Blow up pro les for a quasilinear reaction-diffusion equation with weighted reaction with linear growth
We study the blow up profiles associated to the following second order reaction–diffusion equation with non-homogeneous reaction: ∂tu=∂xx(um)+|x|σu,with σ> 0. Through this study, we show that the non-homogeneous coefficient | x| σ has a strong influence on the blow up behavior of the solutions. F...
| Autores: | , |
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| Tipo de recurso: | artículo |
| Fecha de publicación: | 2019 |
| País: | España |
| Institución: | Universidad Rey Juan Carlos |
| Repositorio: | BURJC-Digital. Repositorio Institucional de la Universidad Rey Juan Carlos |
| OAI Identifier: | oai:burjcdigital.urjc.es:10115/29837 |
| Acceso en línea: | https://hdl.handle.net/10115/29837 |
| Access Level: | acceso abierto |
| Palabra clave: | reaction-di usion equations non-homogeneous reaction blow up critical case self-similar solutions phase space analysis |
| Sumario: | We study the blow up profiles associated to the following second order reaction–diffusion equation with non-homogeneous reaction: ∂tu=∂xx(um)+|x|σu,with σ> 0. Through this study, we show that the non-homogeneous coefficient | x| σ has a strong influence on the blow up behavior of the solutions. First of all, it follows that finite time blow up occurs for self-similar solutions u, a feature that does not appear in the well known autonomous case σ= 0. Moreover, we show that there are three different types of blow up self-similar profiles, depending on whether the exponent σ is closer to zero or not. We also find an explicit blow up profile. The results show in particular that global blow up occurs when σ> 0 is sufficiently small, while for σ> 0 sufficiently large blow up occurs only at infinity, and we give prototypes of these phenomena in form of self-similar solutions with precise behavior. This work is a part of a larger program of understanding the influence of non-homogeneous weights on the blow up sets and rates. © 2019, Springer Science+Business Media, LLC, part of Springer Nature. |
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