Calculus of Variations and Geometric Measure Theory

M. Muratori - T. Petitt - F. Quirós

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

created by petitt on 06 Nov 2024

[BibTeX]

preprint

Inserted: 6 nov 2024

Year: 2024

ArXiv: 2403.12854 PDF

Abstract:

We investigate the asymptotic behavior as t+ of solutions to a weighted porous medium equation in RN, whose weight ρ(x) behaves at spatial infinity like xγ with subcritical power, namely γ[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 u0(x) are not globally integrable with respect to the weight and behave at infinity like xα, for α(0,Nγ). In the special case ρ(x)=xγ and u0(x)=xα 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 Lp spaces for p[1,) and even globally in L 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.