exponential attractors for a class of reaction difusion problems with time delay

More precisely, we demonstrate the existence of an exponential attractor, provided that a bounded invariant region exists and the past history decays exponentially fast.. Key words: Reac

© 2007 Birkh¨auser Verlag, Basel

1424-3199/07/040649-19, published online August 09, 2007

DOI 10.1007/s00028-007-0326-7

Exponential attractors for a class of reaction-diffusion problems with time delays

Maurizio Grasselli and Dalibor Pra ˇz ´ak

Abstract We consider a reaction-diffusion system subject to homogeneous Neumann boundary conditions on a

given bounded domain The reaction term depends on the population densities as well as on their past histories in

a very general way This class of systems is widely used in population dynamics modelling Due to its generality, the longtime behavior of the solutions can display a certain complexity Here we prove a qualitative result which can be considered as a common denominator of a large family of specific models More precisely, we demonstrate the existence of an exponential attractor, provided that a bounded invariant region exists and the past history decays

exponentially fast This result will be achieved by means of a suitable adaptation of the
-trajectory method coming

back to the seminal paper of M´alek and Neˇcas.

1 Introduction

A large number of mathematical models in population dynamics have the following form (see, e.g., [1, 2, 3, 12, 18, 19, 20, 21, 26, 37, 38, 42, 43, 44, 45] and references therein)

∂ t u − Duuu = FFF (uuu,uuu t ), in  × (0, ∞) , (1.1)

where  is a bounded, open and connected subset ofRh , h ∈ N Here uuu = (u1, , u M ):

 × R → [0, ∞) M represents the population density vector and D = diag[d1, , dM]

is a diffusion matrix (di > 0, i = 1, , M) Moreover, FFF is a reaction function which depends not only on u u u(x, t ), but also includes a general functional dependence on the past

history up to t, denoted by

We suppose that the populations are isolated, so that system (1.1) is subject to the Neumann homogeneous boundary conditions

Mathematics Subject Classifications (2000): 35B41, 45K05, 92D25.

Key words: Reaction-diffusion equations, nonlocal effects, invariant regions,
-trajectory method, exponential

attractors.

The first author was partially supported by the Italian PRIN 2006 research project Problemi a frontiera libera,

transizioni di fase e modelli di isteresi The second author was supported by the research project M ˇSM 0021620839

and by the project LC06052 (Jindˇrich Neˇcas Center for Mathematical Modeling).

In addition, in view of (1.2), the initial datum is the past history up to t= 0, i.e.,

Among the main goals in the investigation of such kind of systems there are, in particular,

the stability of the equilibria and the existence of (stable) periodic solutions However, if M

is very large even the determination of the equilibria might be an extremely difficult task Therefore, dealing with very general situations, it seems convenient to interpret the model

as a dissipative dynamical system in a suitable phase space, trying to prove the existence of

sufficiently small (i.e., finite fractal dimensional) compact invariant sets which characterize

the longtime dynamics of the model itself Typical mathematical objects possessing the mentioned features are the global attractor (see, e.g., [9, 23, 40]) and the exponential attractor (cf [13, 14, 15]) It is also worth seeing [31] for an updated review of such notions The key step of such an analysis is a suitable choice of the underlying phase space (cf the pioneering contribution [24]) One possibility is to introduce a new variable, the so-called summed past history, which accounts for the memory effects, and obeys certain first order dissipative equation (see [22], or [6, 7] for equations like (1.1)) This approach is useful when one wants to analyze the stability of the attractor with respect to certain parameters

of the given system (e.g., in [6], the relaxation times) The limitation of the summed past history approach is that the delay has to be described in terms of a convolution with rather restricted class of kernels

Another approach is the construction of the so-called trajectory attractor (see [9], cf also [8].) Here the phase space consists of negative semi-trajectories and the dynamical system

is defined by means of a translation semigroup Note that this is a natural setting for the problems with delay, and no additional variables are needed A detailed comparison of the past history approach and the concept of trajectory attractor is given in [8] It turns out that

if both methods are applicable, they are equivalent regarding the notion of global attractors, but the advantages of the former stand out

