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The impact of Robin boundary condition on a chemotaxis-consumption-growth model

Piotr Knosalla

TL;DR

The paper investigates a parabolic-elliptic chemotaxis-consumption-growth system on a bounded domain with a non-homogeneous Robin boundary condition for the chemoattractant. It employs a transformation and fixed-point methods (Leray-Schauder) to establish the existence of a positive stationary state $(U,V)$, along with uniform bounds and, for small boundary data, uniqueness. Global existence and uniform boundedness of solutions are proven via the Moser-Alikakos iteration, enabling the analysis of long-time behavior. Under small Robin data, the solutions converge exponentially to the positive stationary state in appropriate norms, establishing robust asymptotic stability across dimensions. This work extends known results to Robin boundary conditions and provides explicit dependence of the stationary and convergence properties on the boundary parameter $\gamma$.

Abstract

We investigate a parabolic-elliptic chemotaxis-consumption-growth system with a Robin boundary condition imposed on the signal. First, we analyse the steady state problem, then we show that the solutions of the system are global and uniformly bounded in time in any space dimension. Next, under smallness assumption on the boundary data, we show that the solutions converge to non-constant steady states as time tends to infinity.

The impact of Robin boundary condition on a chemotaxis-consumption-growth model

TL;DR

The paper investigates a parabolic-elliptic chemotaxis-consumption-growth system on a bounded domain with a non-homogeneous Robin boundary condition for the chemoattractant. It employs a transformation and fixed-point methods (Leray-Schauder) to establish the existence of a positive stationary state , along with uniform bounds and, for small boundary data, uniqueness. Global existence and uniform boundedness of solutions are proven via the Moser-Alikakos iteration, enabling the analysis of long-time behavior. Under small Robin data, the solutions converge exponentially to the positive stationary state in appropriate norms, establishing robust asymptotic stability across dimensions. This work extends known results to Robin boundary conditions and provides explicit dependence of the stationary and convergence properties on the boundary parameter .

Abstract

We investigate a parabolic-elliptic chemotaxis-consumption-growth system with a Robin boundary condition imposed on the signal. First, we analyse the steady state problem, then we show that the solutions of the system are global and uniformly bounded in time in any space dimension. Next, under smallness assumption on the boundary data, we show that the solutions converge to non-constant steady states as time tends to infinity.
Paper Structure (5 sections, 10 theorems, 66 equations)

This paper contains 5 sections, 10 theorems, 66 equations.

Table of Contents

  1. Introduction
  2. Main results
  3. Stationary problem
  4. Global existence
  5. Asymptotic behavior of solutions

Key Result

Theorem 2.2

There exists a positive solution $(U,V)\in(C^{2,\alpha}(\overline{\Omega}))^2$ of (systSPE). Any such solution satisfy Moreover there exists a constant $\gamma^\star$ such that for $\gamma\in(0,\gamma^\star)$ the pair $(U,V)$ is a unique positive solution of systSPE.

Theorems & Definitions (21)

  • Remark 2.1
  • Theorem 2.2
  • Theorem 2.3
  • Theorem 2.4
  • Lemma 3.1
  • Lemma 3.2
  • proof
  • proof : Proof of Theorem \ref{['systSPE:ex:thm']}
  • Lemma 3.3
  • proof
  • ...and 11 more