Anomalous self-similar solutions of exponential type for the subcritical fast diffusion equation with weighted reaction
We prove existence and uniqueness of the branch of the so-called anomalous eternal solutions in exponential self-similar form for the subcritical fast-diffusion equation with a weighted reaction term ∂ t u = Δ u m + | x | σ u p , posed in R N with N ⩾ 3, where 1,$> 0 < m < m c = N − 2 N , p...
| Autores: | , |
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| Tipo de documento: | artigo |
| Data de publicação: | 2022 |
| País: | España |
| Recursos: | Universidad Rey Juan Carlos |
| Repositório: | BURJC-Digital. Repositorio Institucional de la Universidad Rey Juan Carlos |
| OAI Identifier: | oai:burjcdigital.urjc.es:10115/29838 |
| Acesso em linha: | https://hdl.handle.net/10115/29838 |
| Access Level: | Acceso aberto |
| Palavra-chave: | anomalous solutions fast diffusion equation exponential self-similar solutions weighted reaction phase plane analysis critical exponents |
| Resumo: | We prove existence and uniqueness of the branch of the so-called anomalous eternal solutions in exponential self-similar form for the subcritical fast-diffusion equation with a weighted reaction term ∂ t u = Δ u m + | x | σ u p , posed in R N with N ⩾ 3, where 1,$> 0 < m < m c = N − 2 N , p > 1 , and the critical value for the weight σ = 2 ( p − 1 ) 1 − m . The branch of exponential self-similar solutions behaves similarly as the well-established anomalous solutions to the pure fast diffusion equation, but without a finite time extinction or a finite time blow-up, and presenting instead a change of sign of both self-similar exponents at m = m s = ( N − 2)/( N + 2), leading to surprising qualitative differences. In this sense, the reaction term we consider realizes a perfect equilibrium in the competition between the fast diffusion and the reaction effects. |
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