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

摘要

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), u_t -1/k(x)(p{x)u_x)_x =f(u(x_0,t)) for(x,t)∈(0,1)×(0,T), u(0,t) = 0 = u(1,t)for t∈(0,T), u(x,0) = u_0(x)for x ∈[0,1], where k, p, f and u_0 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.