Complete blow-up for a semilinear parabolic equation

Abstract

Let T <= [infinity], a and x[subscript 0] be constants with a > 0 and 0 < x[subscript 0] < a. We establish the unique solution u for the following semilinear parabolic initial-boundary value problem: u[subscript t] (x,t)-u[subscript xx] (x,t) = [the integral of] (u(X[subscript 0],t)) for 0 < x < a, 0 < t < T, u(x,0) = [or {} is an empty set (x) >= 0 for 0 <= x <= a, u (0,t) = u[subscript x] (a,t) = 0 for 0 < t < T, Where [the integral of] and [or {} is an empty set] are given functions. We also show that under certain conditions, u blows up in a finite time, and the set of the blow-up points is the entire interval [0, a].