Local well-posedness for a novel nonlocal model for cell-cell adhesion via receptor binding

TL;DR

This paper proves local well-posedness for a complex nonlocal cell adhesion model involving integro-PDEs and a novel nonlinear integral equation, using fixed point methods and the Kantorovich-Rubinstein norm.

Contribution

It introduces a new analytical framework for a nonlocal cell adhesion model, including the first use of the Kantorovich-Rubinstein norm in this context.

Findings

Established local well-posedness for the nonlocal model.

Decoupled the system and applied Banach's fixed point theorem.

First application of Kantorovich-Rubinstein norm to a nonlinear integral equation.

Abstract

Local well-posedness is established for a highly nonlocal nonlinear diffusion-adhesion system for bounded initial values with small support. Macroscopic systems of this kind were previously obtained by the authors through upscaling in [32] and can account for the effect of microscopic receptor binding dynamics in cell-cell adhesion. The system analysed here couples an integro-PDE featuring degenerate diffusion of the porous media type and nonlocal adhesion with a novel nonlinear integral equation. The approach is based on decoupling the system and using Banach's fixed point theorem to solve each of the two equations individually and subsequently the entire system. The main challenge of the implementation lies in selecting a suitable framework. One of the key results is the local well-posedness for the integral equation with a Radon measure as a parameter. The analysis of this equation utilizes the Kantorovich-Rubinstein norm, marking the first application of this norm in handling a nonlinear integral equation.