UPPER SEMICONTINUITY OF THE ATTRACTOR FOR LATTICE DYNAMICAL SYSTEMS OF PARTLY DISSIPATIVE REACTION-DIFFUSION SYSTEMS

UPPER SEMICONTINUITY OF THE ATTRACTOR FOR LATTICE DYNAMICAL SYSTEMS OF PARTLY

DISSIPATIVE REACTION-DIFFUSION SYSTEMS

AHMED Y. ABDALLAH

Received 12 July 2004 and in revised form 7 December 2004

We investigate the existence of a global attractor and its upper semicontinuity for the infinite-dimensional lattice dynamical system of a partly dissipative reaction-diffusion system in the Hilbert spacel²×l². Such a system is similar to the discretized FitzHugh- Nagumo system in neurobiology, which is an adequate justification for its study.

1. Introduction

Lattice dynamical systems arise in various application fields, for instance, in chemical reaction theory, material science, biology, laser systems, image processing and pattern recognition, and electrical engineering (cf. [6,7,13]). In each field, they have their own forms, but in some other cases, they appear as spatial discretizations of partial differential equations (PDEs). Recently, many authors studied various properties of the solutions for several lattice dynamical systems. For instance, the chaotic properties have been investi- gated in [1,7,8,10,11,17], and the travelling solutions have been carefully studied in [2,3,7,8,9,22].

From [18], we know that it is difficult to estimate the attractor of the solution semi- flow generated by the initial value problem of dissipative PDEs on unbounded domains because, in general, it is infinite dimensional. Therefore, it is significant to study the lat- tice dynamical systems corresponding to the initial value problem of PDEs on unbounded domains because of the importance of such systems and they can be regarded as an ap- proximation to the corresponding continuous PDEs if they arise as spatial discretizations of PDEs.

The main idea of this work is originated from [4,16]. In [4,19,20], the researchers proved the existence of global attractors for different lattice dynamical systems and they investigated the finite-dimensional approximations of these global attractors. In fact, Bates et al. [4] studied first-order lattice dynamical systems, and Zhou [20] gave a gen- eralization of the result given by [4]. Zhou [19] studied a second-order lattice dynamical system and investigated the upper semicontinuity of the global attractor.

Copyright©2005 Hindawi Publishing Corporation Journal of Applied Mathematics 2005:3 (2005) 273–288 DOI:10.1155/JAM.2005.273

For a positive integerr, consider the Hilbert space l²=

u=

ui

i∈Z^r:i=

i1,i2,. . .,ir

∈Z^r,ui∈R,

i∈Z^r

u²_i <∞

(1.1)

whose inner product and norm are given by, for allu=(u_i)i∈Z^r,v=(v_i)i∈Z^r∈l²,

u,v =

i∈Z^r

uivi, u = u,u^1/2=

i∈Z^r

u²_i 1/2

. (1.2)

We will study the following lattice dynamical system of a partly dissipative reaction- diffusion system:

˙

u_i+ν(Au)i+f1

u_i+g1

u_i+αh1

u_i,v_i=q_1,i,

˙ v_i+f2

v_i+g2

v_i−βh2

u_i,v_i=q_2,i, (1.3)

i=(i1,i2,. . .,ir)∈Z^r,t >0, with the initial conditions

u_i(0)=u_i,0, v_i(0)=v_i,0, (1.4) whereνis a positive constant,αandβare real constants such that

αβ >0, (1.5)

the operatorA:l²→l²is defined by, for allu=(ui)i∈Z^r∈l²,i=(i1,i2,. . .,ir), (Au)i=2ru(i1,i2,...,ir)−u(i1−1,i2,...,ir)−u(i1,i2−1,...,ir)− ··· −u(i1,i2,...,ir−1)

−u(i1+1,i2,...,ir)−u(i1,i2+1,...,ir)− ··· −u(i1,i2,...,ir+1), (1.6) and forj=1, 2,s,s1,s2∈R,

qj= qj,i

i∈Z^r∈l², (1.7)

fj(s),gj(s)∈C¹(R,R), fj(0)=gj(0)=0, (1.8) hj

s1,s2

∈C¹R²,R

, hj(0, 0)=0, (1.9)

εj≤f_j(s), (1.10)

gj

ss≥0, (1.11)

h1

s1,s2

s1=h2

s1,s2

s2, (1.12)

whereε_jis a positive constant.

