REACTION-DIFFUSION SYSTEM WITH DIFFERENT DIFFUSION COEFFICIENTS

REACTION-DIFFUSION SYSTEM WITH DIFFERENT DIFFUSION COEFFICIENTS

L. W. SOMATHILAKE AND J. M. J. J. PEIRIS

Received 30 January 2004 and in revised form 30 August 2004

We deal with a mathematical model for a four-component chemical reaction-diffusion process. The model is described by a system of strongly coupled reaction-diffusion equa- tions with different diffusion rates. The existence of the global solution of this reaction- diffusion system in unbounded domain is proved by using semigroup theory and esti- mates on the growth of solutions.

1. Introduction

In this paper, we prove the existence of a global solution in an unbounded domain of the reaction-diffusion system

∂u1

∂t ⁼a1∆u1−f1

u1,u2,v1,v2

, x∈Rⁿ,t >0,

∂u2

∂t ⁼a2∆u2−f2

u1,u2,v1,v2

, x∈Rⁿ,t >0,

∂v1

∂t ⁼b1∆u1+d1∆v1+ f3

u1,u2,v1,v2

, x∈Rⁿ,t >0,

∂v2

∂t ⁼b2∆u2+d2∆v2+ f4

u1,u2,v1,v2

, x∈Rⁿ,t >0,

(RDS1)

with the initial conditions

ui(x, 0)=u⁰_i(x), vi(x, 0)=v_i⁰(x) (i=1, 2),x∈Rⁿ. (IC1) Here f1(u1,u2,v1,v2)=m[k2u^m1v^r2−k1u^r2v1^m], f2(u1,u2,v1,v2)= −r[k2u^m1v^r2−k1u^r2v^m1], k1≥0,k2≥0,m≥0,r≥1, f3=ρ f1, and f4=ρ f2,ρ >0.

The constantsai,bi(i=1, 2) are such thatai>0,bi=0 (i=1, 2), and 4aidi> b²_i (i= 1, 2) which reflects the parabolicity of the system.∆is the Laplace operator inRⁿ. More- over we assume that the functionsu⁰_i (i=1, 2) andv⁰_i (i=1, 2) are uniformly bounded, continuous, and nonnegative. This reaction-diffusion system is a mathematical model for

Copyright©2005 Hindawi Publishing Corporation Journal of Applied Mathematics 2005:1 (2005) 23–36 DOI:10.1155/JAM.2005.23

a chemical reaction of the form

mA+rB¯^k1

k2

rB+mA.¯ (1.1)

u1,u2,v1, andv2represent the concentrations of ¯A, ¯B,A, andB, respectively (see [3]).

We remark that the system

∂u

∂t ⁼a∆u−uh(v), x∈Ω, t >0,

∂v

∂t ⁼b∆u+d∆v+uh(v), x∈Ω, t >0,

(RDS2)

with the initial conditions

u(x, 0)=u0(x), v(x, 0)=v0(x), x∈Ω (IC2) on a bounded domainΩ⊂Rⁿwith Neumann boundary conditions,b >0,a=d,v0≥ (b/(a−d))u0≥0, andh(s), a differentiable nonnegative function onR, has been stud- ied by Kirane [4]. The existence of global solutions for system (RDS2) on unbounded domains has been studied by Badraoui [1]. The existence of global solutions inRⁿfor (RDS2) withh(s)=v^mhas been studied by Collet and Xin [2].

The quasilinear system of reaction-diffusion equations

∂u

∂t ^{= ∇ ·}

a(u)∇u−uh(u)v, x∈Ω, t >0,

∂v

∂t ^{= ∇ ·}

b(v)∇v+uh(u)v−λv, x∈Ω, t >0,

(RDS3)

with the initial conditions

u(x, 0)=u0(x), v(x, 0)=v0(x), x∈Ω, (IC3) and with Neumann or Dirichlet boundary conditions, is studied by Kirane [5] where in particular the existence of a globally bounded solution is shown. Also he has discussed large time behavior of the solution.

Our aim is to investigate the existence of a global solution for system (RDS1)–(IC1) in an unbounded domain.

Throughout this paper the following notations are used.

(1) · is the supremum norm onRⁿ, that is,u =sup_x∈Rn|u(x)|.

(2)Cub(Rⁿ) is the space of uniformly bounded continuous functions onRⁿequipped with the supnorm.

