Blow up for a completely coupled Fujita type reaction-diffusion system
Volume 92 / 2002
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.
