Existence and supercontractive estimates for parabolic-elliptic systems

In this paper we study existence and regularity of the solutions of the following parabolic-elliptic system of partial differential equations {\renewcommand{\arraystretch}{1.2} $$ \left\{ \begin{array}{cl} u_{t} - {\rm div}(A(x,t)\D u) = -{\rm div}(u M(x) \D \psi) & \mbox{in $\Omega \times (0,T)$,} \\ -{\rm div}( M(x) \D \psi) = |u|^{\theta} & \mbox{in $\Omega \times (0,T)$,}\\ \psi(x,t) = 0 & \mbox{on $\partial\Omega \times (0,T)$,}\\ u(x,t) = 0 & \mbox{on $\partial\Omega \times (0,T)$,} \\ u(x,0) = u_{0}(x) & \mbox{in $\Omega$} \end{array} \right. $$} where $\theta \in (0,1)$, $\Omega$ 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 $u M(x) \D \psi$ in the divergence term of the first equation is not regular enough, there exist solutions that immediately improve their regularity and belong to every Lebesgue space. Finally, we study the behavior in time of such regular solutions and we prove estimates that describe their blow-up for $t$ near zero.