Approximate Controllability of a Reaction-Diffusion System with a Cross-Diffusion Matrix and Fractional Derivatives on Bounded Domains

Volume 2010, Article ID 281238,14pages doi:10.1155/2010/281238

Research Article

Approximate Controllability of

a Reaction-Diffusion System with

a Cross-Diffusion Matrix and Fractional

Derivatives on Bounded Domains

Salah Badraoui

Laboratoire LAIG, Universit´e du 08 Mai 1945, BP. 401, Guelma 24000, Algeria

Correspondence should be addressed to Salah Badraoui,[email protected]

Received 11 July 2009; Accepted 5 January 2010

Academic Editor: Ugur Abdulla

Copyrightq2010 Salah Badraoui. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

We study the following reaction-diffusion system with a cross-diffusion matrix and fractional derivativesuta1Δua2Δv−c1−Δα1u−c2−Δα2v1ωf1x, tinΩ×0, t∗,vtb1Δub2Δv−

d1−Δβ1u−d2−Δβ2v1ωf2x, tinΩ× 0, t∗,uv0 on∂Ω×0, t∗,ux,0 u0x,vx,0

v0xinx∈Ω,whereΩ⊂RNN≥1is a smooth bounded domain,u0, v0∈L2Ω, the diffusion

matrixM a1a2

b1b2

has semisimple and positive eigenvalues 0 < ρ1 ≤ ρ2, 0 < α1, α2, β1, β2< 1,

ω⊂Ωis an open nonempty set, and 1ωis the characteristic function ofω. Specifically, we prove that under some conditions over the coefficientsai, bi, ci, dii1,2, the semigroup generated by the linear operator of the system is exponentially stable, and under other conditions we prove that for allt∗>0 the system is approximately controllable on0, t∗ .

1. Introduction

In this paper we prove controllability for the following reaction-diffusion system with cross diffusion matrix:

uta1Δua2Δv−c1−Δα1u−c2−Δα2v1ωf1x, t inΩ× 0, t∗,

vtb1Δub2Δv−d1−Δβ1u−d2−Δβ2v1ωf2x, t inΩ× 0, t∗,

uv0 on∂Ω× 0, t∗,

ux,0 u0x, vx,0 v0x inx∈Ω,

1.1

We assume the following assumptions.

H1 Ωis a smooth bounded domain inRN _N≥1.

H2The diffusion matrixMa1a2 b1 b2

has semisimple and positive eigenvalues 0< ρ1≤

ρ2.

H3cj, dj j 1,2are real constants,αj, βj j1,2are real constants belonging to the interval 0,1.

H4u0, v0∈L2Ω.

H5The distributed controlsf1, f2∈L20, t∗ ;L2Ω.

Specifically, we prove the following statements.

iIf c2d10 and min{c1λ₁1−α1ρ1, d2λ₁1−β2ρ1}>0, whereλ1is the first eigenvalue

of−Δwith Dirichlet condition, or if c2/0, d1/0, c1 ≥0, andd2 ≥ 0; then, under

the hypothesesH1–H3, the semigroup generated by the linear operator of the system is exponentially stable.

iiIf c2d10 and under the hypothesesH1–H5, then, for allt∗>0 and all open

nonempty subset ωofΩthe system is approximately controllable on 0, t∗ .

This paper has been motivated by the work done in1 and the work done by H. Larez and H. Leiva in2 . In the work1 , the auther studies the asymptotic behavior of the solution of the system

uta ∂2_u

∂x2 β

∂u ∂xb

∂2_v

∂x2 ft, u, v, x∈R, t >0,

vtc ∂2_u

∂x2 d

∂2_v

∂x2 β

∂v

∂xgt, u, v, x∈R, t >0

1.2

supplemeted with the initial conditions

ux,0 u0x, vx,0 v0x, x∈R. 1.3

The author proved that in the Banach spaceX×X where XCubRis the space of bounded

uniformly continuous real valued functions onR, iff andg are locally Lipshitz and under some conditions over the coefficients a, b, c, d, β, and if u0, v0 ∈ C {u ∈ CubR :

limx→∞uxexist},then ut, vt ∈ C for all t < tmax.Moreover, Ut limx→∞ux andVt limx→∞vxsatisfy the system of ordinary differential equations

Ut ft, Ut, Vt,

with the initial data

U0 lim

x→∞u0x, V0 xlim→∞v0x. 1.5

The same result holds forC−{u∈CubR: limx→ −∞uxexist}.

