The Cauchy problem for quasi-linear parabolic systems revisited

We study a class of parabolic quasilinear systems, in which the diffusion matrix is not uniformly elliptic, but satisfies the Petrovskii condition (positivity of eigenvalues’ real part). Local well-posedness is known since the work of Amann in the 90s, by a semi-group method. We first revisit these results in the context of Sobolev spaces modelled on <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msup> <mml:mi>L</mml:mi> <mml:mn>2</mml:mn> </mml:msup> </mml:math> and then explore the endpoint Besov case <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msubsup> <mml:mi>B</mml:mi> <mml:mrow> <mml:mi>p</mml:mi> <mml:mo>,</mml:mo> <mml:mn>1</mml:mn> </mml:mrow> <mml:mrow> <mml:mi>d</mml:mi> <mml:mo>/</mml:mo> <mml:mi>p</mml:mi> </mml:mrow> </mml:msubsup> </mml:math> . We also exemplify our method on the SKT system, showing the existence of local, non-negative, strong solutions.