# Seminario di Equazioni Differenziali e Applicazioni: “Large time behavior of unbounded solutions of first order Hamilton-Jacobi equations in R^N”

## Martedì 19 Settembre 2017, ore 11:30 - Sala Riunioni VII piano - Thi Tuyen Nguyen

ARGOMENTI: Seminari

Martedì 19 Settembre 2017 alle ore 11:30 in Sala Riunioni VII piano, Thi Tuyen Nguyen (Università di Padova) terrà un seminario dal titolo “Large time behavior of unbounded solutions of first order Hamilton-Jacobi equations in $\mathbb{R}^N$”.

Abstract
We study the large time behavior of solutions of first-order convex Hamilton-Jacobi Equations of Eikonal type $u_t+H(x,Du)=l(x),$ set in the whole space $\mathbb{R}^N\times [0,\infty).$ We assume that $l$ is bounded from below but may have arbitrary growth and therefore the solutions may also have arbitrary growth. A complete study of the structure of solutions of the ergodic problem $H(x,Dv)=l(x)+c$ is provided : contrarily to the periodic setting, the ergodic constant is not anymore unique, leading to different large time behavior for the solutions.
We establish the ergodic behavior of the solutions of the Cauchy problem (i) when starting with a bounded from below initial condition and (ii) for some particular unbounded from below initial condition, two cases for which we have different ergodic constant which play a role. When the solution is not bounded from below, we provide a counter-example showing that the convergence may fail in general.
The work is joint with Guy Barles (Tours), Olivier Ley (Rennes) and Thanh Viet Phan (HCM, VietNam).