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Bounded weak solutions for Keller-Segel equations with generalized diffusion and logistic source via an unbalanced Optimal Transport splitting scheme

Kyungkeun Kang, Hwa Kil Kim, Geuntaek Seo

TL;DR

The paper develops a global existence theory for bounded weak solutions to a Keller-Segel system with generalized diffusion and logistic damping by employing an unbalanced optimal transport splitting scheme. By decomposing the dynamics into a Wasserstein gradient flow for diffusion and a Fisher-Rao gradient flow for the logistic reaction, the authors construct approximate solutions via alternating JKO steps and obtain uniform $L^{\infty}$ bounds under a chi-threshold ${\chi_*}$ that depends on the diffusion-drift balance and the initial density. They prove the limit passage to a weak solution, establish $1/2$-Hölder continuity in the Wasserstein-Fisher-Rao metric, and derive gradient estimates that ensure regularity and boundedness globally in time; the chi-threshold is quantified in terms of the degradation exponent $r$, the logistic parameters $\alpha,\beta$, and the initial max-density. The analysis extends the gradient-flow methodology for KS-type systems to the unbalanced OT setting and demonstrates how logistic damping can prevent blow-up under precise parameter regimes, with potential implications for related reaction-diffusion systems. The approach also accommodates variations of the elliptic equation linking $\rho$ and the chemoattractant $c$, highlighting the robustness of the unbalanced JKO framework for parabolic-elliptic KS models.

Abstract

We consider a parabolic-elliptic type of Keller-Segel equations with generalized diffusion and logistic source under homogeneous Neumann-Neumann boundary conditions. We construct bounded weak solutions globally in time in an unbalanced optimal transport framework, provided that the magnitude of the chemotactic sensitivity can be restricted depending on parameters. In the case of subquadratic degradation of the logistic source, we quantify the chemotactic sensitivity, in particular, in terms of the power of degradation and the pointwise bound of the initial density.

Bounded weak solutions for Keller-Segel equations with generalized diffusion and logistic source via an unbalanced Optimal Transport splitting scheme

TL;DR

The paper develops a global existence theory for bounded weak solutions to a Keller-Segel system with generalized diffusion and logistic damping by employing an unbalanced optimal transport splitting scheme. By decomposing the dynamics into a Wasserstein gradient flow for diffusion and a Fisher-Rao gradient flow for the logistic reaction, the authors construct approximate solutions via alternating JKO steps and obtain uniform bounds under a chi-threshold that depends on the diffusion-drift balance and the initial density. They prove the limit passage to a weak solution, establish -Hölder continuity in the Wasserstein-Fisher-Rao metric, and derive gradient estimates that ensure regularity and boundedness globally in time; the chi-threshold is quantified in terms of the degradation exponent , the logistic parameters , and the initial max-density. The analysis extends the gradient-flow methodology for KS-type systems to the unbalanced OT setting and demonstrates how logistic damping can prevent blow-up under precise parameter regimes, with potential implications for related reaction-diffusion systems. The approach also accommodates variations of the elliptic equation linking and the chemoattractant , highlighting the robustness of the unbalanced JKO framework for parabolic-elliptic KS models.

Abstract

We consider a parabolic-elliptic type of Keller-Segel equations with generalized diffusion and logistic source under homogeneous Neumann-Neumann boundary conditions. We construct bounded weak solutions globally in time in an unbalanced optimal transport framework, provided that the magnitude of the chemotactic sensitivity can be restricted depending on parameters. In the case of subquadratic degradation of the logistic source, we quantify the chemotactic sensitivity, in particular, in terms of the power of degradation and the pointwise bound of the initial density.
Paper Structure (11 sections, 13 theorems, 215 equations)

This paper contains 11 sections, 13 theorems, 215 equations.

Table of Contents

  1. Introduction
  2. Keller-Segel models & gradient flows approach
  3. Main results
  4. Preliminaries
  5. Splitting JKO scheme
  6. Wasserstein JKO step
  7. Fisher-Rao JKO step
  8. Approximate weak solutions: Construction and $L^\infty$-estimates
  9. Proof of Theorem \ref{['main']}
  10. Appendix
  11. Proofs of Proposition \ref{['CS_theorem']}

Key Result

Theorem 1.5

Suppose that ${ \Omega }$ is a strictly convex bounded domain in ${ \mathbb{R} }^d$ with smooth boundary and assume that $U$ and $F$ are functions satisfying Assumption assumptionsUF. Let $\rho_0 \in L^\infty({ \Omega })$. If $0< \chi < \chi_*$, there exists a weak solution of KSL, namely $\rho$. where $C>0$ is a constant independent of $T$.

Theorems & Definitions (38)

  • Remark 1.2
  • Remark 1.3
  • Definition 1.4
  • Theorem 1.5
  • Remark 1.6
  • Remark 1.7
  • Remark 1.8
  • Remark 1.9
  • Definition 2.1
  • Theorem 2.2: Benamou-Brenier formula
  • ...and 28 more