subspace S ⊆ X × Lp (−T, 0; X), there is a unique solution u(x, f, ·) of (N N DE), (2) the on the initial values, i.e., if a sequence ¡ xn ¢solutions depend continuously ¡x¢ in S converges to ∈ S, then u(xn , fn , t) converges to u(x, f, t) f fn uniformly for t in compact intervals. It is now our purpose to investigate the well-posedness of (N N DE). As we said before, we restate equation (N N DE) as an abstract Cauchy problem on the space E := X × Lp (−T, 0; X) using the operator G from Definition 2.5. Definition 3.3. We define the operator C by the matrix µ ¶ B Φ C := , 0 G defined on the domain ©¡ ¢ ª D(C) := fx ∈ D(B) × D(G) : f (0) = x
⊆
E = X × Lp (−T, 0; X).
It is easy to show that this operator is closed and densely defined on E, provided that Φ is closed and densely defined on Lp (−T, 0; X). Let us introduce the abstract Cauchy problem associated to C: ˙ t ≥ 0, U(t) = CU(t), Ã ! (N ACP ) x . U(0) = f In [17] it was showed that a linear equation of the form (N N DE) and the associated abstract Cauchy problem (N ACP ) are ¡”equivalent”, in the sense that ¢ (N N DE) has a unique global solution for every fx ∈ D(C) depending continuously on the initial value if and only if (N ACP ) is well-posed (in the usual sense). Therefore, well-posedness of (N N DE) is obtained by proving well-posedness of (N ACP ), which is done by showing that the operator C is a generator of a strongly continuous semigroup. The proof given in [17] can be followed to prove the equivalence between problems (N N DE) and (N ACP ) also in the nonlinear case. Of course, in this case we must deal with local solutions and local semigroups, as the following result shows. Theorem 3.4. The nonlinear delay equation (N N DE) is well-posed if and only if the operator (C, D(C)) is the generator of a local semigroup ¡ ¢ (T (t))t≥0 on E. In this case, (N N DE) has a unique local solution u for every fx ∈ D(C), given by ( ¡ ¡ ¢¢ π1 T (t) fx , t ∈ [0, T ), u(t) = f (t), a.e. t ∈ [−T, 0], where π1 is the projection onto the first component of E.
NONLINEAR DELAY EQUATIONS WITH NONAUTONOMOUS PAST
13
4. The Generator In view of Theorem 3.4, we now give sufficient conditions on C so that it generates a nonlinear local strongly continuous semigroup on E. First, we write C in the form µ ¶ µ ¶ B 0 0 Φ (20) C = C0 + F, where C0 := and F := , 0 G 0 0 with domain D(C0 ) = D(C) and F : E → E. If the linear operator (C0 , D(C0 )) generates a strongly continuous semigroup, we can apply Theorem 2.16 to C0 and F to obtain conditions such that the operator (C, D(C)) generates a nonlinear strongly continuous semigroup. First of all, we need to compute the inverse (λ − C0 )−1 of (λ − C0 ) for any λ ∈ R (such an inverse will be used later). This means that λ > ω0 (T0 (·)), where ω0 (T0 (·)) is the growth bound of the semigroup (T0 (t))t≥0 defined in (10). We recall the following result, considered when ω0 (T0 (·)) = −∞. Lemma 4.1 (Lemma 4.1, [17]). For any λ ∈ R define the bounded operator ²λ : X → E by (21)
(²λ x)(s) := eλs U (s, 0)x,
s ∈ [−T, 0], x ∈ X.
