Electronic Journal of Qualitative Theory of Differential Equations 2005, No. 24, 1-10; http://www.math.u-szeged.hu/ejqtde/

On the uniform boundedness of the solutions of systems of reaction-diffusion equations

Lamine MELKEMI, Ahmed Zerrouk MOKRANE, and Amar YOUKANA

Abstract. We consider a system of reaction-diffusion equations for which the uniform boundedness of the solutions can not be derived by existing methods. The system may represent, in particular, an epidemic model describing the spread of an infection disease within a population. We present an L^{p} argument allowing to establish the global existence and the uniform boundedness of the solutions of the considered system.

1. INTRODUCTION

In this work we consider the following class of reaction diffusion systems

(1)  \{ \begin{matrix}\frac{∂u}{∂t} − d_{1}Δu = c − f(u, v) − αu in ℝ^{+} × Ω \\ \frac{∂v}{∂t} − d_{2}Δv = g(u, v) − σv  in ℝ^{+} × Ω \end{matrix}

with Neuman boundary conditions

(2)  \frac{∂u}{∂ν} = \frac{∂v}{∂ν} = 0 on ℝ^{+} × Γ

where Ω is an open, bounded domain in ℝ^{n} with boundary Γ = ∂Ω of class C^{1}, d_{1}, d_{2}, α, σ are positive constants, c ≥ 0 and f, g are nonnegative functions of class C^{1}(ℝ^{+} × ℝ^{+}) satisfying the following assumptions

(A1)  f(0, .) = 0, g(., 0) ≥ 0,
(A2)  ξ ≥ 0, η ≥ 0 ⟹ 0 ≤ f(ξ, η) ≤ (1 + η)^{β}φ(ξ)

and

(A3)  (∀ξ, η ≥ 0) g(ξ, η) ≤ ψ(η)f(ξ, η) with lim_{η→+∞} \frac{ψ(η)}{η} = 0

where β ≥ 1 and φ, ψ are nonnegative functions of class C(ℝ^{+}). EJQTDE, 2005 No. 24, p. 1

2000 Mathematics Subject Classification. 35K57, 35K45.

Key words and phrases. Reaction-Diffusion, Uniform Boundedness, Lyapunov function.

We are interested in the problem of the existence of global solutions to the system (1)-(2) in the class C(Ω̄) with initial data as well as their uniform boundedness

(3)  u(0, x) = u_{0}, v(0, x) = v_{0}

with u_{0}, v_{0} ∈ L^{∞}(Ω) and u_{0} ≥ 0, v_{0} ≥ 0.

We observe that the system (1)-(3) generalizes that when g = f since the assumption (A3) obviously holds in this case. In practice, when g = f , the system (1)-(3) may represent an epidemic model describing the spread of an infection disease within a population in which case c is the recruitment ratio into the susceptible class, α is the natural death rate and \frac{1}{σ} denotes the average infectious period (for details see for example [12], [3], [6],[8]). When c = 0 and g(u, v) = f(u, v) + αu + σv we get a system of reaction-diffusion equations with balance law initially posed by R. H. Martin for which the problem of the existence of global solutions and their uniform boundedness was studied by many authors. The question was positively answered by Alikakos [1] and Masuda [10] under the assumption (A2) and later on by Haraux and Youkana [4], Youkana [13], Barabanova [2] and Kanel and Kirane [9] when the vector field may be a function with exponential growth.

When f = g and c > 0, it is clear that the problem of the uniform boundedness of the solutions of (1)-(3) is not as immediate as the global existence. In [7], Hollis, Martin and Pierre used duality arguments on L^{p} techniques allowing under certain conditions to derive the uniform boundedness of solutions of (1) with mixed boundary conditions from uniform estimates. However we do not see how the method in [7] may be applied to the problem (1)-(3). In order to establish the uniform boundedness of the solutions of (1)-(3) in this case (that is when g = f and c > 0) one may propose the method developed by Morgan in [11]. Indeed the method in [11] assures the uniform boundedness of the solutions of (1)-(3) once it is verified that

(i) The vector field of (1) (i.e. the right hand side in (1)) is polynomial in v )

(ii) c − f(u, v) − αu,

(4)  λ(c − f(u, v) − αu) + μ(g(u, v) − σv)

EJQTDE, 2005 No. 24, p. 2 for some positive constants λ, μ > 0 are uniformly upper bounded and

(iii) ∫_{Ω}(u + v) ≤ C independent of t.

