Seminario de Ecuaciones en Derivadas Parciales y Análisis Numérico

28 de marzo de 2025
12:30
Aula 520, Módulo 17

Facultad de Ciencias,
Universidad Autónoma de Madrid.

The anisotropic slow diffusion equation:
asymptotic behaviour of solutions and free boundaries
Filomena Feo

Dipartimento di Ingegneria,
Università degli Studi di Napoli “Parthenope”

Abstract
In this talk, we expose several recent results concerning the study of the nonnegative solutions
of the following anisotropic equation
ut =
XN
i=1
(umi )xixi in RN × (0,+∞)
with N ≥ 2 in the slow diffusion range, i.e. mi > 1 for all i = 1, . . . ,N. We focus on the
existence and uniqueness of a self-similar fundamental solution, the asymptotic behaviour of
nonnegative solutions with L1 initial data and the asymptotic behaviour of the support and
free boundary of bounded solutions with compactly supported initial data.
Based on a joint work with J. L. Vázquez and B. Volzone.

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Finite Element Approximation
of fractional nonlinear PDEs
Stefano Fronzoni
Mathematical Institute,
University of Oxford

Abstract
We present a finite element method for the numerical solution of the fractional porous
medium equation on a domain Ω ⊂ Rd. After introducing the fractional Laplacian operator, in
its several definitions, we present a rigorous passage to the limit in the fully discrete approximation
of the fractional porous medium equation. As the spatial and temporal discretization
parameters tend to zero, we show convergence for a subsequence of finite element approximations
to a weak solution of the initial boundary-value problem, with an argument based on
compactness techniques for nonlinear partial differential equations.
We then show the algorithm used for the implementation of the method and its application
to other problem of interest, involving the fractional Laplacian in interaction potentials,
including the fractional Keller-Segel model and blow-up of its solution.