Improved conditions for single-point blow-up in reaction-diffusion systems

TL;DR

This paper establishes new conditions under which solutions to certain reaction-diffusion systems blow up at a single point, without assuming type I blow-up or equal diffusion rates, and for a broad class of nonlinearities.

Contribution

It proves single-point blow-up for positive solutions in a ball without assuming type I blow-up or equidiffusivity, extending previous results to more general systems.

Findings

No type I blow-up assumption needed

Allows any positive diffusion coefficient delta

Handles a broad class of nonlinearities beyond pure power laws

Abstract

We study positive blowing-up solutions of the system: $$u_{t}-\delta\Delta u=v^p,\,\,\, v_{t}-\Delta v=u^{q},$$ as well as of some more general systems. For any $p,\,q>1$, we prove single-point blow-up for any radially decreasing, positive and classical solution in a ball. This improves on previously known results in 3 directions: (i) no type I blow-up assumption is made (and it is known that this property may fail); (ii) no equidiffusivity is assumed, i.e. any $\delta>0$ is allowed; (iii) a large class of nonlinearities $F(u,v)$, $G(u,v)$ can be handled, which need not follow a precise power behavior. As side result, we also obtain lower pointwise estimates for the final blow-up profiles.