Blow-up for degenerate parabolic equations with nonlocal source

Abstract

This paper deals with the blow-up properties of the solution to the degenerate nonlinear reaction diffusion equation with nonlocal source $x^{q}u_{t}-(x^{\gamma }u_{x})_{x}=\int _{0}^{a}u^{p}dx$ in $(0,a)\times (0,T)$ subject to the homogeneous Dirichlet boundary conditions. The existence of a unique classical nonnegative solution is established and the sufficient conditions for the solution exists globally or blows up in finite time are obtained. Furthermore, it is proved that under certain conditions the blow-up set of the solution is the whole domain.