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Manuscript submitted to Website: http://AIMsciences.orgAIMS’ JournalsVolume X, Number 0X, XX 200X pp. X–XX

NONLINEAR DELAY EQUATIONSWITH NONAUTONOMOUS PAST

Genni Fragnelli

Dipartimento di Ingegneria dell’InformazioneUniversita di Siena

Via Roma 56, 53100 Siena, Italy

Dimitri Mugnai

Dipartimento di Matematica e InformaticaUniversita di Perugia

Via Vanvitelli 1, 06123 Perugia, Italy

(Communicated by the associate editor name)

Abstract. Inspired by a biological model on genetic repression proposed byP. Jacob and J. Monod, we introduce a new class of delay equations withnonautonomous past and nonlinear delay operator. With the aid of some newtechniques from functional analysis we prove that these equations, which coverthe biological model, are well–posed.

1. Introduction and motivations. In 1961, Francois Jacob and Jacques Monodpresented a visionary gene control model, for which they received the Nobel Prizein Physiology or Medicine in 1965. In their model the gene is transcribed into aspecific RNA species, the messenger RNA (mRNA). Nowadays the mathematicalmodel they introduced to study genetic repression in eucharyotic cells (which, asopposed to bacteria, have well-defined cell nuclei), see, e.g., [19] or [21], is wellknown.

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. Inthe same year the Nobel Prize in Chemistry was awarded to Roger D. Kornberg forhis fundamental studies concerning the transfer of information stored in the genesto those parts of the cells that produce proteins.

However, already four decades ago Goodwin suggested that time delays caused bythe processes of transcription and translation as well as spatial diffusion of reactantscould play a role in the behavior of the system ([20]). Later studies on these modelsincluded either time delays (see, e.g., [3], [23] or [33]) or spatial diffusion (see, e.g.

2000 Mathematics Subject Classification. Primary: 34G20, 47A10; Secondary: 47D06, 47H20,47N60.

Key words and phrases. nonlinear delay equations, evolution family, local semigroup, geneticrepression.

The research of the second author is supported by the MIUR National Project Metodi Varia-zionali ed Equazioni Differenziali Nonlineari.

1

2 GENNI FRAGNELLI AND DIMITRI MUGNAI

[25]). The fundamental models which include time delays and spatial diffusion areproposed in [6], [24] and [36], where the following system of equations is considered:

du1(t)dt

= h(v1(t− r1))− b1u1(t) + a1(u2(t, 0)− u1(t)), t ≥ 0,

dv1(t)dt

= −b2v1(t) + a2(v2(t, 0)− v1(t)), t ≥ 0,

∂u2(t, x)∂t

= D1∂2u2(t, x)

∂x2− b1u2(t, x), t ≥ 0, x ∈ (0, 1],

∂v2(t, x)∂t

= D2∂2v2(t, x)

∂x2− b2v2(t, x) + c0u2(t− r2, x), t ≥ 0, x ∈ (0, 1],

(1)with boundary conditions

∂u2(t, 0)∂x

= −β1(u2(t, 0)− u1(t)), t ≥ 0,

∂v2(t, 0)∂x

= −β∗1(v2(t, 0)− v1(t)), t ≥ 0,

∂u2(t, 1)∂x

=∂v2(t, 1)

∂x= 0, t ≥ 0,

(2)

and initial conditions

u1(s) = f1(s),v1(s) = g1(s),u2(s, x) = f2(s, x),v2(s, x) = g2(s, x),

u1(0) = u1,0,

v1(0) = v1,0,

u2(0, x) = u2,0(x),v2(0, x) = v2,0(x),

(3)

for x ∈ (0, 1], f1, g1 : [−r1, 0] → R and f2, g2 : [−r2, 0] → L1[0, 1]. The functionsfi, gi for i = 1, 2 describe the prehistory of the system and they have to satisfy thefollowing 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 denotethe rates of transfer between ω and Ω \ ω and they are directly proportional tothe concentration gradient. The constants Di are the diffusivity coefficients andthe constant c0 is the production rate for the repressor. The nonlinear function happearing in (1) is a decreasing function and represents the production of mRNA(messenger ribo nuclein acid). It is of the form

h(x) =1

1 + kxρ, (4)

where k is a kinetic constant and ρ is the Hill coefficient. The delay r1 > 0 is thetranscription time, i.e., the time necessary to the transcription reaction, and r2 > 0is the translation time. The constants β1 and β∗1 are the constants of Fick’s law(see, e.g., [1, Chapter VI]). We underline the fact that all biological constants arepositive. Concerning the Hill coefficient, generally it results ρ > 1 if more than onemolecule of type v1 is needed to repress a molecule of type u1 and ρ ≤ 1 if everymolecule of type v1 interacts only with one molecule of type u1.

According to this model, the eucharyotic cell Ω consists of two compartmentswhere the most important chemical reactions take place. Such compartments areenclosed 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

NONLINEAR DELAY EQUATIONS WITH NONAUTONOMOUS PAST 3

the nucleus where mRNA is produced. The second compartment, denoted by Ω\ω,is the cytoplasm in which the ribosomes are randomly dispersed. The process oftranslation and the production of the repressor take place here.

We denote by ui and vi the concentrations of mRNA and of the repressor, re-spectively, in ω if i = 1 and in Ω \ ω if i = 2. These two species interact to controleach other’s production. In the nucleus ω, mRNA is transcribed from the gene at arate depending on the concentration of the repressor v1. The mRNA leaves ω andenters the cytoplasm Ω \ ω where it diffuses and reacts with ribosomes. Throughthe delayed process of translation, a sequence of enzymes is produced, which inturn produce a repressor v2. Such a repressor goes back to ω, where it inhibits theproduction of u1.

But, according to this model, the repressor in the cytoplasm at time t and positionx depends on the mRNA that was at time t − r2 at the same position x. Thisassumption, however, is unrealistic.

