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

Hyung Jun Choi, Seonghak Kim, Youngwoo Koh

Research output: Contribution to journalArticlepeer-review

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 Müller & Šverák [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. TE check: Please check the reference citation in abstract.

Keywords

  • adhesion and volume filling
  • convex integration
  • forward-backward-forward type
  • partial differential inclusion
  • Population model

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