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 x (0, T), and Lu = xqut - uxx. This article considers the following degenerate semilinear parabolic initial-boundary value problem, Lu = xp f(u) in Q, u(x,0)=0 on D, ux (0, t)=0=ux(a, t) fort>0, where f(0) > 0, f' > 0, f ≥ 0, and limu→c- f(u) = 00 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 < q, then the only quenching point is x = 0. If p = q, then the quenching set is D.