For this reason in [16] the author presented a system of modified equations, whichtake 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, forsimplicity, 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, sheconsiders the Laplacian ∆D with domain

D(∆D) := f ∈ W 2,1[0, 1] : f(0) = f(1) = 0 (5)

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

U(t, s) := T (s− t), −1 ≤ t ≤ s ≤ 0, (6)

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 u2(t− r2, x) bethe modification of u2(t− r2, x) governed by (U(t, s))−1≤t≤s≤0, i.e.

u2(t− r2, x) :=

U(−r2, 0)u2(t− r2, x), 0 ≤ t− r2,U(−r2, t− r2)f2(t− r2, x), 0 ≥ t− r2,

=

T (r2)u2(t− r2, x), 0 ≤ t− r2,T (t)f2(t− r2, x), 0 ≥ t− r2.

(7)

Then, system (1) becomes

du1(t)dt

= h(v1(t− r1))− b1u1(t) + a1(u2(t, 0)− u1(t)), t ≥ 0,

dv1(t)dt

= −b2v1(t) + a2(v2(t, 0)− v1(t)), t ≥ 0,

∂u2(t, x)∂t

= D1∂2u2(t, x)

∂x2− b1u2(t, x), t ≥ 0, x ∈ (0, 1],

∂v2(t, x)∂t

= D2∂2v2(t, x)

∂x2− b2v2(t, x) + c0u2(t− r2, x), t ≥ 0, x ∈ (0, 1].

(8)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 simplifiedand linearized system around the steady-state solutions of (8) (see [24, Section 5]).

4 GENNI FRAGNELLI AND DIMITRI MUGNAI

She rewrites (8) as a delay equation with nonautonomous past of the form

(NDE)

u(t) = Bu(t) + Φut, t ≥ 0,

u(0) = x ∈ X,

u0 = 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 modifiedhistory function ut : R− → X is given by

(MHF ) ut(τ) :=

U(τ, t + τ)f(t + τ) for t + τ ≤ 0,

U(τ, t + τ)u(t + τ) for t + τ > 0,

for some backward evolution family (U(t, s))t≤s on X (see, e.g., [9]). Thereforeproblem (8) describes the behaviour of systems where the history function is mod-ified 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-posednessand qualitative properties of equations of the form (NDE) (see [13] and [17]). Inparticular, they showed that solving (NDE) is equivalent to solving the abstractCauchy problem

(ACP )

U(t) = CU(t), t ≥ 0,

U(0) =

(x

f

)

on the product space E := X × Lp(R−, X), where C is defined by the operatormatrix

C :=(

B Φ0 G

)

on the domain

D(C) :=(

xf

)∈ D(B)×D(G) : f(0) = x

,

for a suitable operator (G,D(G)). Using perturbation theory for C0–semigroupsthey proved the generator property of C and then obtained results on the asymptoticbehavior of (ACP ), and hence of (NDE).

Such partial functional differential equations with ”nonautonomous past” wereintroduced by S. Brendle and R. Nagel in [5], where the existence of mild solutionswas shown by constructing an appropriate semigroup on a space of continuousfunctions. Under appropriate conditions, classical solutions were found in [15].

For the general theory of (NDE) with semigroups when (U(t, s))t≤s ≡ Id, onecan see, e.g., [4], [12].

The aim of this paper is to study a nonlinear version of (NDE) by a localsemigroup approach. To be more precise, we consider the nonlinear problem

(NNDE)

u(t) = Bu(t) + Φ(ut), t ∈ [0, Tmax(x, f)),u(0) = x ∈ X,

u0 = f ∈ Lp(−T, 0; X), p ≥ 1,

on some Banach space X where T > 0 is fixed, possibly ∞, (B,D(B)) and ut aredefined as before, and the delay operator Φ is nonlinear.

NONLINEAR DELAY EQUATIONS WITH NONAUTONOMOUS PAST 5

The main result concerning the well–posedness in the sense of Hadamard forproblem (NNDE) 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(NNDE) 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

(NNDE) as a nonlinear abstract Cauchy problem (NACP );• in Section 4 we prove that the nonlinear abstract Cauchy problem (NACP )

is well–posed;• in Section 5 we apply the results proved in the previous sections to the bio-

logical model (8).

2. Preliminaries.

2.1. Well-posedness of Nonautonomous Cauchy Problems. In this subsec-tion 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 theproblem

(NCP )

u(t) = −A(t)u(t), −T ≤ t ≤ s ≤ 0,

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 theBanach space X, the nonautonomous Cauchy problem (NCP ) is said well-posedwith 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 (NCP ) ⊂ D(A(s))

is dense in X.(ii) (Uniqueness) For every x ∈ Ys the solution us(·, x) of (NCP ) 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

‖usn(t, xn)− us(t, x)‖ → 0

uniformly for t in compact subsets of [−T, 0], where

us(t, x) :=

us(t, x) if s ≥ t,

x if s < t.

If, in addition, there exist constants Mω > 0 and ω ∈ R such that

‖us(t, x)‖ ≤ Mωeω(s−t)‖x‖for all x ∈ Ys and t ≥ s, then (NCP ) is called well-posed with exponentiallybounded solutions.

6 GENNI FRAGNELLI AND DIMITRI MUGNAI

As in [27, Proposition 2.5], we can show that for each well-posed (NCP ) thereexists a unique backward evolution family (U(t, s))−T≤t≤s≤0 solving (NCP ), i.e.,the function t 7→ u(t) := U(t, s)x is a classical solution of (NCP ) for s ∈ [−T, 0]and x ∈ Ys.

2.2. Evolution Families and Semigroups on Lp(−T, 0; X). We first reviewthe basic notations and results on backward evolution families in order to describethe modification of the history function given in (MHF ).

Definition 2.2. A family (U(t, s))−T≤t≤s≤0 of bounded linear operators on a Ba-nach space X is called an (exponentially bounded, backward) evolution familyif

(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) ‖U(t, s)‖ ≤ Mωeω(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 referto, e.g., [7], [11, Section VI.9], [22], [26], [32], [34]. To this purpose, we first ex-tend (U(t, s))−T≤t≤s≤0 to an evolution family (U(t, s))−T≤t≤s, and from now now,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 toan evolution family (U(t, s))−T≤t≤s by setting

U(t, s) :=

U(t, s) for − T ≤ t ≤ s ≤ 0,

U(t, 0) for − T ≤ t ≤ 0 ≤ s,

U(0, 0) = Id otherwise.

