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arXiv · 2608.22844

Bifurcation structure and mesa pattern formation in a one-component nonlocal adhesion model with population pressure and degenerate mobility

Shimpei Makida·Hideki Murakawa
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Abstract

We analyze pattern formation from a homogeneous steady state in a one-component nonlocal adhesion model with population pressure and degenerate mobility. First, using linear stability analysis, we derive the instability threshold and a selection rule for the fastest-growing mode, and elucidate the mechanism by which the selected wavenumber shifts toward lower wavenumbers as the mean density increases. We then perform a weakly nonlinear analysis near the critical adhesion strength and derive an explicit expression for the Landau coefficient in the Stuart--Landau equation. This expression shows that the critical bifurcation is classified as supercritical or subcritical according to the mean density, the nonlinear exponent, and the second-harmonic response ratio of the kernel. Furthermore, we show that a large nonlinear exponent promotes a transition to subcriticality and confirm, through numerical bifurcation analysis and time-dependent simulations, a bifurcation structure with a fold point and the formation of mesa patterns. Finally, through the energy limit as \(m\to\infty\), we relate the observed mesa profiles to a capacity-constrained limiting structure.

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Shimpei Makida, Hideki Murakawa. 2026-08-24. Bifurcation structure and mesa pattern formation in a one-component nonlocal adhesion model with population pressure and degenerate mobility. https://arxiv.org/abs/2608.22844

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