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

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 method, and carefully combining various estimations.