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: (Formula presented.) Here, Ω is an open and bounded subset of RN, N>2, θ∈(0,2N) and 0<+∞. We prove existence results for data f∈L1(ΩT) and a corresponding increase in summability that obeys the Lp-regularity theorems for parabolic equations found in [1] and [5]. Moreover, despite the term uM(x)∇ψ not being regular enough, the solution u belongs to Ls(ΩT)∩Lq(0,T;W01,q(Ω)) for suitable s>1 and q>1.