Nonetheless, there are models to which the past history approach does not apply, while the trajectory one does For instance, in the case of discrete state-dependent delays, a recent result about the existence of the global trajectory attractor is proven in [36] (see also references therein) There the author also treats the case of distributed (but finite) delays by means of a more conventional approach which regards the delays as additional variables

In the case of infinite delays, a meaningful example can be easily given by exploiting the generality of (1.1) Indeed, hereditary effects can be modelled, for instance, as follows (see, e.g., [39])

0

−∞

K(u u u(t + s), s)ds

In the present paper, we want to cover such general classes of models; hence the summed past history approach is not applicable As a phase space, we take the set of the historiesη,

defined on (−∞, 0] An additional requirement is that η solves (1.1) on [−
, 0], where

>0 is some fixed number This has certain similarity to the trajectory approach discussed above, but also goes back to the ideas of [28], see also [29] or [34], where a similar approach was applied to the class of problems with bounded delay The main advantage of such a

choice is that the phase space can be endowed with a more convenient L2-norm rather then

the L∞-norm; see also the discussion below Theorem 1.1 In particular, we easily establish

the smoothing property of the dynamics, yielding the existence of an exponential attractor The main requirement of our analysis is the existence of a bounded invariant region This is a natural assumption for the reaction-diffusion systems (without delay), and has been used by many authors; see e.g [5, 10, 11, 30] However, in presence of delays, things become more delicate As shown in [16, 27], the simple problem

∂ t u(x, t ) − µu(x, t) = u(x, t)1− u(x, t − τ)

admits solutions that grow exponentially, despite a natural sign condition of the right-hand side Nonetheless, there are numerous models governed by systems of type (1.1), for which the invariant regions exist (cf., for instance, [6, 7, 19, 32, 33, 37, 38, 39, 42, 43, 44, 45]) Some meaningful examples are considered in the last section

To be more specific, we assume that there exist ai ≥ 0 and bi > 0, i = 1, , M, such

that, settingI =  M

i=1[ai , b i], ifη ∈ I almost everywhere in  for any s ∈ (−∞, 0], then

u(x, t ) ∈ I for t ≥ 0 as well Here u is a suitable concept of solution, defined on a maximal time interval [0, Tmax ) The immediate consequence is that Tmax= +∞, and, trivially, the asymptotic boundedness of the dynamics

At our abstract level of presentation, we will assume that there exists a closed, bounded setK ⊂ L2(; RM ) , such that, if η ∈ C((−∞, 0]; K), then u(·, t) ∈ K for all t ≥ 0 In

concrete examples, however, one always has

K =v ∈ L2(; RM ) : v v v(x) ∈ I, a.e x ∈ ,

thus the solutions are even uniformly bounded As a phase space of our problem, we choose

B = C((−∞, 0]; K)

Before introducing a metric inB, we first indicate by ·2the standard norm in L2(; RM )

induced by the natural scalar product (·, ·), and then we state the basic assumption on FFF ,

namely,

FFF (uuu1,η1) − FFF (uuu2,η2)2

2≤ 



uuu1− uuu22

2+

0

−∞

η1(s) − η2(s)2

2eγ s ds



 ,

for any u u1 , u u2 ∈ K and η1,η2∈ B Observe that, the only requirement on the past history

dependence is the exponential decay This generality allows us to consider a wide class of memory kernels in concrete cases (see, e.g., Section 6, compare also with [12])

We can now endowB with the norm given by

η2

0

−∞

η(s)2

where γ > 0 is the same as in (1.5) Observe that B is not complete (in fact it is not closed)

with respect to the X-norm.

