Zeitschrift für angewandte Mathematik und Physik ZAMP

, Volume 53, Issue 2, pp 197-210

First online:

Critical blowup exponents for a system of reaction-diffusion equations with absorption

  • N. BedjaouiAffiliated withDépartement de Mathématiques, INSSET, Université de Picardie, 02109 St-Quentin, France, and Centre de Mathématiques Appliquées UMR CNRS 7641, Ecole Polytechnique, 91128 Paliaseau, France, e-mail: bedjaoui@cmapx.polytechnique.fr
  • , P. SoupletAffiliated withDépartement de Mathématiques, INSSET, Université de Picardie, 02109 St-Quentin, France, and Laboratoire de Mathématiques, UMR CNRS 7641 Université de Versailles, 45 avenue des Etats-Unis, 78035 Versailles, Fance, e-mail: souplet@math.uvsq.fr

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Abstract.

We consider a system of two reaction-diffusion equations with absorption, under Dirichlet boundary conditions, modelling the cooperative interaction of two diffusing biological species. We obtain optimal conditions on the exponents of the reaction and absorption terms for the existence of blowing up solutions, or for the global existence and boundedness of all solutions. The proofs of blowup rely on comparison with carefully constructed self-similar weak subsolutions with compact support.

Key words. Parabolic systems, blowup, global existence, critical exponents, reaction-diffusion, absorption.