  We study a class of parabolic quasilinear systems, in which the diffusion
matrix is not uniformly elliptic, but satisfies a Petrovskii condition of
positivity of the real part of the eigenvalues. Local wellposedness is known
since the work of Amann in the 90s, by a semi-group method. We revisit these
results in the context of Sobolev spaces modelled on L^2 and exemplify our
method with the SKT system, showing the existence of local, non-negative,
strong solutions.