Then 1. for every x ∈ X, ²λ x is an eigenvector of G with eigenvalue λ. Moreover k²λ kL(X,E) ≤ Mω max{1, e(ω−λ)T } with ω ∈ R and Mω ≥ 1 is given according to the definition of growth bound in (10). 2. If λ ∈ ρ(B), then λ ∈ ρ(C0 ), and the resolvent Rλ := (λI − C0 )−1 is given by µ ¶ R(λ, B) 0 (22) Rλ = . ²λ R(λ, B) R(λ, G0 ) As a second step, we determine explicitly the semigroup generated by C0 . Proposition 2 (Proposition 4.2, [17]). The operator (C0 , D(C0 )) is the generator of a strongly continuous semigroup (T0 (t))t≥0 on E given by µ ¶ S(t) 0 (23) T0 (t) := , St T0 (t) where T0 is given in (2.6) and St : X → Lp (−T, 0; X) is defined by ( U (τ, 0)S(t + τ )x, τ + t > 0, (24) (St x)(τ ) := 0, otherwise. As an immediate consequence of the Generation Theorem by Feller–Miyadera– Phillips (see [11, Proposition II.3.8]), we have the next corollary. Corollary 1. The operator (C0 , D(C0 )) is closed and densely defined in E. Now, let C : Lp (−T, 0; X) → X be given by Cz(t, ·) := Φ(e zt (·)) for all t ≥ 0 and z ∈ Lp (−T, 0; X). We then define the operators Cν , ν ∈ N, as Cν z(t, ·) := Φν (e zt (·))
∀ t ≥ 0,
14
GENNI FRAGNELLI AND DIMITRI MUGNAI
where Φν is a whatever Lipschitz continuous extension of Φ|{kzk≥ ν1 } on B 1/ν ; for example, one could take ( 1 Φ(z), ³ ´ kzk ≥ ν , Φν (z) := z νkzkΦ ν1 kzk , 0 ≤ kzk ≤ ν1 . By General Assumption 2, it is not difficult to show that Φν is globally Lipschitz continuous for any ν ∈ N. Proposition 3. The operators Cν are globally Lipschitz continuous. Proof. Let r and m the functions defined as r : u(s) ∈ X 7→ ut (s)X, where ut is the history function defined as ut (s) := u(t + s), and m : ut (s) ∈ X 7→ u et (s) ∈ X, where u et is defined in (3.1). The functions r and m are linear by definition (in fact the evolution family which characterized the modified history function u et is a family of linear operators). Then m ◦ r is linear and thus m ◦ r is globally Lipschitz continuous. Since Φν is locally Lipschitz continuous for all ν, we obtain that each Cν is globally Lipschitz continuous as well. ¤ As an immediate consequence of the previous proposition and of the definitions we have the following corollary. Corollary 2. The operator C is approximated by the globally Lipschitz operator Cν . Now set (25)
Fν :=
µ 0 0
¶ Cν . 0
Since the operator C is approximated by the globally Lipschitz operators Cν , the operator µ ¶ 0 C F := 0 0 is approximated by Fν , which are again globally Lipschitz continuous. By Proposition 1, the next result is immediate. Proposition 4. The operator F generates a local semigroup S D = X. t t>0
(V (t), Dt )t≥0 with
Now, we give an explicit expression of the local semigroup (V (t), Dt ). Consider the Cauchy problem ( ˙ V(t) = FV(t), t ∈ [0, Tmax (x, f )), ¡ ¢ (N ACP )1 V(0) = fx ∈ E, µ ¶ v(t) where f (0) = x and V(t) := ∈ E for all t ∈ [0, Tmax (x, f )). Then (N ACP )1 z(·, t) is equivalent to the system v(t) ˙ = Cz := Φ(e zt ), z ˙ = 0, v(0) = x, z(0, ·) = f, z(0, 0) = f (0) = x.
NONLINEAR DELAY EQUATIONS WITH NONAUTONOMOUS PAST
15
