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

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 |x|-γ 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 |x|-α, for α∈(0,N-γ). In the special case ρ(x)=|x|-γ and u₀(x)=|x|-α 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.