Using standard manipulations, one can check that concerning the system (1)-(3) considered here, the conditions (i)-(iii) are satisfied provided that ψ is bounded. Unfortunately, when ψ is not bounded, the condition (ii) is not satisfied.

In this paper our main concern is to establish the global existence and the uniform boundedness of the solutions of the system (1)-(3) under the assumptions (A1)-(A3). To this end we make use of the Lyapunov function techniques and present an approach similar to that developed by Haraux and Youkana [4] and Barabanova [2].

2. STATEMENT AND PROOF OF THE MAIN RESULT

It is classical that for nonnegative u_{0}, v_{0} ∈ L^{∞}(Ω) there exists a unique local nonnegative solution (u, v) of class C(Ω̄) of (1)-(3) on ]0, T^{∗}[, where T^{∗} is the eventual blowing-up time in L^{∞}(Ω) (see [5]). By the comparison principle one may also show that

(5)  0 ≤ u(t, x) ≤ max(||u_{0}||_{∞}, \frac{c}{α}) =: K.

As a consequence our problem amounts to establish the uniform boundedness of v. To do so, we will make use of the result established in [5] from which the uniform boundedness of v is derived once

||g(u, v) − σv||_{p} ≤ C

(where C is a nonnegative constant independent of t) for some p > \frac{n}{2}. Because of the assumptions (A2) and (A3), we are led to establish the uniform boundedness of the ||v||_{p} on ]0, T^{∗}[ in order to get that of ||v||_{∞} on ]0, T^{∗}[.

For p ≥ 2, we let

γ = \frac{(d_{1} − d_{2})^{2}}{4d_{1}d_{2}} , γ(p) = \frac{pγ + 1}{p − 1}, M_{p} = K + \frac{c}{γ(p)σ} .

The main result of this paper is stated in what follows:

Proposition 1. Assume p ≥ 2 and let

(6)  G_{b}(t) = ∫_{Ω}[bu + exp(−\frac{p − 1}{pγ + 1} ln(γ(p)[M_{p} − u]))v^{p}]dt

where (u, v) is the solution of (1)-(3) on ]0, T^{∗}[. Then under the assumptions (A1)-(A3) there exist two positive constants a > 0 and b > 0 such that

(7)  \frac{d}{dt}G_{b} ≤ −(p − 1)σG_{b} + a .

Before proving this theorem we first need the following two technical lemmas:

Lemma 1. Let (u, v) be a solution of (1)-(3). Then

(8)  \frac{d}{dt} ∫_{Ω} udx + ∫_{Ω} f(u, v)dx + α ∫_{Ω} udx = c|Ω|.

Proof. We integrate the both sides of the equation

\frac{du}{dt} − d_{1}Δu = c − f(u, v) − αu

satisfied by u on Ω. We find

∫_{Ω} f(u, v)dx = c|Ω| − α ∫_{Ω} u − \frac{d}{dt} ∫_{Ω} u(t, x)dx.

Lemma 2. Assume that p ≥ 2. Then under the assumptions (A1)-(A3) there exists b_{1} > 0 such that

(9)  [pη^{p−1}g(ξ, η) − \frac{1}{γ(p)M_{p}} η^{p}f(ξ, η)] ≤ b_{1}f(ξ, η)

for all 0 ≤ ξ ≤ K and η ≥ 0.

Proof. According to the assumption (A3), we have

[pη^{p−1}g(ξ, η) − \frac{1}{γ(p)M_{p}} η^{p}f(ξ, η)] ≤ [p\frac{ψ(η)}{η} − \frac{1}{γ(p)M_{p}}]η^{p}f(ξ, η).

