Extinction in Finite Time of Solutions to Degenerate Parabolic Equations with Nonlinear Boundary Conditions1

This work concerns a nonlinear diffusion]absorption equation with nonlinear boundary flux. The main topic of interest is the problem of finite time extinction, i.e., the solutions vanish after a finite time. The sufficient and necessary conditions for occurrence of extinction are established. It is shown that extinction is caused by either strong absorption in the interior of the domain or fast diffusion combined with strong absorption through the boundary of the domain. Extinction results are also obtained for the mixed boundary value problem. In contrast to the nonlinear Neumann problem, the absorption on the boundary is no longer important, i.e., the occurrence of extinction in this case is completely determined by the effects of diffusion and interior absorption.