An inhomogeneous porous medium equation with non-integrable data: asymptotics
Fuente:
arXiv
Saved in:
| Main Authors: | , , |
|---|---|
| Format: | Preprint |
| Published: |
2024
|
| Subjects: | |
| Online Access: | |
| Tags: |
Add Tag
No Tags, Be the first to tag this record!
|
| _version_ | 1866909142540615680 |
|---|---|
| author | Muratori, Matteo Petitt, Troy Quirós, Fernando |
| author_facet | Muratori, Matteo Petitt, Troy Quirós, Fernando |
| contents | We investigate the asymptotic behavior as $t\to+\infty$ of solutions to a weighted porous medium equation in $ \mathbb{R}^N $, whose weight $ρ(x)$ behaves at spatial infinity like $ |x|^{-γ} $ with subcritical power, namely $ γ\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|^{-α} $, for $α\in(0,N-γ)$. In the special case $ ρ(x)=|x|^{-γ} $ and $ u_0(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 $ 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2403_12854 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | An inhomogeneous porous medium equation with non-integrable data: asymptotics Muratori, Matteo Petitt, Troy Quirós, Fernando Analysis of PDEs Functional Analysis 35B40 (Primary) 35K55, 35B45, 34D05 (Secondary) We investigate the asymptotic behavior as $t\to+\infty$ of solutions to a weighted porous medium equation in $ \mathbb{R}^N $, whose weight $ρ(x)$ behaves at spatial infinity like $ |x|^{-γ} $ with subcritical power, namely $ γ\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|^{-α} $, for $α\in(0,N-γ)$. In the special case $ ρ(x)=|x|^{-γ} $ and $ u_0(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 $ 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. |
| title | An inhomogeneous porous medium equation with non-integrable data: asymptotics |
| topic | Analysis of PDEs Functional Analysis 35B40 (Primary) 35K55, 35B45, 34D05 (Secondary) |
| url | https://arxiv.org/abs/2403.12854 |