Since \frac{ψ(η)}{η} goes to 0 as η → +∞, there exists η_{0} > 0 such that

(0 ≤ ξ ≤ K, η ≥ η_{0}) ⟹ [p\frac{ψ(η)}{η} − \frac{1}{γ(p)M_{p}}]η^{p}f(ξ, η) ≤ 0

EJQTDE, 2005 No. 24, p. 4 On the other hand

(ξ, η) ⟼ pη^{p−1}ψ(η) − \frac{1}{γ(p)M_{p}} η^{p}

being continuous is bounded on the compact interval [0, η_{0}] so that (9) immediately follows.

We proceed now to the proof of Proposition 1

Proof of Proposition 1. Let

h(u) = −\frac{p − 1}{pγ + 1} ln(γ(p)(M_{p} − u))

so that

G_{b}(t) = b ∫_{Ω} udx + L(t)

where

L(t) = ∫_{Ω} e^{h(u)}v^{p}dx.

Differentiating L(t) with respect to t and using the Green formula one obtains

\frac{d}{dt}L = H + S

where

H = −d_{1} ∫_{Ω}(h^{′2}(u) + h′′(u))e^{h(u)}v^{p}(∇u)^{2}dx
−p(d_{1} + d_{2}) ∫_{Ω} h′(u)e^{h(u)}v^{p−1}∇u∇vdx
−d_{2} ∫_{Ω} p(p − 1)e^{h(u)}v^{p−2}(∇v)^{2}dx

and

S = c ∫_{Ω} h′(u)e^{h(u)}v^{p}dx + ∫_{Ω}[pv^{p−1}g(u, v) − h′(u)v^{p}f(u, v)]e^{h(u)}dx
−α ∫_{Ω} h′(u)ue^{h(u)}v^{p}dx − σ ∫_{Ω} pe^{h(u)}v^{p}dx.

We observe that H involves a quadratic form with respect to ∇u and ∇v

Q = d_{1}(h^{′2}(u)+h′′(u))v^{p}(∇u)^{2}+p(d_{1}+d_{2})h′(u)v^{p−1}∇u∇v+d_{2}p(p−1)v^{p−2}(∇v)^{2} EJQTDE, 2005 No. 24, p. 5 which is nonnegative if

(10) [p(d_{1} + d_{2})h′(u)v^{p−1}]^{2} − 4d_{1}d_{2}p(p − 1)(h^{′2}(u) + h′′(u))v^{2p−2} ≤ 0.

We have chosen h(u) in such a way that

h′(u) = \frac{1}{γ(p)(M_{p} − u)}, h′′(u) = \frac{γ(p)}{[γ(p)(M_{p} − u)]^{2}} .

The left hand side of (10) can be written

v^{2p−2}{p^{2}[(d_{1} − d_{2})^{2}h^{′2}(u) − 4d_{1}d_{2}h′′(u)] + 4d_{1}d_{2}p(h^{′2}(u) + h′′(u))}

= 4d_{1}d_{2}pv^{2p−2}\{p[γ \frac{1}{[γ(p)(M_{p} − u)]^{2}} − \frac{γ(p)}{[γ(p)(M_{p} − u)]^{2}} ]

+\frac{1 + γ(p)}{[γ(p)(M_{p} − u)]^{2}} \}

= 0

since pγ−pγ(p)+1+γ(p) = 0. Therefore (10) holds, Q ≥ 0 and consequently

H = − ∫_{Ω} Qe^{h(u)}dx ≤ 0.

