Extinction of solutions of semilinear higher order parabolic equations with degenerate absorption potential

Abstract

We study the first vanishing time for solutions of the Cauchy-Dirichlet problem to the semilinear $2m$-order ($m\ge 1$) parabolic equation $u_t+Lu+a(x)|u{|}^{q-1}u=0$, $0<q<1$ with $a(x)\ge 0$ bounded in the bounded domain $\Omega \subset {\mathbb{R}}^N$. We prove that if $N>2m$ and ${\int }_0^1{s}^{-1}\text{meas}\{x\in \Omega :|a(x)|\le s{\}}^{\frac{2m}{N}}ds<+\infty$, then the solution $u$ vanishes in a finite time. When $N=2m$, the condition becomes ${\int }_0^1{s}^{-1}(\text{meas}\{x\in \Omega :|a(x)|\le s\})(-\ln \text{meas}\{x\in \Omega :|a(x)|\le s\})ds<+\infty$.