Asymptotic behavior for the critical nonhomogeneous porous medium equation in low dimensions

TL;DR

This paper investigates the long-term behavior of solutions to a critical nonhomogeneous porous medium equation in low dimensions, revealing new asymptotic profiles and extending previous results to dimensions one and two.

Contribution

It provides the first detailed analysis of asymptotic profiles for the equation in low dimensions, identifying new behaviors and completing the understanding for dimensions below three.

Findings

In dimension 2, asymptotic profiles are self-similar solutions depending on initial data.

In dimension 1, solutions exhibit a mixture of self-similar and traveling wave behaviors.

The study extends the understanding of the equation's asymptotics to low dimensions, previously known only for higher dimensions.

Abstract

We deal with the large time behavior for a porous medium equation posed in nonhomogeneous media with singular critical density $$ |x|^{-2}\partial_tu(x,t)=\Delta u^m(x,t), \quad (x,t)\in \real^N\times(0,\infty), \ m\geq1, $$ posed in dimensions $N=1$ and $N=2$, which are also interesting in applied models according to works by Kamin and Rosenau. We deal with the Cauchy problem with bounded and continuous initial data $u_0$. We show that in dimension $N=2$, the asymptotic profiles are self-similar solutions that vary depending on whether $u_0(0)=0$ or $u_0(0)=K\in(0,\infty)$. In dimension $N=1$, things are strikingly different, and we find new asymptotic profiles of an unusual mixture between self-similar and traveling wave forms. We thus complete the study performed in previous recent works for the bigger dimensions $N\geq3$.