Concerning the second term S we observe that

S ≤ ∫_{Ω}(ch′(u) − σp)e^{h(u)}v^{p}dx + ∫_{Ω}[pv^{p−1}g(u, v) − h′(u)v^{p}f(u, v)]e^{h(u)}dx
≤ −(p − 1)σ ∫_{Ω} e^{h(u)}v^{p}dx + ∫_{Ω}[pv^{p−1}g(u, v) − h′(u)v^{p}f(u, v)]e^{h(u)}dx

since

h′(u) = \frac{1}{γ(p)(M_{p} − u)} ≤ \frac{1}{γ(p)(M_{p} − K)} = \frac{σ}{c}

On the other hand

−h′(u) = −\frac{1}{γ(p)(M_{p} − u)} ≤ −\frac{1}{γ(p)M_{p}}
h(u) ≤ −\frac{1}{γ(p)} ln \frac{c}{σ} .

EJQTDE, 2005 No. 24, p. 6 Therefore by virtue of Lemma 2 (in particular (9)) and the fact that v ≥ 0

pv^{p−1}g(u, v) − h′(u)v^{p}f(u, v) ≤ pv^{p−1}g(u, v) − \frac{1}{γ(p)M_{p}} v^{p}f(u, v)
≤ b_{1}f(u, v).

As a consequence

S ≤ −(p − 1)σL + b_{1} ∫_{Ω} e^{h(u)}f(u, v)dx
≤ −(p − 1)σL + b_{1}e^{− \frac{1}{γ(p)} ln \frac{c}{σ}} ∫_{Ω} f(u, v)dx.

Let

b = b_{1}e^{− \frac{1}{γ(p)} ln \frac{c}{σ}}

where b_{1} > 0 is a positive constant satisfying (9). Using (8) one obtains

S ≤ −(p − 1)σL + b(c|Ω| − \frac{d}{dt} ∫_{Ω} u(t, x)dx)
≤ −(p − 1)σL + bc|Ω| − b\frac{d}{dt} ∫_{Ω} u(t, x)dx.

Since G_{b}(t) = b ∫_{Ω} udx + L(t), it follows that

S ≤ −(p − 1)σG_{b} + (p − 1)σb ∫_{Ω} udx + bc|Ω| − b\frac{d}{dt} ∫_{Ω} u(t, x)dx
≤ −(p − 1)σG_{b} + b((p − 1)σK + c)|Ω| − b\frac{d}{dt} ∫_{Ω} u(t, x)dx

from which we conclude that

\frac{d}{dt}G_{b}(t) ≤ −(p − 1)σG_{b}(t) + a

with a = b((p − 1)σK + c)|Ω|.

We are now ready to establish the global existence and uniform boundedness of the solutions of (1)-(3)

Theorem 1. Under the assumptions (A1)-(A3), the solutions of (1)-(3) are global and uniformly bounded on [0, +∞[.

EJQTDE, 2005 No. 24, p. 7

Proof. By a multiplication of the inequality (7) by e^{(p−1)σt} and then integrating, we deduce that there exists a positive constant C_{1} > 0 independent of t such that:

G_{b}(t) ≤ C_{1}.

Since

e^{h(u)} ≥ e^{− \frac{1}{γ(p)} ln γ(p)M_{p}}

it follows that for all p ≥ 2

∫_{Ω} v^{p}dx ≤ e^{\frac{1}{γ(p)} ln[Kγ(p)+ \frac{c}{σ}]}G_{b}(t)
≤ C_{1}.[Kγ(p) + \frac{c}{σ}]^{\frac{1}{γ(p)}} =: C(p).

Select now p > \frac{n}{2} and proceed to bound ||g(u, v) − σv||_{p}. Beforehand let

A = max_{0≤η≤η_{0}} ψ(η), B = max_{0≤ξ≤K} φ(ξ)

where η_{0} is such that

η ≥ η_{0} ⟹ ψ(η) ≤ η

According to (A2)-(A3), we have

g(u, v) ≤ ψ(v)f(u, v)
≤ ψ(v)φ(u)(1 + v)^{β}
≤ Bψ(v)(1 + v)^{β}

since 0 ≤ u ≤ K. Therefore

∫_{Ω} g(u, v)^{p} ≤ B^{p} ∫_{Ω} ψ(v)^{p}(1 + v)^{βp}
= B^{p}(∫_{v≤η_{0}} ψ(v)^{p}(1 + v)^{βp} + ∫_{v≥η_{0}} ψ(v)^{p}(1 + v)^{βp})

as a consequence

∫_{Ω} g(u, v)^{p} ≤ (AB(1 + η_{0})^{β})^{p}|Ω| + B^{p} ∫_{Ω} v^{p}(1 + v)^{βp}.

