Electronic Journal of Qualitative Theory of Differential Equations 2009, No. 73, 1-7; http://www.math.u-szeged.hu/ejqtde/

EXISTENCE OF GLOBAL SOLUTIONS FOR A SYSTEM OF REACTION-DIFFUSION EQUATIONS WITH EXPONENTIAL

NONLINEARITY

EL HACHEMI DADDIOUAISSA

Abstract. We consider the question of global existence and uniform boundedness of nonnegative solutions of a system of reaction-diffusion equations with exponential nonlinearity using Lyapunov function techniques.

1. Introduction

In this paper we consider the following reaction−diffusion system

\frac{∂u}{∂t} − aΔu = Π − f(u, v) − αu (x, t) ∈ Ω × R_{+}  (1.1)
\frac{∂v}{∂t} − bΔv = f(u, v) − σκ(v) (x, t) ∈ Ω × R_{+}  (1.2)

with the boundary conditions

\frac{∂u}{∂η} = \frac{∂v}{∂η} = 0 on ∂Ω × R_{+},  (1.3)

and the initial data

u(0, x) = u_{0}(x) ≥ 0; v(0, x) = v_{0}(x) ≥ 0 in Ω,  (1.4)

where Ω is a smooth open bounded domain in R^{n}, with boundary ∂Ω of class C^{1} and η is the outer normal to ∂Ω. The constants of diffusion a, b are positive and such that a ≠ b and Π, α, σ are positive constants, κ and f are nonnegative functions of class C^{1}(R_{+}) and C^{1}(R_{+} × R_{+}) respectively.

The reaction-diffusion system (1.1) − (1.4) arises in the study of physical, chemical, and various biological processes including population dynamics (especially AIDS, see C. Castillo-Chavez et al. [3], for further details see [5, 7, 12, 16, 17]).

The case Π = 0, α = 0, σ = 0 and f(u, v) = h(u)T (v), with h(u) = u (for simplicity), has been studied by many authors. Alikakos [1] established the existence of global solutions when T (v) ≤ C(1 + |v|^{(n+2)/n}). Then Massuda [13] obtained a positive result for the case T (v) ≤ C(1 + |v|^{α}) with arbitrary α > 0. The question when T (v) = e^{αv^{β}}, 0 < β < 1, α > 0 was positively answered by Haraux and Youkana [9], using Lyapunov function techniques, see also Barabanova [2] for β = 1, with some conditions and later on by Kanel and Kirane [11], using useful properties inherent to the Green function. The idea behind the Lyapunov functional stems from Zelenyak’s article [18], which has also been used by Crandall et al. [4] for other purposes. The goal of this work is to generalize the existing results of L. Melkemi et al. [14],

EJQTDE, 2009 No. 73, p. 1 where they established the existence of global solutions, when f(ξ, τ) ≤ ψ(ξ)φ(τ) such that

2000 Mathematics Subject Classification. 35K57, 35K45.

Key words and phrases. Reaction-diffusion systems; Lyapunov functional; Global solution.

lim_{τ→+∞} \frac{ln(1 + φ(τ))}{τ} = 0.

Hence, the main purpose of this paper is to give a positive answer, concerning the global existence and the uniform boundedness in time, of solutions of system (1.1) − (1.4), with exponential nonlinearity, such that f satisfies

(A1) ∀τ ≥ 0, f(0, τ) = 0,

(A2) ∀ξ ≥ 0, ∀τ ≥ 0, 0 ≤ f(ξ, τ) ≤ φ(ξ)(τ + 1)^{λ}e^{rτ}, (A3) κ(τ) = τ^{μ}, μ ≥ 1,

where r, λ are positive constants, such that λ ≥ 1, φ is a nonnegative function of class C(R^{+}). Our aim in this work, is to establish the global existence of solutions of (1.1)−(1.4), with exponential nonlinearity expressed by the condition (A2), for arbitrary v_{0} and u_{0} satisfying

max ( ∥ u_{0} ∥_{∞}, \frac{Π}{α}) < \frac{θ^{2}}{2 − θ} \frac{8ab}{rn(a − b)^{2}} ,  (1.5)

where θ < 1 is a positive real number very close to 1. For this end we use comparison principle and Lyapunov function techniques.

2. Existence of local solutions

The usual norms in spaces L^{p}(Ω), L^{∞}(Ω) and C(\overline{Ω}) are respectively denoted by

∥ u ∥_{p}^{p}= \frac{1}{| Ω |} ∫_{Ω} | u(x) |^{p} dx, ∥ u ∥_{∞}= max_{x∈Ω} | u(x) | .