It follows that z(t, ·) ≡ z(0, ·) = f (·) for all t > 0. Moreover ( ( U (τ, 0)z(t + τ, τ ), t + τ ≥ 0, U (τ, 0)f (τ ), t + τ ≥ 0, zet (τ ) = = U (τ, t + τ )f (t + τ ), t + τ < 0, U (τ, t + τ )f (t + τ ), t + τ < 0. Thus
( Φ(U (·, 0)f (·)), t + · ≥ 0, v(t) ˙ = Φ(˜ zt ) Φ(U (·, t + ·)f (t + ·)), t + · < 0,
(of course v depends also on τ , the time variable in the past). Integrating, we obtain t+·≥0 tΦ(U (·, 0)f (·)), Z t v(t) = x + Φ(U (·, σ + ·)f (σ + ·))dσ, t + · < 0. 0
Hence, the unique solution of (N ACP )1 is à ! x + tΦ(U (·, 0)f (·)) , t + · ≥ 0, µ ¶ f x à ! (26) t 7→ V(t) = =: VΦ (t) , Rt f x + 0 Φ(U (·, σ + ·)f (σ + ·))dσ , t+·<0 f for all t ∈ [0, Tmax (x, f )). Then, the local semigroup (V (t), Dt )t∈[0,Tmax (x,f )) generated by F is given by (27)
(V (t)Y)(s) := VΦ (t)(Y(s))
for all t ∈ [0, Tmax (x, f )) and Y ∈ E. Remark 3. If the delay operator Φ and the evolution family (U (t, s))−T ≤t≤s≤0 are positive, then it is clear that the local semigroup (V (t), Dt )t∈[0,Tmax (x,f )) is positive as well. The following proposition is the essential tool for the main results. The idea goes back to Webb ([35]), where a new norm is introduced in order to make a strongly continuous semigroup also a quasi contractive semigroup. Proposition 5. Assume tha [0, +∞) ⊂ ρ(B). Then there exists a norm on E equivalent to the original one and ω ≥ 0 such that C0 − ωI is m–dissipative, i.e. C0 is quasi m–dissipative. Proof. The thesis follows if we prove that there exist a suitable norm on E and ω ≥ 0 such that (a) R(I − α0 (C0 − ωI)) = E for some α0 > 0; (b) (I − α(C0 − ωI))−1 is a function for all α > 0 and k(I − α(C0 − ωI))−1 k ≤ 1 for any α > 0, since k · k = k · kLip for linear operators. 0ω (a) Let α0 > 0 and ω ≥ 0. Put γ := 1+α α0 . Then γ > 0 and, consequently, γ ∈ ρ(B). By Lemma 4.1, (γI − C0 ) is invertible and so (a) is proved. (b) To invert (I − α(C0 − ωI)) is equivalent to invert α(γI − C0 ) with γ := ω + 1/α. But α > 0 and ω ≥ 0, so, proceeding as in the proof of (a), we get that (γI − C0 ) is invertible and by Lemma 4.1 the inverse is µ ¶ R(γ, B) 0 R(γ, C0 ) = , ²γ R(γ, B) R(γ, G0 )
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GENNI FRAGNELLI AND DIMITRI MUGNAI
where ²γ is defined in Lemma 4.1 as well. Now, the claim follows if we prove that in a suitable equivalent norm we have k(I − α(C0 − ωI))−1 k ≤ 1. First, let us note that ¡ ¢−1 1 (28) k(I − α(C0 − ωI))−1 k = k α(γI − C0 ) k = kR(γ, C0 )k. α Moreover, since C0 generates a strongly continuous semigroup on E (see Proposition 2), by [11, Proposition I.5.5] there exist ω ¯ ∈ R and Mω¯ ≥ 1 such that kT0 (t)k ≤ Mω¯ eω¯ t
∀ t ≥ 0.
By Lemma 4.1, if λ > 0 then λ ∈ ρ(C0 ), and so we can apply the Renorming Lemma (see [2, Lemma 3.5.4] or [29, Lemma I.5.1]); in this way, since γ > 0, we can find a norm in E which is equivalent to the original one and such that 1 (29) kR(γ, C0 )k ≤ . γ−ω ¯ Combining (28) and (29) we finally get 1 1 = ≤ 1 ∀α > 0 k(I − α(C0 − ωI))−1 k ≤ α(γ − ω ¯) 1 + α(ω − ω ¯) as soon as ω ≥ max{0, ω ¯ }. The claim follows.
