Agenda de la FDP

Séminaire d'Analyse

Le lundi à 10h30 - Salle 1180 (Bât E2)(Tours)
Responsable :
Extinction in a finite time for solutions of a class of Parabolic Equations involving $p$-Laplacian
Yves Belaud
jeudi 17 octobre 2019 - 10h30 - Salle 1180 (Bât E2)(Tours)

Résumé :
We study the property of extinction in a finite time for nonnegative solutions of\\
$\displaystyle \frac{1}{q} \frac{\partial}{\partial t}(u^q) - \nabla (|\nabla u|^{p-2} \nabla u) + a(x) u^\lambda = 0$
for the Dirichlet Boundary Conditions when $q > \lambda > 0$, $p \geq 1+q$, $p \geq 2$, $a(x) \geq 0$ and
$\Omega$ a bounded domain of $\mathbb{R}^N$ ($N \geq 1)$. We give conditions of extinction of solutions in terms of
character of degeneration of $a(x)$, i.e., in dependence of asymptotic of absorption potential near to the set where $a(x)=0$.\\
When $p>1+q$, the threshold is for power functions but for $p=1+q$, it happens extinction in a finite time for very flat absorption potential $a(x)$.\\
The first part of the talk is abstracts results on Hilbert spaces : we give a sufficient condition under the integral form for solutions to vanish in a finite time. Then we give a necessary condition involving a limit.
The second part tackles with applications of these results to second order parabolic equations and leads to sharp result.