In the work done in2 , the authers studied the system1.1withc2d10, c1d2,

and α1 β2 1/2. They proved that if the diffusion matrix

_{a b}

c d

has semi-simple and positive eigenvalues 0< ρ1≤ρ2,f1, f2∈L20, τ;L2Ω,then ifλ1₁/2ρ1β >0λ1is the first

eigenvalue of−Δ, the system is approximately controllable on 0, τ for all open nonempty subset ωofΩ.

2. Notations and Preliminaries

In the following we denote by

M2Rthe set of 2×2 matrices with entries fromR,

L2_{Ωthe set of all measurable functionsu:Ω → Rsuch that}

Ω|u|2dx <∞,

H1_{Ω the set of all the functions u ∈ L}2_{Ω that have generalized derivatives}

∂u/∂xj∈L2Ωfor allj1, . . . , N, H1

0Ωthe closure of the setC∞0 Ωin the Hilbert spaceH1Ω,

H2_{Ω the set of all the functions u ∈ L}2_{Ω that have generalized derivatives}

∂u/∂xj, ∂2u/∂xj∂xk∈L2Ωfor allj, k1, . . . , N.

We will use the following results.

Theorem 2.1cf.3 . Let us consider the following classical boundary-eigenvalue problem for the

laplacien:

−Δuλu, onΩ,

u0, on∂Ω, 2.1

whereΩis a nonempty bounded open set inRN_{andD−Δ H}2_Ω∩H1 0Ω.

This problem has a countable system of eigenvalues 0< c ≤λ1 < λ2 <· · · < λj < · · · and λj → ∞asj → ∞.

iAll the eigenvalues λj have finite multiplicity mj equal to the dimension of the corresponding eigenspaceSj.

iiLet{ϕjk} mj

k1be a basis of the Sjfor everyj,then the eigenvectors {ϕjk}

mj,∞

k1,j1form

a complete orthonormal system in the space L2_{Ω.Hence for all u ∈ L}2_Ωwe

haveu ∞_j1mj

k1u, ϕjkϕjk.If we putEju

mj

k1u, ϕjkϕjk then we get u

_∞

j1Eju.

ivFor allu∈D−Δwe have −Δu∞_j1λjEju.

vThe operatorΔgenerates an analytic semigroup {T_Δt}on L2_{Ωdefined by}

TΔtu

∞

j1

e−λjtEju. 2.2

Definition 2.2. Let 0< α <1 a real number, the operator−Δαis defined by

−Δα_:D−Δα_⊂L2_Ω−→L2_Ω,

D−Δα

⎧ ⎨

⎩u∈L2Ω|

∞

j1

mj

k1

λα_ju, ϕjk

2

<∞

⎫ ⎬ ⎭,

−Δα_u∞ j1

mj

k1

λα_jϕjk, u

ϕjk.

2.3

In particular, we obtain ϕjk ∈ D−Δα and −Δαϕjk λα_jϕjk. Since {ϕjk} mj,∞

k1,j1 form a

complete orthonormal system in the spaceL2_{Ω,then it is dense inL}2_{Ω, and henceD−Δ}α is dense in L2_Ω.

Proposition 2.3cf.4 . LetXbe a Hilbert separable space and{Aj}_j≥1and {Pj}_j≥1two families

of bounded linear operators inX, with{Pj}_j≥1a family of complete orthogonal projections such that AjPjPjAj, j ≥1.