(2) Setting I := [−T,∞), on the space E := Lp(I;X), p ≥ 1, we then define thecorresponding evolution semigroup (T (t))t≥0 by

(T (t)f)(s) := U(s, s + t)f(s + t) =

U(s, s + t)f(s + t) −T ≤ s ≤ s + t ≤ 0,

U(s, 0)f(s + t) −T ≤ s ≤ 0 ≤ s + t,

f(s + t) otherwise,

for all f ∈ E, s ∈ R, t ≥ 0.

As in [28], we have that the semigroup (T (t))t≥0 is strongly continuous on E. Wedenote its generator by (G,D(G)). Note that we do not assume any differentiabilityfor (U(t, s))−T≤t≤s, and hence the precise description of the domain D(G) is difficult(see Section 2.1 below). However, in [30, Proposition 2.1] the following importantproperty of D(G) is proved.

Lemma 2.4. The domain D(G) of G, the generator of (T (t))t≥0 on E, is a densesubset of

C0([−T,∞), X) := f : [−T,∞) → X : f is continuous and limt→∞

f(t) = 0.

Since (G,D(G)) is a local operator (see [11, Proposition 2.3] and [30, Theorem2.4]), we can restrict it to the space E := Lp(−T, 0;X), p ≥ 1, by the followingdefinition.

NONLINEAR DELAY EQUATIONS WITH NONAUTONOMOUS PAST 7

Definition 2.5. Take

D(G) := f|[−T,0] : f ∈ D(G)and define

Gf := (Gf)|[−T,0] for f = f|[−T,0] ∈ D(G).

The operator G is not a generator on E. However, if we identify E with thesubspace f ∈ E : f(s) = 0 ∀ s ≥ 0, then E is invariant under (T (t))t≥0. As aconsequence, we obtain the following lemma.

Lemma 2.6. The semigroup (T0(t))t≥0 induced by (T (t))t≥0 on E is given by

(T0(t)f)(s) =

U(s, s + t)f(t + s), −T ≤ s + t ≤ 0,

0, otherwise,(9)

for any f ∈ E.

Let us recall that the growth bound ω0(T0(·)) of the semigroup (T0(t))t≥0 isdefined as

ω0(T0(·)) := inf

ω ∈ R : ∃Mω ≥ 1 s.t. ‖T0(t)‖ ≤ Mωeωt ∀ t ≥ 0

, (10)

which implies that for all ω1 > ω0(T0(·)) there is Mω1 ≥ 1 s.t.

‖T0(t)‖ ≤ Mω1eω1t ∀ t ≥ 0. (11)

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

D(G0) = f ∈ D(G) ∩ E : f(0) = 0, G0f = 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.

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 forsemigroups, we set

ω0(U) := inf

ω ∈ R : ∃Mω ≥ 1 s.t. ‖U(t, s)‖ ≤ Mωeω(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

‖R(λ,G0)‖ ≤ Mω1

λ− ω1, (12)

where Mω1 and ω1 are as in (11) (see, e.g. [11, Proposition II.3.8]).Therefore, we end up with operators (G0, D(G0)) ⊂ (G,D(G)) ⊂ (G,D(G)),

where only the first and the third are generators. Since the domain D(G) of thegenerator 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 thesemigroup (T0(t))t≥0. To this aim we recall the following result.

8 GENNI FRAGNELLI AND DIMITRI MUGNAI

Lemma 2.8 ([14], Lemma 4.4). The set

D :=

f ∈ W 1,p(−T, 0; X); f(0) ∈ D(B),

f(s) ∈ Ys, s 7→ A(s)f(s) ∈ Lp(−T, 0;X)

is a core of G. Moreover,

Gf = f ′ + A(·)f a.e.

for every f ∈ D.

Here (A(·), D(A(·))) are the given unbounded operators which appear in (NCP ).

2.3. Local Semigroup. A typical phenomenon in nonlinear problems is that thesolution of an abstract Cauchy problem may exist only locally. Thus the notion oflocal semigroup is fundamental. Here we recall some definitions, for which we referto [8], [18].

Definition 2.9. A real Banach lattice X is said ordered if it contains a closedsubset 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 generatingcone. 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 negativepart 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 amulti-valued function from X to X.

Definition 2.10. The function A ⊂ X×X is defined as Af = A(f) := g : (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 thereis a constant L such that ‖Af −Ag‖ ≤ L‖f − g‖ for all f, g ∈ D(A). The smallestconstant L is called the Lipschitz seminorm of A and is denoted by ‖A‖Lip. Ofcourse, if A is linear ‖A‖Lip is simply ‖A‖.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‖(I − αA)−1‖Lip ≤ 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 Xif the following conditions hold:

NONLINEAR DELAY EQUATIONS WITH NONAUTONOMOUS PAST 9

1. Dt ⊆ Ds ⊆ X for every t ≥ s ≥ 0;2.

⋃t>0 Dt is dense in X;

3. V (t) : Dt → X is continuous for t ≥ 0;4. 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;

5. limt0 ‖V (t)f − f‖ = 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

Cf := limt0

V (t)f − f

t,

where f ∈ D(C) and

D(C) :=

f ∈

⋃t>0

Dt : limt0

V (t)f − f

texists

.

The local semigroup (V (t), Dt)t≥0 is called positive if it is defined on an orderedBanach 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 thatthe operator C is approximated by globally Lipschitz operators Cν , ν ∈ N, ifthere exists a family of globally Lipschitz operators Cν : X → X such that

Cνf = Cf

for all f ∈ X satisfying ‖f‖ ≥ 1/ν.

Remark 1. In [8] an analogous approximation method at infinity was introducedto show that the sum of two nonlinear operators can be a generator of a local semi-group (see Theorem 2.16 below). Of course the following results still hold true forapproximations at infinity. Here we need the approximation at 0 for the biolog-ical application (see Section 5), since the function under consideration is Holdercontinuous, but not Lipschitz continuous near 0.

Proposition 1. Every operator C : X → X on a Banach space X approximatedby globally Lipschitz 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 considerthe following sequence of abstract Cauchy problems

u(t) = Cνu(t),u(0) = f ∈ X.

