Partial mass concentration for fast-diffusions with non-local aggregation terms
We study well-posedness and long-time behaviour of aggregation-diffusion equations of the form ∂ρ∂t=Δρm+∇⋅(ρ(∇V+∇W∗ρ)) in the fast-diffusion range, 0<m<1, and V and W regular enough. We develop a well-posedness theory, first in the ball and then in Rd, and characterise the long-time asymptotic...
| Autores: | , , |
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| Formato: | artículo |
| Fecha de publicación: | 2023 |
| País: | España |
| Recursos: | Universidad Complutense de Madrid (UCM) |
| Repositorio: | Docta Complutense |
| Idioma: | inglés |
| OAI Identifier: | oai:docta.ucm.es:20.500.14352/73294 |
| Acesso em linha: | https://hdl.handle.net/20.500.14352/73294 |
| Access Level: | acceso abierto |
| Palavra-chave: | 517.95 517.957 Nonlinear parabolic equations Nonlinear diffusion Dirac delta formation Blow-up in infinite time Viscosity solutions Análisis matemático Ecuaciones diferenciales 1202 Análisis y Análisis Funcional 1202.07 Ecuaciones en Diferencias |
| Resumo: | We study well-posedness and long-time behaviour of aggregation-diffusion equations of the form ∂ρ∂t=Δρm+∇⋅(ρ(∇V+∇W∗ρ)) in the fast-diffusion range, 0<m<1, and V and W regular enough. We develop a well-posedness theory, first in the ball and then in Rd, and characterise the long-time asymptotics in the space W−1,1 for radial initial data. In the radial setting and for the mass equation, viscosity solutions are used to prove partial mass concentration asymptotically as t→∞, i.e. the limit as t→∞ is of the form αδ0+ρˆdx with α≥0 and ρˆ∈L1. Finally, we give instances of W≠0 showing that partial mass concentration does happen in infinite time, i.e. α>0. |
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