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arXiv · 2108.09088

Self-similar blow-up patterns for a reaction-diffusion equation with weighted reaction in general dimension

Razvan Gabriel Iagar·Ana I. Muñoz·Ariel Sánchez
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Abstract

We classify the finite time blow-up profiles for the following reaction-diffusion equation with unbounded weight: $$ \partial_tu=Δu^m+|x|^σu^p, $$ posed in any space dimension $x\in\mathbf{R}^N$, $t\geq0$ and with exponents $m>1$, $p\in(0,1)$ and $σ>2(1-p)/(m-1)$. We prove that blow-up profiles in backward self-similar form exist for the indicated range of parameters, showing thus that the unbounded weight has a strong influence on the dynamics of the equation, merging with the nonlinear reaction in order to produce finite time blow-up. We also prove that all the blow-up profiles are \emph{compactly supported} and might present two different types of interface behavior and three different possible \emph{good behaviors} near the origin, with direct influence on the blow-up behavior of the solutions. We classify all these profiles with respect to these different local behaviors depending on the magnitude of $σ$. This paper generalizes in dimension $N>1$ previous results by the authors in dimension $N=1$ and also includes some finer classification of the profiles for $σ$ large that is new even in dimension $N=1$.

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Razvan Gabriel Iagar, Ana I. Muñoz, Ariel Sánchez. 2021-08-20. Self-similar blow-up patterns for a reaction-diffusion equation with weighted reaction in general dimension. https://arxiv.org/abs/2108.09088

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