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 ($m \geq 1$) parabolic equation $u_t+Lu+a(x) |u|^{q-1}u=0$, $0<q<1$ with $a(x) \geq 0$ bounded in the bounded domain $\Omega \subset \R^N$. We prove that if $N>2m$ and $\displaystyle \int_0^1 s^{-1} \text{meas} \{x \in \Omega : |a(x)| \leq s \}^\frac{2m}{N} ds < + \infty$, then the solution $u$ vanishes in a finite time. When $N=2m$, the condition becomes $\displaystyle \int_0^1 s^{-1} (\text{meas} \{x \in \Omega : |a(x)| \leq s \}) (-\ln \text{meas} \{x \in \Omega : |a(x)| \leq s \}) ds < + \infty$.