Define the following family of linear operatorsStw∞_j1eAjt_P

jw, w∈X, t≥0.Then aStis a linear and bounded operator if eAjt ≤_{gt, j} ≥_1withgt≥_{0,continiuous for}

t≥0,

bunder the above condition (a),{St}_t≥0is a strongly continiuous semigroup in the Hilbert spaceX,whose infinitesimal generatorAis given by

Aw ∞

j1

AjPjw, w∈DA, DA

⎧ ⎨ ⎩w∈X|

∞

j1

_AjPjw2

<∞

⎫ ⎬

⎭. 2.4

Theorem 2.4cf.5 . SupposeΩis connected,fis a real function inΩ, andf 0on a nonempty

open subset ofΩ. Thenf≡0inΩ.

3. Abstract Formulation of the Problem

In this section we consider the following notations.

iXL2_Ω×L2_{Ω. Xis a Hilbert space with the inner product}

iiWe define

A11u, v a1Δua2Δv−c1−Δα1u−c2−Δα2v,

A12u, v b1Δub2Δv−d1−Δβ1u−d2−Δβ2v.

3.2

iiiLetw u, v,then we can define the linear operator

A:DA⊂X−→X,

DA H2_Ω;R∩H1 0Ω;R

2

,

Aw−MΔ−c1B1−Δα1−c2B2−Δα2−d1B3−Δβ1−d2B4−Δβ2

w, 3.3 where M

a1 a2

b1 b2

, B1

1 0 0 0

, B2

0 1 0 0 , B3 0 0 1 0

, B4

0 0 0 1 . 3.4

Therefore, for allw∈DA

A11u, v a1

∞

j1

λjEjua2

∞

j1

λjEjvc1

∞

j1

λα1_j Ejuc2

∞

j1

λα2_j Ejv,

A12u, v b1

∞

j1

λjEjub2

∞

j1

λjEjvd1

∞

j1

λβ1_j Ejud2

∞

j1

λβ2_j Ejv.

3.5

If we put

Pj

Ej 0 0 Ej

, j1,2, 3.6

then3.3can be written as

Aw≡

A11u, v

A12u, v

∞ j1

λjMλα1_j c1B1λα2_j c2B2λβ1_j d1B3λβ2_j d2B4

Pjw, 3.7

and we have for allw∈X

w ∞

j1

Pjw, w2 ∞

j1

Consequently, system1.1can be written as an abstract differential equation in the Hilbert spaceXin the following form:

•

w−AwBωft, inΩ× 0, t∗,

w0, on 0, t∗×∂Ω,

w0 w0, inx∈Ω,

3.9

wheref ≡colf1, f2∈L20, T ;X andBω

₁

ω 0

0 1ω

is a bounded linear operator fromU intoX.

4. Main Results

4.1. Generation of a

C

-Semigroup

Theorem 4.1. Ifc2 d1 0, then, under hypotheses (H1)–(H3), the linear operator−Adefined by

3.3is the infinitesimal generator of strongly continuous semigroup{St}_t≥0given by

Stw ∞

j1

eAjt_P

jw, w∈X, 4.1

where

Mj −λjM−λα1_j c1B1−λα2_j c2B2−λβ1_j d1B3−λβ2_j d2B4, 4.2

AjMjPj. 4.3

Moreover, if

minc1λ1₁−α1ρ1, d2λ1₁−β2ρ1

>0, 4.4

then theC0-semigoup{St}t≥0is exponentially stable, that is, there exist two positives constantsc, δ such that

St ≤ce−δt, for allt≥0. 4.5

Proof. In order to apply theProposition 2.3, we observe that−Acan be written as follows:

−Aw∞ j1

where

Aj−

λjMλα1j c1B1λα2j c2B2λβ1j d1B3λβ2j d2B4

Pj. 4.7

Therefore,AjMjPjandAjPjPjAj.

