On some singular parabolic-elliptic systems with convection term

In this paper we study the existence and regularity of the solutions to the following singular parabolic-elliptic system of partial differential equations given by {ut−div(A(x,t)∇u)=−div(uM(x)∇ψ)+h(u)fin Ω×(0,T),−div(M(x)∇ψ)=uθin Ω×(0,T),ψ(x,t)=0on ∂Ω×(0,T),u(x,t)=0on ∂Ω×(0,T),u(x,0)=u0(x)in Ω,where Ω is an open and bounded subset of RN,N>2,θ∈(0,1),T>0, 0≤u0∈L1(Ω) and f is a positive function that belongs to L1(Ω×(0,T)); and h is a positive continuous function possibly singular at the origin. In particular, we show how the coupling between the equations in the system (i.e. the first singular parabolic equation and the second elliptic equation) gives rise to a regularizing effect, producing the existence of solutions to the system.