preprint
Inserted: 6 nov 2024
Year: 2024
Abstract:
We investigate the asymptotic behavior as of solutions to a weighted porous medium equation in , whose weight behaves at spatial infinity like with subcritical power, namely . Inspired by some results by Alikakos-Rostamian and Kamin-Ughi from the 1980s on the unweighted problem, we focus on solutions whose initial data are not globally integrable with respect to the weight and behave at infinity like , for . In the special case and 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 spaces for and even globally in 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.