Existence and regularity of solutions to parabolic-elliptic nonlinear systems

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 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, $Ω$ is an open and bounded subset of $\mathbb R^N$, $N>2$, $θ\in(0,\frac{2}{N})$, $0<T<+\infty$ and $Ω_T=Ω\times(0,T)$. We prove existence results for data $f\in L^1(Ω_T)$ 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 $u M(x)\nablaψ$ not being regular enough (since it only belongs to $L^2(Ω_T)$), the solution $u$ belongs to $L^s(Ω_T)\cap L^q(0,T;W^{1, q}_0(Ω))$ for suitable $s\>1$ and $q>1$.

• The paper introduces strong existence and boundedness results under high summability conditions, providing C^0, L^2, and L∞ estimates for u and ψ.
• The paper establishes a weak solution framework for intermediate data, detailing precise thresholds where solution regularity shifts from strong to distributional formulations.
• The paper develops an entropy solution theory for minimal integrability cases, extending the existence framework to handle singular data and degenerate nonlinear couplings.

Existence and Regularity in Parabolic-Elliptic Nonlinear Systems

Problem Statement and Context

The paper "Existence and regularity of solutions to parabolic-elliptic nonlinear systems" (2604.16100) addresses the analytical properties of a class of nonlinear parabolic-elliptic PDE systems with discontinuous coefficients. The prototype system is given by

$$\begin{cases} u_t - \operatorname{div}(A(x, t) \nabla u) = -\operatorname{div}(u M(x) \nabla \psi) + f(x, t) & \text{in } \Omega_T, \ -\operatorname{div}(M(x) \nabla \psi) = |u|^\theta & \text{in } \Omega_T, \ \psi(x, t) = 0,\ u(x, t) = 0 & \text{on } \partial\Omega \times (0, T), \ u(x, 0) = 0 & \text{in } \Omega, \end{cases}$$

where $\Omega \subset \mathbb{R}^N$ ($N > 2$), $T > 0$, and $\theta \in (0, \frac{2}{N})$. The coefficients $A$ and $M$ are measurable, uniformly elliptic matrix-valued functions, and $f$ is a source term in $L^1(\Omega_T)$. The structure is motivated by degenerate Keller-Segel type chemotaxis models, but with parabolic-elliptic coupling, sublinear nonlinearity, discontinuous coefficients, and possibly singular data.

The work systematically details existence, uniqueness, and regularity thresholds under minimal integrability assumptions on the data, with particular attention to both standard weak solutions and entropy-type solutions for highly singular forcing terms.

Main Results and Methodological Innovations

Existence and Regularity Results

The central contributions of the paper can be summarized as follows:

• Strong existence and boundedness for high summability data: If $f \in L^m(\Omega_T)$ with $\Omega \subset \mathbb{R}^N$0, then for suitable exponents, solutions $\Omega \subset \mathbb{R}^N$1 exist in high regularity classes, specifically $\Omega \subset \mathbb{R}^N$2 and $\Omega \subset \mathbb{R}^N$3. The strong form of the initial condition is satisfied.
• Weak solution framework for intermediate summability: For data $\Omega \subset \mathbb{R}^N$4 in $\Omega \subset \mathbb{R}^N$5 with $\Omega \subset \mathbb{R}^N$6 in $\Omega \subset \mathbb{R}^N$7, existence is retained but the regularity of $\Omega \subset \mathbb{R}^N$8 correspondingly weakens, with $\Omega \subset \mathbb{R}^N$9 for suitable exponents. The initial condition is only attained in a weak sense.
• Threshold for distributional solutions: For $N > 2$0, one regains enough regularity ($N > 2$1) for the convection term $N > 2$2 to belong to $N > 2$3, making the passage from approximate to weak solutions rigorous in the distributional setting.
• Entropy solution regime with minimal summability: For $N > 2$4 with $N > 2$5, distributional solutions cannot be constructed directly due to lack of integrability; instead, the entropy solution framework is applied. Here, $N > 2$6 for every $N > 2$7.

These results systematically chart the threshold phenomena as integrability of the forcing drops, with precise relations to the parabolic $N > 2$8-regularity theory (Aronson-Serrin and subsequent developments) adapted to the degenerate parabolic-elliptic, strongly coupled, and discontinuous setting.

A Priori Estimates and Approximation Methodology

The backbone of the proofs is a refined approximation scheme based on truncations and fixed-point theory (Schauder), coupled with delicate a priori estimates, including:

• Uniform $N > 2$9 bounds for approximations, yielding positivity and weak integrability for all data $T > 0$0.
• Stampacchia truncation techniques and parabolic Gagliardo-Nirenberg inequalities, generating higher summability for nonlinear terms.
• Strong $T > 0$1-type bounds in the data-regular regime, and breakdown of uniform bounds as the data becomes more singular.
• Propagation of regularity from the elliptic subproblem for $T > 0$2 to the full parabolic-elliptic system.
• The passage to the limit is conducted via compactness theorems in $T > 0$3 spaces (Simon’s lemma), leveraging strong convergence of truncates (Porretta-type compactness) and lower semicontinuity.

Notably, the approach accounts for the lack of coercivity and lower-order terms' degeneracy, handling the ill-posedness introduced by $T > 0$4 for low regularity $T > 0$5.

Entropy Solution Theory

For singular data, the entropy formulation is rigorously developed, allowing solutions even when the convection/diffusive flux cannot be defined distributionally. The proof adapts arguments from the theory of scalar conservation laws and degenerate parabolic equations, ensuring the sequence of approximations converges almost everywhere and in $T > 0$6, and that the entropy inequalities are closed in the limit.

Furthermore, the entropy solution framework is shown to generalize the weak/distributional solutions as the data becomes more regular, thus providing a unified existence theory spanning the full range of $T > 0$7 data.

Implications and Theoretical Significance

The analytic framework introduced in this work bridges gaps in the existence theory for parabolic-elliptic systems with discontinuous coefficients and nonlinear, possibly degenerate, couplings. The results clarify the regularity threshold for passage from entropy solutions to strong distributional solutions, contingent on the integrability of the source.

This comprehensive analysis is particularly relevant for chemotaxis-type systems, degenerate diffusion models, and cross-diffusion models in mathematical biology and physics, where source data may naturally be measures or only integrable and coefficients may be non-smooth. The methods extend to broader classes of degenerate parabolic PDEs with sublinear or critical growth, and suggest directions for future research on systems with further singular nonlinear interactions or inhomogeneous boundary data.

The entropy solution concept employed for the highly singular regime offers a robust analytical foundation that potentially informs the study of nonlinear evolution in random media, problems with rough initial data, and PDEs with measure data, where standard distributional solution techniques fail.

Future directions suggested by this work include the investigation of uniqueness, stability, and qualitative behavior of solutions under weaker structural conditions, as well as extensions to nonlocal or anisotropic systems and multiple interacting species.

Conclusion

This paper establishes a comprehensive existence and regularity theory for parabolic-elliptic nonlinear systems with minimal data regularity, discontinuous coefficients, and nonlinear couplings. Through technically sophisticated approximation and compactness arguments, a complete map of solution regularity as a function of data integrability is provided, including rigorous introduction of entropy solutions for the most singular regime. These results deepen the understanding of regularity phenomena in degenerate, strongly coupled PDE systems and provide foundational tools for further study in nonlinear parabolic analysis and related applied fields.