## Journal of the Mathematical Society of Japan

### Optimal condition for non-simultaneous blow-up in a reaction-diffusion system

#### Abstract

We study the positive blowing-up solutions of the semilinear parabolic system: $u_{t}-\Delta u=v^{p}+u^{r},$ $v_{t}-\Delta v=u^{q}+v^{s}$, where $t\in(0,T),$ $x\in R^{N}$ and $p,$ $q,$ $r,$ $s>1$. We prove that if $r>q+1$ or $s>p+1$ then one component of a blowing-up solution may stay bounded until the blow-up time, while if $r<q+1$ and $s<p+1$ this cannot happen. We also investigate the blow up rates of a class of positive radial solutions. We prove that in some range of the parameters $p,$ $q,$ $r$ and $s$, solutions of the system have an uncoupled blow-up asymptotic behavior, while in another range they have a coupled blow-up behavior.

#### Article information

Source
J. Math. Soc. Japan Volume 56, Number 2 (2004), 571-584.

Dates
First available in Project Euclid: 3 October 2007

Permanent link to this document
http://projecteuclid.org/euclid.jmsj/1191418646

Digital Object Identifier
doi:10.2969/jmsj/1191418646

Mathematical Reviews number (MathSciNet)
MR2048475

Zentralblatt MATH identifier
1059.35049

#### Citation

SOUPLET, Philippe; TAYACHI, Slim. Optimal condition for non-simultaneous blow-up in a reaction-diffusion system. J. Math. Soc. Japan 56 (2004), no. 2, 571--584. doi:10.2969/jmsj/1191418646. http://projecteuclid.org/euclid.jmsj/1191418646.