Adhesion and volume filling in one-dimensional population dynamics under no-flux boundary condition

Abstract

We study the (generalized) one-dimensional population model developed by Anguige and Schmeiser [J. Math. Biol. 58 (3) (2009), 395–427], which reflects cell–cell adhesion and volume filling under no-flux boundary condition. In this generalized model, depending on the adhesion and volume filling parameters $\alpha ,\beta \in [0,1]$, the resulting equation is classified into six types. Among these, we focus on the type exhibiting strong effects of both adhesion and volume filling, which results in a class of advection–diffusion equations of the forward–backward–forward type. For five distinct cases of initial maximum, minimum, and average population densities, we derive the corresponding patterns for the global behavior of weak solutions to the initial and no-flux boundary value problem. Due to the presence of a negative diffusion regime, we indeed prove that the problem is ill-posed and admits infinitely many global-in-time weak solutions, with the exception of one specific case of the initial datum. This nonuniqueness is inherent in the method of convex integration that we use to solve the Dirichlet problem of a partial differential inclusion arising from the ill-posed problem.