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: \begin{equation*} \begin{cases} u_t - \operatorname{div}(A(x, t) \nabla u) = -\operatorname{div}(u M(x) \nabla ψ) + f(x, t) & $\text{in }$ Ω_T, \ -\operatorname{div}(M(x) \nabla ψ) = |u|^θ& $\text{in }$ Ω_T, \ ψ(x, t) = 0 & $\text{on }$ \partial Ω\times (0, T), \ u(x, t) = 0 & $\text{on }$ \partial Ω\times (0, T), \ u(x, 0) = 0 & $\text{in }$ Ω. \end{cases} \end{equation*} Here, $\mathrm{\Omega }$ is an open and bounded subset of ${\mathbb{R}}^N$, $N>2$, $\theta \in (0,\frac{2}{N})$, $0<T<+\mathrm{\infty }$ and ${\mathrm{\Omega }}_T=\mathrm{\Omega }\times (0,T)$. We prove existence results for data $f\in L^1({\mathrm{\Omega }}_T)$ and a corresponding increase in summability that obeys the $L^p$-regularity theorems for parabolic equations proved by Aronson-Serrin and by Boccardo-Dall’Aglio-Gallouët-Orsina. In particular, despite the term $uM(x)\mathrm{\nabla }\psi$ not being regular enough (since it only belongs to $L^2({\mathrm{\Omega }}_T)$), the solution $u$ belongs to $L^s({\mathrm{\Omega }}_T)\cap L^q(0,T;W_0^{1,q}(\mathrm{\Omega }))$ for suitable $s>1$ and $q>1$.