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Boundary symmetry breaking via logistic damping in a chemotaxis-growth system

Yiren Chen, Padi Fuster Aguilera, Vincent Martinez, Kun Zhao

Abstract

We establish global stability for a chemotaxis-growth model with logarithmic sensitivity under dynamic Dirichlet boundary conditions on a 1D domain. We analyze both parabolic-parabolic and parabolic-hyperbolic systems. The key challenge is handling time-dependent boundary data for the unknown functions. We overcome this by introducing dynamic reference profiles which suitably interpolate boundary values. Using an expanded entropy functional measuring deviation from these profiles, we prove energy estimates the uniform boundedness of solutions and global asymptotic stability of perturbations.

Boundary symmetry breaking via logistic damping in a chemotaxis-growth system

Abstract

We establish global stability for a chemotaxis-growth model with logarithmic sensitivity under dynamic Dirichlet boundary conditions on a 1D domain. We analyze both parabolic-parabolic and parabolic-hyperbolic systems. The key challenge is handling time-dependent boundary data for the unknown functions. We overcome this by introducing dynamic reference profiles which suitably interpolate boundary values. Using an expanded entropy functional measuring deviation from these profiles, we prove energy estimates the uniform boundedness of solutions and global asymptotic stability of perturbations.
Paper Structure (19 sections, 4 theorems, 200 equations)

This paper contains 19 sections, 4 theorems, 200 equations.

Table of Contents

  1. Introduction
  2. Parabolic-Parabolic System under Dynamic Dirichlet B.C.
  3. Proof of Theorem \ref{['thm1']} when $\gamma=1$
  4. Entropy-based estimates
  5. $L^2$-estimates
  6. $H^1$-estimates
  7. Asymptotic stability
  8. Proof of Theorem \ref{['thm1']} when $\gamma >1$
  9. Entropy-based estimates
  10. $L^2$-estimates
  11. Parabolic-Hyperbolic System under Dynamic Dirichlet B.C.
  12. Proof of Theorem \ref{['thm2']} when $\gamma=1$
  13. Proof of Theorem \ref{['thm2']} when $\gamma\geqslant 2$
  14. Recovery of regular dissipation for ${\tilde{u}}$
  15. $H^1$-estimates
  16. ...and 4 more sections

Key Result

Theorem 2.1

Consider the initial-boundary value problem 1.1 with $\gamma\geqslant 1$. Suppose that the initial data satisfy $u_0 > 0$, $u_0,v_0 \in H^1((a,b))$, and are compatible with the boundary conditions. Assume that the boundary data $\alpha_1$, $\alpha_2$, $\beta_1$, $\beta_2$ are smooth functions on $\m where $\underline{\alpha_1}$, $\underline{\alpha_2}>0$ are constants. Then there exists a unique so

Theorems & Definitions (7)

  • Theorem 2.1
  • Remark 2.1
  • Lemma 2.2
  • proof
  • Theorem 3.1
  • Remark 3.1
  • Lemma 3.2: ZLMZ