¤
From now on, though not explicitly stated again, we will assume that E is endowed with the norm found in Proposition 5, so that C0 is quasi m-dissipative. Now, let us set n¡ ¢ o x E+ := f ∈ E : x ∈ X+ , f (τ ) ∈ X+ a.e. τ ∈ [−T, 0] . ¡ ¢ ¡ ¢ ¡ ¢ ¡ ¢ With the writing fx ≤ yg we mean that yg − fx ∈ E+ . The next theorem gives conditions under which the operator (C, D(C)) is a generator, so that problem (N ACP ), and then (N N DE), is well-posed. Such a Theorem is a corollary of Theorem 2.16, taking A = C0 , X = E, S(t) = T0 (t) and F = F, which generates the positive local semigroup (V (t), Dt )t≥0 . Theorem 4.2. Assume the following conditions: (1) B generates a positive semigroup S(t); (2) [0, +∞) ⊆ ρ(B); (3) the evolution family U := (U (t, s))−T ≤t≤s≤0 associated to (T0 (t))t≥0 defined in Lemma 2.6 is positive; (4) the delay operator Φ is positive; (5) (V (t), Dt )t≥0 (see (27)) leaves D(C0 ) invariant; ¡ ¢ ¡ ¢ (6) for every fx ∈ D(C0 ) ∩ E+ there exists a constant t0 fx > 0 such that the commutator inequality µ ¶ µ ¶ x x (30) V (t)T0 (t) ≤ T0 (t)V (t) f f ¡ ¢ holds for any t ∈ [0, t0 fx ]. ¡ ¢ Then the nonlinear Lie-Trotter product formula holds, i.e. for every fx ∈ D(C0 ), µ ¶ · µ ¶ µ ¶¸n µ ¶ t t x x V T (t) := lim T0 f f n→+∞ n n · µ ¶ µ ¶¸n µ ¶ (31) t t x T0 = lim V f n→+∞ n n
NONLINEAR DELAY EQUATIONS WITH NONAUTONOMOUS PAST
exists for every t ∈ [0, t0
¡x¢ f
17
] and defines a (local) positive semigroup (T (t))t≥0
with generator (C, D(C)) (see (20)) and D(C) = D(C0 ). Moreover the estimate µ ¶ µ ¶ µ ¶ x x x (32) V (t)T0 (t) ≤ T (t) ≤ T0 (t)V (t) f f f ¡ ¢ ¡ ¢ holds true for any fx ∈ D(C0 ) ∩ E+ and any t ∈ [0, t0 fx ]. µ ¶ B 0 and that C0 generates a strongly continuous semiProof. Recall that C0 = 0 G group (T0 (t))t≥0 by Proposition 2. Now, (T0 (t))t≥0 is positive, since U is positive (see (9)); B generates a positive semigroup (S(t))t≥0 which induces a positive family St defined in (24). Then T0 (t) is positive by (23). Moreover, since [0, +∞) ⊂ ρ(B), by Proposition 5 C0 is quasi m–dissipative. Therefore, by Proposition 4, F generates a local semigroup (V (t), Dt )t≥0 which is positive by Remark 3; by assumption, (V (t), Dt )t≥0 leaves D(C0 ) invariant. Hence Theorem 2.16 can be applied. ¤ 5. The biological application In this section we want to apply the theory developed in the previous sections to the model of genetic repression presented in the introduction. First, we rewrite (8) as a (N N DE) and, for the sake of simplicity, we assume r1 = r2 = T (if r1 6= r2 nothing changes, except for some notations). Moreover, we take X := R2+ × (L1 [0, 1])2 and as (B, D(B)) the operator −b1 − a1 0 a1 δ0 0 0 −b2 − a2 0 a2 δ0 (33) B := 0 0 D1 ∆ − b1 0 0 0 0 D2 ∆ − b2 with domain ½µ x ¶ ¾ ¡f ¢ ¡x¢ y 2 2,1 2 0 0 D(B) := ∈ R × (W [0, 1]) : L = and f (1) = g (1) = 0 , y f g + g
where δ0 f (t, x) := f (t, 0) for any continuous functions f (i.e. δ0 is the Dirac measure d2 2,1 in the x–variable), ∆ := dx [0, 1])2 → R2+ is defined by 2 and the operator L : (W ! µ ¶ Ã f 0 (0) f β1 + f (0) , (34) L = g0 (0) g + g(0) ∗ β1
d where, with abuse of notation, we have set ”0 = dx ”. Our purpose is to apply Theorem 4.2. In order to do that, an essential fact will be that C0 is quasi m-dissipative. As already observed in the previous Section, this can be obtained by choosing an equivalent norm in the domain E. Without any further comment, we assume this fact. As in [16], we can prove the following theorem.
Theorem 5.1. The operator (B, D(B)) generates on X a positive analytic semigroup (S(t))t≥0 .
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GENNI FRAGNELLI AND DIMITRI MUGNAI
Thus it is well known (see for example [7, Theorem 2.7]) that ω0 (S(·)) = s(B), where ω0 (S(·)) and s(B) are the growth bound of (S(t))t≥0 and the spectral bound of (B, D(B)), respectively. In order to apply Theorem 4.2 we have to compute ρ(B). To this aim we can proceed as in [16] and consider the matrix operator B on X of the form µ ¶ A D B := , 0 C where the operators A, C, D are diagonal matrices, i.e. µ ¶ −b1 − a1 0 A := , 0 −b2 − a2 µ ¶ D1 ∆ − b1 0 C := 0 D2 ∆ − b2 and µ ¶ a 1 δ0 0 D := , 0 a2 δ0 with ©¡ ¢ ª D(A) := R2 , D(C) := fg ∈ (W 2,1 [0, 1])2 : f 0 (1) = g 0 (1) = 0 , D(D) := (W 2,1 [0, 1])2
and
D(B) = R2 × D(C).
