Existence versus explosion instantanée pour des équations de la chaleur linéaires avec potentiel singulier
In this Note, we consider the linear heat equation ut - Au= a(x)u in (0,T)xW,u=0 on (0,T)xaW, and u(0)=uº on W, where W C RN is a smooth bounded domain. We assume that a€L^loc(W) a >=0 and u>=. A simple condition on the potential a is necessary and suficient for the existence of positive weak...
| Autores: | , |
|---|---|
| Tipo de recurso: | artículo |
| Fecha de publicación: | 1999 |
| País: | España |
| Institución: | Universitat Politècnica de Catalunya (UPC) |
| Repositorio: | UPCommons. Portal del coneixement obert de la UPC |
| Idioma: | inglés |
| OAI Identifier: | oai:upcommons.upc.edu:2117/764 |
| Acceso en línea: | https://hdl.handle.net/2117/764 |
| Access Level: | acceso abierto |
| Palabra clave: | Parabolic partial differential equations Partial differential equations linear heat equations singular potentials Equacions en derivades parcials Classificació AMS::35 Partial differential equations::35K Parabolic equations and systems Classificació AMS::35 Partial differential equations::35B Qualitative properties of solutions |
| Sumario: | In this Note, we consider the linear heat equation ut - Au= a(x)u in (0,T)xW,u=0 on (0,T)xaW, and u(0)=uº on W, where W C RN is a smooth bounded domain. We assume that a€L^loc(W) a >=0 and u>=. A simple condition on the potential a is necessary and suficient for the existence of positive weak solutions that are global in time and grow at most exponentially in time. We show that this condition, based on the existence of a Hardy type inequality with weight a(x), is "almost" necessary for the local existence in time of positive weak solutions. Applying these results to some "critical" potentials, we find new results on existence and on instantaneous and complete blow-up of solutions. |
|---|