(13)

10 GENNI FRAGNELLI AND DIMITRI MUGNAI

By Crandall–Liggett Theorem (see [18]), problem (13) has a unique local solutionVν(·)f and every Vν(t) : X → X is a nonlinear strongly continuous semigroup with‖Vν(t)‖Lip ≤ eωνt for some ων ∈ R. By definition of Cν , Vν(·)f also solves

u(t) = Cu(t),u(0) = f ∈ X,

(14)

provided that ‖Vν(t)f‖ ≥ 1/ν for any t ∈ [0, Tν), for a certain Tν > 0. By uniquenessof solutions for (13), the local semigroup is well defined by setting, analogously towhat 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 ‖V (t)f‖ = ‖Vν(t)f‖ ≥ 1/ν and

V (t)f = limν→∞

Vν(t)f if 0 ≤ t ≤ Tmax(f),

and, by the definition of local semigroup (V (t), Dt)t≥0, we immediately get that⋃t>0 Dt = X.

The next theorem states that if we have a sum of a quasi m-dissipative operatorand a generator of a local semigroup, the Lie-Trotter product formula holds and thesum is a generator of a local semigroup; this theorem corresponds to [8, Theorem15] for approximations at infinity, but the proof can be restated almost word byword for approximations at 0.

Theorem 2.16. Let A be a quasi m–dissipative operator on an ordered Banachlattice 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 localsemigroup (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) > 0such that the commutator inequality

V (t)S(t)f ≤ S(t)V (t)f (15)

holds for any t ∈ [0, t0(f)]. Then the nonlinear Lie-Trotter product formula holds,i.e. for every f ∈ D(A)

U(t)f := limn→+∞

[S

(t

n

)V

(t

n

)]n

f = limn→+∞

[V

(t

n

)S

(t

n

)]n

f (16)

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

V (t)S(t)f ≤ U(t)f ≤ S(t)V (t)f (17)

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 AbstractNonlinear Cauchy Problems. In this section we want to rewrite the nonlineardelay equation with nonautonomous past

(NNDE)

u(t) = Bu(t) + Φ(ut), t ∈ [0, Tmax(x, f)),u(0) = x ∈ X,

u0 = f ∈ Lp(−T, 0; X), p ≥ 1,

where the nonlinear delay operator Φ acts on a modified history function ut (seebelow), as a nonlinear abstract Cauchy problem

(NACP )

U(t) = CU(t), t ∈ [0, Tmax(x, f)),

U(0) =

(x

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) : ‖z‖ ≥ 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 reg-ularity 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

‖S(t)‖ ≤ Mω2eω2t, for all t ≥ 0. (18)

Moreover if α > ω2, then α ∈ ρ(B), where ρ(B) is the resolvent set of (B, D(B)),and by (12)

‖R(α, B)‖ ≤ Mω2

α− ω2, (19)

where Mω2 and ω2 are as in (18) (see, e.g., [11, Proposition II.3.8]).

Definition 3.1. The modified history function ut : [−T, 0] → X in (NDE) isdefined as

ut(τ) : =

U(τ, t + τ)u(t + τ) for t + τ ≥ 0,

U(τ, t + τ)f(t + τ) for t + τ ≤ 0,

=

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,

where the evolution family (U(t, s))−T≤t≤s is the extension (as in Definition 2.3) of(U(t, s))−T≤t≤s≤0.

Definition 3.2. A function u : [−T, Tmax(x, f))) → X is said a classical solutionof (NNDE) if

1. u ∈ C([−T, Tmax(x, f)), X) ∩ C1([0, Tmax(x, f)), X),2. u(t) ∈ D(B), ut ∈ D(Φ), t ∈ [0, Tmax(x, f)),

12 GENNI FRAGNELLI AND DIMITRI MUGNAI

3. u satisfies (NNDE) for all t ∈ [0, Tmax(x, f)).We say that (NNDE) is well-posed if

1. for every( x

f

)in a dense subspace S ⊆ X × Lp(−T, 0; X), there is a unique

solution u(x, f, ·) of (NNDE),2. the solutions depend continuously on the initial values, i.e., if a sequence

( xn

fn

)in S converges to

( xf

) ∈ S, then u(xn, fn, t) converges to u(x, f, t) uniformlyfor t in compact intervals.

It is now our purpose to investigate the well-posedness of (NNDE). As we saidbefore, we restate equation (NNDE) as an abstract Cauchy problem on the spaceE := X × Lp(−T, 0;X) using the operator G from Definition 2.5.

Definition 3.3. We define the operator C by the matrix

C :=(

B Φ0 G

),

defined on the domain

D(C) :=( x

f

) ∈ 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 , providedthat Φ is closed and densely defined on Lp(−T, 0; X).

Let us introduce the abstract Cauchy problem associated to C:

(NACP )

U(t) = CU(t), t ≥ 0,

U(0) =

(x

f

).

In [17] it was showed that a linear equation of the form (NNDE) and the as-sociated abstract Cauchy problem (NACP ) are ”equivalent”, in the sense that(NNDE) has a unique global solution for every

( xf

) ∈ D(C) depending continu-ously on the initial value if and only if (NACP ) is well-posed (in the usual sense).Therefore, well-posedness of (NNDE) is obtained by proving well-posedness of(NACP ), which is done by showing that the operator C is a generator of a stronglycontinuous semigroup.

The proof given in [17] can be followed to prove the equivalence between problems(NNDE) and (NACP ) also in the nonlinear case. Of course, in this case we mustdeal with local solutions and local semigroups, as the following result shows.

Theorem 3.4. The nonlinear delay equation (NNDE) is well-posed if and only ifthe operator (C, D(C)) is the generator of a local semigroup (T (t))t≥0 on E. In thiscase, (NNDE) has a unique local solution u for every

( xf

) ∈ D(C), given by

u(t) =

π1

(T (t)( x

f

)), t ∈ [0, T ),

f(t), a.e. t ∈ [−T, 0],

where π1 is the projection onto the first component of E.