For example, one can consider f_j(s)=δ_js, where δ_j is a positive constant, g_j(s)= nj

i=0γi js²ⁱ⁺¹, wherenjis a nonnegative integer andγi jis a nonnegative constant for each i=0, 1,. . .,n_j,h1(s1,s2)=s^m₁¹⁻¹s^m₂², andh2(s1,s2)=s^m₁¹s^m₂²⁻¹, wherem1,m2≥2 are con- stants. In fact, for f1(s)=λs, f2(s)=δs,g2(s)=0,h1(s1,s2)=s2, andh2(s1,s2)=s1, where λ andδ are positive constants, the system given by (1.3) and (1.4) can be regarded as a spatial discretization of the following partly dissipative reaction-diffusion system with continuous spatial variablex∈R^randt∈R⁺:

ut−ν∆u+λu+g1(u) +αv=q1, vt+δv−βu=q2. (1.13) The system (1.13) describes the signal transmission across axons and is a model of FitzHugh-Nagumo equations in neurobiology, (cf. [5,12,15] and the references therein).

The existence of global attractors of the system given by (1.13) has been proved in a bounded domain (cf. [14]) and inR^r(cf. [16]).

2. Preliminaries

We can write the operatorAin the following form:

A=A1+A2+···+Ar (2.1)

such that for j=1, 2,. . .,r, andu=(u_i)_i∈Z^r∈l²,

Aju_i=2u(i1,i2,...,ir)−u(i1,i2,...,ij−1,ij−1,ij+1,...,ir)−u(i1,i2,...,ij−1,ij+1,ij+1,...,ir). (2.2) Forj=1, 2,. . .,r, define the operatorsBj,B^∗_j :l²→l²as follows: foru=(ui)i∈Z^r∈l²,

Bju_i=u(i1,i2,...,ij−1,ij+1,ij+1,...,ir)−u(i1,i2,...,ir),

B^∗_ju_i=u(i1,i2,...,ij−1,ij−1,ij+1,...,ir)−u(i1,i2,...,ir). (2.3) Then, it follows that forj=1, 2,. . .,r,

Aj=B^∗_jBj=BjB^∗_j, (2.4)

there exists a constantC0=C0(r) such that

Au²≤C0u², B_ju ²= B^∗_ju ²≤4u², ∀u∈l², (2.5) Bju,v=

u,B^∗_jv, ∀u,v∈l². (2.6) It is clear that A,A_j,B_j, and B^∗_j, j=1, 2,. . .,r, are bounded linear operators froml² intol².

We can represent the initial value problem, (1.3) and (1.4), in the following form:

˙

u+νAu+f1(u) +g1(u) +αh1(u,v)=q1,

˙

v+f2(v) +g2(v)−βh2(u,v)=q2, u(0)=

ui,0

i∈Z^r, v(0)=

vi,0

i∈Z^r,

(2.7)

where u=(u_i)i∈Z^r, v=(v_i)i∈Z^r, Au=(Au_i)i∈Z^r, and for j=1, 2, f_j(u)=(f_j(u_i))i∈Z^r, g_j(u)=(g_j(u_i))_i∈Z^r,h_j(u,v)=(h_j(u_i,v_i))_i∈Z^r,q_j=(q_j,i)_i∈Z^r.

Consider the Hilbert spaceE:=l²×l², endowed with the inner product and norm as follows: forϕj=(u^(j),v^(j))^T=((u⁽_i^j)), (v_i^(j)))^T_i∈Zr∈E,j=1, 2,

ϕ1,ϕ2

E=

u⁽¹⁾,u⁽²⁾+v⁽¹⁾,v⁽²⁾,

ϕE= ϕ,ϕ^1/2_E , ∀ϕ∈E. (2.8)

Now, the system given by (2.7) is equivalent to the following initial value problem in the Hilbert spaceE:=l²×l²:

