Self-similar solutions preventing finite time blow-up for reaction-diffusion equations with singular potential

Abstract

We prove existence and uniqueness of a global in time self-similar solution growing up as $t\to\infty$ for the following reaction-diffusion equation with a singular potential $$ u_t=\Delta u^m+|x|^{\sigma}u^p, $$ posed in dimension $N\geq2$, with $m>1$, $\sigma\in(-2,0)$ and $1<p<1-\sigma(m-1)/2$. For the special case of dimension $N=1$, the same holds true for $\sigma\in(-1,0)$ and similar ranges for $m$ and $p$. The existence of this global solution prevents finite time blow-up even with $m>1$ and $p>1$, showing an interesting effect induced by the singular potential $|x|^{\sigma}$. This result is also applied to reaction-diffusion equations with general potentials $V(x)$ to prevent finite time blow-up via comparison.