Volume X, Number 0X, XX 200X pp. X–XX
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
(Communicated by the associate editor name)
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.
2000 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 Varia- zionali ed Equazioni Differenziali Nonlineari.
1
[25]). The fundamental models which include time delays and spatial diffusion are proposed 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 = −β1∗(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 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 biare the kinetic rates of decay, aidenote 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
h(x) = 1
1 + kxρ, (4)
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, 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, re- spectively, 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 ∆Dwith domain
D(∆D) := {f ∈ W2,1[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≥0is 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 eu2(t − r2, x) be the modification of u2(t − r2, x) governed by (U (t, s))−1≤t≤s≤0, i.e.
e
u2(t − r2, x) :=
½ U (−r2, 0)u2(t − r2, x), 0 ≤ t − r2, U (−r2, t − r2)f2(t − r2, x), 0 ≥ t − r2,
=
½ 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) + c0u˜2(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 simplified and linearized system around the steady-state solutions of (8) (see [24, Section 5]).
She rewrites (8) as a delay equation with nonautonomous past of the form
(N DE)
˙u(t) = Bu(t) + Φeut, t ≥ 0, u(0) = x ∈ X,
e
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 modified history function eut: R−→ X is given by
(M HF ) u˜t(τ ) :=
(U (τ, t + τ )f (t + τ ) for t + τ ≤ 0,e U (τ, t + τ )u(t + τ ) for t + τ > 0,e
for some backward evolution family ( eU (t, s))t≤s on X (see, e.g., [9]). Therefore problem (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-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
(ACP )
U(t) = CU(t),˙ t ≥ 0, U(0) =
Ãx f
!
on the product space E := X × Lp(R−, X), where C is defined by the operator matrix
C :=
µB Φ
0 G
¶
on the domain
D(C) :=
½µx f
¶
∈ D(B) × D(G) : f (0) = x
¾ ,
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].
For the general theory of (N DE) with semigroups when ( eU (t, s))t≤s ≡ Id, one 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
(N N DE)
˙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 eut are defined as before, and the delay operator Φ is nonlinear.
The main result concerning the well–posedness in the sense of Hadamard for problem (N N DE) is described in Theorem4.2. 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 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 the problem
(N CP ) (
˙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 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
kˆusn(t, xn) − ˆus(t, x)k → 0 uniformly for t in compact subsets of [−T, 0], where
ˆ
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 kus(t, x)k ≤ Mωeω(s−t)kxk
for all x ∈ Ys and t ≥ s, then (N CP ) is called well-posed with exponentially bounded solutions.
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 Ba- nach 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 ≤ 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 refer to, 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 ( eU (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 to an evolution family ( eU (t, s))−T ≤t≤s by setting
U (t, s) :=e
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 eE := Lp(I; X), p ≥ 1, we then define the corresponding evolution semigroup ( eT (t))t≥0 by
( eT (t) ef )(s) := eU (s, s + t) ef (s + t) =
U (s, s + t) ef (s + t) −T ≤ s ≤ s + t ≤ 0, U (s, 0) ef (s + t) −T ≤ s ≤ 0 ≤ s + t, f (s + t)e otherwise,
for all ef ∈ eE, s ∈ R, t ≥ 0.
As in [28], we have that the semigroup ( eT (t))t≥0is strongly continuous on eE. We denote its generator by ( eG, D( eG)). Note that we do not assume any differentiability for ( eU (t, s))−T ≤t≤s, and hence the precise description of the domain D( eG) is difficult (see Section2.1 below). However, in [30, Proposition 2.1] the following important property of D( eG) is proved.
Lemma 2.4. The domain D( eG) of eG, the generator of ( eT (t))t≥0on eE, is a dense subset of
C0([−T, ∞), X) := {f : [−T, ∞) → X : f is continuous and lim
t→∞f (t) = 0}.
Since ( eG, D( eG)) is a local operator (see [11, Proposition 2.3] and [30, Theorem 2.4]), we can restrict it to the space E := Lp(−T, 0; X), p ≥ 1, by the following definition.
Definition 2.5. Take
D(G) := { ef|[−T,0] : ef ∈ D( eG)}
and define
Gf := ( eG ef )|[−T,0] for f = ef|[−T,0]∈ D(G).
The operator G is not a generator on E. However, if we identify E with the subspace {f ∈ eE : f (s) = 0 ∀ s ≥ 0}, then E is invariant under ( eT (t))t≥0. As a consequence, we obtain the following lemma.