The basic existence result we need is the following

THEOREM 1.1 Let (1.5) hold For any given η ∈ B and any fixed T > 0, there exists

a unique u u u solution to (1.1), (1.3), (1.4) on [0, T ] such that

u ∈ C((−∞, T ]; L2(; RM )) ∩ L2( 0, T ; W 1,2 (; RM )),

Moreover, u u u(t ) ∈ K for all t ∈ [0, T ] and there exists K0> 0, independent of T , such that,

for any fixed r ∈ (0, T ),

sup

t ∈[0,T −r]

t +r

t

∂t u2

−1,2 + uuu2

1,2

Proof Using the Lipschitz continuity in (1.5), it is easy to construct u u on some small

interval [0, T0], via the standard fixed point argument Note that, since u(t) ∈ K for all

t ∈ (−∞, T0], it follows that

T0

0 ∂t u2

−1,2 + uuu2

1,2

≤ K Observe that K and T0only depend onK and the constants in (1.5), but are independent of

η = uuu|( −∞,0] This fact allows us to repeat the argument on the time interval [T0, 2T0] and

so on, reaching eventually the given time T Uniqueness follows by the usual Gronwall-type

As a consequence, we can now define a solution operator S(t) : B → B by S(t)η = uuu t,

where u uis given by the previous theorem (recall the notation (1.2).) Our aim is to construct

an exponential attractor for a suitable interpretation of the dynamical system (S(t), B).

Problem (1.1), (1.3), (1.4) has a unique solution for any initial datum inB However,

note that we do not have continuity of the solution with respect to the X-norm of the initial

datum (compare with [6, 7]) One would need to use the sup-norm ofB, which, however,

is much less convenient for the further analysis To circumwent this difficulty, we follow

the
-trajectory approach (see [28, 29]), and set

B −
= S(
)B ,

where
> 0 is an arbitrary fixed number Simply said, B −
consists of those elements

point of view of the large time dynamics it is irrelevant if we restrict our study toB −

The advantage is that S(t) is Lipschitz continuous on B −
with respect to the norm X,

as we shall prove below (see Lemma 3.2) Note that alsoB −
is not complete (even less

compact) in X, but we will show that it is asymptotically compact This is enough to apply

the standard tools from the theory of attractors

REMARK 1.2 By the same scheme one can cover the more general case where there are also discrete delays (see, e.g., [32]), i.e.,

∂ t u − Duuu = FFF (uuu(t),uuu(t − τ1), , uu u(t − τk ), u u t ),

with some τi >0 Moreover, the linear diffusion operator might also be replaced by a non-linear elliptic operator in divergence form− ∇·A A A( ∇uuu) satisfying appropriate assumptions.

2 Preliminaries from the theory of attractors

LetX be a bounded metric space By N χ ( A, ρ) we denote the smallest number of sets

with diameter≤ 2ρ that cover A ⊂ X The fractal dimension is defined by

d X f ( A) = lim sup

ε→0+

lnN χ ( A, ε)

− ln ε .

X → X be a continuous semigroup of operators For the reader’s

conve-nience, we recall that

(1) A is compact;

A = A, ∀t > 0;

(3) distX B B, A) → 0 as t → ∞, for any B B ⊂ X bounded.

Of course, if a global attractor exists, then it is unique The setE ⊂ X is called exponential attractor if

(1) E is compact;

E ⊂ E, ∀t > 0;

(3) d X f ( E) < ∞;

(4) ∃ σ > 0 such that, for any B B ⊂ X bounded, dist X B B, E) ≤ C(B B B)e−σt.

It is customary to work in a setting where the underlying phase spaceX is either compact

or, at least, complete space (typically, a closed subset of a Banach space with some higher

regularity) In the present case it is convenient to relax this assumption to the asymptotic

compactness, see the definition in Theorem 2.1 here below.

The construction of an exponential attractor is based on several ideas that are nowadays well-known and widely used in the literature Because our setting is somewhat unusual (the phase space is not complete), we present the following theorem with the proof

The key step of the construction normally consists in certain squeezing or smoothing property of the semigroup (see, e.g., [13, 14, 15]) To keep the things simple for now, we replace it by a more abstract “iterated covering property” – see (2.3) below – which will later be deduced from the smoothing property in the spirit of [15] (cf Lemma 5.1 below) Remark also that (2.3) is only slightly stronger that (2.4) Also, we recall that the latter property is equivalent to the existence of an exponential attractor in the discrete case (see [35])

THEOREM 2.1 Let

semigroup We assume:

(1) asymptotic compactness: for arbitrary sequences {ηn} ⊂ X and tn → ∞, there is

a subsequence (not relabeled) {ηn} and η0 n )η n → η0.