(3) For any f ∈Cub(Rⁿ), f =

Rⁿf(x)dxif the integral exists.

(4) For f ∈L^p(Rⁿ) (p≥1),f_p=(|f|^p)^{1/ p}.

2. Existence of a local solution

We convert system (RDS1)–(IC1) to an abstract first-order system in the Banach space X=(Cub(Rⁿ))⁴of the form

d dt

u(t)=Au(t) +Fu(t), t >0, u(0)=u0∈X,

(2.1)

whereu(t)=(u1(t),u2(t),v1(t),v2(t))^T. The operatorAis defined as

A=







a1∆ 0 0 0

0 a2∆ 0 0

b1∆ 0 d1∆ 0 0 b2∆ 0 d2∆





 (2.2)

with domainD(A)= {u=(u1,u2,v1,v2)^T∈X, (∆u1,∆u2,∆v1,∆v2)^T∈X}. Moreover the functionFis defined as

Fu(t)=

−f1

u(t),−f2

u(t),f3

u(t),f4

u(t)^T, (2.3)

where

f_iu(t)=f_iu1(t),u2(t),v1(t),v2(t), i=1, 2, 3, 4. (2.4) Note that forλ >0 the operatorλ∆generates an analytic semigroupG(t) in the spaceX given by

G(t)u(x)= 1 (4πλt)^n/2

Rⁿexp

−(x−y)² 4λt

u(y)dy, t >0,x∈Rⁿ. (2.5)

LetS1(t),S2(t),S3(t), andS4(t) be the semigroups generated bya1∆,a2∆,d1∆, andd2∆, respectively. Then one can show thatAgenerates an analytic semigroupS(t) given by

S(t)=







S1(t) 0 0 0

0 S2(t) 0 0

S5(t) 0 S3(t) 0 0 S6(t) 0 S4(t)





, (2.6)

where

S5(t)= b1

a1−d1

S1(t)−S3(t), (2.7)

S6(t)= b2

a2−d2

S2(t)−S4(t). (2.8)

Assume thatFis locally Lipschitz inuin the spaceX. Then there exist classical solutions on maximal existence interval [0,T0] (see [6]).

3. Existence of global solutions

For proving the existence of a global solution we assume that the solutions are nonnega- tive.

Theorem3.1. Consider the reaction-diffusion system (RDS1) with nonnegative initial con- ditions(u⁰1(x),u⁰2(x),v⁰1(x),v2⁰(x))∈(Cub(Rⁿ))⁴,ai>0,di> ai, andbi<0. Then there exist global in-time classical solutions such that

u1,u2,v1,v2

∈

C[0,∞);Cub

Rⁿ

∩C¹(0,∞);Cub

Rⁿ4

. (3.1)

Lemma3.2. Let(u1,u2,v1,v2)be a classical solution of (RDS1). Define the functionals Fⁱu_i,v_i=

α_i+u_i+u²_ie^ivi (i=1, 2)with_i>0,α_i>0. (3.2) Then for any smooth nonnegative functionψ=ψ(x,t) (x∈Rⁿ)with exponential spatial decay at infinity,

d dt

ψFⁱ= ψ_t+d_i∆ψFⁱ+ d_i−a_iF1ⁱ−b_iF2ⁱ

∇ψ∇u_i

−

ψaiF11ⁱ +biF12ⁱ ∇ui²+ai+di

F12ⁱ +biF22ⁱ

∇ui∇vi

+diF22ⁱ

∇vi2 +

ψρF2ⁱfi−F1ⁱfi

, i=1, 2,

(3.3)

where

F1ⁱ=∂Fⁱ

∂u_i, F2ⁱ=∂Fⁱ

∂v_i, F11ⁱ =∂²Fⁱ

∂u²_i , F22ⁱ =∂²Fⁱ

∂v_i², F12ⁱ = ∂²Fⁱ

∂ui∂vi, i=1, 2.

(3.4)

Proof. Fori=1, 2, we have d

dt

ψFⁱ=

ψtFⁱ+

ψ ∂Fⁱ

∂ui

∂u_i

∂t +∂Fⁱ

∂vi

∂v_i

∂t

=

ψtFⁱ+

ψ

F1ⁱ

∂ui

∂t +F2ⁱ

∂vi

∂t

=

ψ_tFⁱ+

ψF1ⁱ

a_i∆ui−f_i+F2ⁱ

b_i∆ui+d_i∆vi+ρ f_i

=

ψtFⁱ+ai

ψF1ⁱ∆ui+bi

ψF2ⁱ∆ui+di

ψF2ⁱ∆vi

−

ψF1ⁱfi+ρ

ψF2ⁱfi.

