Existence and regularity of solutions to parabolic-elliptic nonlinear systems

TL;DR

This paper establishes existence and regularity results for solutions to a nonlinear parabolic-elliptic PDE system with discontinuous coefficients, demonstrating increased summability despite limited regularity of certain terms.

Contribution

It proves existence and enhanced summability of solutions to a complex PDE system with discontinuous coefficients, extending regularity results for such systems.

Findings

Solutions exist for data in L^1( abla ext{Omega}_T)

Solutions belong to L^s and L^q(W^{1,q}_0) spaces for suitable s, q

Despite limited regularity of the nonlinear term, solutions exhibit improved regularity

Abstract

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 \psi) + f(x, t) & \text{in } \Omega_T, \\ -\operatorname{div}(M(x) \nabla \psi) = |u|^\theta & \text{in } \Omega_T, \\ \psi(x, t) = 0 & \text{on } \partial \Omega \times (0, T), \\ u(x, t) = 0 & \text{on } \partial \Omega \times (0, T), \\ u(x, 0) = 0 & \text{in } \Omega. \end{cases} \end{equation*} Here, $\Omega$ is an open and bounded subset of $\mathbb R^N$, $N>2$, $\theta\in(0,\frac{2}{N})$, $0<T<+\infty$ and $\Omega_T=\Omega\times(0,T)$. We prove existence results for data $f\in L^1(\Omega_T)$ 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-Gallou\"et-Orsina. In particular, despite the term $u M(x)\nabla\psi$ not being regular enough (since it only belongs to $L^2(\Omega_T)$), the solution $u$ belongs to $L^s(\Omega_T)\cap L^q(0,T;W^{1, q}_0(\Omega))$ for suitable $s>1$ and $q>1$.