Now, we have to verify condition a of theProposition 2.3. We shall suppose that 0< ρ1< ρ2.Then, there exists a set{Q1, Q2} ∈M2R 2of complementary projections onR2

such that

eMt_eρ1t_Q

1eρ2tQ2. 4.8

IfG g11_g21g12_g22is the matrix passage from the canonical basis ofR2 _{to the basis composed}

with the eigenvectors ofM, then

Q1

1 ρ1ρ2

g11g22 −g11g12

g21g22 −g12g21

, Q2

1 ρ1ρ2

−g12g21 g11g12

−g21g22 g11g22

. 4.9

Hence,

e−λjMt_e−λjρ1t_Q

1e−λjρ2tQ2. 4.10

We have also

e−λαj1c1B1t

⎛ ⎝e−λ

α1

j c1t ₀

0 1

⎞

⎠_{, e}−λα2

j c2B2t

1 −λα2_j c2t

0 1

,

e−λβj1d1B3t

⎛

⎝ 1 0

−λβ1 j d1t 1

⎞

⎠_{, e}−λβ1

j d2B4t

⎛ ⎝1 0

0 e−λβj2d2t

⎞ ⎠_.

4.11

From4.10-4.11into4.7we obtain

eAjt_e−λjρ1t_Q

1e−λjρ2tQ2

KjtPj, 4.12

where

Kjt

⎛ ⎜ ⎝e

−λα1

j c1t_λα2β1

j c2d1t2e−λ α1

j c1t −λα2

j c2te−λ α1

j c1λ β2

j d2t

−λβ1

j d1t e−λ

β2

j d2t

⎞ ⎟

Asc2d10 we get

Kjt

⎛ ⎝e−λ

α1

j c1t ₀

0 e−λβj2d2t

⎞

⎠_{. 4.14}

Asλj → ∞asj → ∞,then this implies the existence of a positive numbercand a real number δ such that eAjt ≤ _ceδt_{, for every j} ≥ _1.Therefore−A_{is a strongly continious}

semigroup{St}_t≥0given by4.1. We can even estimate the constantscandδas follows. i If min{c1 λ1₁−α1ρ1, d2 λ₁1−β2ρ1} ≤ 0. As limj→ ∞{−λα1j c1 λ

1−α1

j ρ1} limj→ ∞{−λβ2j c1λj1−β2ρ1}−∞, then there exist constants

δ1max

−λα1 j

c1λ1j−α1ρ1

|λα1_j c1λ1j−α1ρ1

≤0, j ≥1,

δ2max

−λβ2 j

d2λ1j−β2ρ1

|λβ2_j d2λ1j−β2ρ1

≤0, j ≥1,

4.15

hence, if we put

δmax{δ1, δ2} ≥0, 4.16

c0

1 ρ1ρ2 max

_{g11g22,g11g12,g21g22,g12g21, 4.17}

we easily obtain

eAjt≤_4c

0e−δt, j≥1. 4.18

iiIf min{c1λ1₁−α1ρ1, d2λ1₁−β2ρ1}>0. If we put

δminλα₁1c1λ1₁−α1ρ1

, λβ2₁ d2λ1₁−β2ρ1

>0, 4.19

then we find that

eAjt≤_4c

0e−δt, j≥1. 4.20

Therefore, the linear operator−Agenerates a strongly continuous semigroup{St}_t≥0 onX given by expression4.1.

Finally, if min{c1 λ11−α1ρ1, d2 λ11−β2ρ1} > 0,we have already proved 4.20. Using

4.20into4.1we get that theC0-semigoup{St}t≥0is exponentially stable. The expression

Theorem 4.2. If

c2/0, d1/0, c1≥0, d2≥0, 4.21

then, under the hypotheses (H1)–(H3), the linear operator−A defined by3.3 is the infinitesimal generator of strongly continuous semigroup exponentially stable{St}_t≥0defined by4.1. Specially, there exist two positives constantsc, δsuch that

St ≤ce−δt, ∀t≥0. 4.22

To prove this result, we need the following lemma.

