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

Noureddine Igbida ¹   ; Mokhtar Kirane ²

¹ Faculté de Mathématiques et d'Informatique LAMFA, CNRS FRE 2270 Université de Picardie–Jules Verne 33 rue Saint Leu 80038 Amiens, France
² Faculté 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

Colloquium Mathematicum, Tome 92 (2002) no. 1, pp. 87-96

Résumé

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.

Détail

DOI : 10.4064/cm92-1-8

Keywords: paper provides blow results fujita type reaction diffusion system equations form mit delta mit delta mit delta mit delta mit delta mit delta mathbb v under initial conditions mathbb where nonnegative continuous bounded functions subject conditions dependence parameters growth functions infinity prove finite blow time every solution above system generalizing results fujita scalar cauchy problem escobedo herrero fila levine uda renc lawowicz systems