Existence and regularity of solutions to parabolic–elliptic nonlinear systems

In this paper, we study the existence and summability of the solutions to the following parabolic–elliptic system of partial differential equations with discontinuous coefficients: $${aligned} \{ {array}{cc} u_t-div(A(x, t) ∇ u)=-div(u M(x) ∇ ψ )+f(x,t) & in Ω _T, \\ -div(M(x) ∇ ψ )=|u|^θ & in Ω _T, \\ ψ (x, t)=0 & on ∂ Ω × (0, T), \\ u(x, t)=0 & on ∂ Ω × (0, T), \\ u(x, 0)=0 & in Ω {array}. {aligned}$$ u t - div ( A ( x , t ) ∇ u ) = - div ( u M ( x ) ∇ ψ ) + f ( x , t ) in Ω T , - div ( M ( x ) ∇ ψ ) = | u | θ in Ω T , ψ ( x , t ) = 0 on ∂ Ω × ( 0 , T ) , u (