Concerning a local existence, we can conclude directly from the theory of abstract semilinear equations (see A. Friedman [6], D. Henry [10], A. Pazy [15]), that for nonnegative functions u_{0} and v_{0} in L^{∞}(Ω), there exists a unique local nonnegative solution (u, v) of system (1.1) − (1.4) in C(\overline{Ω}) on ]0, T^{∗}[, where T^{∗} is the eventual blowing-up time.

3. Existence of global solutions

Using the comparison principle, one obtains

0 ≤ u(t, x) ≤ max(∥ u_{0} ∥_{∞}, \frac{Π}{α}),  (3.1)

from which it remains to establish the uniform boundedness of v.

According to the results of [8], it is enough to show that

∥ f(u, v) − σκ(v) ∥_{p}≤ C  (3.2)

(where C is a nonnegative constant independent of t) for some p > \frac{n}{2}. The main result of this paper is

Theorem 3.1. Under the assumptions (A1) − (A3) and (1.5), the solutions of (1.1) − (1.4) are global and uniformly bounded on [0, +∞[.

EJQTDE, 2009 No. 73, p. 2 Let be ω, β, γ and M positive constants such that ω ≥ 1,

β = θ \frac{4ab}{(a − b)^{2}} , γ = max (λ, μ, \frac{(β + 1)(2 − θ)Mr}{βθ(1 − θ)} )  (3.3)

and

M = max ( ∥ u_{0} ∥_{∞}, \frac{Π}{α}) < \frac{θ^{2}}{2 − θ} \frac{8ab}{rn(a − b)^{2}} .  (3.4)

We can choose

p = \frac{θ^{2}}{2 − θ} \frac{4ab}{(a − b)^{2}Mr}  (3.5)

as consequence of (3.4), we observe that p > \frac{n}{2}. The key result needed to prove the theorem 3.1 is the following Proposition 3.2. Assume that (A1) − (A3) hold and let (u, v) be a solution of (1.1) − (1.4) on ]0, T^{∗}[, with arbitrary v_{0} and u_{0} satisfying (1.5). Let

R_{ρ}(t) = ρ ∫_{Ω} udx + ∫_{Ω} (\frac{M}{(2 − θ)M − u})^{β}(v + ω)^{γp}e^{prv}dx.  (3.6)

Then, there exist p > n/2 and positive constants s and Γ such that

\frac{dR_{ρ}}{dt} ≤ −sR_{ρ} + Γ.  (3.7)

It’s very important to state a number of lemmas, before proving this proposition. Lemma 3.3. If (u, v) is a solution of (1.1) − (1.4) then

∫_{Ω} f(u, v)dx ≤ Π|Ω| − \frac{d}{dt} ∫_{Ω} u(t, x)dx.  (3.8)

Proof. We integrate both sides of (1.1),

f(u, v) = Π − αu − \frac{d}{dt}u(t, x) − aΔu

satisfied by u, which is positive and then we find (3.8). □ Lemma 3.4. Let be ψ a nonnegative function of class C(R^{+}), such that

lim_{τ→+∞} \frac{ψ(τ)}{τ + ω} = 0

and let A be positive constant. Then there exists N_{1} > 0, such that

[\frac{ψ(τ)}{τ + ω} − A](τ + ω)^{γp}e^{prτ} f(ξ, τ) ≤ N_{1}f(ξ, τ),  (3.9)

for all 0 ≤ ξ ≤ M and τ ≥ 0. Proof. Since

lim_{τ→+∞} \frac{ψ(τ)}{τ + ω} = 0,

there exists τ_{0} > 0, such that for all 0 ≤ ξ ≤ K, τ > τ_{0}, we have

[\frac{ψ(τ)}{τ + ω} − A](τ + ω)^{γp}e^{prτ} f(ξ, τ) ≤ 0.

