Rigorous stability theory for segregated travelling waves

Establish a fully rigorous one-dimensional stability theory for segregated travelling waves beyond the formal argument presented in the paper, and develop a nonlinear stability theory for these waves in higher spatial dimensions.

Background

The paper studies segregated travelling waves in a pressure-based model of heterogeneous cell populations containing proliferative and non-proliferative cells with distinct mobility coefficients. It provides variational estimates for the one-dimensional wave speed and a formal perturbative argument indicating that stability depends on the relative mobilities of the two cell types.

The authors explicitly identify the absence of a fully rigorous one-dimensional stability theory as well as the lack of a nonlinear stability theory in higher dimensions. Resolving these issues would turn the formal and linear analyses presented in the paper into mathematically rigorous results and would clarify the stability properties of segregated waves beyond the settings treated analytically.