Abstract
We are interested in quasi-linear parabolic systems, in which the diffusion matrix is not uniformly elliptic, but verifies a condition, known as the Petrovskii condition, of positivity of the real part of the eigenvalues. The locally well-posedness in $W^{1,p}$ has been known since Amann's work in the years 90, by a method of semi-groups. We will revisit these results in the context of Sobolev spaces modeled on $L^2$ : in particular, we will see that while the Petrovskii condition may not be sufficient to ensure exponential decay in time for systems of ordinary differential equations, the quasilinear structure nevertheless ensures the well-posedness of the system. This work is in collaboration with Ayman Moussa.
