Abstract Results on the Finite Extinction Time Property: Application to a Singular Parabolic Equation

We start by studying the finite extinction time for solutions of the abstract Cauchy problem

where

A A

is a maximal monotone operator and

B B

is a positive operator on a Hilbert space

H H

. We use a suitable spectral energy method to get some sufficient conditions which guarantee this property. As application we consider a singular semilinear parabolic equation:

Au=-\Delta u A u = − Δ u

,

Bu=a(x)u^q B u = a ( x ) u q

,

a(x) \geq 0 a ( x ) ≥ 0

bounded and

-1<q<1 − 1 < q < 1

, on a regular bounded domain

\Omega Ω

and Dirichlet boundary conditions.