Single-point blow-up for parabolic systems with exponential nonlinearities and unequal diffusivities

TL;DR

This paper establishes conditions under which solutions to certain parabolic systems with exponential nonlinearities and unequal diffusivities blow up at a single point, extending previous results that only applied to equal diffusivities.

Contribution

It proves single-point blow-up for a broader class of solutions with unequal diffusivities, answering an open question from prior research.

Findings

Proves single-point blow-up for systems with unequal diffusivities.

Extends previous results limited to equal diffusivities.

Addresses an open problem from Friedman and Giga (1987).

Abstract

We study positive blowing-up solutions of systems of the form: $$u_t=\delta_1 \Delta u+e^{pv},\quad v_t= \delta_2\Delta v+e^{qu},$$ with $\delta_1,\delta_2>0$ and $p, q>0$. We prove single-point blow-up for large classes of radially decreasing solutions. This answers a question left open in a paper of Friedman and Giga~(1987), where the result was obtained only for the equidiffusive case $\delta_1=\delta_2$ and the proof depended crucially on this assumption.