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arXiv · 2402.12455

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

Razvan Gabriel Iagar·Ariel Sánchez
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Abstract

We establish the existence of self-similar solutions presenting finite time blow-up to the quasilinear reaction-diffusion equation $$ u_t=Δu^m + u^p, $$ 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.

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Razvan Gabriel Iagar, Ariel Sánchez. 2024-02-19. Existence of blow-up self-similar solutions for the supercritical quasilinear reaction-diffusion equation. https://arxiv.org/abs/2402.12455

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