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A. Carrillo, José and Fernández-Jiménez, Alejandro and Gómez-Castro, D.
(2023)
*Partial mass concentration for fast-diffusions with non-local aggregation terms.*
(Unpublished)

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## Abstract

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.

Item Type: | Article |
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Uncontrolled Keywords: | Nonlinear parabolic equations ; Nonlinear diffusion ; Dirac delta formation ; Blow-up in infinite time ; Viscosity solutions |

Subjects: | Sciences > Mathematics > Mathematical analysis Sciences > Mathematics > Differential equations |

ID Code: | 77670 |

Deposited On: | 24 Apr 2023 10:15 |

Last Modified: | 08 May 2023 06:34 |

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