Adhesion and volume filling in one-dimensional population dynamics under Dirichlet boundary condition

TL;DR

This paper extends a 1D population model to include adhesion and volume filling, classifies the resulting equations, and proves the existence of infinitely many global solutions with complex behaviors under certain conditions.

Contribution

It introduces a new classification of the generalized model and establishes the existence of multiple weak solutions using convex integration techniques.

Findings

Existence of infinitely many global weak solutions.

Solutions exhibit fine-scale density mixtures and eventual decay.

Solutions become smooth and decay exponentially over time.

Abstract

We generalize the one-dimensional population model of Anguige \& Schmeiser [1] reflecting the cell-to-cell adhesion and volume filling and classify the resulting equation into the six types. Among these types, we fix one that yields a class of advection-diffusion equations of forward-backward-forward type and prove the existence of infinitely many global-in-time weak solutions to the initial-Dirichlet boundary value problem when the maximum value of an initial population density exceeds a certain threshold. Such solutions are extracted from the method of convex integration by M\"uller \& \v Sver\'ak [12]; they exhibit fine-scale density mixtures over a finite time interval, then become smooth and identical, and decay exponentially and uniformly to zero as time approaches infinity.