Analysis of a Cross-Nonlinear Porous-Medium System Modeling Pressure-Driven Cell Population Dynamics

TL;DR

This paper introduces a complex cross-diffusion model for cell population dynamics driven by pressure, combining nonlinear interactions and porous-medium diffusion, with rigorous analysis of solutions and pattern formation.

Contribution

It presents a novel fully nonlinear cross-diffusion model with theoretical proofs of well-posedness, non-negativity, and support invariance, advancing understanding of tissue growth and pattern formation.

Findings

Proved local well-posedness for nonnegative solutions.

Established conditions for solution regularity and non-negativity.

Identified finite-time blow-up scenarios and invariant support domains.

Abstract

In this work, we introduce a cross-diffusion model that couples population density and occupied area to investigate how internal pressure drives growth and motility. By blending nonlinear nonlocal interactions with porous-medium diffusion and an antidiffusive pressure term, the model captures the two-way feedback between local density fluctuations and tissue expansion or contraction. Building on Shraiman's area-growth paradigm, we enrich the framework with density-dependent spreading at the population boundary and a novel cross-diffusion term, yielding fully nonlinear transport in both equations. We prove local well-posedness for nonnegative solutions in Sobolev spaces and, under higher regularity, show both density and area remain nonnegative. Uniqueness follows when the initial density's square root lies in $H^2$, even if density vanishes on parts of the domain. We also exhibit initial data that induce finite-time blow-up, highlighting potential singularity formation. Finally, we establish that the density's spatial support remains invariant and characterize the co-evolution of occupied area and population density domains, offering new insights into pattern formation and mass transport in biological tissues.