Bounded solutions to a quasilinear and singular parabolic equation with pp-Laplacian

In this paper, we study the existence of a positive local in time solution for the following singular nonlinear problem with homogeneous Dirichlet boundary conditions: {∂tu−Δpu=u−δ+f(x,u,∇u)in  (0,T)×Ω=QT,u=0  on  (0,T)×∂Ω,u>0in  QT,u(0,x)=u0≥0in  Ω, where ΩΩ stands for a regular bounded domain of RNRN, ΔpuΔpu is the pp-Laplacian defined by Δpu=div(|∇u|p−2|∇u|), 2≤p<∞2≤p<∞, δ>0δ>0 and T>0T>0. The nonlinear term f:Ω×R×RN⟶Rf:Ω×R×RN⟶R is a Carathéodory function satisfying the growth condition f(x,s,ξ)≤(asq−1+b)+c|ξ|p−pqfor a.a.  x∈Ω,s∈R+  and  |ξ|≥M where a,c,M>0a,c,M>0 and b≥0b≥0 are some constants and q∈[p,p∗)q∈[p,p∗) where p∗=pNN−p if p