Nonnegative solutions for a class of singular parabolic problems involving p-Laplacian

Abstract

We deal with the existence of nonnegative solutions to parabolic problems which are singular in the u variable whose model is

u_t−Δ_pu=f(x,t)(1/u^θ+1) in Ω×(0,T),

u(x,t)=0 on ∂Ω×(0,T),

u(x,0)=u₀(x) in Ω.

Here Ω is a bounded open subset of R^N, N≥2, 0<T<+∞, θ>0, Δ_pu=div (|∇u|^p−2∇u) with p>1.

As far as the data are concerned, we assume f(x,t)∈L^r(0,T;L^m(Ω)), with 1/r+N/pm<1, f(x,t)≥0 a.e. in Ω×(0,T) and u₀(x)≥0 a.e. in Ω.

We consider also the case where the right-hand side depends on the gradient of the solution. In this last case the model of the right-hand side is F(x,t,u,∇u)=(f(x,t)+D|∇u|^q)/u^θ, with θ>0, D>0, 1<q<p and f(x,t) as before.