A+ CATEGORY SCIENTIFIC UNIT

Blow up for a completely coupled Fujita type reaction-diffusion system

Volume 92 / 2002

Noureddine Igbida, Mokhtar Kirane Colloquium Mathematicum 92 (2002), 87-96 MSC: Primary 35K57, 35B33. DOI: 10.4064/cm92-1-8

Abstract

This paper provides blow up results of Fujita type for a reaction-diffusion system of 3 equations in the form $u_t -{\mit \Delta } (a_{11}u)=h(t,x)| v | ^p,$ $v_t -{\mit \Delta } (a_{21}u)-{\mit \Delta } (a_{22}v)=k(t,x)| w | ^q,$ $w_t -{\mit \Delta } (a_{31}u)-{\mit \Delta }(a_{32}v) -{\mit \Delta } (a_{33}w)=l(t,x)| u | ^r, $ for $x\in {\mathbb R}^N$, $t>0$, $p>0$, $q>0,$ $r>0$, $a_{ij}=a_{ij}(t,x,u,v)$, under initial conditions $u(0,x)= u_{0}(x), v(0,x)= v_{0}(x), w(0,x)= w_{0}(x)$ for $x\in {\mathbb R}^N$, where $u_{0}, v_{0}, w_{0}$ are nonnegative, continuous and bounded functions. Subject to conditions on dependence on the parameters $p, q, r, N$ and the growth of the functions $h, k, l$ at infinity, we prove finite blow up time for every solution of the above system, generalizing results of H. Fujita for the scalar Cauchy problem, of M. Escobedo and M. A. Herrero, of Fila, Levine and Uda, and of J. Renc/lawowicz for systems.

Authors

  • Noureddine IgbidaFaculté de Mathématiques et d'Informatique
    LAMFA, CNRS FRE 2270
    Université de Picardie–Jules Verne
    33 rue Saint Leu
    80038 Amiens, France
    e-mail
  • Mokhtar KiraneFaculté de Mathématiques et d'Informatique
    LAMFA, CNRS FRE 2270
    Université de Picardie–Jules Verne
    33 rue Saint Leu
    80038 Amiens, France
    and
    Laboratoire de Mathématiques
    Pôle Sciences et Technologies
    Université de La Rochelle
    Avenue Michel Crépeau
    17042 La Rochelle Cedex, France
    e-mail

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