Complete blow-up for degenerate semilinear parabolic equations

Let T⩽∞, a, q and x0 be constants with a>0, q⩾0 and 0<x0<a. Existence of a unique solution u is established for the following degenerate semilinear parabolic initial-boundary value problem:xqut−uxx=f(u(x0,t)), 0<x<a, 0<t<T,u(x,0)=u0(x)⩾0, 0⩽x⩽a,u(0,t)=u(a,t)=0, 0<t<T,where f and u0 are given functions. We show that under certain conditions, u blows up in a finite time, and the set of blow-up points is the entire interval [0,a].