Existence, uniqueness, and quenching properties of solutions for degenerate semilinear parabolic problems with second boundary conditions

Let q 0, p 0, T ∞, D = (0, a), D¯ = [0, a], Ω = D × (0,T ), and Lu = xqut − uxx. This article considers the following degenerate semilinear parabolic initial-boundary value problem, Lu = xpf (u) in Ω, u(x, 0) =0 onD¯ , ux (0, t) = 0 = ux(a, t) for t > 0, where f (0) > 0, f > 0, f 0, and limu→c− f (u)=∞for some positive constant c. Existence of a unique classical solution is proved. It is shown that if p >q, then quenching occurs only at the boundary point x = a while if p