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

We establish the existence of self-similar solutions presenting finite time blow-up to the quasilinear reaction-diffusion equation $$u_t=\mathrm{\Delta }{u}^m+u^p,$$ posed in dimension $N\ge 3$, $m>1$. More precisely, we show that there is always at least one solution in backward self-similar form if $p>p_s=m(N+2)\mathrm{/}(N-2)$. 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.