# 2nd IMI one-day workshop on PDEs

• #### Fechas:

Del 24/02/20 al 24/02/20

Web del evento

## Descripción ↑ subir

Organized by Instituto de Matemática Interdisciplinar

Organizing Comittee:

F. del Teso (UCM)

D. Gómez-Castro (UCM)

J.L. Vázquez (UAM, UCM)

## Programa ↑ subir

24
Feb 2020
• 09:30 - 10:30
Raúl Ferreira: Grow-up for a localized reaction-diffusion equation
We study the behaviour of  global solutions to the semilinear heat equation  with a reaction localized in a ball
$$u_t=\Delta u+\chi_{B_L}u^p,$$
for $0<p\le1$, $0<L<\infty$. We study when global solutions are bounded or unbounded. In particular we show that the precise value of the length $L$ plays a crucial role in the critical case $p=1$ for  $N\ge3$. We also obtain the asymptotic
behaviour of unbounded solutions and prove that the grow-up rate is different to the one obtained when $L=\infty$.
• 10:30 - 11:30
Luz Roncal: Hardy's inequalities, an extension problem and Helgason conjecture
We will introduce Hardy's inequality from several points of view and we will turn into its fractional version. One of the approaches to prove fractional Hardy's inequality leads to the study of solutions of the extension problem. It happens that these solutions are related with the eigenfunctions of certain Laplace-Beltrami operators. Eventually, the problem of characterising such eigenfunctions, which is part of the so-called Helgason conjecture, remains open in several contexts.

The aim of this talk is to give an overview of the above topics and report recent progress in several contexts, putting an emphasis on the case of the Heisenberg group.

Joint work Sundaram Thangavelu (Indian Institute of Sciences in Bangalore, India).
• 11:45 - 12:45
Xavier Ros-Oton: Nonlocal equations in C^{k,alpha} domains and applications to obstacle problems
We study the boundary regularity of solutions to integro-differential equations in C^{k,\alpha} domains. Our main result establishes new fine estimates, which are optimal and extend known results from Grubb (for k = \infty) and from the author and Serra (for k = 1).

As a consequence of our results, we deduce the higher regularity of free boundaries in obstacle problems for nonlocal operators.

This is a joint work with Nicola Abatangelo.

• 12:45 - 13:45
Jørgen Endal: The one-phase fractional Stefan problem
We study the existence, properties of solutions, and free boundaries of the one-phase Stefan problem with fractional diffusion posed in $\R^N$. The equation for the enthalpy $h$  reads $\partial_t h+ (-\Delta)^{s}\Phi(h) =0$ where the temperature $u:=\Phi(h):=\max\{h-L,0\}$ is defined for some constant $L>0$  called the latent heat, and $(-\Delta)^{s}$ is the fractional Laplacian with exponent $s\in(0,1)$. We prove the existence of a continuous and bounded selfsimilar solution of the form $h(x,t)=H(x\,t^{-1/(2s)})$ which exhibits a free boundary at the change-of-phase level $h(x,t)=L$ located at $x(t)=\xi_0 t^{1/(2s)}$ for some $\xi_0>0$. This special solution will be an important tool to obtain that the temperature has finite speed of propagation while the enthalpy has infinite speed, and that the support of the temperature never recedes. Other interesting properties like e.g. $L\to0^+$ and $L\to\infty$ will also be discussed, and the theory itself is illustrated by convergent finite-difference schemes.

• 16:00 - 17:00
María del Mar González: ODE methods for non-local equations
Non-local equations cannot be treated using classical ODE theorems.
Nevertheless, several new methods have been introduced: first, from the
explicit symbol of the conformal fractional Laplacian, a variation of
constants formula is obtained for fractional Hardy operators. We thus
develop, in addition to a suitable extension in the spirit of
Caffarelli–Silvestre, an equivalent formulation as an infinite system of
second order constant coefficient ODEs. Classical ODE quantities like
the Hamiltonian and Wronskian may then be utilized.

We give some applications to the the non-degeneracy of the fast-decay
singular solution of the fractional Lane–Emden equation and, if time
permits, to the uniqueness of ground states for the fractional plasma
problem.

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