  Stochastic partial differential equations (SPDEs) have become a key modelling
tool in applications. Yet, there are many classes of SPDEs, where the existence
and regularity theory for solutions is not completely developed. Here we
contribute to this aspect and prove the existence of mild solutions for a broad
class of quasilinear Cauchy problems, including - among others -
cross-diffusion systems as a key application. Our solutions are local-in-time
and are derived via a fixed point argument in suitable function spaces. The key
idea is to combine the classical theory of deterministic quasilinear parabolic
partial differential equations (PDEs) with recent theory of evolution
semigroups. We also show, how to apply our theory to the
Shigesada-Kawasaki-Teramoto (SKT) model. Furthermore, we provide examples of
blow-up and ill-posed operators, which can occur after finite-time showing that
solutions can only be local-in-time for general quasilinear SPDEs, while they
might be global-in-time for special subclasses of problems.

Probability