(3.5)

However,

ψF1ⁱ∆ui=

F1ⁱψ∆ui

= −

∇

Fⁱψ∇u_i

= − F1ⁱ∇ψ+ψ∇F1ⁱ

∇ui

= −

F1ⁱ∇ψ∇u_i−

ψ∇F1ⁱ∇u_i

= −

F1ⁱ∇ψ∇ui−

ψF11ⁱ ∇ui+F12ⁱ ∇vi

∇ui,

(3.6)

that is,

ψF1ⁱ∆ui= −

F1ⁱ∇ψ∇u_i−

ψF11ⁱ ∇u_i²−

ψF12ⁱ ∇u_i∇v_i. (3.7)

Similarly

ψF2ⁱ∆ui= −

F2ⁱ∇ψ∇vi−

ψF12ⁱ ∇ui²−

ψF22ⁱ ∇ui∇vi,

ψF2ⁱ∆vi= −

F2ⁱ∇ψ∇v_i−

ψF22ⁱ ∇v_i²−

ψF12ⁱ ∇u_i∇v_i.

(3.8)

Also

Fⁱ∆ψ= −

∇Fⁱ∇ψ= −

F1ⁱ∇u_i∇ψ−

F2ⁱ∇v_i∇ψ. (3.9)

Using (3.5)–(3.9) we get

d dt

ψFⁱ= ψt+di∆ψFⁱ−ai

F1ⁱ∇ψ∇ui−ai

ψF11ⁱ ∇ui²

−a_i

ψF12ⁱ ∇u_i∇v_i−b_i

F2ⁱ∇ψ∇u_i−b_i

ψF12ⁱ ∇u_i²

−bi

ψF22ⁱ ∇ui∇vi+di

F1ⁱ∇ui∇ψ−d

ψF22ⁱ ∇vi²

−di

ψF12ⁱ ∇vi· ∇vi−

ψF1ⁱfi+ρ

ψF2ⁱfi

= ψ_t+d_i∆ψFⁱ+ d_iF1ⁱ−a_iF1ⁱ−b_iF2ⁱ

∇ψ∇u_i

− aiF11ⁱ +biF12ⁱ ∇ui²−

ψaiF12ⁱ +bF22ⁱ +diF12ⁱ

∇ui· ∇vi

−d_i

ψF22ⁱ ∇v_i²−

ψF1ⁱf_i+ρ

ψF2ⁱf_i

= ψt+di∇ψFⁱ+ di−ai

F1ⁱ−biF2ⁱ

∇ψ∇ui

−

ψaiF11ⁱ +biF12ⁱ ∇ui²+ai+di

F12ⁱ +biF22ⁱ

∇ui∇vi

+d_iF22ⁱ ∇v_i²+

ψρF2ⁱf_i−F1ⁱf_i.

(3.10)

Lemma 3.3. There exist four positive constantsαi=αi(ai,b,di,u⁰_i) (i=1, 2)andi= i(ai,bi,di,u⁰_i) (i=1, 2)such that

d dt

ψFⁱ≤ ψt+di∇ψFⁱ+ di−ai

F1ⁱ−biF2ⁱ

∇ψ∇ui

−1 2

ψ ai

2F11ⁱ ∇u_i²+d_iF22ⁱ ∇v_i²

−1 2

ψF1ⁱf_i, i=1, 2.

(3.11)

Proof. For any (ui,vi)∈[0,u⁰_i]×R⁺(i=1, 2), we chooseαiandiinLemma 3.2such that

ρF2ⁱ≤1

2F1ⁱ, i=1, 2, (3.12)

a_i+d_i²F12ⁱ

2

+b²_iF22ⁱ

2

+b_i2a_i+d_iF12ⁱ

F22ⁱ

−a_id_iF11ⁱ F22ⁱ ≤0, i=1, 2, (3.13) F1ⁱ

2

F11ⁱ

≤Fⁱ, i=1, 2, (3.14)

F12ⁱ ≤ ai

2b_iF11ⁱ , i=1, 2. (3.15)

We verify these conditions as follows Fⁱ=

αi+ui+u²_ie^ivi, i=1, 2, F1ⁱ=

1 + 2u_ie^ivi, i=1, 2, F11ⁱ =2e^ivi, i=1, 2,

F2ⁱ=i

αi+ui+u²_ie^ivi, i=1, 2, F22ⁱ =²_i

α_i+u_i+u²_ie^ivi, i=1, 2, F12=F21ⁱ =i

1 + 2u_ie^ivi, i=1, 2.