Lemma 2.6. The semigroup (T0(t))t≥0 induced by ( eT (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 is defined as
ω0(T0(·)) := infn
ω ∈ R : ∃ Mω≥ 1 s.t. kT0(t)k ≤ Mωeωt ∀ t ≥ 0o
, (10) which implies that for all ω1> ω0(T0(·)) there is Mω1≥ 1 s.t.
kT0(t)k ≤ 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( eG) ∩ 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 invertibleª .
In addition, if U := (U (t, s))−T ≤t≤s≤0, analogously to the growth bound for semigroups, we set
ω0(U) := infn
ω ∈ R : ∃ Mω≥ 1 s.t. kU (t, s)k ≤ Mωeω(s−t) ∀ t ≤ so
; 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 kR(λ, G0)k ≤ 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)) ⊂ ( eG, D( eG)), 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.
Lemma 2.8 ([14], Lemma 4.4). The set D :=n
f ∈ W1,p(−T, 0; X); f (0) ∈ D(B), f (s) ∈ Ys, s 7→ A(s)f (s) ∈ Lp(−T, 0; X)
o
is a core of G. Moreover,
Gf = f0+ 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 : (f, g) ∈ A}. The domain D(A) of A is {f : Af 6= ∅}. The range of L is R(A) := S
{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)−1kLip ≤ 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≥0a local semigroup on a Banach space X if the following conditions hold:
1. Dt⊆ Ds⊆ X for every t ≥ s ≥ 0;
2. S
t>0Dtis 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. limt&0kV (t)f − f k = 0 for all f ∈S
t>0Dt.
Definition 2.14. The infinitesimal generator of a local semigroup (V (t), Dt)t≥0
is defined as
Cf := lim
t&0
V (t)f − f
t ,
where f ∈ D(C) and
D(C) :=
( f ∈ [
t>0
Dt: lim
t&0
V (t)f − f t exists
) .
The local semigroup (V (t), Dt)t≥0is 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
Cνf = 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 semi- group (see Theorem2.16below). Of course the following results still hold true for approximations at infinity. Here we need the approximation at 0 for the biolog- ical 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 operators Cν, ν ∈ N, is the generator of a local semigroup (V (t), Dt)t≥0 withS
t>0Dt= X.
Proof. Let C be approximated by globally Lipschitz operators Cν. Let us consider the following sequence of abstract Cauchy problems
(˙u(t) = Cνu(t),
u(0) = f ∈ X. (13)
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≤ eωνt for some ων ∈ R. By definition of Cν, Vν(·)f also solves
(˙u(t) = Cu(t),
u(0) = f ∈ X, (14)
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
ν→∞Vν(t)f if 0 ≤ t ≤ Tmax(f ),
and, by the definition of local semigroup (V (t), Dt)t≥0, we immediately get that S
t>0Dt= 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
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 := lim
n→+∞
· S
µt n
¶ V
µt n
¶¸n
f = lim
n→+∞
· V
µt n
¶ S
µt 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 )].
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
(N N DE)
˙u(t) = Bu(t) + Φ(eut), t ∈ [0, Tmax(x, f )), u(0) = x ∈ X,
e
u0= f ∈ Lp(−T, 0; X), p ≥ 1,
where the nonlinear delay operator Φ acts on a modified history function eut (see below), as a nonlinear abstract Cauchy problem
(N ACP )
U(t) = CU(t),˙ t ∈ [0, Tmax(x, f )), U(0) =
à 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≥0on 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 reg- ularity subspaces Yt(see Definition2.1).
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
kS(t)k ≤ 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)
kR(α, B)k ≤ 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 eut: [−T, 0] → X in (N DE) is defined as
e ut(τ ) : =
(U (τ, t + τ )u(t + τ )e for t + τ ≥ 0, U (τ, t + τ )f (t + τ )e 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 ( eU (t, s))−T ≤t≤sis the extension (as in Definition2.3) of (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
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 )),
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 ¡x
f
¢in a dense subspace S ⊆ X × Lp(−T, 0; X), there is a unique solution u(x, f, ·) of (N N DE),
2. the solutions depend continuously on the initial values, i.e., if a sequence¡xn
fn
¢ in S converges to ¡x
f
¢∈ S, then u(xn, fn, t) converges to u(x, f, t) 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 Definition2.5.
