EXISTENCE, UNIQUENESS AND QUENCHING OF THE SOLUTION FOR A NONLOCAL DEGENERATE SEMILINEAR PARABOLIC PROBLEM

Let a and T be positive constants, D = (0, a), D¯ = [0, a], Ω = D × (0, T], and Lu = x qut − uxx, where q is a nonnegative number. This article studies the following problem, Lu(x,t) = Zx 0 k(y)f(u(y,t))dy in Ω, where k is a positive function on D¯, f > 0, f 0≥ 0, f 00≥ 0, and limu→1− f(u) = ∞, subject to the initial condition u(x, 0) = 0 on D¯, and the boundary conditions u(0,t) = 0 = u(a,t) for 0 < t ≤ T. Existence of a unique solution, the critical length, and the quenching behavior of the solution are studied.