Séminaire d'Analyse

Extinction in a finite time for solutions of a class of Parabolic Equations involving $p$-Laplacian
Yves Belaud
Thursday 17 October 2019 10:30 -  Tours -  Salle 1180 (Bât E2)

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.

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