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

TL;DR

This paper investigates conditions under which solutions to certain higher-order semilinear parabolic equations with degenerate absorption potential vanish in finite time, extending known results to higher-order and degenerate cases.

Contribution

It establishes new criteria involving the absorption potential's measure for finite-time extinction of solutions in higher-order parabolic equations.

Findings

Solutions vanish in finite time under specified measure conditions.

Derived conditions depend on the dimension relative to the order of the PDE.

Extended extinction results to degenerate absorption potentials.

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 $\displaystyle \int_0^1 s^{-1} (\text{meas} \{x \in \Omega : |a(x)| \leq s \}) (-\ln \text{meas} \{x \in \Omega : |a(x)| \leq s \}) ds < + \infty$.