2024/03/19 by Matteo Muratori, Muratori, Matteo, Troy Petitt +3
Computer Science · Mathematics · #Advanced Mathematical Modeling in Engineering #Differential Equations and Numerical Methods #Differential Equations and Boundary Problems
paper · pdf · doi:10.48550/arxiv.2403.12854
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.