4. The Generator. In view of Theorem 3.4, we now give sufficient conditions onC so that it generates a nonlinear local strongly continuous semigroup on E . First,we write C in the form

C = C0 + F , where C0 :=(

B 00 G

)and F :=

(0 Φ0 0

), (20)

NONLINEAR DELAY EQUATIONS WITH NONAUTONOMOUS PAST 13

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 Fto obtain conditions such that the operator (C, D(C)) generates a nonlinear stronglycontinuous semigroup.

First of all, we need to compute the inverse (λ − C0)−1 of (λ − C0) for anyλ ∈ R (such an inverse will be used later). This means that λ > ω0(T0(·)), whereω0(T0(·)) is the growth bound of the semigroup (T0(t))t≥0 defined in (10). We recallthe following result, considered when ω0(T0(·)) = −∞.

Lemma 4.1 (Lemma 4.1, [17]). For any λ ∈ R define the bounded operator ελ :X → E by

(ελx)(s) := eλsU(s, 0)x, s ∈ [−T, 0], x ∈ X. (21)Then

1. for every x ∈ X, ελx is an eigenvector of G with eigenvalue λ. Moreover‖ελ‖L(X,E) ≤ Mω max1, e(ω−λ)T with ω ∈ R and Mω ≥ 1 is given accordingto 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

Rλ =(

R(λ,B) 0ελR(λ,B) R(λ,G0)

). (22)

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 generatorof a strongly continuous semigroup (T0(t))t≥0 on E given by

T0(t) :=(

S(t) 0St T0(t)

), (23)

where T0 is given in (2.6) and St : X → Lp(−T, 0;X) is defined by

(Stx)(τ) :=

U(τ, 0)S(t + τ)x, τ + t > 0,

0, otherwise.(24)

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(zt(·))for all t ≥ 0 and z ∈ Lp(−T, 0; X). We then define the operators Cν , ν ∈ N, as

Cνz(t, ·) := Φν(zt(·)) ∀ t ≥ 0,

where Φν is a whatever Lipschitz continuous extension of Φ|‖z‖≥ 1ν on B1/ν ; for

example, one could take

Φν(z) :=

Φ(z), ‖z‖ ≥ 1

ν ,

ν‖z‖Φ(

1ν

z‖z‖

), 0 ≤ ‖z‖ ≤ 1

ν .

By General Assumption 2, it is not difficult to show that Φν is globally Lipschitzcontinuous for any ν ∈ N.

Proposition 3. The operators Cν are globally Lipschitz continuous.

14 GENNI FRAGNELLI AND DIMITRI MUGNAI

Proof. Let r and m the functions defined as r : u(s) ∈ X 7→ ut(s)X, where ut isthe history function defined as ut(s) := u(t + s), and m : ut(s) ∈ X 7→ ut(s) ∈ X,where ut is defined in (3.1). The functions r and m are linear by definition (infact the evolution family which characterized the modified history function ut is afamily of linear operators). Then m r is linear and thus m r is globally Lipschitzcontinuous. Since Φν is locally Lipschitz continuous for all ν, we obtain that eachCν is globally Lipschitz continuous as well.

As an immediate consequence of the previous proposition and of the definitionswe have the following corollary.

Corollary 2. The operator C is approximated by the globally Lipschitz operatorCν .

Now set

Fν :=(

0 Cν

0 0

). (25)

Since the operator C is approximated by the globally Lipschitz operators Cν , theoperator

F :=(

0 C0 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 (V (t), Dt)t≥0 with⋃t>0 Dt = X.

Now, we give an explicit expression of the local semigroup (V (t), Dt). Considerthe Cauchy problem

(NACP )1

V(t) = FV(t), t ∈ [0, Tmax(x, f)),V(0) =

( xf

) ∈ E ,

where f(0) = x and V(t) :=(

v(t)z(·, t)

)∈ E for all t ∈ [0, Tmax(x, f)). Then (NACP )1

is equivalent to the system

v(t) = Cz := Φ(zt),z = 0,

v(0) = x,

z(0, ·) = f,

z(0, 0) = f(0) = x.

It follows that z(t, ·) ≡ z(0, ·) = f(·) for all t > 0. Moreover

zt(τ) =

U(τ, 0)z(t + τ, τ), t + τ ≥ 0,

U(τ, t + τ)f(t + τ), t + τ < 0,=

U(τ, 0)f(τ), t + τ ≥ 0,

U(τ, t + τ)f(t + τ), t + τ < 0.

Thus

v(t) = Φ(zt)

Φ(U(·, 0)f(·)), t + · ≥ 0,

Φ(U(·, t + ·)f(t + ·)), t + · < 0,

NONLINEAR DELAY EQUATIONS WITH NONAUTONOMOUS PAST 15

(of course v depends also on τ , the time variable in the past). Integrating, we obtain

v(t) = x +

tΦ(U(·, 0)f(·)), t + · ≥ 0∫ t

0

Φ(U(·, σ + ·)f(σ + ·))dσ, t + · < 0.

Hence, the unique solution of (NACP )1 is

t 7→ V(t) =

(x + tΦ(U(·, 0)f(·))

f

), t + · ≥ 0,

(x +

∫ t

0Φ(U(·, σ + ·)f(σ + ·))dσ

f

), t + · < 0

=: VΦ(t)(

xf

), (26)

for all t ∈ [0, Tmax(x, f)). Then, the local semigroup (V (t), Dt)t∈[0,Tmax(x,f)) gener-ated by F is given by

(V (t)Y)(s) := VΦ(t)(Y(s)) (27)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 arepositive, then it is clear that the local semigroup (V (t), Dt)t∈[0,Tmax(x,f)) is positiveas well.

The following proposition is the essential tool for the main results. The idea goesback to Webb ([35]), where a new norm is introduced in order to make a stronglycontinuous semigroup also a quasi contractive semigroup.