Lemma 4.3. For every two real positives constantscandλ, one has for every0< δ < λ/c

cte−λt≤ 1 eλ/c−δe

−δct_{, ∀t≥0, 4.23}

and for every0< δ < λ/√c

ct2e−λt ≤ 4

e2_λ/√_c−δe

−δ√ct_{, ∀t≥0. 4.24}

Proof ofLemma 4.3.It is easy to verify that for every ε >0 :te−εt _{≤1/eε, for allt≥0.} Let 0< δ < λ/candελ/c−δ >0, then we get

te−λ/ct≤ 1 eλ/c−δe

−δt_{, ∀t≥0. 4.25}

Hence, we get4.23.

Also, it is easy to verify that for every ε > 0 : t2_e−εt _{≤ 4/e}2_ε2_{, for all t ≥ 0. Let}

0< δ < λ/√candελ/√c−δ >0, then we get

t2e−λ√ct≤ 4

e2_λ/√_c−δ2e

−δt_{, ∀t≥0.}

4.26

Hence, from 4.26we get ct2_e−λt √_ct2_e−λ/√c√ct _{≤ 4/e}2_λ/√_c−δ2_e−δ√ct_{for allt≥ 0} and 0< δ < λ/√c, which gives4.24.

With the same manner we can prove that for every 0< δ < λc−1/n_{and every n∈N}∗ we have

tne−λc−1/nt≤ n n

eεne

and consequently, for every two real positives constantscandλand every n∈N∗we have

ctne−λt ≤ n n

eεne −δc−1/n_t

, for allt≥0 and every 0< δ < λ. 4.28

Now, we are ready to proveTheorem 4.2.

Proof ofTheorem 4.2. By applyingProposition 2.3we start from formula4.12and we put

Kjt

K11,jt K12,jt

K21,jt K22,,jt

, 4.29

where

K11,jt e−λ

α1

j c1t_λα2β1

j c2d1t

2_e−λα1

j c1t_{, K}

12,jt −λα2_j c2te−λ α1

j c1λ β2

j d2t_,

K21,jt −λβ1_j d1t, K22,,jt e−λ

β2

j d2t_, ∀j≥_1.

4.30

To estimatee−λjρ1t_K

11,jtwe have in taking into accountc1 ≥0

e−λjρ1λαj1c1t≤_e−λ1ρ1t_, ∀t≥_0, 4.31

and applying theLemma 4.3cλ_jα2β1|c2d1|we get

λα2_j β1c2d1t2e−λjρ1λ α₁ j c1t

≤ 4

e2_λ1−α2β1/2

j /

|c2d1|

ρ1

λ_jα1−α2β1/2/|c2d1|

c1−γ1

e−γ1λαj2β1/2 √

|c2d1|t_,

4.32

for allt ≥ 0 and 0 < γ1 < λj1−α2β1/2/

|c2d1|ρ1 λjα1−α2β1/2/

|c2d1|c1. But we have

λ_j1−α2β1/2/|c2d1|ρ1λ_jα1−α2β1/2/

|c2d1|c1≥λ₁1−α2β1/2/

|c2d1|ρ1, for allj≥1.Then

we get for every 0< γ1<λ₁1−α2β1/2/

|c2d1|ρ1that

λα2_j β1c2d1t2e−λjρ1λ α₁ j c1t

≤ 4

e2_λ1−α2β1/2

1 /

|c2d1|

ρ1−δ1

e−γ1λ1α2β1/2 √

|c2d1|t_{, ∀t≥0.} 4.33

From4.31-4.33we get

e−λjρ1t_K

where

σ114

⎛

⎝λ11−α2β1/2

|c2d1|

ρ1−δ1

⎞ ⎠

−1

, δ1min λ1ρ1, γ1λ₁α2β1/2

!

|c2d1|

"

, 4.35

and 0< γ1 <λ₁1−α2β1/2/

|c2d1|ρ1.

