Asymptotic L1-decay of solutions of the porous medium equation to self-similarity
We consider the flow of gas in an N -dimensional porous medium with initial density v0 (x) ≥ 0. The density v(x, t) then satisfies the nonlinear degenerate parabolic equation vt = ∆v m where m > 1 is a physical constant. Assuming that (1 + $2 )v0 (x) dx < ∞, we prove that v(x, t) behaves asympto...
| Autores: | , |
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| Tipo de recurso: | artículo |
| Fecha de publicación: | 2000 |
| País: | España |
| Institución: | Universitat Autònoma de Barcelona |
| Repositorio: | Dipòsit Digital de Documents de la UAB |
| Idioma: | inglés |
| OAI Identifier: | oai:ddd.uab.cat:115374 |
| Acceso en línea: | https://ddd.uab.cat/record/115374 https://dx.doi.org/urn:doi:10.1512/iumj.2000.49.1756 |
| Access Level: | acceso abierto |
| Sumario: | We consider the flow of gas in an N -dimensional porous medium with initial density v0 (x) ≥ 0. The density v(x, t) then satisfies the nonlinear degenerate parabolic equation vt = ∆v m where m > 1 is a physical constant. Assuming that (1 + $2 )v0 (x) dx < ∞, we prove that v(x, t) behaves asymptotically, as t → ∞, like the Barenblatt-Pattle solution V ( $, t). We prove that the L1 -distance decays at a rate t 1/((N+2)m−N) . Moreover, if N = 1, we obtain an explicit time decay for the L∞ distance at a suboptimal rate. The method we use is based on recent results we obtained for the Fokker-Planck equation [2], [3]. |
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