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Nonlinear Stability in a Free Boundary Model of Active Locomotion
Leonid Berlyand1, C Alex Safsten2, Lev Truskinovsky3
1Department of Mathematics and Huck Institute for Life Sciences, The Pennsylvania State University, University Park, USA.
Archive for Rational Mechanics and Analysis
|December 15, 2025
Summary
This study proves the stability of cell movement models. It establishes the asymptotic nonlinear stability of both static and dynamic solutions in active matter systems, advancing our understanding of cell self-propulsion.
Area of Science:
- Mathematical Biology
- Active Matter Physics
- Nonlinear Dynamics
Background:
- Cellular self-propulsion is often modeled using Keller-Segel systems with free boundaries.
- These active systems exhibit complex behaviors, including stationary and traveling wave solutions.
- Traveling waves mimic autonomous cell locomotion, representing propagating pulses.
Purpose of the Study:
- To provide the first proof of asymptotic nonlinear stability for both stationary and traveling wave solutions in a Keller-Segel model.
- To analyze the stability of static cells and dynamic, self-propelling cells.
- To develop a novel methodology applicable to non-self-adjoint operators in active matter.
Main Methods:
- Linear stability analysis using the spectral theorem for stationary solutions.
- Spectral methods combined with the Gearhart-Prüss-Greiner (GPG) theorem for traveling waves.
- Nonlinear stability proof via dominance of the linear part and Grönwall inequality.
Main Results:
- Established asymptotic nonlinear stability for stationary solutions using eigenvalue analysis.
- Proved linear stability for traveling waves despite non-self-adjoint linearized problems, using the GPG theorem.
- Demonstrated nonlinear stability for both solution types under appropriate parameter values.
Conclusions:
- The study offers the first rigorous proof of stability for both static and dynamic solutions in this class of cell movement models.
- The developed spectral and inequality-based methods are applicable to other active matter systems with non-self-adjoint operators.
- This work enhances the mathematical understanding of autonomous cell locomotion and active matter dynamics.
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