ApplyingLemma 4.3and taking into account4.21we get with the same manner that for every 0< δ2<λ1₁−α2/|c2|ρ1

e−λjρ1t_K

12,jt≤σ2e−δ2λ α2

1 |c2|t, ∀t≥0, j≥1, 4.36

where

σ2

1 eλ1₁−α2/|c2|

ρ1−δ2

, _4.37

and or every 0< δ3<λ1₁−β1/|d1|ρ1

e−λjρ1t_K

21,jt≤σ3e−δ3λ β1

1 |d1|t, ∀t≥0, j≥1, 4.38

where

σ3

1 eλ1₁−β1/|d1|

ρ1−δ3

, _4.39

e−λjρ1t_K

22,jt≤e−λ1ρ1t, ∀t≥0, j ≥1. 4.40

From4.34-4.40into4.12we get

eAjt≤_4c

0σe−δt, ∀t≥0, j ≥1, 4.41

where c0is defined by4.17and

σ1σ1σ2σ3, 0< δ <min

δ1, δ2λα2₁ |c2|, δ3λβ1₁|d1|, λ1ρ1

. 4.42

Using4.41into4.1we get that theC0-semigoup{St}t≥0generated by−Ais exponentially

4.2. Approximate Controllability

Befor giving the definition of the approximate controllabiliy for the sytem3.9, we have the following known result: for allw0 ∈Xandf ∈ L2 0, T;U,the initial value problem3.9

admits a unique mild solution given by

wt Stw0

#_t

0

St−τBωfτdτ, t∈0, T . 4.43

This solution is denoted bywt;f.

Definition 4.4. System3.9is said to beapproximately controllableat timet∗whenever the set Ft∗{wt∗;f| ∀f ∈L2 0, t∗;U}is densely embedded inX; that is,

∀w0, w1 ∈X, ∀ε >0; ∃f∈L2 0, t∗;U:w

t∗;f−w1< ε. 4.44

The following criteria for approximate controllability can be found in6 .

Criteria 1. System3.9is approximately controllable on0, t∗ if and only if

B∗S∗tw0, ∀t∈0, t∗ ⇒w0. 4.45

Now, we are ready to formulate the third main result of this work.

Theorem 4.5. If the following condition

c2d10 4.46

is satisfied; then, under hypotheses (H1)–(H5), for all t∗>0and all open subsetω⊂Ω,system3.9

is approximately controllable on0, t∗ .

Proof. The proof of this theorem relies on the Criteria1and the following lemma.

Lemma 4.6. Let{α1j}_j≥1, {β1j}_j≥1 and {α2j}_j≥1,{β2j}_j≥1 be sequences of real numbers such that α11 > α12 > α13 >· · ·,α21 > α22 > α23 >· · · andα1j > α2j, for allj ≥0, then for any t∗ ∈R∗one

has

∞

j1

eα1jt_β

1jeα2jtβ2j

0, ∀t∈0, t∗ ⇒β1jβ2j 0, ∀j≥1. 4.47

Proof ofLemma 4.6.By analyticity we get∞_j1eα1jt_β

1jeα2jtβ2j 0, ∀t≥0 and from this we getβ11

_∞

j2eα1j−α11tβ1j

_∞

j1eα2j−α11tβ2j0, ∀t≥0. Under the assumptions of the lemma we get∞_j2eα1j−α11t_β

1j∞j1eα2j−α11tβ2j → 0 as t → ∞and so β11 0.Ifα12 > α21, we

divide∞_j2eα1jt_β 1j

_∞

j1eα2jtβ2j0 byeα12tand we passt → ∞we getβ120. Ifα21> α12,we

divide∞_j2eα1jt_β

we divide∞_j2eα1jt_β

1j∞j1eα2jtβ2j 0 byeα12tand we passt → ∞and getβ12β21 0.But

in this we case we can integrate under the symbol of sommation over the intervall0, t and we getβ12eα21tβ21eα12t0. Henceβ12β21 0. Continuing this way we see thatβ1j β2j0, for all j≥1.

