Stability for nonlinear diffusive PDEs

This thesis focuses on well-posedness and stability estimates (i.e. continuous dependence with respect to the nonlinearities) for a family of cross-diffusion systems that includes several models used to describe cell diffusion in Mathematical Biology for one or more species. We discuss rigorous quantitative results for a class of (possibly degenerate) parabolic PDEs, including scalar equations as well as systems. The approach is typical of Analysis of PDEs, hence the concepts of weak solutions in Sobolev spaces, a priori estimates and well-posedness are crucial. Considering scalar, nonlinear diffusion, we extend the existing solvability, uniqueness, boundedness and stability results. Regarding non-degenerate models for multiple species, we work in the framework of systems with an entropy structure and a limited set of other assumptions that allow us to prove stability estimates, improving some of the existing results. Finally, we have been working on the multi-species, degenerate case, providing a continuous dependence estimate under suitable (but rather restrictive) assumptions. Two different types of numerical simulations are presented in order to complement the abstract results.