Proposition 5. Assume tha [0, +∞) ⊂ ρ(B). Then there exists a norm on Eequivalent 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 andω ≥ 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 ‖(I − α(C0 − ωI))−1‖ ≤ 1

for any α > 0, since ‖ · ‖ = ‖ · ‖Lip for linear operators.(a) Let α0 > 0 and ω ≥ 0. Put γ := 1+α0ω

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

R(γ, C0) =(

R(γ,B) 0εγR(γ, B) R(γ, G0)

),

where εγ is defined in Lemma 4.1 as well.Now, the claim follows if we prove that in a suitable equivalent norm we have

‖(I − α(C0 − ωI))−1‖ ≤ 1. First, let us note that

‖(I − α(C0 − ωI))−1‖ = ‖(α(γI − C0))−1‖ =

1α‖R(γ, C0)‖. (28)

Moreover, since C0 generates a strongly continuous semigroup on E (see Proposition2), by [11, Proposition I.5.5] there exist ω ∈ R and Mω ≥ 1 such that

‖T0(t)‖ ≤ Mωeωt ∀ t ≥ 0.

16 GENNI FRAGNELLI AND DIMITRI MUGNAI

By Lemma 4.1, if λ > 0 then λ ∈ ρ(C0), and so we can apply the RenormingLemma (see [2, Lemma 3.5.4] or [29, Lemma I.5.1]); in this way, since γ > 0, wecan find a norm in E which is equivalent to the original one and such that

‖R(γ, C0)‖ ≤ 1γ − ω

. (29)

Combining (28) and (29) we finally get

‖(I − α(C0 − ωI))−1‖ ≤ 1α(γ − ω)

=1

1 + α(ω − ω)≤ 1 ∀α > 0

as soon as ω ≥ max0, ω. The claim follows.

From now on, though not explicitly stated again, we will assume that E is en-dowed with the norm found in Proposition 5, so that C0 is quasi m-dissipative.

Now, let us set

E+ :=( x

f

) ∈ E : x ∈ X+, f(τ) ∈ X+ a.e. τ ∈ [−T, 0]

.

With the writing( x

f

) ≤ ( yg

)we mean that

( yg

)− ( xf

) ∈ E+.The next theorem gives conditions under which the operator (C, D(C)) is a gener-

ator, so that problem (NACP ), and then (NNDE), is well-posed. Such a Theoremis 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

( xf

) ∈ D(C0) ∩ E+ there exists a constant t0( x

f

)> 0 such that the

commutator inequality

V (t)T0(t)(

xf

)≤ T0(t)V (t)

(xf

)(30)

holds for any t ∈ [0, t0( x

f

)].

Then the nonlinear Lie-Trotter product formula holds, i.e. for every( x

f

) ∈ D(C0),

T (t)(

xf

):= lim

n→+∞

[T0

(t

n

)V

(t

n

)]n (xf

)

= limn→+∞

[V

(t

n

)T0

(t

n

)]n (xf

) (31)

exists for every t ∈ [0, t0( x

f

)] 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

V (t)T0(t)(

xf

)≤ T (t)

(xf

)≤ T0(t)V (t)

(xf

)(32)

holds true for any( x

f

) ∈ D(C0) ∩ E+ and any t ∈ [0, t0( x

f

)].

NONLINEAR DELAY EQUATIONS WITH NONAUTONOMOUS PAST 17

Proof. Recall that C0 =(

B 00 G

)and that C0 generates a strongly continuous semi-

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 familySt 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 de-veloped in the previous sections to the model of genetic repression presented in theintroduction. First, we rewrite (8) as a (NNDE) 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

B :=

−b1 − a1 0 a1δ0 00 −b2 − a2 0 a2δ0

0 0 D1∆− b1 00 0 0 D2∆− b2

(33)

with domain

D(B) :=( x

yfg

)∈ R2

+ × (W 2,1[0, 1])2 : L(

fg

)=

( xy

)and f ′(1) = g′(1) = 0

,

where δ0f(t, x) := f(t, 0) for any continuous functions f (i.e. δ0 is the Dirac measurein the x–variable), ∆ := d2

dx2 and the operator L : (W 2,1[0, 1])2 → R2+ is defined by

L

(fg

)=

(f ′(0)β1

+ f(0)g′(0)β∗1

+ g(0)

), (34)

where, with abuse of notation, we have set ”′ = ddx”.

Our purpose is to apply Theorem 4.2. In order to do that, an essential fact willbe that C0 is quasi m-dissipative. As already observed in the previous Section, thiscan be obtained by choosing an equivalent norm in the domain E . Without anyfurther 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 semi-group (S(t))t≥0.

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 boundof (B, D(B)), respectively.

In order to apply Theorem 4.2 we have to compute ρ(B). To this aim we canproceed as in [16] and consider the matrix operator B on X of the form

B :=(

A D0 C

),

where the operators A, C, D are diagonal matrices, i.e.

A :=(−b1 − a1 0

0 −b2 − a2

),

18 GENNI FRAGNELLI AND DIMITRI MUGNAI

C :=(

D1∆− b1 00 D2∆− b2

)

and

D :=(

a1δ0 00 a2δ0

),

with

D(A) := R2, D(C) :=(

fg

) ∈ (W 2,1[0, 1])2 : f ′(1) = g′(1) = 0

,

D(D) := (W 2,1[0, 1])2 and D(B) = R2 ×D(C).

Of courseB ⊆ B.

Moreover, since the operator B is one-sided coupled (see [10, Definition 1.1]), forthe 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 letE ⊂ C with D(E) = KerL and L0 := (L|ker C

)−1 : R2 → ker(C) ⊆ (L1[0, 1])2. Thenthe 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

B :=(

A 00 E

)(Id 0−L0 Id

)+

(0 D0 0

).

Now, let E1 be the operator E1 ⊆ D1∆− b1 with domain

D(E1) := f ∈ W 2,1[0, 1] : f ′(1) = 0 and f ′(0) = −β1f(0) (35)

and E2 the operator E2 ⊆ D2∆− b2 with domain

D(E2) := f ∈ W 2,1[0, 1] : f ′(1) = 0 and f ′(0) = −β∗1f(0). (36)

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 followingproperty:

s(E) < 0 ⇔ s(E1) < 0 and s(E2) < 0.

As a consequence of Proposition 6 and Proposition 7, we can compute the resol-vent set of the operator B:

Theorem 5.2. Assume that s(E1), s(E2), and s(A + DL0) are negative. Thens(B) < 0 and, consequently, [0, +∞) ⊂ ρ(B).

