Global solution of reaction diffusion system with non diagonal matrix

Abstract

Abstract The purpose of this paper is to prove the global existence in time of solutions for the coupled reaction-diffusion system: { ∂ u ∂ t − a Δ u − b Δ v = f ( u , v ) in ] 0 , + ∞ [ × Ω ∂ u ∂ t − c Δ v = g ( u , v ) in ] 0 , + ∞ [ × Ω * * 487 * * $$\left\{ {\matrix{{{{\partial u} \over {\partial t}} - a\Delta u - b\Delta v = f\,(u,\,v)} \hfill & {{\rm in}\,]0,\, + \infty [ \times \Omega } \hfill \cr {{{\partial u} \over {\partial t}} - c\Delta v = g(u,v)} \hfill & {{\rm in}\,]0,\, + \infty [ \times \Omega } \hfill } } \right.$$ with triangular matrix of diffusion coefficients. By combining the Lyapunov functional method with the regularizing effect, we show that global solutions exist. Our investigation applied for a wide class of the nonlinear terms f and g.