EJQTDE, 2009 No. 73, p. 3 Now if τ is in the compact interval [0, τ_{0}], then the continuous function

χ(ξ, τ) = [ψ(τ)(τ + ω)^{γp−1} − A(τ + ω)^{γp}]e^{prτ}

is bounded. □ Lemma 3.5. For all τ ≥ 0 we have

[\frac{Πβ}{(1 − θ)M} − σpκ(τ)(\frac{γ}{τ + ω} + r)](τ + ω)^{γp}e^{prτ} ≤ −s(τ + ω)^{γp}e^{prτ} + B_{1}, (3.10)

where B_{1} and s are positive constants. Proof. Let us put

ξ = \frac{Πβ}{(1 − θ)M} + s

\begin{matrix}\frac{Πβ}{(1 − θ)M} (τ + ω)^{pγ}e^{prτ} − σpκ(τ)[γ(τ + ω)^{γp−1} + r(τ + ω)^{γp}]e^{prτ} = \\ (\frac{Πβ}{(1 − θ)M} − ξ)(τ + ω)^{pγ}e^{prτ} + (\frac{ξ}{κ(τ)} − σrp)κ(τ)(τ + ω)^{γp}e^{prτ} , \end{matrix}

then, using Lemma 3.4 we can conclude the result. □ Proof. (of Proposition 3.2) Let

g(u) = (\frac{M}{(2 − θ)M − u})^{β},

so that

R_{ρ}(t) = ρ ∫_{Ω} udx + G(t),

where

G(t) = ∫_{Ω} g(u)(v + ω)^{γp}e^{prv}dx.

Differentiating G with respect to t and a simple use of Green’s formula gives

G′(t) = I + J,

where

I = − a ∫_{Ω} g′′(u)(v + ω)^{γp}e^{prv}|∇u|^{2}dx
− (a + b) ∫_{Ω} g′(u)[γp(v + ω)^{γp−1} + pr(v + ω)^{γp}]e^{prv}∇u∇vdx

− b ∫_{Ω} g(u)[γp(γp − 1)(v + ω)^{γp−2} + 2γp^{2}r(v + ω)^{γp−1} + p^{2}r^{2}(v + ω)^{γp}]e^{prv}|∇v|^{2}dx,

J =  ∫_{Ω} Πg′(u)(v + ω)^{γp}e^{prv}dx − ∫_{Ω} αg′(u)u(v + ω)^{γp}e^{prv}dx

+ ∫_{Ω} (g(u)[γp(v + ω)^{γp−1} + rp(v + ω)^{γp}] − g′(u)(v + ω)^{γp})f(u, v)e^{prv}dx

− ∫_{Ω} σ[γp(v + ω)^{γp−1} + rp(v + ω)^{γp}]κ(v)g(u)e^{prv}dx.

EJQTDE, 2009 No. 73, p. 4 We can see that I involves a quadratic form with respect to ∇u and ∇v, which is nonnegative if

δ = (p(a + b)g′(u)[γ(v + ω)^{γp−1} + r(v + ω)^{γp}])^{2}
− 4abγp(γp − 1)g′′(u)g(u)(v + ω)^{2γp−2}
− 4abg′′(u)g(u)(v + ω)^{γp}[2γp^{2}r(v + ω)^{γp−1} + p^{2}r^{2}(v + ω)^{γp}] ≤ 0.

Indeed

\begin{aligned}δ = & [(pγ)^{2}(a + b)^{2}β^{2} − 4abβ(β + 1)pγ(pγ − 1)]\frac{g(u)^{2}(v + ω)^{2pγ−2}}{((2 − θ)M − u)^{2}} \\ & + [(a + b)^{2}β^{2} − 4abβ(β + 1)]\frac{rp^{2}g(u)^{2}(v + ω)^{2pγ−1}}{((2 − θ)M − u)^{2}} [2γ + r(v + ω)], \end{aligned}

the choice of β and γ gives

\begin{aligned}δ ≤ & [β + 1 − pγ(1 − θ)]\frac{4abβpγg(u)^{2}(v + ω)^{2pγ−2}}{((2 − θ)M − u)^{2}} \\ & + 4ab(θ − 1)\frac{rpβg(u)^{2}(v + ω)^{2pγ−1}}{((2 − θ)M − u)^{2}} [2 + (rp)(v + ω)] ≤ 0, \end{aligned}

it follows that

I ≤ 0.

