Abstract
In this paper we generalize a result obtained in (15) concerning uniform boundedness and so global existence of solutions for reaction-diusion systems with a general 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 or of weak exponential growth..
Highlights
We consider the following reaction-diffusion system (1.1) ∂u ∂t − a∆u b∆v =−σf (u, v) in R+ × Ω, (1.2)∂v ∂t c∆u − d∆v ρf (u, v) with the boundary conditions in R+ × Ω,(1.3) and the initial data ∂u ∂η∂v ∂η on R+ × ∂Ω
The constants a, b, c and d are supposed to be positive and b + c)2 ≤ 4ad which reflects the parabolicity of the system and implies at the same time that the matrix of diffusion
The initial data are assumed to be in the following region
Summary
U(0, x) = u0(x), v(0, x) = v0(x) in Ω, where Ω is an open bounded domain of class C1 in Rn, with boundary ∂Ω, and. ∂Ω, σ and ρ are positive constants. The constants a, b, c and d are supposed to be positive and b + c)2 ≤ 4ad which reflects the parabolicity of the system and implies at the same time that the matrix of diffusion. 1 is positive definite; that is the eigenvalues λ1 and λ2 (λ1 < λ2) of its transposed are positive. The initial data are assumed to be in the following region
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