(3.16)

Denote

_i⁽¹⁾= 1

2ραi+u⁰_i+u⁰_i²,

_i⁽²⁾= 1

b_iα_i+u⁰_i+u⁰_i²,

_i⁽³⁾= 1

bi1 + 2u⁰_i, α⁽¹⁾_i =

a_i+d_i²1 + 2u⁰_i²+2a_i+d_i1 + 2u⁰_i

2a_id_i ,

α⁽²⁾_i =1 + 2u⁰_i²+ 2u⁰_i

2 ⁼

1 +u⁰_i²+u⁰_i²

2 .

(3.17)

If we choosei(i=1, 2) such thati≤⁽¹⁾_i (i=1, 2) then condition (3.12) is satisfied.

Condition (3.13) is satisfied if a_i+d_i²²_i

1 + 2u_i²e^2ivi+b_i²⁴_i

α_i+u_i+u²_i²e^2ivi+b_i2a_i+d_i³_i 1 + 2u_i

×

αi+ui+u²_i²e^2ivi−2aidi²_i

αi+ui+u²_ie^2ivi≤0, i=1, 2, (3.18) that is,

ai+di2

1 + 2ui2

+b_i²²_i

αi+ui+u²_i²

+b_i2a_i+d_i1 + 2u_iα_i+u_i+u²_ii−2a_id_iα_i+u_i+u²_i≤0. (3.19) If we choosei(i=1, 2) andαi (i=1, 2) such thati≤_i⁽²⁾(i=1, 2) andαi≤α⁽¹⁾_i (i= 1, 2), then (3.19) is satisfied. In other words, condition (3.13) is satisfied.

Condition (3.14) is satisfied if 1 + 2ui

2

2 ^≤

αi+ui+u²_i, (3.20)

that is,

α_i>1 + 2u²_i+ 2u_i

2 . (3.21)

If we chooseαi(i=1, 2) such thatαi≤α⁽²⁾_i (i=1, 2), then (3.21) is satisfied. Hence con- dition (3.14) is satisfied.

Similarly we can show that condition (3.15) is satisfied if_i(i=1, 2) is chosen such thati≤_i⁽³⁾.

Now select

αi≥maxα⁽¹⁾_i ,α⁽²⁾_i (i=1, 2), i≤min⁽¹⁾_i ,_i⁽²⁾,_i⁽³⁾

(i=1, 2). (3.22)

Then conditions (3.12)–(3.15) are satisfied.

Then, from (3.13) and (3.15), we get

a_iF11ⁱ +b_iF12ⁱ ∇u_i²+a_i+d_iF12ⁱ +b_iF22ⁱ

∇u_i∇v_i+d_iF22ⁱ ∇v_i²

≥1 2

a_iF11ⁱ +b_iF12ⁱ ∇u_i²+d_iF22ⁱ ∇v_i²

≥1 2

a_i

2F11ⁱ ∇ui²+diF22ⁱ ∇vi² .

(3.23)

From (3.12), we get

ψρF2ⁱfi−F1ⁱfi

≤ −1 2

ψF1ⁱfi, i=1, 2. (3.24) From (3.3), (3.23), (3.24), we get (3.11).

The proof ofLemma 3.3is completed.

Theorem3.4. Ifαi,i(i=1, 2)satisfy (3.22), then there exist a test functionψand real positive constantsβ_i(i=1, 2)andσ_isuch that

ψF_i≤β_ie^σit, ∀t >0,i=1, 2. (3.25) Proof. We define the test functionψ(x) as

ψ(x)= 1

1 +x−x0²n, x∈Rⁿ, (3.26)

andx0∈Rⁿis an arbitrary point.

Thenψis a smooth function with exponential decay at infinity and satisfies|∆ψ| ≤ K1ψ,|∇ψ| ≤K2ψ. LetK=max(K1,K2) for some positive constantK.