˙

ϕ+C(ϕ)=F(ϕ), ϕ(0)= u0,v0

_T

, (2.9)

whereϕ=(u,v)^T,C(ϕ)=(νAu, 0)^T,F(ϕ)=(G1(ϕ),G2(ϕ))^T, G1(ϕ)=q1−f1(u)−g1(u)−αh1(u,v),

G2(ϕ)=q2−f2(v)−g2(v) +βh2(u,v). (2.10) For a given function f(s1,s2)∈C¹(R²,R), letD1f(a,b) represent the partial derivative of f with respect to the first independent variable,s1, at (s1,s2)=(a,b), and letD2f(a,b) represent the partial derivative of f with respect to the second independent variable,s2, at (s1,s2)=(a,b). From (1.9) and the mean value theorem, it follows that givenu=(ui)i∈Z^r, v=(v_i)_i∈Z^r∈l², there existξ_1i,ξ_2i∈(0, 1), for eachi∈Z^r, such that

h1(u,v) ²=

i∈Z^r

h1

u_i,v_i²=

i∈Z^r

D1h1

ξ_1iu_i,v_iu_i+D2h1

0,ξ_2iv_iv_i²

≤2

|amax|≤u max

|b|≤v

D1h1(a,b)²

u² + 2

|bmax|≤v

D2h1(0,b)²

v².

(2.11)

Thus, foru,v∈l², we haveh1(u,v)∈l². Similarly one can show that foru,v∈l²,h2(u,v)

∈l². By using (1.8) and the mean value theorem, it is easy to show that f1, f2,g1, andg2

mapl²intol². From the above discussion, it is obvious thatFmapsEintoE.

3. The existence of an absorbing set

First we will prove that there exists a unique local solution of the system given by (2.9) inE. Suppose that (1.7), (1.8), and (1.9) are satisfied. LetGbe a bounded set inE, and

ϕj=(u^(j),v^(j))=((u^(j)_i ), (v^(j)_i ))i∈Z^r∈G,j=1, 2, then Fϕ1

−Fϕ2 ²

E

= G1

ϕ1

−G1

ϕ2 ²+ G2

ϕ1

−G2

ϕ2 ²

≤2 f1

u⁽¹⁾−f1

u^{(2) 2}+ 4 g1

u⁽¹⁾−g1

u^{(2) 2} + 4α² h1

u⁽¹⁾,v⁽¹⁾−h1

u⁽²⁾,v^{(2) 2}+ 2 f2

v⁽¹⁾−f2

v^{(2) 2} + 4 g2

v⁽¹⁾−g2

v^{(2) 2}+ 4β² h2

u⁽¹⁾,v⁽¹⁾−h2

u⁽²⁾,v^{(2) 2}.

(3.1)

Using (1.9) and the mean value theorem, it follows that there existξ_3i,ξ_4i∈(0, 1), for each i∈Z^r, andL1=L1(G) such that

h1

u⁽¹⁾,v⁽¹⁾−h1

u⁽²⁾,v^{(2) 2}

=

i∈Z^r

h1

u⁽¹⁾_i ,v_i⁽¹⁾−h1

u⁽²⁾_i ,v_i⁽²⁾²

=

i∈Z^r



D1h1

u⁽¹⁾_i +ξ3i

u⁽²⁾_i −u⁽¹⁾_i ,v⁽¹⁾_i u⁽¹⁾_i −u⁽²⁾_i +D2h1

u⁽²⁾_i ,v⁽¹⁾_i +ξ4i

v⁽²⁾_i −v_i⁽¹⁾v⁽¹⁾_i −v_i⁽²⁾





2

≤2

|a|≤u⁽¹⁾+maxu⁽²⁾,_|b|≤v⁽¹⁾

D1h1(a,b)² u⁽¹⁾−u^{(2) 2}

+ 2

|a|≤u⁽²⁾,|maxb|≤v⁽¹⁾+v⁽²⁾

D2h1(a,b)² v⁽¹⁾−v^{(2) 2}

≤L1 u⁽¹⁾−u^{(2) 2}+ v⁽¹⁾−v^{(2) 2}.