We are now ready to proveTheorem 4.5. For this purpose, we observe that

B∗ωBω, S∗tw ∞

j1

eM∗jt_P∗

jw, w∈X, t≥0, 4.48

where{St}_t≥0is the C0-semigroup generated by−A.

Without lose of generality, we suppose that 0< ρ1< ρ2.Hence

B_ω∗S∗tw ∞

j1

B∗_ωeM∗jt_P∗

jw ∞

j1

B_ω∗eM∗jt_P∗

jw ∞

j1 2

s1

B∗_ωK∗_jte−λjρst_P∗

sj

w, 4.49

wherePsjQsPjPjQs, s1,2.

Now, suppose forw∈XthatB_ω∗S∗tw0, for allt∈0, t∗ .Then

B∗_ωS∗tw0⇐⇒ ∞

j1 2

s1

B_ω∗K∗_jte−λjρst_P∗

sj

wx 0, ∀x∈Ω. 4.50

If4.46is satisfied, then4.50take the form

∞

j1 2

s1

⎛ ⎝e−

λjρsλαj1c1t ₀

0 e−λjρsλβj2d2t

⎞

⎠B_ω∗P_sj∗wx 0, ∀x∈Ω. 4.51

Then, from lemma4.6we obtain that for s1,2 and allx∈ω

B∗_ωQ∗_sP_j∗wx Q∗_s

⎛ ⎜ ⎜ ⎜ ⎜ ⎜ ⎝ mj

k1

u, ϕjk

1ωϕjkx

mj

k1

v, ϕjk

1ωϕjkx

⎞ ⎟ ⎟ ⎟ ⎟ ⎟ ⎠ 0 0

, j≥1. 4.52

SinceQ1Q2IR2,we get that allx∈ω

⎛ ⎜ ⎜ ⎜ ⎜ ⎜ ⎝ mj

k1

u, ϕjk

1ωϕjkx

mj

k1

v, ϕjk

1ωϕjkx

⎞ ⎟ ⎟ ⎟ ⎟ ⎟ ⎠ 0 0

On the other hand, fromTheorem 2.4we know that ϕjkare analytic functions, which implies the analticity ofEjum_kj₁u, ϕjkϕjkandEjvm_kj₁v, ϕjkϕjk.Then we can conclude that fors1,2 and allx∈Ω

⎛ ⎜ ⎜ ⎜ ⎜ ⎜ ⎝

mj

k1

u, ϕjk

ϕjkx

mj

k1

v, ϕjk

ϕjkx

⎞ ⎟ ⎟ ⎟ ⎟ ⎟ ⎠

0 0

, j≥1. 4.54

Hence Pjw 0, for all j ≥ 1, which implies that w 0. This completes the proof of

Theorem 4.5.

References

1 S. Badraoui, “Asymptotic behavior of solutions to a 2×2 reaction-diffusion system with a cross diffusion matrix on unbounded domains,”Electronic Journal of Differential Equations, vol. 2006, no. 61, pp. 1–13, 2006.

2 H. Larez and H. Leiva, “Interior controllability of a 2×2 a reaction-diffusion system with cross diffusion matrix,” to appear inBoundary Value Problems.

3 E. Zeidler,Applied Functional Analysis, vol. 109 ofApplied Mathematical Sciences, Springer, New York, NY, USA, 1995.

4 H. Leiva, “A lemma on C0-semigroups and applications,”Quaestiones Mathematicae, vol. 26, no. 3, pp.

247–265, 2003.

5 S. Axler, P. Bourdon, and W. Ramey,Harmonic Function Theory, vol. 137 ofGraduate Texts in Mathematics, Springer, New York, NY, USA, 1992.