UNIFORM BOUNDEDNESS AND GLOBAL EXISTENCE OF SOLUTIONS FOR REACTION-DIFFUSION SYSTEMS WITH A BALANCE LAW AND A FULL MATRIX OF DIFFUSION COEFFICIENTS.

The purpose of this paper is to prove uniform boundedness and so global existence of solutions for reaction-diusion systems with a full matrix of diusion coecien ts satisfying a balance law. Our techniques are based on invariant regions and Lyapunov functional methods. The nonlinearity of the reaction term which we take positive in an invariant region has been supposed to be polynomial..


INTRODUCTION
We consider the following reaction-diffusion system ∂u ∂t -a∆u-b∆v = -σf (u, v) in R + × Ω (1.1) with the boundary conditions ∂u ∂η = ∂v ∂η = 0 on R + × ∂Ω, (1.3) and the initial data u(0, x) = u 0 (x), v(0, x) = v 0 (x) in Ω, (1.4) where Ω is an open bounded domain of class C 1 in R n , with boundary ∂Ω, and ∂ ∂η denotes the outward normal derivative on ∂Ω, σ, ρ, a, b and c are positive constants satisfying the condition 2a > (b + c) which reflects the parabolicity of the system.The initial data are assumed to be in the following region The function f (r, s) is continuously differentiable, nonnegative on Σ with (1.6) The system (1.1)-(1.2) may be regarded as a perturbation of the simple and trivial case where b = c = 0; for which nonnegative solutions exist globally in time.
When the coefficient of −∆u in equation (1.1) is different from the one of −∆v in equation (1.2), N. Alikakos [1] established global existence and L ∞ -bounds of solutions for positive initial data for f (u, v) = uv β and 1 < β < (n+2) n and K. Masuda [16] showed that solutions to this system exist globally for every β > 1 and converge to a constant vector as t → +∞. A. Haraux and A. Youkana [6] have generalized the method of K.Masuda to handle nonlinearities uF (v) that are form a particular case of ours; since the hypothesis (1.5) is replaced automatically by f (0, s) = 0 for any s ≥ 0 .Recently S. Kouachi and A. Youkana [14] have generalized the method of A. Haraux and A. Youkana to the case c > 0 and the limit (1.6) is a small number strictly positive , hypothesis that is in fact, weaker than the last one.

Invariant regions.
Proposition 1. Suppose that the function f is nonnegative on the region Σ and that the conditions (1.5) and (1.6) are satisfied, then for any (u 0 , v 0 ) in Σ the solution (u(t, .),v(t, .)) of the problem (1.1)-(1.4)remains in Σ for any time and there exists a positive constant M such that ( and the initial data where, for any (t, x) in ]0, T * [ × Ω, and First, let's notice that the condition of parabolicity of the system (1.1)-(1.2) implies the one of the (2.2)-(2.3)system; since 2a > (b + c) ⇒ a − √ bc > 0. Now, it suffices to prove that the region = 0 for all z ≥ 0 and all v ≥ 0, then w(t, x) ≥ 0 for all (t, x) ∈ ]0, T * [ × Ω, thanks to the invariant region's method ( see Smoller [19] ) and because F (w, z) ≥ 0 for all (w, z) in IR + × IR + and z 0 (x) ≥ 0 in Ω, we can deduce by the same method applied to equation (2.3), that then Σ is an invariant region for the system (1.1)-(1.3).At the end, to show that w(t, x) is uniformly bounded on ]0, T * [ × Ω, it is sufficient to apply the maximum's principle directly to equation (2.2).
For the case b c > σ ρ , the same reasoning with equations with the same boundary condition (2.4) implies the invariance of IR + × IR + and the uniform boundeness of w(t, x) on ]0, T * [ × Ω, where in this case we take Once, invariant regions are constructed, one can apply Lyapunov technique and establish global existence of unique solutions for (1.1)-(1.4).

Global existence.
As the determinant of the linear algebraic system (2.6) or (2.6) , with regard to variables u and v, is different from zero, then to prove global existence of solutions of problem (1.1)-(1.4)comes back in even to prove it for problem (2.2)-(2.5).To this subject, it is well known that (see Henry [7]) it suffices to derive an uniform estimate of F (w, z) p on [0, T * [ for some p > n/2.
The main result and in some sense the heart of the paper is: ) and w(t, x) and z(t, x) are given by (2.6) (respectively (2.6) ).
By simple use of Green's formula, we get where The discriminant of T is given by: EJQTDE, 2001 No. 7, p. 5 Theses two last inequalities can be written as follows: Using the following inequality where ξ and ρ are two negative constants and σ ∈ R, we can show that where the positive constants m 1 and m 2 are given by: where C(β, γ, λ, µ, M ) is a positive constant.Hence, EJQTDE, 2001 No. 7, p. 6 , the same reasoning with w and z given by (2.6) and using the limit (1.6) we deduce the same result.

3. Remarks and comments Remark 1 . 2 1 + C 2 2 = 0, α 1 > 1 and α 2 > 1 . 3 .
In the case when b c = global existence of solutions of problem (1.1)-(1.4)with no condition on the constants or on the growth of the function f to part its positivity and f (r, c b r) = 0, for all r ≥ 0. To verify this, it suffices to apply the maximum principle directly to equations (2.2) −(2.3) .Remark 2. By application of the comparison's principle to equation (2.3), blowingup in finite time can occur in the case where b c = σ ρ , especially when the reaction term satifies an inequality of the form:|f (u, v)| ≥ C 1 |u| α1 + C 2 |v| α2 ,where C 1 , C 2 , α 1 and α 2 are positive constants such that C Remark One showed the global existence for functions f (u, v) of polynomial growth (condition 1.6), but our results remain valid for functions of exponential growth (but small) while replacing the condition 1.6 by: Suppose that the function f (r, s) is continuously differentiable, nonnegative on Σ and satisfying conditions (1.5) and(1.6).Then all solutions of (1.1)-(1.3)with initial data in Σ are global in time and uniformly bounded on (0, +∞)×Ω.derive an uniform estimate of F (w, z) p on [0, T * [ for some p > n/2.Since, for u and v in Σ, w ≥ 0 and z ≥ 0, and as w + z = 2 √ bv with w uniformly bounded on [0, T * [ × Ω by M , then (1.6) is equivalent to By the preliminary remarks, we conclude that the solution is global and uniformly bounded on [0, +∞[ × Ω.