On the global solutions to a class of strongly degenerate parabolic equations

We are concerned with degenerate parabolic equations in nondivergence form of the type ∂tu=a(δ(x))up(Δu+λg(u))∂tu=a(δ(x))up(Δu+λg(u)) in Ω×(0,∞)Ω×(0,∞), where Ω⊂RN(N≥1)Ω⊂RN(N≥1) is a smooth bounded domain, δ(x)=dist(x,∂Ω)δ(x)=dist(x,∂Ω), λ>0λ>0, p≥1p≥1, and gg is either a nondecreasing function having a sublinear growth or g(u)=ug(u)=u. The degenerate character of our problem is also given by the potential a(δ(x))a(δ(x)) which may vanish at the boundary ∂Ω∂Ω. Under some suitable assumptions on gg, aa, and λλ, we establish the existence and uniqueness of a classical solution and we determine its asymptotic profile as t→∞t→∞. If g(u)=ug(u)=u and aa satisfies ∫01s/a(s)ds<∞, we also provide a blow-up result as λλ approaches the first eigenvalue λ1λ1 of the Laplace operator −Δ−Δ.