(3.2)

Thus

h1

u⁽¹⁾,v⁽¹⁾−h1

u⁽²⁾,v^{(2) 2}≤L1 ϕ1−ϕ2 ². (3.3) We can obtain similar results, as (3.3), for f1, f2,g1,g2, andh2by using (1.8), (1.9), and the mean value theorem. In such a case, using (3.1), there existsL2=L2(G) such that

Fϕ1

−Fϕ2 ²_E≤L2 ϕ1−ϕ2 ². (3.4) ThusF is locally Lipschitz fromEintoE. In such a case from the standard theory of ordinary differential equations, we get the following lemma.

Lemma 3.1. If (1.7), (1.8), and (1.9) are satisfied, then for any initial data ϕ(0)=(u0, v0)^T ∈E, there exists a unique local solutionϕ(t)=(u(t),v(t))^T of (2.9) such thatϕ∈ C¹([0,T),E)for someT >0. IfT <∞, thenlimt→T⁻ϕ²E= ∞.

Assume that (1.5), (1.7), (1.8), (1.9), (1.10), (1.11), and (1.12) are satisfied. Now we are ready to prove that the solution of the system given by (2.9) exists globally and there exists an absorbing set.

Letϕ=(u,v)^T∈Ebe a solution of (2.9). If we consider the inner product of (2.9) with ϕinE, taking into account (2.4) and (2.6), we obtain

1 2

d

dtu²+ν^r

j=1

B_ju ²+f1(u),u+g1(u),u+αh1(u,v),u=

q1,u, (3.5) 1

2 d

dtv²+f2(v),v+g2(v),v−βh2(u,v),v=

q2,v, (3.6)

If we multiply (3.5) by|β|, (3.6) by|α|, and we sum up the results, taking into account (1.5) and (1.12), we find that

|β| 2

d

dtu²+^|α| 2

d

dtv²+|β|ν^r

j=1

B_ju²+|β|

f1(u),u+|α|

f2(v),v +|β|

g1(u),u+|α|

g2(v),v= |β|

q1,u+|α| q2,v.

(3.7)

By using (1.8), (1.10), and the mean value theorem, for eachi∈Z^r, there exists a constant ξ5i∈(0, 1) such that

f1(u),u=

i∈Z^r

f1

ui

ui

=

i∈Z^r

f₁ξ5iui

u²_i≥ε1u². (3.8)

Thus we have

f1(u),u≥ε1u², f2(v),v≥ε2v². (3.9)

From (1.11), we obtain

g1(u),u=

i∈Z^r

g1

ui ui

≥0, g2(v),v≥0. (3.10)

Now if we substitute (3.9) and (3.10) into (3.7), we find that

|β| 2

d

dtu²+^|α| 2

d

dtv²+ε1|β|u²+ε2|α|v²

≤ |β|

q1,u+|α|

q2,v≤|β| 2ε1

q1 ²+ε1|β|

2 u²+^|α| 2ε2

q2 ²+ε2|α| 2 v².

(3.11)

Thus if we choose

σ=min|β|,|α|,ε1|β|,ε2|α|

, (3.12)

then it follows that d

dtϕ²_E+ϕ²_E≤ β

σ q1 2

+ α

σ q2 2

. (3.13)

From the Gronwall lemma, we obtain ϕ²E≤e^−t ϕ0 ²

E+1−e^−t σ²

β q1 ²+α q2 ²

, (3.14)

tlim−→∞ϕ²_E≤ β

σ q1 2

+ α

σ q2 2

. (3.15)

Inequality (3.15) implies that the solution semigroup{S(t)}t≥0of (2.9) exists globally and possesses a bounded absorbing set inE. In such a case, the maps

S(t) :ϕ(0)∈E−→S(t)ϕ(0)=ϕ(t)∈E, t≥0, (3.16) generate a continuous semigroup{S(t)}t≥0onE. Now fromLemma 3.1and (3.15), we are ready to present the following lemma.

