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

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