  For a class of quasilinear parabolic systems with nonlinear Robin boundary
conditions we construct a compact local solution semiflow in a nonlinear phase
space of high regularity. We further show that a priori estimates in lower
norms are sufficient for the existence of a global attractor in this phase
space. The approach relies on maximal $L_p$-regularity with temporal weights
for the linearized problem. An inherent smoothing effect due to the weights is
employed for gradient estimates. In several applications we can improve the
convergence to an attractor by one regularity level.
