Stability and uniqueness of bounded weak solutions to triangular degenerate cross-diffusion systems

TL;DR

This paper proves the continuous dependence and uniqueness of bounded weak solutions for a class of triangular degenerate cross-diffusion systems, including models for bacterial nutrient response, despite degeneracy challenges.

Contribution

It establishes uniqueness and continuous dependence for solutions to degenerate cross-diffusion systems using novel Lipschitz conditions and the $H^{-1}$ method.

Findings

Proved continuous dependence on initial data.

Established uniqueness of bounded weak solutions.

Applied the method to nutrient taxis models.

Abstract

The continuous dependence on the initial data and consequently the uniqueness of bounded weak solutions to a class of triangular reaction-cross-diffusion equations is shown. The class includes two-species doubly degenerate equations for nutrient taxis models describing the response of bacteria to nutrient conditions. The key difficulty is the lack of a gradient bound for the difference of the first component of the solution, due to the degeneracy. This issue is overcome by assuming a nonstandard Lipschitz-type condition, applying the $H^{-1}$ method, and carefully combining various estimations.