Definition 3.3. We define the operator C by the matrix C :=
µB Φ
0 G
¶ , defined on the domain
D(C) :=©¡x
f
¢∈ D(B) × D(G) : f (0) = xª
⊆ 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:
(N ACP )
U(t) = CU(t),˙ t ≥ 0, U(0) =
Ãx f
! .
In [17] it was showed that a linear equation of the form (N N DE) and the as- sociated abstract Cauchy problem (N ACP ) are ”equivalent”, in the sense that (N N DE) has a unique global solution for every ¡x
f
¢ ∈ D(C) depending continu- ously 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 ¡x
f
¢∈ 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 Theorem3.4, 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
C = C0+ F, where C0:=
µB 0
0 G
¶
and F :=
µ0 Φ 0 0
¶
, (20)
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 Theorem2.16to C0and 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≥0defined 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
(²λx)(s) := eλsU (s, 0)x, s ∈ [−T, 0], x ∈ X. (21) Then
1. for every x ∈ X, ²λx is an eigenvector of G with eigenvalue λ. Moreover k²λkL(X,E)≤ Mωmax{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 Rλ=
µ R(λ, B) 0
²λR(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 generator of a strongly continuous semigroup (T0(t))t≥0 on E given by
T0(t) :=
µS(t) 0 St 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(ezt(·))
for all t ≥ 0 and z ∈ Lp(−T, 0; X). We then define the operators Cν, ν ∈ N, as Cνz(t, ·) := Φν(ezt(·)) ∀ t ≥ 0,
where Φν is a whatever Lipschitz continuous extension of Φ|{kzk≥1
ν} on B1/ν; for example, one could take
Φν(z) :=
(Φ(z), kzk ≥ 1ν, νkzkΦ
³1 ν z
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→ eut(s) ∈ X, where eut 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 eutis 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
Fν :=
µ0 Cν
0 0
¶
. (25)
Since the operator C is approximated by the globally Lipschitz operators Cν, the operator
F :=
µ0 C 0 0
¶
is approximated by Fν, which are again globally Lipschitz continuous.
By Proposition1, the next result is immediate.
Proposition 4. The operator F generates a local semigroup (V (t), Dt)t≥0 with S
t>0Dt= X.
Now, we give an explicit expression of the local semigroup (V (t), Dt). Consider the Cauchy problem
(N ACP )1
(˙V(t) = FV(t), t ∈ [0, Tmax(x, f )), V(0) =¡x
f
¢∈ E,
where f (0) = x and V(t) :=
µv(t) z(·, t)
¶
∈ E for all t ∈ [0, Tmax(x, f )). Then (N ACP )1
is equivalent to the system
˙v(t) = Cz := Φ(ezt),
˙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 e
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,
(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 Z t
0
Φ(U (·, σ + ·)f (σ + ·))dσ, t + · < 0.
Hence, the unique solution of (N ACP )1 is
t 7→ V(t) =
Ãx + tΦ(U (·, 0)f (·)) f
!
, t + · ≥ 0,
à x +Rt
0Φ(U (·, σ + ·)f (σ + ·))dσ f
!
, t + · < 0
=: VΦ(t) µx
f
¶ , (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≤0are 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))−1k ≤ 1 for any α > 0, since k · k = k · kLip for linear operators.
(a) Let α0 > 0 and ω ≥ 0. Put γ := 1+αα0ω
0 . Then γ > 0 and, consequently, γ ∈ ρ(B). By Lemma4.1, (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 Lemma4.1the inverse is
R(γ, C0) =
µ R(γ, B) 0
²γR(γ, B) R(γ, G0)
¶ , where ²γ is defined in Lemma4.1as well.
Now, the claim follows if we prove that in a suitable equivalent norm we have k(I − α(C0− ωI))−1k ≤ 1. First, let us note that
k(I − α(C0− ωI))−1k = k¡
α(γI − C0)¢−1 k = 1
αkR(γ, C0)k. (28) Moreover, since C0generates 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¯eωt¯ ∀ 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
kR(γ, C0)k ≤ 1
γ − ¯ω. (29)
Combining (28) and (29) we finally get k(I − α(C0− ωI))−1k ≤ 1
α(γ − ¯ω) = 1
1 + α(ω − ¯ω)≤ 1 ∀ α > 0 as soon as ω ≥ max{0, ¯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 Proposition5, so that C0is quasi m-dissipative.