(2) continuity: for all η, ˜η ∈ X and t,˜t ∈ [0, T ]

with some a ∈ (0, 1].

(3) iterated covering property: there exist N ≥ 1, θ ∈ (0, 1) and t∗> 0 such that, for

any B ⊂ X with diam X B ≤ 2ρ, one has

Then there exists an exponential attractor

d X f ( E) ≤ 1

a

ln N

− ln θ + 1 .

Proof We split the argument into two main steps.

I We begin with an exponential attractorE∗ n , X ),

∗) Pick ρ > 0 such that diam X ≤ 2ρ Applying (2.3) repeatedly, we

arrive at

N χ n X , θ n ρ) ≤ N n , ∀n ∈ N (2.4)

This implies the existence ofE∗ n , X ) – together with the

estimate (see [35, Theorem 1])

d X f ( E∗)≤ ln N

A certain care is needed, since we assume thatX is neither compact nor complete However,

asymptotic compactness is enough for the existence of a global compact attractorA, which

is defined as the ω-limit set of X The exponential attractor is constructed as

m,n≥0

m E n ,

where En n X such that dist X n X , E n ) ≤ 2θ n ρ The existence of

E nsis guaranteed by (2.4) The double union is not compact, but thanks to the asymptotic

compactness, any sequence that intersects infinitely many Enshas an accumulation point

inA Hence E∗is compact.

II We define

(2.1)–(2.2), from which one deduces thatE = F(E∗× [0, t∗]) is the desired exponential

X ) Regarding the dimension estimate, one has

d X f ( E) ≤ 1

a d

f

X ×[0,∞) ( E∗× [0, t∗])≤ 1

a

ln N

− ln θ + 1 .

3 Continuity and the smoothing property

We start with some simple estimates about the continuity of the time shift Recall the convention (1.2)

LEMMA 3.1 Let u u:R → L2(; RM ) be a measurable function Then

uuu t2

X≤ e−γ t uuu02

uuu t2

X≤ e−γ t uuu02

X+

t

0

uuu2

0

t

uuu2

2≤ e−γ t uuu02

Proof One has

uuu t2

0

−∞

uuu(t + s)2

2eγ s ds =

t

−∞uuu(τ)2

2eγ (τ −t) dτ

= e−γ t
t

−∞uuu(τ)2

2eγ τ dτ

Hence (3.1) follows, while for t≥ 0 one writes

uuu t2

X= e−γ t
0

−∞uuu(τ)2

2eγ τ dτ+

t

0

uuu(τ)2

2eγ (τ −t) dτ

≤ e−γ t uuu02

X+

t

0

uuu(τ)2

2dτ

The following lemma is a standard estimate of the difference of solutions, but it has

important consequences: the continuity of S(t) as well as the smoothing property, which is

the key step toward the existence of an exponential attractor

LEMMA 3.2 Let u u u, v v v be solutions on[−
, T ] corresponding to initial data η, ˜η˜η in B−
Then w w = uuu − vvv satisfies

sup

t ∈[0,T ] w w(t )2

2+ d0

T

0

w w2

1,2 ≤ CT η − ˜η˜η2

where d0= mini=1, ,M d i and C T > 0 depends on T

Proof Testing the equation written for w w by w wgives

∂ t w w2

2+

M



i=1

d i ∇w i2

2= (FFF (uuu,uuu t ) − FFF (vvv,vvv t ), w w)

Using (1.5) and adding d0w w2

2to both sides give

∂ t w w(t )2

2+ d0w w(t )2

1,2 ≤ c w w(t )2

2+ w w t2

X

, ∀t ∈ [−
, T ] Note that, from now on, the constant c may change from line to line We fix s ∈ [−
, 0],

τ ∈ [0, T ] and integrate on (s, τ) to get

w w(τ )2

2+ d0

τ

s

w w2

1,2 ≤ w w(s)2

2+ c


τ

s

w w2

2+

τ

s

w w t2

X dt



.

