Global existence and blow up in a parabolic problem with nonlocal dynamical boundary conditions

Abstract

We consider an initial and boundary value problem for a class of possibly degenerate second-order parabolic PDEs. The condition prescribed at the lateral boundary of the domain is of non local type, requiring the time derivative of the unknown to equal a given function of the total flux through the boundary itself (such conditions arise in problems involving contact with a body where diffusivity is infinite). We give results of local-in-time existence of (weak) solutions, and then we investigate the existence of solutions defined for all positive times. We show that, due to certain nonlinearities appearing in the problem, blow up in a finite time may occur, and we give, by means of suitable a~priori estimates, criteria for the global existence of solutions, or for occurrence of blow up.