# Seminario di Equazioni Differenziali e Applicazioni: “Regularity and long time behaviour of viscosity solutions for integro-differential equations”

## Venerdì 7 Luglio 2017, ore 11:00 - Aula 2BC30 - Adina Ciomaga

ARGOMENTI: Seminars

Venerdì 7 Luglio 2017 alle ore 11:00 in Aula 2BC30, Adina Ciomaga (Université Paris Diderot) terrà un seminario dal titolo “Regularity and long time behaviour of viscosity solutions for integro-differential equations”.

Abstract
In this talk, I will talk about the long time behaviour of viscosity solutions for nonlocal PDEs, associated with Levy-Ito operators. These equations have the form $u_t -I[x,u] + H(x,Du) = f(x)$, where $I[x,u]$ is the integro-differential operator and $H(x,Du)$ is the Hamiltonian.
The standard program is to show that, in the periodic setting, solutions of such equations $u(t,x)$ behave, for large times as $\lambda t + v(x)$ where $\lambda$ is a constant and $v$ is the unique periodic solution of the ergodic problem associated to $\lambda$. This type of results rely on two key ingredients: regularity of solutions and strong maximum principle. I will explain the interplay between the nonlocal term and the Hamiltonian, with an emphasis on the loss of ellipticity of the nonlocal diffusion and the loss of coercivity of the Hamiltonian, both when establishing the regularity and long time behavior.