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

Abstract

We study positive blowing-up solutions of systems of the form: $$u_t={\delta }_1\Delta u+e^{pv},v_t={\delta }_2\Delta v+e^{qu},$$ 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.

arXiv admin note: text overlap with arXiv:1501.05112