Invariant regions and global existence of solutions for reaction-diffusion systems with a general full matrix

K Boukerrioua

doi:10.1186/1029-242x-2014-24

Abstract

The aim of this study is to prove the global existence in time of solutions for reaction-diffusion systems. We make use of the appropriate techniques which are based on invariant regions and Lyapunov functional methods. We consider a full matrix of diffusion coefficients and we show the global existence of the solutions. MSC:35K45, 35K57.

Highlights

1 Introduction We are mainly interested in the global existence in time of solutions to a reaction-diffusion system of the form
To prove that is an invariant region for system ( . )-( . ) it suffices to prove that the region
We conclude by noting that the study of the global existence of strongly coupled systems has been a major development, and several articles are devoted to this subject

Summary

Introduction

∂ ∂η denotes the outward normal derivative on ∂ , denotes the Laplacian operator with respect to the x variable, a, b, c, d, σ are positive constants satisfying the condition (b + c) < ad, which reflects the parabolicity of the system and implies at the same time that the matrix of diffusion is positive definite, ≥. Melkemi et al [ ] established the existence of global solutions (eventually uniformly bounded in time) using a novel approach that involved the use of a Lyapunov function for system We note that the condition of parabolicity implies that det(A) = ad – bc > , where A is the matrix of diffusion. ) follow from the basic existence theory for abstract semilinear differential equations (see Henry [ ] and Pazy [ ]).

Multiplying first through by

Then for η