Then from (3.11), we obtain d

dt

ψFⁱ≤Kdi

ψFⁱ+Kdi−ai +1

2bi

F1ⁱψ∇ui

−ai

4

ψF11ⁱ |∇u|²−di

2

ψF22ⁱ ∇vi²−1 2

ψF1ⁱfi

≤Kdi

ψFⁱ+K² ai

di−ai +1

2bi2 ψ

F1ⁱ

2

F11ⁱ

≤

Kd_i+K² a_i

d_i−a_i+1 2b_i²

2

ψFⁱ, i=1, 2.

(3.27)

Let

σ_i=Kd_i+K² a_i

d_i−a_i+1 2b_i²

2

, i=1, 2, β_i≥

α_i+u⁰_i+u⁰_i²e^vi0ψ1, i=1, 2.

(3.28)

Then

d dt

ψFⁱ≤σ_i

ψFⁱ fori=1, 2, (3.29)

which implies (3.25).

Lemma3.5. For any unit cubeQand any finitep≥1,

Q

v_i^pdx≤2ⁿ β_i

αi_i^pe^σit(p+ 1)^p+1 fori=1, 2. (3.30) Proof. Using the results inTheorem 3.4, for any nonnegative integerpwe have

βie^σit≥

ψFⁱ≥αi

ψe^ivi≥αi_i^p

Qψv_i^p

p!, t >0,i=1, 2. (3.31) By takingx0at the center ofQ, we get

β_ie^σit≥αi_i^p p!

Q

v_i^p 2^{n ≥}

αi_i^p 2ⁿ(p+ 1)^p+1

Qv_i^p, i=1, 2. (3.32)

This implies (3.30).

Lemma3.6. There exist constantsc_i=c_i(n,λ,u⁰_i,v3⁰₋i,t),i=1, 2such that G(t−s)∗u^r_i(x,s)v^m3₋i(x,s)= 1

4πλ(t−s)^n/2

e^{−|x−y|2/4λ(t−s)}u^r_i(y,s)v3^m₋i(y,s)dy

≤c_i(t−s)^n/2q+ (t−s)^−n/2p, i=1, 2,

(3.33)

for anyp >max{1,n/2},1/ p+ 1/q=1. HereG(t)is the semigroup generated by the operator λ∆,(λ >0)on the spaceCub(Rⁿ).

Proof. Let{Q_j}, j=0, 1, 2,..., be the tilling ofRⁿby unit cubesQ_j’s such thatxis at the center ofQ0. Then

e^{−(x−y)2/4λ(t−s)}u^r_i(y,s)v3^m−i(y,s)dy=

Qj

Qj

e^{−(x−y)2/4λ(t−s)}u^r_i(y,s)v3^m−i(y,s)dy. (3.34)

Fory∈Qjwe have the inequality e^{−(x−y)2/8λ(t−s)}≤sup

y∈Qj

e^{−(x−y)2/8λ(t−s)}=e^{−dist2(x,Qj)/8λ(t−s)}. (3.35)

Also there exists a positive constantc(n) such that ify∈Q_j,j=0, we have

c(n) dist²(x,Qj)≥(x−y)². (3.36) LetI1=

e^{−(x−y)2/8λ(t−s)}u^r_i(y,s)v3^m₋i(y,s)dy. Then applying H¨older’s inequality with p≥ n/2 and its conjugateq, we get

I1≤

Qj

e^{−q(x−y)2/8λ(t−s)} 1/q

Qj

u^{r p}_i (y,s)v^mp3−i(y,s)dy 1/ p

≤

Qj

e^{−qdist2(x,Qj)/8λ(t−s)} 1/q

Qj

u^{r p}_i (y,s)v3^mp₋i(y,s)dy 1/ p

≤(8πλ)^n/2q(t−s)^n/2q q

Qj

u^{r p}_i (y,s)v^mp3−i(y,s)dy 1/ p

≤(8πλ)^n/2q(t−s)^n/2q q

Qj

u_i^{r p}v3−i^mpdy 1/ p

≤(8πλ)^n/2q(t−s)^n/2q q u⁰_i^r

Qj

v3₋i^mpdy 1/ p

≤(8πλ)^n/2q(t−s)^n/2q q u⁰_i^r

2^mp β_i

αi^mp_i e^σt(mp+ 1)^mp+1 1/ p

=(8πλ)^n/2q(t−s)^n/2q

q u⁰_i^r2^m β_i

α_i 1/ p

e^{σt/ p−}_i^m(mp+ 1)^{m+1/ p}.

(3.37)