Long-time extinction of solutions of some semilinear parabolic equations
Résumé
We study the long time behaviour of solutions of semi-linear parabolic equation of the following type $\partial_t u-\Delta u+a_0(x)u^q=0$ where $a_0(x) \geq d_0 \exp\left( \frac{\omega(|x|)}{|x|^2}\right)$, $d_0>0$, $1>q>0$ and $\omega$ a positive continuous radial function. We give a Dini-like condition on the function $\omega$ by two different method which implies that any solution of the above equation vanishes in a finite time. The first one is a variant of a local energy method and the second one is derived from semi-classical limits of some Schrödinger operators.
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