Of course B ⊆ B. Moreover, since the operator B is one-sided coupled (see [10, Definition 1.1]), for the spectral bound of B the following proposition holds. Proposition 6. Let L : (W 2,1 [0, 1])2 → R2 be the operator defined in (34) and let E ⊂ C with D(E) = KerL and L0 := (L|ker C )−1 : R2 → ker(C) ⊆ (L1 [0, 1])2 . Then the spectral bounds of the operators B, E and A + DL0 satisfy s(B) < 0 ⇐⇒ s(E) < 0 and s(A + DL0 ) < 0. The proof of this proposition follows again by [10, Theorem 4.1], rewriting B as µ ¶µ ¶ µ ¶ A 0 Id 0 0 D B := + . 0 E −L0 Id 0 0 Now, let E1 be the operator E1 ⊆ D1 ∆ − b1 with domain (35)
D(E1 ) := {f ∈ W 2,1 [0, 1] : f 0 (1) = 0 and f 0 (0) = −β1 f (0)}
and E2 the operator E2 ⊆ D2 ∆ − b2 with domain (36)
D(E2 ) := {f ∈ W 2,1 [0, 1] : f 0 (1) = 0 and f 0 (0) = −β1∗ f (0)}.
The following result follows at once, as in [16, Proposition 5.4] Proposition 7. The spectral bounds of the operators E and Ei satisfy the following property: s(E) < 0 ⇔ s(E1 ) < 0 and s(E2 ) < 0. As a consequence of Proposition 6 and Proposition 7, we can compute the resolvent set of the operator B: Theorem 5.2. Assume that s(E1 ), s(E2 ), and s(A + DL0 ) are negative. Then s(B) < 0 and, consequently, [0, +∞) ⊂ ρ(B).
NONLINEAR DELAY EQUATIONS WITH NONAUTONOMOUS PAST
19
Now, let us go back to the biological model, defining the nonlinear delay operator Φ : D(Φ) = W 1,p (−T, 0; X) → X as 0 h◦δ −T 0 0 0 0 0 0 , (37) Φ := 0 0 0 0 0 0 c0 δ −T 0 where δ −T f (·, x) = f (−T, x) for any continuous function f (i.e. δ −T is the Dirac measure in the t–variable) and the function h is defined as follows: 1 if ϑ ≤ M, 1 + kϑρ 1 1 ρ−1 (38) h(ϑ) = − (ϑ − M ) + if M < ϑ ≤ Mρ , (1 + kM ρ )2 kρM 1 + kM ρ 0 if ϑ ≥ Mρ , where Mρ = M +
1 + KM ρ and M is a large constant. kρM ρ−1
Remark 4. With respect to the function h described in (4), the nonlinearity has now been changed only for large values of the variable ϑ, starting with the tangent line in M until it reaches 0 and then considering the null function. In this way, this function h is still continuous in R+ , globally Lipschitz continuous in any set of the form [ε, +∞) and convex if ρ ≤ 1, as the function of (4). However, this change is 1 not so relevant from a biological point of view, since, if M is large enough, 1 + kM ρ is so small that it has no biological meaning. Remark 5. Since Φ is bounded, it is clear that the solution of (8) is defined for any t ≥ 0. We will consider the following evolution family IdR+ 0 0 0 0 IdR+ 0 0 (39) U (t, s) := 0 0 T (s − t) 0 0 0 0 IdL1 [0,1]
for − T ≤ t ≤ s ≤ 0,