Using (3.1) and (3.2), we deduce

τ

s w w2

2+

τ

s w w t2

X dt ≤ c



w w02

X+

τ

0 w w2 2



.

Hence

w w(τ )2

2+ d0

τ

w w2

1,2 ≤ w w(s)2

2+ c



w w02

X+

τ

w w2 2



.

Integrating over s ∈ (−
, 0) and using (3.3) eventually yields

w w(τ )2

2+ d0

τ

0 w w2

1,2 ≤ c



η − ˜η˜η2

X+

τ

0 w w2 2



.

Since τ ∈ [0, T ] is arbitrary, we have the conclusion from the Gronwall lemma.
Recalling (3.2), we have, as immediate corollary,

S(t)η − S(t) ˜η˜η X ≤ CT η − ˜η˜η X , ∀t ∈ [0, T ], ∀η, ˜η˜η ∈ B −
, (3.5)

i.e., the local Lipschitz continuity of solution operators Note that CT grows at most

exponentially with respect to T

A second important by-product of Lemma 3.2 is





T

0

w w2

1,2





1/2

In fact, we can now follow the idea of [28] (see also [29]) to employ the last estimate

to obtain a smoothing property of the flow in the space of L2( 0, T ; L2(; RM )) This

is based on the compactness from the well-known Aubin-Lions lemma Thus we need a similar estimate for the time derivative

LEMMA 3.3 Let u u u, v v v be solutions on[−
, T ] corresponding to initial data η, ˜η˜η in B−
Then there holds





T

0

∂t w2

−1,2





1/2

Proof The left-hand side of (3.7) equals to

sup

ϕ ∈B1(0)

T

0

∂t w, w ϕ ,

where·, · is the duality between W 1,2 (; RM ) and (W 1,2 (; RM ))∗, while B1(0)is the

unit ball of L2( 0, T ; W 1,2 (; RM ))centered at 0.

From the equation for w wwe have

T

0

∂t w, w ϕ = −a

T

0

( ∇w w, ∇ϕ) +

T

0

(F F F (u u u, u u t ) − FFF (vvv,vvv t ), w w)

= I1+ I2.

By (3.6) and the H¨older inequality, it follows

|I1| ≤ c

T

0

w w1,2 ϕ 1,2≤





T

0

w w2

1,2





1/2

≤ CT η − ˜η˜η X

An analogous estimate holds for I2, in virtue of (1.5) and (3.4)

4 Asymptotic smoothness and compactness

In order to construct an exponential attractor, we need to restrict ourselves to a smoother subset ofB −
In particular, we need more regularity with respect to time An interesting

feature of the
-trajectory approach is that the regularity of weak solutions (1.7) is enough.

Due to the hyperbolic nature of our problem, sets of higher regularity cannot be absorbing Instead we prove the existence of smooth sets which are exponentially attracting (compare with [7])

LEMMA 4.1 There exists B1⊂ B −
with the following properties:

(1) S(t) B1⊂ B1for ∀t > 0;

(2) there exists K1> 0 such that

sup

t≤0

t

t−1 ∂t η2

−1,2 + η2

1,2

(3) B1attracts B −
exponentially fast.

Proof Let η ∈ B −
be arbitrary and fix T ≥ 1 Denote by uuu the solution on [0, T ] with

u0= η Thanks to (1.8) we have

sup

t ∈[1,T ]

t

t−1 ∂t u2

−1,2 + uuu2

1,2

Note that K0 is independent of T , because the solution on [t − 1, t] can be seen as the solution on [0, 1] with the initial condition u u t−1∈ B −

By (4.2) there exists t0∈ (0, 1) such that uuu(t0)2

1,2 ≤ K0 We define ˜ψ ∈ B by

˜ψψψ(t) =



u u(t + T ), t ∈ [t0− T , 0] ,

u u(t0), t ≤ t0− T

It follows that

sup

t≤0

t

t−1 ∂t ˜ψψψ2

−1,2 +  ˜ψ ψ2

1,2

This implies the existence of< i>E∗ n , X ) – together with the

estimate... t ), w w)

= I1+ I2.

By... w2 2



.

Integrating over s ∈ (−
, 0) and using