Spectral stability of shock-fronted travelling waves under viscous relaxation

TL;DR

This paper rigorously proves the spectral stability of shock-fronted travelling waves in reaction-nonlinear diffusion PDEs with viscous relaxation, using geometric singular perturbation theory and eigenvalue problem analysis.

Contribution

It provides a rigorous stability proof for shock-fronted waves under viscous relaxation, extending geometric singular perturbation methods to eigenvalue problem control.

Findings

Shock-fronted travelling waves are spectrally stable for small relaxation parameter.

Eigenvalue problem of the regularised system is controlled by a reduced problem at zero relaxation.

The approach clarifies the relationship between geometric construction and wave stability analysis.

Abstract

Reaction-nonlinear diffusion partial differential equations can exhibit shock-fronted travelling wave solutions. Prior work by Yi et. al. (2021) has demonstrated the existence of such waves for two classes of regularisations, including viscous relaxation. Their analysis uses geometric singular perturbation theory: for sufficiently small values of a parameter $\varepsilon > 0$ characterising the `strength' of the regularisation, the waves are constructed as perturbations of a singular heteroclinic orbit. Here we show rigorously that these waves are spectrally stable for the case of viscous relaxation. Our approach is to show that for sufficiently small $\varepsilon>0$, the `full' eigenvalue problem of the regularised system is controlled by a (reduced) slow eigenvalue problem defined for $\varepsilon = 0$. In the course of our proof, we examine the ways in which this geometric construction complements and differs from constructions of other reduced eigenvalue problems that are known in the wave stability literature.