  We investigate the abstract Cauchy problem for a quasilinear parabolic
equation in a Banach space of the form \( du_t -L_t(u_t)u_t dt = N_t(u_t)dt +
F(u_t)\cdot d\mathbf X_t \), where \( \mathbf X\) is a \( \gamma\)-H\"older
rough path for \( \gamma\in(1/3,1/2)\). We explore the mild formulation that
combines functional analysis techniques and controlled rough paths theory which
entail the local well-posedness of such equations. We apply our results to the
stochastic Landau-Lifshitz-Gilbert and Shigesada-Kawasaki-Teramoto equation. In
this framework we obtain a random dynamical system associated to the
Landau-Lifshitz-Gilbert equation.