Now, let us go back to the biological model, defining the nonlinear delay operatorΦ : D(Φ) = W 1,p(−T, 0; X) → X as

Φ :=

0 hδ−T 0 00 0 0 00 0 0 00 0 c0δ

−T 0

, (37)

NONLINEAR DELAY EQUATIONS WITH NONAUTONOMOUS PAST 19

where δ−T f(·, x) = f(−T, x) for any continuous function f (i.e. δ−T is the Diracmeasure in the t–variable) and the function h is defined as follows:

h(ϑ) =

11 + kϑρ

if ϑ ≤ M,

− 1(1 + kMρ)2

kρMρ−1(ϑ−M) +1

1 + kMρif M < ϑ ≤ Mρ,

0 if ϑ ≥ Mρ,

(38)

where Mρ = M +1 + KMρ

kρMρ−1and M is a large constant.

Remark 4. With respect to the function h described in (4), the nonlinearity hasnow been changed only for large values of the variable ϑ, starting with the tangentline in M until it reaches 0 and then considering the null function. In this way, thisfunction h is still continuous in R+, globally Lipschitz continuous in any set of theform [ε, +∞) and convex if ρ ≤ 1, as the function of (4). However, this change is

not so relevant from a biological point of view, since, if M is large enough,1

1 + kMρ

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 forany t ≥ 0.

We will consider the following evolution family

U(t, s) :=

IdR+ 0 0 00 IdR+ 0 00 0 T (s− t) 00 0 0 IdL1[0,1]

for − T ≤ t ≤ s ≤ 0, (39)

where (T (t))t≥0 denotes the heat semigroup on L1[0, 1] generated by the Laplacianwith 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 areclearly positive.

Now it is easy to prove that system (8) is equivalent to the delay equation withnonautonomous past

W (t) = BW (t) + Φ(Wt), t ≥ 0,W (0) = x ∈ X,

W0 = f ∈ Lp(−T, 0; X),(40)

where

W (t) :=

u1(t)v1(t)u2(t)v2(t)

, f =

f1

g1

f2

g2

, x =

u1,0

v1,0

u2,0

v2,0

(41)

and the modified history function Wt : [−T, 0] → X is defined by

Wt(τ) :=

U(τ, 0)W (t + τ) for 0 ≤ t + τ,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)).

20 GENNI FRAGNELLI AND DIMITRI MUGNAI

Finally, define the operator (G,D(G)) as the matrix

G :=

ddσ 0 0 00 d

dσ 0 00 0 G 00 0 0 d

dσ

,

with domain

D(G) := (W 1,p[−T, 0])2 ×D(G)×W 1,p(−T, 0; L1[0, 1]).

Here ddσ denotes the weak derivative and (G, D(G)) is the closure of

Af = f ′ + f ′′, (42)

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 problemassociated to the operator

C :=(

B Φ0 G

), (43)

with domainD(C) := ( x

f

) ∈ D(B)×D(G) : f(0) = x (44)

on the product space E := X × Lp(−T, 0; X) and initial value( x

f

).

In order to apply Theorem 4.2 to the concrete model, we will make the followinghypotheses, which will be assumed throughout the rest of the paper.

Main Assumption: The Hill coefficient ρ is less or equal to 1 (the biologicalmeaning of the previous assumption is described in the Introduction).

This assumption lets us say that Φ satisfies General Assumption 2 in Section 3without 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 forinequality (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 the local positive

semigroup solving the Cauchy problem (NACP )1. Then for any(

xf

)∈ D(C0)∩E+

there exists t0( x

f

)> 0 such that the commutator condition

V (t)T0(t)(

xf

)≤ T0(t)V (t)

(xf

)

holds for all t ∈ [0, t0( x

f

)]. Here (T0(t))t≥0 is the semigroup generated by (C0, D(C0)

(see Proposition 2).

Proof. Let ζ :=(

xf

)∈ D(C0) ∩ E+ and t ≥ 0. Since the semigroup (T0(t))t≥0 is

linear, by the Riesz Representation Theorem (see [31, Chapter 2]), there exists ameasure P (·, t, s), independent of ζ, such that

(T0(t)ζ)(s) =∫ 0

−T

ζ(y)P (dy, t, s).

Now, let us suppose that there exists t0(ζ) such that

P (Ω, t, ·) ≥ 1 ∀ t ∈ [0, t0(ζ)]. (45)

NONLINEAR DELAY EQUATIONS WITH NONAUTONOMOUS PAST 21

Since Φ is convex, it is apparent that the function( x

f

) 7→ VΦ(t)( x

f

)is convex

component by component for any t. Then Jensen inequality implies

(V (t)T0(t)ζ)(s) = VΦ(t)(T0(t)ζ)(s) = VΦ(t)∫ 0

−T

ζ(y)P (dy, t, s)

≤ 1P (Ω, t, s)

∫ 0

−T

VΦ(t)(P (Ω, t, s)ζ(y))P (dy, t, s)

and since P (Ω, t, s) ≥ 1, the last integral is

≤∫ 0

−T

(V (t)P (Ω, t, s)ζ)(y)P (dy, t, s),

where all the previous inequalities are intended component by component. Now, letus recall that Φ is either linear or nonlinear, but that its nonlinear part (given bythe function h) is decreasing on positive functions, so that

∫ 0

−T

(V (t)P (Ω, t, s)ζ)(y)P (dy, t, s) ≤∫ 0

−T

(V (t)ζ)(y)P (dy, t, s)

= (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 factthat all the operators are positive and that f is positive, as well.

Finally, let us prove (45). By Riesz representation Theorem, the measure P issuch that P (Ω, t, ·) = ‖T0(t)‖, where ‖T0(t)‖ represents the norm of the operatorT0(t).

Now, the operator T0 is as in (23), where U is given in (39), so that (45) isimmediate.

Remark 7. In [8, Corollary 20] it is asserted that, under the setting of Theorem2.16, 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) whenA is linear, the authors use Riesz Representation Theorem, but they assume to dealwith a probability measure, or equivalently, with a semigroup (S(t))t≥0 generatedby A such that ‖S(t)‖ = 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 ‖S(t)‖ = 1 for any t ≥ 0.

