Existence of blow-up self-similar solutions for the supercritical quasilinear reaction-diffusion equation

TL;DR

This paper proves the existence of self-similar solutions with finite-time blow-up for a supercritical quasilinear reaction-diffusion equation, showing multiple solutions and challenging previous critical exponent assumptions.

Contribution

It demonstrates the existence of self-similar blow-up solutions in the supercritical regime and shows non-optimality of the Lepin critical exponent for this class of equations.

Findings

Existence of at least one self-similar blow-up solution for p > p_s.

Multiple solutions exist when the dimension N is sufficiently large.

Contrasts with the semilinear case where the critical exponent is optimal.

Abstract

We establish the existence of self-similar solutions presenting finite time blow-up to the quasilinear reaction-diffusion equation $$ u_t=\Delta u^m + u^p, $$ posed in dimension $N\geq3$, $m>1$. More precisely, we show that there is always at least one solution in backward self-similar form if $p>p_s=m(N+2)/(N-2)$. In particular, this establishes \emph{non-optimality of the Lepin critical exponent} introduced in \cite{Le90} in the semilinear case $m=1$ and extended for $m>1$ in \cite{GV97, GV02}, for the existence of self-similar blow-up solutions. We also prove that there are multiple solutions in the same range, provided $N$ is sufficiently large. This is in strong contrast with the semilinear case, where the Lepin critical exponent has been proved to be optimal.