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Convergence, concentration and critical mass phenomena in a model of cell motion with boundary signal production

Nicolas Meunier, Philippe Souplet

TL;DR

This work analyzes a nonlinear, nonlocal boundary-driven PDE for cytoplasmic signaling c in a bounded cell-domain Ω, modeling actin-driven advection coupled to boundary cue production. The authors establish a comprehensive solvability theory, detailing local existence in low-regularity spaces, mass-conserving structure, and entropy-based Liapunov functionals. They prove a sharp, dimension-independent critical mass phenomenon for the quadratic case (f(c) ~ c) akin to Keller–Segel, and show global existence with exponential convergence to a steady state under subquadratic or small-data regimes, while revealing finite-time blow-up when the nonlinearity is quadratic or superquadratic and mass concentrates near the boundary. The results include precise blow-up profiles and asymptotics, a detailed epsilon-regularity framework, and a cylinder-specific local theory, all reinforcing the model’s relevance to persistent versus Brownian-like cell migration modes and supporting future studies of traveling waves with saturated nonlinearities.

Abstract

We consider a model of cell motion with boundary signal production which describes some aspects of eukaryotic cell migration. Generic polarity markers located in the cell are transported by actin which they help to polymerize. This leads to a problem whose mathematical novelty is the nonlinear and nonlocal destabilizing term in the boundary condition. We provide a detailed study of the qualitative properties of this model, namely local and global existence, convergence and blow-up of solutions. We start with a complete analysis of local existence-uniqueness in Lebesgue spaces. This turns out to be particularly relevant, in view of the mass conservation property and of the existence of $L^p$ Liapunov functionals, also obtained in this paper. With the help of this local theory, we next study the global existence and convergence of solutions. In particular, in the case of quadratic nonlinearity, for any space dimension, we find an explicit, sharp mass threshold for global existence vs.~finite time blow-up of solutions. The proof is delicate, based on the possiblity to control the solution by means of the entropy function via an $\eps$-regularity type argument. This critical mass phenomenon is somehow reminiscent of the well-known situation for the $2d$ Keller-Segel system. For nonlinearitities with general power growth, under a suitable smallness condition on the initial data, we show that solutions exist globally and converge exponentially to a constant. As for the possibility of blow-up for large initial data, it turns out to occur only for nonlinearities with quadratic or superquadratic growth, whereas all solutions are shown to be global and bounded in the subquadratic case, thus revealing the existence of a sharp critical exponent for blow-up. Finally, we analyse some aspects of the blow-up asymptotics of solutions in time and space.

Convergence, concentration and critical mass phenomena in a model of cell motion with boundary signal production

TL;DR

This work analyzes a nonlinear, nonlocal boundary-driven PDE for cytoplasmic signaling c in a bounded cell-domain Ω, modeling actin-driven advection coupled to boundary cue production. The authors establish a comprehensive solvability theory, detailing local existence in low-regularity spaces, mass-conserving structure, and entropy-based Liapunov functionals. They prove a sharp, dimension-independent critical mass phenomenon for the quadratic case (f(c) ~ c) akin to Keller–Segel, and show global existence with exponential convergence to a steady state under subquadratic or small-data regimes, while revealing finite-time blow-up when the nonlinearity is quadratic or superquadratic and mass concentrates near the boundary. The results include precise blow-up profiles and asymptotics, a detailed epsilon-regularity framework, and a cylinder-specific local theory, all reinforcing the model’s relevance to persistent versus Brownian-like cell migration modes and supporting future studies of traveling waves with saturated nonlinearities.

Abstract

We consider a model of cell motion with boundary signal production which describes some aspects of eukaryotic cell migration. Generic polarity markers located in the cell are transported by actin which they help to polymerize. This leads to a problem whose mathematical novelty is the nonlinear and nonlocal destabilizing term in the boundary condition. We provide a detailed study of the qualitative properties of this model, namely local and global existence, convergence and blow-up of solutions. We start with a complete analysis of local existence-uniqueness in Lebesgue spaces. This turns out to be particularly relevant, in view of the mass conservation property and of the existence of Liapunov functionals, also obtained in this paper. With the help of this local theory, we next study the global existence and convergence of solutions. In particular, in the case of quadratic nonlinearity, for any space dimension, we find an explicit, sharp mass threshold for global existence vs.~finite time blow-up of solutions. The proof is delicate, based on the possiblity to control the solution by means of the entropy function via an -regularity type argument. This critical mass phenomenon is somehow reminiscent of the well-known situation for the Keller-Segel system. For nonlinearitities with general power growth, under a suitable smallness condition on the initial data, we show that solutions exist globally and converge exponentially to a constant. As for the possibility of blow-up for large initial data, it turns out to occur only for nonlinearities with quadratic or superquadratic growth, whereas all solutions are shown to be global and bounded in the subquadratic case, thus revealing the existence of a sharp critical exponent for blow-up. Finally, we analyse some aspects of the blow-up asymptotics of solutions in time and space.
Paper Structure (21 sections, 28 theorems, 352 equations)

This paper contains 21 sections, 28 theorems, 352 equations.

Table of Contents

  1. Introduction
  2. Motivations and aims of the paper
  3. Main results
  4. Local and global existence and convergence results
  5. Blow-up
  6. Asymptotic behavior of blow-up solutions
  7. Contributions of this work
  8. Outline
  9. Model construction and first properties
  10. Biological justification - Model construction
  11. Conservation of the marker content, nondimensionalization and scaling
  12. Preliminaries to the existence-uniqueness proofs
  13. Proof of Theorem \ref{['thm1']}
  14. Proof of Theorem \ref{['thm1']} and Proposition \ref{['thm2prop']}: existence
  15. Proof of Theorem \ref{['thm1']} and Proposition \ref{['thm2prop']}: uniqueness
  16. ...and 6 more sections

Key Result

Theorem 1

Let $1\leq m\leq p<\infty$ and assume $f(c)=|c|^{m-1}c$ and $c_0\in L^p(\Omega)$.

Theorems & Definitions (68)

  • Definition 1.1
  • Theorem 1
  • Remark 1.1
  • Theorem 2
  • Theorem 3
  • Theorem 4
  • Remark 1.2
  • Theorem 5
  • Remark 1.3
  • Theorem 6
  • ...and 58 more