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Interface behavior for the solutions of a mass conserving free boundary problem modeling cell polarization

Anna Logioti, Barbara Niethammer, Matthias Röger, Juan J. L. Velázquez

Abstract

We consider a parabolic non-local free boundary problem that has been derived as a limit of a bulk-surface reaction-diffusion system which models cell polarization. In previous papers, we have established well-posedness of this problem and derived conditions on the initial data that imply continuity of the free boundary as $t\to 0$. In this paper we extend the qualitative study of the free boundary by considering axisymmetric data. Under additional monotonicity assumptions on the data we prove global continuity of the free boundary. On the other hand, if the initial data violate a "no-fattening" condition we show that the free boundary can oscillate as $t \to 0$.

Interface behavior for the solutions of a mass conserving free boundary problem modeling cell polarization

Abstract

We consider a parabolic non-local free boundary problem that has been derived as a limit of a bulk-surface reaction-diffusion system which models cell polarization. In previous papers, we have established well-posedness of this problem and derived conditions on the initial data that imply continuity of the free boundary as . In this paper we extend the qualitative study of the free boundary by considering axisymmetric data. Under additional monotonicity assumptions on the data we prove global continuity of the free boundary. On the other hand, if the initial data violate a "no-fattening" condition we show that the free boundary can oscillate as .
Paper Structure (8 sections, 13 theorems, 127 equations, 3 figures)

This paper contains 8 sections, 13 theorems, 127 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Background
  3. Preliminaries
  4. Global continuity result for axisymmetric solutions
  5. Monotonicity and non-degeneracy
  6. Proof of Theorem \ref{['main_thm']}
  7. An oscillatory solution to the classical parabolic obstacle problem
  8. Oscillatory solutions for a nonlocal obstacle problem on the real line

Key Result

Lemma 1.2

Consider $\Gamma=\mathcal{S}^2$, $1\leq p<\infty$ and an axisymmetric function $U\in W^{2,1}_p(\mathcal{S}^2\times (0,T))$ represented by some $u\in W^{2,1}_{p,\text{axs}}((-1,1)\times (0,T))$ as $U=u_{\text{axs}}$. Then $U$ solves 2eq:ob2 (with $u$ replaced by $U$) if and only if $u$ solves Moreover, for given axisymmetric data $U_0=u_{0,\text{axs}}\in H^2(\mathcal{S}^2)$ and $G=g_{\text{axs}}\i

Figures (3)

  • Figure 1: Main building block of the construction. The boundary of the support of $U$ is indicated by the oval line.
  • Figure 2: Points on the level $n$ (circles) and $n+1$ (triangles)
  • Figure 3: Graph of $g$ (solid) and of $u_0^-$ (dashed)

Theorems & Definitions (27)

  • Remark 1.1: Axisymmetric data
  • Lemma 1.2
  • proof
  • Theorem 2.2
  • Lemma 2.3
  • proof
  • Lemma 2.4
  • proof
  • Lemma 2.5
  • proof
  • ...and 17 more