Near Field Asymptotic Behavior for the Porous Medium Equation on the Half-Line
Kamin and Vázquez [11] proved in 1991 that solutions to the Cauchy-Dirichlet problem for the porous medium equation ut = (um)xx, m > 1, on the half-line with zero boundary data and nonnegative compactly supported integrable initial data behave for large times as a dipole-type solution to the equa...
| Autores: | , , |
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| Tipo de recurso: | artículo |
| Estado: | Versión publicada |
| Fecha de publicación: | 2017 |
| País: | Argentina |
| Institución: | Consejo Nacional de Investigaciones Científicas y Técnicas |
| Repositorio: | CONICET Digital (CONICET) |
| Idioma: | inglés |
| OAI Identifier: | oai:ri.conicet.gov.ar:11336/182625 |
| Acceso en línea: | http://hdl.handle.net/11336/182625 |
| Access Level: | acceso abierto |
| Palabra clave: | ASYMPTOTIC BEHAVIOR MATCHED ASYMPTOTICS POROUS MEDIUM EQUATION ON THE HALF-LINE https://purl.org/becyt/ford/1.1 https://purl.org/becyt/ford/1 |
| Sumario: | Kamin and Vázquez [11] proved in 1991 that solutions to the Cauchy-Dirichlet problem for the porous medium equation ut = (um)xx, m > 1, on the half-line with zero boundary data and nonnegative compactly supported integrable initial data behave for large times as a dipole-type solution to the equation having the same first moment as the initial data, with an error which is o(t-1/m). However, on sets of the form 0 < x < g(t), with g(t) = o(t1/(2m)) as t →, in the so-called near field, a scale which includes the particular case of compact sets, the dipole solution is o( t-1/m), and their result gives neither the right rate of decay of the solution nor a nontrivial asymptotic profile. In this paper, we will improve the estimate for the error, showing that it is o(t-(2m+1)/(2m2) (1+x)1/m). This allows in particular to obtain a nontrivial asymptotic profile in the near field limit, which is a multiple of x1/m, thus improving in this scale the results of Kamin and Vázquez. |
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