Existence and supercontractive estimates for parabolic-elliptic systems

In this paper we study existence and summability of the solutions of the following parabolic-elliptic system of partial differential equations {ut - div(A(x, t)del u) = -div(uM(x)del psi) in Omega x (0, T), -div(M(x)del psi) = |u|theta in Omega x (0, T), psi(x, t) = 0 on partial derivative Omega x (0, T), u(x, t) = 0 on partial derivative Omega x (0, T), u(x, 0) = u(0)(x) in Omega where theta is an element of (0, 1), Omega is a bounded subset of R-N, N > 2, and T > 0. We will prove existence results for initial data u(0) in L-1(Omega). Moreover, despite the datum u(0) is assumed to be only a summable function and although the function uM(x)del psi in the divergence term of the first equation is not regular enough, there exist solutions that immediately improve their summability and belong to every Lebesgue space. Finally, we study the behavior in time of such solutions and we prove estimates that describe their blow-up for t near zero. (c) 2022 Elsevier Ltd. All rights reserved.