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

We study the first vanishing time for solutions of the Cauchy-Dirichlet problem to the semilinear $2m$-order ($m\ge 1$) parabolic equation $u_t+Lu+a(x)\mathrm{\mid }u{\mathrm{\mid }}^{q-1}u=0$, $0<q<1$ with $a(x)\ge 0$ bounded in the bounded domain $\mathrm{\Omega }\subset {\mathbb{R}}^N$. 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$,thenthesolution$u$vanishesinafinitetime\mathrm{.}When$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$.