where (T (t))t≥0 denotes the heat semigroup on L1 [0, 1] generated by the Laplacian with Dirichlet boundary conditions. Note that by Lemma 2.7 we get ω0 (U) = −∞, where U := (U (t, s))−T ≤t≤s≤0 . Remark 6. The evolution family (U (t, s))−T ≤t≤s≤0 and the delay operator Φ are clearly positive. Now it is easy to prove that nonautonomous past ˙ (t) W W (0) (40) f W0 where (41)
system (8) is equivalent to the delay equation with ft ), t ≥ 0, = BW (t) + Φ(W = x ∈ X, = f ∈ Lp (−T, 0; X),
u1 (t) v1 (t) W (t) := u2 (t) , v2 (t)
f1 g1 f = f2 , g2
u1,0 v1,0 x= u2,0 v2,0
20
GENNI FRAGNELLI AND DIMITRI MUGNAI
ft : [−T, 0] → X is defined by and the modified history function W ½ U (τ, 0)W (t + τ ) for 0 ≤ t + τ, ft (τ ) := W U (τ, t + τ )f (t + τ ) for t + τ ≤ 0 (here (U (t, s))−T ≤t≤s≤0 is the evolution family defined in (39)). Finally, define the operator (G, D(G)) as the matrix d 0 0 0 dσ d 0 0 0 , dσ G := 0 0 G 0 d 0 0 0 dσ with domain D(G) := (W 1,p [−T, 0])2 × D(G) × W 1,p (−T, 0; L1 [0, 1]). Here
d dσ
denotes the weak derivative and (G, D(G)) is the closure of
(42)
Af = f 0 + f 00 ,
for f in an appropriate subspace of D(G) (for details see [14, Proposition 3.1]). Then, as in Section 3, we can rewrite (40) as the nonlinear abstract Cauchy problem associated to the operator µ ¶ B Φ (43) C := , 0 G with domain (44)
¡ ¢ D(C) := { fx ∈ D(B) × D(G) : f (0) = x}
¡ ¢ on the product space E := X × Lp (−T, 0; X) and initial value fx . In order to apply Theorem 4.2 to the concrete model, we will make the following hypotheses, which will be assumed throughout the rest of the paper. Main Assumption: The Hill coefficient ρ is less or equal to 1 (the biological meaning of the previous assumption is described in the Introduction). This assumption lets us say that Φ satisfies General Assumption 2 in Section 3 without being globally Lipschitz continuous (this is the case when ρ > 1). Moreover, by the Main Assumption above, h is convex. Thus, by definition, the operator Φ is convex as well. By this property, one can deduce a criterion for inequality (30) to hold, using Jensen’s inequality, as the following result shows. Theorem 5.3. Let (C0 , D(C0 ) be as in (20) and let (V (t), Dt )t≥0 be µ the ¶ local positive x semigroup solving the Cauchy problem (N ACP )1 . Then for any ∈ D(C0 ) ∩ E+ f ¡x¢ there exists t0 f > 0 such that the commutator condition µ ¶ µ ¶ x x V (t)T0 (t) ≤ T0 (t)V (t) f f ¡ ¢ holds for all t ∈ [0, t0 fx ]. Here (T0 (t))t≥0 is the semigroup generated by (C0 , D(C0 ) (see Proposition 2).
NONLINEAR DELAY EQUATIONS WITH NONAUTONOMOUS PAST
21
µ ¶ x ∈ D(C0 ) ∩ E+ and t ≥ 0. Since the semigroup (T0 (t))t≥0 is f linear, by the Riesz Representation Theorem (see [31, Chapter 2]), there exists a measure P (·, t, s), independent of ζ, such that Z 0 (T0 (t)ζ)(s) = ζ(y)P (dy, t, s).
