In the first chapter, we study two cross-diffusion systems. We first consider a population dynamics model, where cross effects in the interactions between the different species are modeled by non-local operators. We prove the well-posedness of the corresponding system for any space dimension. Then, we focus on a cross-diffusion system which arises as the fast-reaction limit system in a classical system for the chemical reaction $C_1+C_2\rightleftharpoons C_3$. We prove the convergence when $k\rightarrow +\infty$ of the solution of the system with finite reaction speed $k$ to a global solution of the limit system.
The second chapter contains new global existence results for some reaction-diffusion systems. For networks of elementary chemical reactions of the type $C_i+C_j \rightleftharpoons C_k $ and under Mass Action Kinetics assumption, we prove the existence and uniqueness of global strong solutions, for space dimensions $N\leq 5$ in the semi-linear case, and $N\leq 3$ in the quasi-linear case. We also prove the existence of global weak solutions for a class of parabolic quasi-linear systems with at most quadratic non-linearities and with initial data that are only assumed to be nonnegative and integrable.
In the last chapter, we generalize a global well-posedness result for reaction-diffusion systems whose nonlinearities have a ``triangular'' structure, for which we now take into account advection terms and time and space dependent diffusion coefficients. The latter result is then used in a Leray-Schauder fixed point argument to prove the existence of global solutions in a diffusion-electromigration system.
