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

Abstract

We deal with the large time behavior for a porous medium equation posed in nonhomogeneous media with singular critical density $$|x{|}^{-2}{\partial }_{t}u(x,t)=\Delta u^m(x,t),(x,t)\in {\Re }^N\times (0,\infty ),m\ge 1,$$ 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 $N\ge 3$.