Proof. Let ζ :=
−T
Now, let us suppose that there exists t0 (ζ) such that (45)
P (Ω, t, ·) ≥ 1
∀ t ∈ [0, t0 (ζ)]. ¡ ¢ ¡ ¢ Since Φ is convex, it is apparent that the function fx 7→ VΦ (t) fx is convex component by component for any t. Then Jensen inequality implies Z 0 (V (t)T0 (t)ζ)(s) = VΦ (t)(T0 (t)ζ)(s) = VΦ (t) ζ(y)P (dy, t, s) Z
−T 0
1 VΦ (t)(P (Ω, t, s)ζ(y))P (dy, t, s) P (Ω, t, s) −T and since P (Ω, t, s) ≥ 1, the last integral is Z 0 ≤ (V (t)P (Ω, t, s)ζ)(y)P (dy, t, s), ≤
−T
where all the previous inequalities are intended component by component. Now, let us recall that Φ is either linear or nonlinear, but that its nonlinear part (given by the function h) is decreasing on positive functions, so that Z 0 Z 0 (V (t)P (Ω, t, s)ζ)(y)P (dy, t, s) ≤ (V (t)ζ)(y)P (dy, t, s) −T
−T
= (T0 (t)V (t)P (Ω, t, s)ζ)(s) for all t ≥ 0. Of course, to prove the previous inequalities, we also used the fact that all the operators are positive and that f is positive, as well. Finally, let us prove (45). By Riesz representation Theorem, the measure P is such that P (Ω, t, ·) = kT0 (t)k, where kT0 (t)k represents the norm of the operator T0 (t). Now, the operator T0 is as in (23), where U is given in (39), so that (45) is immediate. ¤ Remark 7. In [8, Corollary 20] it is asserted that, under the setting of Theorem 2.16, if A is linear, then (17) is automatically satisfied. Such a statement is based on [8, Remark 17], which turns out to be incorrect. Indeed, in order to prove (17) when A is linear, the authors use Riesz Representation Theorem, but they assume to deal with a probability measure, or equivalently, with a semigroup (S(t))t≥0 generated by A such that kS(t)k = 1 for any t ≥ 0. Of course this is a requirement which is, in general, not verified. Therefore, [8, Corollary 20] is true only if kS(t)k = 1 for any t ≥ 0. As a consequence of Remark 6 and of Theorems 4.2 and 5.3 we can prove that the operator (C, D(C)) defined in (43) is a local generator, the main result of this paper. Theorem 5.4. Assume that • s(Ei ) < 0, i = 1, 2;
22
GENNI FRAGNELLI AND DIMITRI MUGNAI
• s(A + DL0 ) < 0 and • VΦ leaves D(C0 ) invariant. Then the nonlinear operator (C, D(C)) defined in (43) generates a local positive semigroup (T (t))t≥0 , i.e. system (8) with boundary conditions (2) and initial conditions (3), and with h given in (38) is well-posed. Proof. The proof is now immediate by Proposition 6, since the evolution family U = (U (t, s))−T ≤t≤s≤0 associated to the semigroup (T0 (t))t≥0 defined in Lemma 2.6 is positive. ¤ Finally, we will give some sufficient conditions in order to apply Theorem 5.4. In order to verify the condition s(Ei ) < 0, we can restrict the class of the constants appearing in the problem. More precisely, by classical result on rescaled semigroups (see, e.g., [11, II.2.2]), we get (46)
s(Ei ) = D1 s(∆) − bi ,
i = 1, 2.
On the other hand, by [11, VI.4.b], we get the existence of two constants ξ = ξ(β1 ) > 0 and ξ ∗ = ξ ∗ (β1∗ ) > 0 such that (47)
s(E1 ) ⊆ (−∞, ξ]
and
s(E2 ) ⊆ (−∞, ξ ∗ ].
Theorem 5.4 has the following corollary, the main application of our abstract results. Theorem 5.5. Assume that b1 > ξ, b2 > ξ ∗ , s(A+DL0 ) < 0 and that the prehistory functions f2 and g1 in (3) are such that µ ¶ d t∆D e f2 (−s) + β1∗ (et∆D f2 (−s))(0) = 0, ∀ t, s ∈ [0, T ], dx |x=0 ·