Lemma 3.2. If (1.5), (1.7), (1.8), (1.9), (1.10), (1.11), and (1.12) are satisfied, then for any initial data inE, the solutionϕ(t)of (2.9) exists globally for allt≥0. That is, ϕ∈ C¹([0,∞),E). Moreover, there exists a bounded ballO=OE(0,r0)inE, centered at0with radiusr0, such that for every bounded setGofE, there existsT(G)≥0such that

S(t)G⊂O, ∀t≥T(G), (3.17)

wherer₀²>((β/σ)q1)²+ ((α/σ)q2)². Therefore, there exists a constantT0≥0depend- ing onOsuch that

S(t)O_⊂O, _∀t_≥T0. (3.18)

4. The existence of the global attractor

To prove the existence of the global attractor for the solution semigroup{S(t)}t≥0of (2.9), we need to prove the asymptotic compactness of{S(t)}t≥0. Along the lines of [4], the key idea of showing the asymptotic compactness for such a lattice system is to establish uniform estimates on “Tail Ends” of solutions.

Lemma4.1. If (1.5), (1.7), (1.8), (1.9), (1.10), (1.11), and (1.12) are satisfied andϕ(0)= (u0,v0)∈O, whereOis the bounded absorbing ball given byLemma 3.2, then for anyη >0, there exist positive constantsT(η)andK(η)such that the solutionϕ(t)=(u(t),v(t))^T= (ϕi(t))i∈Z^r=((ui(t)), (vi(t)))^T_i∈Zr∈Eof (2.9) satisfies

im≥K(η)

ϕi(t) ²_E=

im≥K(η)

u²_i(t) +v²_i(t)≤η (4.1)

for allt≥T(η), whereim=max1≤j≤r|i_j|fori=(i1,i2,. . .,i_r)∈Z^r.

Proof. Consider a smooth increasing functionθ∈C¹(R⁺,R) such that

θ(s)=0, 0≤s <1, 0≤θ(s)≤1, 1≤s <2,

θ(s)=1, s≥2,

(4.2)

and there exists a constantM0such thatθ(s)≤M0for alls∈R⁺.

Let L be an arbitrary positive integer. Setw_i=θ(im/L)u_i, z_i=θ(im/L)v_i,w= (wi)i∈Z^r,z=(zi)i∈Z^r, andψ=(w,z)^T. Following [4], we take the inner product of (2.9) withψinE, then it follows that

i∈Z^r

θ im

L 1

2 d dtu²_i+f1

ui

ui+g1

ui

ui+αh1

ui,vi

ui

+ν

i∈Z^r

r j=1

Bju_iBjw_i=

i∈Z^r

θ im

L

q1,iui,

(4.3)

i∈Z^r

θ im

L 1

2 d dtv_i²+f2

vi

vi+g2

vi

vi−βh2

ui,vi

vi

=

i∈Z^r

θ im

L

q2,ivi. (4.4) If we multiply (4.3) by|β_|, (4.4) by|α_|, and we sum up the results, taking into account (1.5) and (1.12), we get

i∈Z^r

θ im

L 



|β| 2

d

dtu²_i+^|α| 2

d

dtv²_i+|β|f1

ui

ui

+|α|f2

v_iv_i+|β|g1

u_iu_i+|α|g2

v_iv_i





+|β|ν

i∈Z^r

r j=1

Bju_iBjw_i=

i∈Z^r

θ im

L

|β|q1,iui+|α|q2,ivi .

(4.5)

Using (1.8), (1.10), and the mean value theorem, it follows that for eachi∈Z^r, f1

ui

ui≥ε1u²_i, f2 vi

vi≥ε2v_i². (4.6) From (1.11), we obtain

g1

u_iu_i≥0, g2

v_iv_i≥0. (4.7)

Recalling (69) of [21], taking into account thatBj ≤2,j=1, 2,. . .,r, one can see that

|β|ν

i∈Z^r

r j=1

B_ju_iB_jw_i≥ −16|β|νnM0

L r₀², ∀t≥T0. (4.8)

Now if we substitute (4.6), (4.7), and (4.8), into (4.5), we find that fort≥T0,

i∈Z^r

θ im

L |β|

2 d

dtu²_i +^|α| 2

d

dtv_i²+ε1|β|u²_i+ε2|α|v²_i

≤

i∈Z^r

θ im

L

|β|q1,iui+|α|q2,ivi

+16|β|νnM0

L r₀²

≤

i∈Z^r

θ im

L

|β| ε1

q²_1,i+ε1|β| 4 u²_i +^|α|

ε2

q²_2,i+ε2|α| 4 v²_i

+16|β|νnM0

L r₀².