Using the fact that for x, y ≥ 0 and r ≥ 1

(x + y)^{r} ≤ 2^{r−1}(x^{r} + y^{r})

EJQTDE, 2005 No. 24, p. 8 we obtain

∫_{Ω} g(u, v)^{p} ≤ (AB(1 + η_{0})^{β})^{p}|Ω| + 2^{βp−1}B^{p}(C(p) + C((β + 1)p))
= : R^{p}.

Finally

||g(u, v) − σv||_{p} ≤ ||g(u, v)||_{p} + σ||v||_{p}
≤ R + σ\sqrt[p]{C(p)}.

Using the result of [5, p.39, Thm 1.6.1] we conclude that the solutions of (1)-(3) are indeed global and uniformly bounded on ]0, +∞[×Ω.

Acknowledgement 1. We have benefited from helpful discussions with Professor M.Ziane of South California University. We gratefully thank him for his advises. We also thank the anonymous referee for his valuable comments.

References

[1] N. Alikakos, L^{p}-Bounds of reaction-diffusion equations, Comm. P.D.E 4(1979). 827-828

[2] A. Barabanova, On the global existence of solutions of a reaction-diffusion equation with exponential nonlinearity, Proc. A.M.S vol.122 no3 (1992). 827-831

[3] C. Castillo-Chavez, K. Cook, W. Huang and S. A. Levin, On the role of long incubation periods in the dynamics of acquires immunodeficiency syndrome (AIDS), J. Math. Biol. 27(1989) 373-398

[4] A. Haraux and A. Youkana., On a result of K. Masuda concerning reaction-diffusion equations, Töhoku Math. J. 40(1988) 159-163

[5] D. Henry, Geometric theory of semilinear parabolic equations, Lecture notes in mathematics 840, Springer Verlag, New York (1981)

[6] Y. Hamaya, On the asymptotic behavior of a diffusive epidemic model (AIDS), Nonlinear Analysis 36(1999) 685-696.

[7] S. I. Hollis, R. H. Martin and M. Pierre, Global existence and boundedness in reactiondiffusion systems, SIAM J. Math. Anal. vol. 18 no3 (1987) 744-761

[8] W. E. Fitzgibbon, M. Langlais, F. Marpeau and J. J. Morgan, Modeling the circulation of a disease between two host populations on non coincident spatial domains, to appear in J. Biological Invasions.

[9] J. I. Kanel and M. Kirane, Global solutions of reaction-diffusion systems with a balance law and nonlinearities of exponential growth, J. Diff. Equations 165 no1 (2000) 24-41

[10] K. Masuda, On the global existence and asymptotic behavior of solutions of reaction-

diffusion equations, Hokkaido Math. J. 12(1983) 360-370

EJQTDE, 2005 No. 24, p. 9

[11] J. Morgan, Boundedness and decay results for reaction-diffusion systems, SIAM J. Math. Anal. 21(1990) 1172-1189

[12] G. F. Webb, A reaction diffusion model for a deterministic diffusive epidemic, J. Math. Anal. Appl.84(1981) 150-161

[13] A.Youkana, Thèse de 3^{ème} cycle, Chapitre 3, Université de Paris 6.(1986)

(Received July 1, 2005)

Department of Mathematics, Faculty of Science, University of Batna

E-mail address: lmekmi@yahoo.fr

Department of Mathematics, Faculty of Science, University of Batna

E-mail address: ahmed mokr@hotmail.com

Department of Mathematics, Faculty of Science, University of Batna

E-mail address: youkana amar@yahoo.fr

EJQTDE, 2005 No. 24, p. 10