Perturbative global solutions of a large class of cross diffusion systems in any dimension

This article focuses on a large family of cross-diffusion systems of the form ${\partial }_{t}U-\Delta A\left(U\right)=0$, in dimension $d\in N^{⁎}$, and where $U\in R^2$. We show that under natural conditions on the nonlinearity A, those systems have a unique smooth (nonnegative for all components) solution when the initial data are small enough in a suitable norm.