(4.9)

Thus if we choose

σ=min|β|,|α|,ε1|β|,ε2|α|

, (4.10)

then it follows that fort≥T0,

i∈Z^r

θ im

L d

dt ϕi(t) ²_E+ ϕi(t) ²_E

≤2

i∈Z^r

θ im

L β

σq1,i

2

+ α

σq2,i

2

+32|β|νnM0

L r₀²

≤2

im≥L

β σq1,i

2

+ α

σq2,i

2

+32|β|νnM0

L r0².

(4.11)

Sinceq1,q2∈l², then for a givenη >0, we can fixLsuch that

2

im≥L

β σq1,i

2

+ α

σq2,i

2

+32|β|νnM0

L r₀²≤η

2, (4.12)

and in such a case we get that

i∈Z^r

θ i_m

L d

dt ϕ_i(t) ²_E+ ϕ_i(t) ²_E

≤η

2, ∀t≥T0. (4.13) From the Gronwall lemma, we obtain

i∈Z^r

θ

im

L ϕi(t) ²_E

≤e^−t

i∈Z^r

θ

im

L ϕi(0) ²_E

+η

2, ∀t≥T0. (4.14) Sinceϕ(0)=(u0,v0)^T∈O, we have

ϕ(0) _E≤r0. (4.15)

Therefore,

i∈Z^r

θ

im

L ϕi(t) ²_E

≤r₀²e^−t+η

2, ∀t≥T0. (4.16)

But again forη >0, there exists a constantT1=T1(η) such that r0²e^−t≤η

2, ∀t≥T1. (4.17)

Using (4.16) and (4.17) withK(η)=2L,T(η)=max{T0,T1}, we obtain

im≥K(η)

ϕi ²

E=

im≥K(η)

θ

im

L ϕi ²

E

≤

i∈Z^r

θ

im

L ϕi ²

E

≤η, ∀t≥T(η).

(4.18)

The proof is completed.

Lemma4.2. If (1.5), (1.7), (1.8), (1.9), (1.10), (1.11), and (1.12) are satisfied, then the solu- tion semigroup{S(t)}t≥0of (2.9) is asymptotically compact inE, that is, if{ϕ_n}is bounded inEandt_n→ ∞, then{S(t_n)ϕ_n}is precompact inE.

Proof. By using Lemmas3.2and4.1, above, the proof of this lemma will be similar to

that of [19, Lemma 3.2].

Theorem4.3. If (1.5), (1.7), (1.8), (1.9), (1.10), (1.11), and (1.12) are satisfied, then the solution semigroup{S(t)}t≥0of (2.9) possesses a global attractorᏭinE.

Proof. From the existence theorem of global attractors, (cf. [18, Lemmas 2 and 4]), we

get the result.

5. Upper semicontinuity of the global attractor

Here we will study the upper semicontinuity of the global attractor Ꮽof the solution semigroup{S(t)}t≥0of (2.9), in the sense thatᏭis approximated by the global attractors of finite-dimensional versions of (2.9), as was done in [4] for a simpler system.

Letnbe a positive integer, and Z^r_n=

i∈Z^r:im≤n. (5.1)

Fori=(i1,i2,. . .,ir)∈Z^r_n, considerw=(wi)im≤n∈R^(2n+1)r. For convenience, we reorder the subscripts of components ofwas follows:

w=

w(−n,−n,...,−n,−n),w(−n,−n,...,−n,−n+1),. . .,w(−n,−n,...,−n,n), w(₋n,−n,...,−n+1,−n),w(₋n,−n,...,−n+1,−n+1),. . .,

w(−n,−n,...,−n+1,n),. . .,w(n,n,...,n,−n),w(n,n,...,n,−n+1),. . .,w(n,n,...,n,n)

T

.

(5.2)

Let

X=w= wi

im≤n:w∈R^(2n+1)r, (5.3)

where subscripts of components ofware ordered as in (5.2).