As a consequence of Remark 6 and of Theorems 4.2 and 5.3 we can prove thatthe operator (C, D(C)) defined in (43) is a local generator, the main result of thispaper.

Theorem 5.4. Assume that• s(Ei) < 0, i = 1, 2;• s(A + DL0) < 0 and• VΦ leaves D(C0) invariant.

Then the nonlinear operator (C, D(C)) defined in (43) generates a local positive semi-group (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 familyU = (U(t, s))−T≤t≤s≤0 associated to the semigroup (T0(t))t≥0 defined in Lemma2.6 is positive.

22 GENNI FRAGNELLI AND DIMITRI MUGNAI

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 con-

stants appearing in the problem. More precisely, by classical result on rescaledsemigroups (see, e.g., [11, II.2.2]), we get

s(Ei) = D1s(∆)− bi, i = 1, 2. (46)

On the other hand, by [11, VI.4.b], we get the existence of two constants ξ =ξ(β1) > 0 and ξ∗ = ξ∗(β∗1) > 0 such that

s(E1) ⊆ (−∞, ξ] and s(E2) ⊆ (−∞, ξ∗]. (47)

Theorem 5.4 has the following corollary, the main application of our abstractresults.

Theorem 5.5. Assume that b1 > ξ, b2 > ξ∗, s(A+DL0) < 0 and that the prehistoryfunctions f2 and g1 in (3) are such that(

d

dxet∆Df2(−s)

)

|x=0

+ β∗1(et∆Df2(−s))(0) = 0, ∀ t, s ∈ [0, T ],

[d

dx(et∆Df2(−s))

]

|x=1

= 0, ∀ t, s ∈ [0, T ]

andg1(−T ) ≥ Mρ

(see (38)), where ∆D denotes the Laplacian operator with Dirichlet conditions onL1[0, 1] (see (5)).

Then system (8) with boundary conditions (2), initial conditions (3) and with hgiven in (38) is well-posed.

Remark 8. The assumption g1(−T ) ≥ M , so that h(g1(−T )) = 0, means that atthe beginning of the story, the concentration of the repressor in the nucleus was sohigh that the growth of the mRNA was positively influenced only by the mRNA inthe 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 VΦ leaves D(C0) invariant.

To this aim let( x

f

) ∈ D(C0), where, we recall, x :=

u1,0

v1,0

u2,0

v2,0

∈ D(B) and f :=

f1

g1

f2

g2

∈ D(G). As we have proved in (26),

VΦ(t)( x

f

)=

(x + tΦ(U(·, 0)f(·))

f

), t + · ≥ 0,

(x +

∫ t

0Φ(U(·, σ + ·)f(σ + ·))dσ

f

), t + · < 0.

Since the second component is the identity, then f ∈ D(G) and f(0) = x. It remainsto prove that

NONLINEAR DELAY EQUATIONS WITH NONAUTONOMOUS PAST 23

(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

,

(b) if t− T < 0, then x +∫ t

0Φ(U(·, σ + ·)f(σ + ·))dσ ∈ D(B) and

f(0) =(x +

∫ t

0

Φ(U(·, σ + ·)f(σ + ·))dσ)|t=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 conditionsrequired in the definition of D(B) are satisfied, i.e.

L

(x3

x4

)=

(x1

x2

)and x′3(1) = x′4(1) = 0,

where, by (34),

L

(x3

x4

)=

(x′3(0)

β1+ x3(0)

x′4(0)β∗1

+ x4(0)

).

Here xi, for i = 1, 2, 3, 4, denote the components of x + tΦ(U(·, 0)f(·)) or x +∫ t

0Φ(U(·, σ + ·)f(σ + ·))dσ.First assume that t− T ≥ 0. Then

x1 = u1,0 + th(g1(−T )), x2 = v1,0, x3 = u2,0,

andx4 = v2,0 + c0δ

−T (e−·∆Df2(·)) = v2,0 + c0eT∆Df2(−T ).

Thus, by assumption, x′3(1) = u′2,0(1) = 0 and

x′3(0)β1

+ x3(0) =u′2,0(0)

β1+ u2,0(0) = u1,0 + th(g1(−T )) = u1,0

sinceh(g1(−T )) = 0.

Moreover,

x′4(1) = v′2,0(1) + c0

[d

dx(eT∆Df2(−T ))

]

|x=1

= 0

andx′4(0)β∗1

+ x4(0) =v′2,0(0)

β∗1+ v2,0(0) +

c0

β∗

[d

dx(eT∆Df2(−T ))

]

|x=0

+ c0(eT∆Df2(−T ))(0) = v1,0

by assumption. Analogously we can prove that if t− T < 0, then

x +∫ t

0

Φ(U(·, σ + ·)f(σ + ·))dσ ∈ D(B).

Indeed in this case

x1 = u1,0 + th(g1(−T )), x2 = v1,0, x3 = u2,0

and

x4 = v2,0 + c0

∫ t

0

eσ∆Df2(σ − T )dσ.

24 GENNI FRAGNELLI AND DIMITRI MUGNAI

Hence, as before, x′3(1) = u′2,0(1) = 0 and

x′3(0)β1

+ x3(0) =u′2,0(0)

β1+ u2,0(0) = u1,0 + th(g1(−T )) = u1,0.

Moreover, by assumption,

x′4(1) = v′2,0(1) + c0

[d

dx

(∫ t

0

eσ∆Df2(σ − T )dσ

)]

|x=1

=∫ t

0

(d

dx(eσ∆Df2(σ − T ))

)

|x=1

dσ = 0,

andx′4(0)β∗1

+ x4(0) =v′2,0(0)

β∗1+ v2,0(0) +

c0

β∗1

[d

dx

(∫ t

0

eσ∆Df2(σ − T )dσ

)]

|x=0

+ c0

∫ t

0

(eσ∆Df2(σ − T ))(0)dσ

= u1,0 +c0

β∗1

∫ t

0

[d

dx

(eσ∆Df2(σ − T )

)|x=0

+ β∗1(eσ∆Df2(σ − T ))(0)]dσ = u1,0.

Thus VΦ leaves D(C0) invariant and the theorem is proved.

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