Global and blow‐up solutions for non‐linear degenerate parabolic systems

Abstract

AbstractIn this paper the degenerate parabolic system ut=u(uxx+av). vt=v(vxx+bu) with Dirichlet boundary condition is studied. For $a. b {<} \lambda_{1} (\sqrt {ab} {<} \lambda_{1} {\rm if}\, \alpha_{1} {\neq} \alpha_{2})$, the global existence and the asymptotic behaviour (α1=α2) of solution are analysed. For $a. b \,{>}\, \lambda_{1}\ (\sqrt{ab} {>} \lambda_{1} {\rm if}\, \alpha_{1} {\neq} \alpha_{2})$, the blow‐up time, blow‐up rate and blow‐up set of blow‐up solution are estimated and the asymptotic behaviour of solution near the blow‐up time is discussed by using the ‘energy’ method. Copyright © 2003 John Wiley & Sons, Ltd.