Now, let us set E+:=n¡x
f
¢∈ E : x ∈ X+, f (τ ) ∈ X+a.e. τ ∈ [−T, 0]
o . With the writing¡x
f
¢≤¡y
g
¢we mean that¡y
g
¢−¡x
f
¢∈ E+.
The next theorem gives conditions under which the operator (C, D(C)) is a gener- ator, 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 Lemma2.6 is positive;
4. the delay operator Φ is positive;
5. (V (t), Dt)t≥0 (see (27)) leaves D(C0) invariant;
6. for every ¡x
f
¢∈ D(C0) ∩ E+ there exists a constant t0
¡x
f
¢> 0 such that the commutator inequality
V (t)T0(t) µx
f
¶
≤ T0(t)V (t) µx
f
¶
(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) µx
f
¶
:= lim
n→+∞
· T0
µt n
¶ V
µt n
¶¸nµ x f
¶
= lim
n→+∞
· V
µt n
¶ T0
µt n
¶¸nµ x f
¶ (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)
µx f
¶
≤ T (t) µx
f
¶
≤ T0(t)V (t) µx
f
¶
(32) holds true for any¡x
f
¢∈ D(C0) ∩ E+ and any t ∈ [0, t0
¡x
f
¢].
Proof. Recall that C0=
µB 0
0 G
¶
and that C0generates a strongly continuous semi- group (T0(t))t≥0 by Proposition2. Now, (T0(t))t≥0 is positive, since U is positive (see (9)); B generates a positive semigroup (S(t))t≥0which induces a positive family Stdefined in (24). Then T0(t) is positive by (23).
Moreover, since [0, +∞) ⊂ ρ(B), by Proposition5C0 is quasi m–dissipative.
Therefore, by Proposition4, F generates a local semigroup (V (t), Dt)t≥0which is positive by Remark3; by assumption, (V (t), Dt)t≥0leaves D(C0) invariant.
Hence Theorem2.16can 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 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
B :=
−b1− a1 0 a1δ0 0
0 −b2− a2 0 a2δ0
0 0 D1∆ − b1 0
0 0 0 D2∆ − b2
(33)
with domain D(B) :=
½µx
y fg
¶
∈ R2+× (W2,1[0, 1])2: L¡f
g
¢=¡x
y
¢and f0(1) = g0(1) = 0
¾ , where δ0f (t, x) := f (t, 0) for any continuous functions f (i.e. δ0is the Dirac measure in the x–variable), ∆ := dxd22 and the operator L : (W2,1[0, 1])2→ R2+is defined by
L µf
g
¶
= Ãf0(0)
β1 + f (0)
g0(0) β∗1 + g(0)
!
, (34)
where, with abuse of notation, we have set ”0 =dxd”.
Our purpose is to apply Theorem4.2. 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 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≥0and 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
B :=
µA D
0 C
¶ , where the operators A, C, D are diagonal matrices, i.e.
A :=
µ−b1− a1 0 0 −b2− a2
¶ ,
C :=
µD1∆ − b1 0 0 D2∆ − b2
¶
and
D :=
µa1δ0 0 0 a2δ0
¶ , with
D(A) := R2, D(C) :=©¡f
g
¢∈ (W2,1[0, 1])2: f0(1) = g0(1) = 0ª , D(D) := (W2,1[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 : (W2,1[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 B :=
µA 0
0 E
¶ µ Id 0
−L0 Id
¶ +
µ0 D 0 0
¶ . Now, let E1be the operator E1⊆ D1∆ − b1with domain
D(E1) := {f ∈ W2,1[0, 1] : f0(1) = 0 and f0(0) = −β1f (0)} (35) and E2 the operator E2⊆ D2∆ − b2with domain
D(E2) := {f ∈ W2,1[0, 1] : f0(1) = 0 and f0(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 Eisatisfy the following property:
s(E) < 0 ⇔ s(E1) < 0 and s(E2) < 0.
As a consequence of Proposition6and Proposition7, 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. Then s(B) < 0 and, consequently, [0, +∞) ⊂ ρ(B).
Now, let us go back to the biological model, defining the nonlinear delay operator Φ : D(Φ) = W1,p(−T, 0; X) → X as
Φ :=
0 h◦δ−T 0 0
0 0 0 0
0 0 0 0
0 0 c0δ−T 0
, (37)
