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Asymptotic Stability in a free boundary model of cell motion
We introduce a free boundary model of the onset of motion of a living cell (e.g. a keratocyte) driven by myosin contraction, with focus on a transition from unstable radial stationary states to stable asymmetric moving states. This model generalizes a previous 1D model (Truskinovsky et al.) by combining a Keller-Segel model, a Hele-Shaw boundary condition and the Young-Laplace law with a nonlocal regularizing term, which precludes blow-up or collapse by ensuring that membrane-cortex interaction is sufficiently strong. We show that this model has a family of asymmetric traveling wave solutions bifurcating from a family of stationary states. Our goal is to establish observable steady cell motion with constant velocity. Mathematically, this amounts to proving stability of the traveling wave solutions, which requires generalization of the standard notion of stability. Our main result is establishing nonlinear asymptotic stability of traveling solutions. To this end, we derive an explicit asymptotic formula for the stability-determining eigenvalue from asymptotic expansions in small speed. This formula greatly simplifies the numerical computation of the sign of this eigenvalue and reveals the physics underlying onset of the cell motion and stability of moving states. If time permits, we will discuss work in progress on fingering instability in multicellular tissue spreading