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arXiv · 2609.20397

A porous medium equation with dominating weighted absorption: three types of self-similar solutions

Razvan Gabriel Iagar·Diana-Rodica Munteanu
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Abstract

Self-similar solutions to the porous medium equation with dominating spatially inhomogeneous absorption $$ \partial_tu=Δu^m-|x|^σu^p, \quad (x,t)\in\real^N\times(0,\infty), \quad N\geq1, $$ with exponents $1 0$, are classified. Looking for solutions in the form $$ u(x,t)=t^{-α}f(|x|t^β), \quad α=\frac{σ+2}{σ(m-1)+2(p-1)}, \quad β=\frac{m-p}{σ(m-1)+2(p-1)}, $$ it is shown that all their profiles satisfy the behavior at infinity given by $$ \lim\limits_{ξ\to\infty}ξ^{σ/(p-1)}f(ξ)=\left(\frac{1}{p-1}\right)^{1/(p-1)}, $$ but the solutions strongly differ with respect to their behavior near the origin: there exist a unique solution with $f(0)>0$, $f'(0)=0$, another unique solution such that $f$ presents a \emph{dead-core}; that is, $f\equiv0$ for $ξ\in[0,ξ_0]$ for some $ξ_0>0$, and, finally, there exists $K^*\in(0,\infty)$ such that, for any $K\in(0,K^*)$, there is at least a solution such that $$ \lim\limits_{ξ\to0}ξ^{-(σ+2)/(m-p)}f(ξ)=K. $$ The large time behavior of general solutions, making strong use of these three types of self-similar solutions, will be addressed in a companion work.

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Razvan Gabriel Iagar, Diana-Rodica Munteanu. 2026-09-17. A porous medium equation with dominating weighted absorption: three types of self-similar solutions. https://arxiv.org/abs/2609.20397

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