Complete blow-up for a degenerate semilinear parabolic problem with a localized nonlinear term

Abstract

We here establish the local existence and uniqueness of a continuous solution under certain conditions of a degenerate semilinear parabolic problem with a localized nonlinear term: let T be any positive real number and x 0 be a fixed number in the interval (0,1), ut - 1/k(x)(p(x)ux)x = f(u(x0,t)) for (x,t) ε (0,1) × (0,T), u(0,t) = 0 = u(1,t) for t ε (0,T), u(x,0) = u 0(x) for x ε [0,1], where k, p, f and u0 are given functions. Moreover, the sufficient condition to blow-up in finite time and the blow-up set of a such solution u are shown.