Concerning the second term J, we can observe that

\begin{aligned}J ≤ & & ∫_{Ω} (\frac{Πβ}{(1 − θ)M} − σpκ(v)[\frac{γ}{v + ω} + r])g(u)(v + ω)^{pγ}e^{prv}dx \\ & + & ∫_{Ω} (p[\frac{γ}{v + ω} + r] − \frac{β}{(2 − θ)M − u})f(u, v)g(u)(v + ω)^{γp}e^{prv}dx. \end{aligned}

Using Lemma 3.5, we get

\begin{aligned}J ≤ & & ∫_{Ω}[−s(v + ω)^{pγ}e^{prv} + B_{1}]g(u)dx \\ & + & ∫_{Ω} (p[\frac{γ}{v + ω} + r] − \frac{θ}{2 − θ} \frac{4ab}{(a − b)^{2}M} )f(u, v)g(u)(v + ω)^{γp}e^{prv}dx, \end{aligned}

or

\begin{aligned}J ≤ & & ∫_{Ω}[−s(v + ω)^{pγ}e^{prv} + B_{1}]g(u)dx \\ & + & ∫_{Ω} (\frac{pγ}{v + ω} − \frac{θ(1 − θ)}{2 − θ} \frac{4ab}{(a − b)^{2}M} )f(u, v)g(u)(v + ω)^{γp}e^{prv}dx \\ & + & ∫_{Ω} (pr − \frac{θ^{2}}{2 − θ} \frac{4ab}{(a − b)^{2}M} )f(u, v)g(u)(v + ω)^{γp}e^{prv}dx. \end{aligned}

From Lemma 3.4 and formula (3.5), it follows

\begin{matrix}J ≤ & ∫_{Ω}[−s(v + ω)^{pγ}e^{prv} + B_{1}]g(u)dx \\ +N_{1} & ∫_{Ω} f(u, v)g(u)dx. \end{matrix}

In addition

g(u) ≤ (\frac{1}{1 − θ} )^{β},

then

J ≤ −sG(t)+ | Ω | B_{1}(\frac{1}{1 − θ} )^{β} + N_{1}(\frac{1}{1 − θ} )^{β} ∫_{Ω} f(u, v)dx.

Then if we put

B = B_{1} | Ω | (\frac{1}{1 − θ} )^{β}

and

ρ = N_{1}(\frac{1}{1 − θ} )^{β}.

Then, if we use Lemma 3.3,

\begin{matrix}J ≤ & − sR_{ρ}(t) + sρ ∫_{Ω} u(t, x)dx + B + ρΠ | Ω | −ρ\frac{d}{dt} ∫_{Ω} u(t, x)dx \\ ≤ & − sR_{ρ}(t) + [sM + Π]ρ | Ω | +B − ρ\frac{d}{dt} ∫_{Ω} u(t, x)dx, \end{matrix}

it follows that

\frac{dR_{ρ}}{dt} ≤ −sR_{ρ} + Γ,

where Γ = [sM + Π]ρ | Ω | +B. □

Proof. (of Theorem 3.1) Multiplying (3.7) by e^{st} and integrating the inequality, it implies the existence of a positive constant C > 0 independent of t such that

R_{ρ}(t) ≤ C.

Since

g(u) ≥ (\frac{1}{2 − θ})^{β},

\begin{matrix}∫_{Ω}(v + ω)^{γp}e^{prv}dx ≤ & (2 − θ)^{β}R_{ρ}(t) \\ ≤ & C(2 − θ)^{β}. \end{matrix}

Since ω ≥ 1 and (3.3) we have also,

∫_{Ω}(v + 1)^{λp}e^{prv}dx ≤ ∫_{Ω}(v + ω)^{γp}e^{prv}dx ≤ C(2 − θ)^{β},
∫_{Ω} v^{μp}dx ≤ ∫_{Ω}(v + ω)^{γp}dx ≤ C(2 − θ)^{β}.

We put

A = max_{0≤ξ≤M} φ(ξ),

according to (A1) − (A3), we have

∫_{Ω} f(u, v)^{p}dx ≤ ∫_{Ω} A^{p}(v + 1)^{λp}e^{prv}dx ≤ A^{p}C(2 − θ)^{β} = A^{p}H^{p},

EJQTDE, 2009 No. 73, p. 6 we conclude

∥f(u, v) − σκ(v)∥_{p} ≤ ∥f(u, v)∥_{p} + ∥σκ(v)∥_{p} ≤ H(A + σ).

By the preliminary remarks (introduction of section 3), we conclude that the solution of (1.1) − (1.4) is global and uniformly bounded on [0, +∞[×Ω.

□

Acknowledgments. I want to thank the anonymous referee for his/her fruitful comments that improved the final version of this article.

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(Received August 31, 2009)

El Hachemi Daddiouaissa Department of Mathematics University Kasdi Merbah, UKM Ouargla 30000, Algeria.

E-mail address: dmhbsdj@gmail.com

EJQTDE, 2009 No. 73, p. 7