Asymptotic behavior of the solutions of the inhomogeneous Porous Medium Equation with critical vanishing density
Sofía Nieto, Guillermo Reyes · Communications on Pure & Applied Analysis · 2012
We study the long-time behavior of non-negative, finite-energysolutions to the initial value problem for the Porous MediumEquation with variable density, i.e. solutions of the problem\begin{eqnarray*}\rho (x) \partial_{t} u = \Delta u^{m}, \quad in \quad Q:= R^n \times R_+, \\ u(x,0)=u_{0}(x), \quad in\quad R^n,\end{eqnarray*}where $m>1$, $u_0\in L^1(R^n, \rho(x)dx)$ and $n\ge 3$.We assume that $\rho (x)\sim C|x|^{-2}$ as $|x|\to\infty$ in$R^n$. Such a decay rate turns out to be critical. Weshow that the limit behavior can be described in terms of a familyof source-type solutions of the associated singular equation$|x|^{-2}u_t = \Delta u^{m}$. The latter have a self-similarstructure and exhibit a logarithmic singularity at the origin.