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 $u_t+Au+Bu=0$ where $A$ is a maximal monotone operator and $B$ is a positive operator on a Hilbert space $H$. 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: $Au=-\Delta u$, $Bu=a(x)u^q$, $a(x) \geq 0$ bounded and $-1<q<1$, on a regular bounded domain $\Omega$ and Dirichlet boundary conditions.