Long-time extinction of solutions of some semilinear parabolic equations

Details Long-time extinction of solutions of some semilinear parabolic equations Long-time extinction of solutions of some semilinear parabolic equations

2008-05-15

arxiv.org/abs/0805.2314v1

Journal of Differential Equations (2007) 238, pages 64-86

Mathematics

Analysis of PDEs

Scientific paper

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 $a_0(x) \geq d_0 \exp(\frac{\omega(|x|)}{|x|^2})$, $d_0>0$, $1>q>0$ 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 Schr\"odinger operators.

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