2024/09/07 by Hyung Jun Choi, Choi, Hyung Jun, Seonghak Kim +3
Biochemistry, Genetics and Molecular Biology · Mathematics · Physics and Astronomy · #35D30 #35K59 #35M13 #92D25 #Analysis of PDEs (math.AP) #Diffusion and Search Dynamics #FOS: Mathematics #Stochastic processes and statistical mechanics #Theoretical and Computational Physics
paper · doi:10.48550/arxiv.2409.04689
openalex publication_date 2024/09/07 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
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 & \v Sverák [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.