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arXiv · 2408.02466

Eternal solutions to a porous medium equation with strong nonhomogeneous absorption. Part II: Dead-core profiles

Razvan Gabriel Iagar·Philippe Laurençot·Ariel Sánchez
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Abstract

Existence of a specific family of \emph{eternal solutions} in exponential self-similar form is proved for the following porous medium equation with strong absorption $$\partial_t u-Δu^m+|x|^σu^q = 0 \;\;\text{ in }\;\; (0,\infty)\times\mathbb{R}^N,$$ with $m>1$, $q\in(0,1)$ and $σ=2(1-q)/(m-1)$. Looking for solutions of the form $$ u(t,x)=e^{-αt}f(|x|e^{βt}), \qquad α=\frac{2}{m-1}β,$$ it is shown that, for $m+q>2$, there exists a unique exponent $β_*\in(0,\infty)$ for which there exists a one-parameter family of compactly supported profiles presenting a \emph{dead core}. The precise behavior of the solutions at their interface is also determined. Moreover, these solutions show the optimal limitations for the finite time extinction property of genuine non-negative solutions to the Cauchy problem, studied in previous works.

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Razvan Gabriel Iagar, Philippe Laurençot, Ariel Sánchez. 2024-08-05. Eternal solutions to a porous medium equation with strong nonhomogeneous absorption. Part II: Dead-core profiles. https://arxiv.org/abs/2408.02466

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