d t∆D (e f2 (−s)) dx
¸ = 0,
∀ t, s ∈ [0, T ]
|x=1
and g1 (−T ) ≥ Mρ (see (38)), where ∆D denotes the Laplacian operator with Dirichlet conditions on L1 [0, 1] (see (5)). Then system (8) with boundary conditions (2), initial conditions (3) and with h given in (38) is well-posed. Remark 8. The assumption g1 (−T ) ≥ M , so that h(g1 (−T )) = 0, means that at the beginning of the story, the concentration of the repressor in the nucleus was so high that the growth of the mRNA was positively influenced only by the mRNA in the cytoplasm. Proof of Theorem 5.5. By (46) and (47), it immediately follows that s(Ei ) < 0, i = 1, 2. Then Proposition 7 implies that s(E) < 0. In order to apply Theorem 5.4, we now must prove that D(C0 ) invariant. VΦ leaves u1,0 v1,0 ¡ ¢ To this aim let fx ∈ D(C0 ), where, we recall, x := u2,0 ∈ D(B) and f := v2,0
NONLINEAR DELAY EQUATIONS WITH NONAUTONOMOUS PAST
23
f1 g1 ∈ D(G). As we have proved in (26), f2 g2 à ! x + tΦ(U (·, 0)f (·)) , t + · ≥ 0, ¡x¢ f à ! VΦ (t) f = Rt x + 0 Φ(U (·, σ + ·)f (σ + ·))dσ , t + · < 0. f Since the second component is the identity, then f ∈ D(G) and f (0) = x. It remains to prove that (a) if t − T ≥ 0, then x + tΦ(U (·, 0)f (·)) belongs to D(B) and ¡ ¢ f (0) = x + tΦ(U (·, 0)f (·)) |t=0 , Rt (b) if t − T < 0, then x + 0 Φ(U (·, σ + ·)f (σ + ·))dσ ∈ D(B) and Z t ¡ ¢ f (0) = x + Φ(U (·, σ + ·)f (σ + ·))dσ |t=0 . 0
In both cases, since f (0) = x, it is clear that the second requirement in (a) and (b) is automatically fulfilled. Now, it is sufficient to prove that the boundary conditions required in the definition of D(B) are satisfied, i.e. µ ¶ µ ¶ x ˜ x ˜1 L 3 = and x ˜03 (1) = x ˜04 (1) = 0, x ˜4 x ˜2 where, by (34),
! µ ¶ Ã x˜03 (0) + x ˜ (0) 3 x ˜3 1 L = x˜04β(0) . x ˜4 ˜4 (0) β∗ + x 1
Here x ˜i , for i = 1, 2, 3, 4, denote the components of x + tΦ(U (·, 0)f (·)) or x + Rt Φ(U (·, σ + ·)f (σ + ·))dσ. 0 First assume that t − T ≥ 0. Then x ˜1 = u1,0 + th(g1 (−T )),
x ˜2 = v1,0 ,
x ˜3 = u2,0 ,
and
x ˜4 = v2,0 + c0 δ −T (e−·∆D f2 (·)) = v2,0 + c0 eT ∆D f2 (−T ). Thus, by assumption, x ˜03 (1) = u02,0 (1) = 0 and u02,0 (0) x ˜03 (0) +x ˜3 (0) = + u2,0 (0) = u1,0 + th(g1 (−T )) = u1,0 β1 β1
since h(g1 (−T )) = 0. Moreover,
· x ˜04 (1)
and
=
0 v2,0 (1)
+ c0
d T ∆D (e f2 (−T )) dx
¸ =0 |x=1
· ¸ 0 v2,0 (0) x ˜04 (0) c0 d T ∆D (e f2 (−T )) +x ˜4 (0) = + v2,0 (0) + ∗ β1∗ β1∗ β dx |x=0 + c0 (eT ∆D f2 (−T ))(0) = v1,0
24
GENNI FRAGNELLI AND DIMITRI MUGNAI
by assumption. Analogously we can prove that if t − T < 0, then Z t x+ Φ(U (·, σ + ·)f (σ + ·))dσ ∈ D(B). 0
Indeed in this case x ˜1 = u1,0 + th(g1 (−T )), and
Z
t
x ˜4 = v2,0 + c0
x ˜2 = v1,0 ,
x ˜3 = u2,0
eσ∆D f2 (σ − T )dσ.
0
Hence, as before, x ˜03 (1) = u02,0 (1) = 0 and u02,0 (0) x ˜03 (0) +x ˜3 (0) = + u2,0 (0) = u1,0 + th(g1 (−T )) = u1,0 . β1 β1 Moreover, by assumption, · µZ t ¶¸ d 0 (1) + c0 eσ∆D f2 (σ − T )dσ x ˜04 (1) = v2,0 dx 0 |x=1 ¶ Z tµ d σ∆D = (e f2 (σ − T )) dσ = 0, dx 0 |x=1 and
· µZ t ¶¸ 0 (0) v2,0 c0 d x ˜04 (0) σ∆D + x ˜ (0) = + v (0) + e f (σ − T )dσ 4 2,0 2 β1∗ β1∗ β1∗ dx 0 |x=0 Z t + c0 (eσ∆D f2 (σ − T ))(0)dσ 0 Z · ¢ c0 t d ¡ σ∆D = u1,0 + ∗ e f2 (σ − T ) |x=0 β1 0 dx ¸ + β1∗ (eσ∆D f2 (σ − T ))(0) dσ = u1,0 .
Thus VΦ leaves D(C0 ) invariant and the theorem is proved.
¤
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