An inhomogeneous porous medium equation with non-integrable data: asymptotics

We investigate the asymptotic behavior as $t\to+\infty$ of solutions to a weighted porous medium equation in $\mathbb{R}^N$, whose weight $\rho(x)$ behaves at spatial infinity like $|x|^{-\gamma}$ with subcritical power, namely $\gamma \in [0,2)$. Inspired by some results by Alikakos-Rostamian and Kamin-Ughi from the 1980s on the unweighted problem, we focus on solutions whose initial data $u_0(x)$ are not globally integrable with respect to the weight and behave at infinity like $|x|^{-\alpha}$, for $\alpha\in(0,N-\gamma)$. In the special case $\rho(x)=|x|^{-\gamma}$ and $u_0(x)=|x|^{-\alpha}$ we show that self-similar solutions of Barenblatt type, i.e. reminiscent of the usual source-type solutions, still exist, although they are no longer compactly supported. Moreover, they exhibit a transition phenomenon which is new even for the unweighted equation. We prove that such self-similar solutions are attractors for the original problem, and convergence takes place globally in suitable weighted $L^p$ spaces for $p\in[1,\infty)$ and even globally in $L^\infty$ under some mild additional regularity assumptions on the weight. Among the fundamental tools that we exploit, it is worth mentioning a global smoothing effect for non-integrable data.