Affiliations: Dipartimento di Scienze di Base e Applicate per l'Ingegneria, Sapienza Università di Roma, Via A. Scarpa 16, 00161 Roma, Italy. E-mails: [email protected], [email protected]
Abstract: We deal with the existence of nonnegative solutions to parabolic problems which are singular in the u variable whose model is ut−Δpu=f(x,t)(1/uθ+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 RN, N≥2, 0<T<+∞, θ>0, Δpu=div (|∇u|p−2∇u) with p>1. As far as the data are concerned, we assume f(x,t)∈Lr(0,T;Lm(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)/uθ, with θ>0, D>0, 1<q<p and f(x,t) as before.
Keywords: nonlinear parabolic equations, singular lower order terms, nonnegative solutions, existence
