Self-similar Blow-Up Profiles for a Reaction–Diffusion Equation with Critically Strong Weighted Reaction
We classify the self-similar blow-up profiles for the following reaction–diffusion equation with critical strong weighted reaction and unbounded weight: ∂tu = ∂x x (um) + |x| σ u p, posed for x ∈ R, t ≥ 0, where m > 1, 0 < p < 1 such that m+ p = 2 and σ > 2 completing the analysis perfor...
| Autores: | , |
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| Tipo de recurso: | artículo |
| Fecha de publicación: | 2022 |
| País: | España |
| Institución: | Universidad Rey Juan Carlos |
| Repositorio: | BURJC-Digital. Repositorio Institucional de la Universidad Rey Juan Carlos |
| OAI Identifier: | oai:burjcdigital.urjc.es:10115/29827 |
| Acceso en línea: | https://hdl.handle.net/10115/29827 |
| Access Level: | acceso abierto |
| Palabra clave: | Reaction–diffusion equations Weighted reaction Blow-up Self-similar solutions Phase space analysis Strong reaction |
| Sumario: | We classify the self-similar blow-up profiles for the following reaction–diffusion equation with critical strong weighted reaction and unbounded weight: ∂tu = ∂x x (um) + |x| σ u p, posed for x ∈ R, t ≥ 0, where m > 1, 0 < p < 1 such that m+ p = 2 and σ > 2 completing the analysis performed in a recent work where this very interesting critical case was left aside. We show that finite time blow-up solutions in self-similar form exist for σ > 2. Moreover all the blow-up profiles have compact support and their supports are localized: there exists an explicit η > 0 such that any blow-up profile satisfies supp f ⊆ [0, η]. This property is unexpected and contrasting with the range m+ p > 2. We also classify the possible behaviors of the profiles near the origin. |
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