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SEMINARIO DOBLE DE ECUACIONES EN DERIVADAS PARCIALES - Convexity and PDEs // Blow-up by Aggregation in Chemotaxis

Speaker:  Julio Rossi, Universidad de Buenos Aires (Argentina); y Manuel del Pino, University of Bath (Reino Unido) ()
Date:  Friday, 03 December 2021 - 10:00
Place:  Aula 520, Dpto. de Matemáticas (UAM) y online - https://us06web.zoom.us/j/83702845622?pwd=bGNHclRibExnSmpDRHJ0R1RuNFRIUT09

Abstract:

"Convexity and PDEs", Julio Rossi (Universidad de Buenos Aires)

Abstract:  We deal with PDEs given in terms of eigenvalues of the Hessian and their relation with concave/convex functions. We will also include a fractional version of the involved ideas. In the second part we will describe notions of convexity for functions defined on a regular tree (a graph in which each node  -- except one -- is connected with a fixed number of successors and one predecessor).

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"Blow-up by Aggregation in Chemotaxis", Manuel del Pino (University of Bath)

Abstract: The classical model for chemotaxis is the planar Keller-Segel system \($ u_t = \Delta u - \nabla\cdot ( u\nabla v ), \quad v(\cdot, t) = \frac 1{2\pi} \log 1{|\cdot |} * u(\cdot ,t) . $\) in \(R^2\times (0,\infty)\). Blow-up of finite mass solutions is expected to take place by aggregation, which is a concentration of bubbling type, common to many geometric flows. We build with precise profiles solutions in the critical-mass case \(8\pi\), in which blow-up in infinite time takes place. We establish stability of the phenomenon detected under arbitrary mass-preserving small perturbations and present new constructions in the finite time blow-up scenario.