We discuss analytic time-regularity of parabolic evolution problems. Boundary incompatibility of initial data and nonlinearity is well known to cause a loss of time analyticity.

We describe weighted Gevrey classes, the weight describing the loss of time-regularity.

Time semidiscretization by hp time-stepping is shown to give exponential rates of convergence.

We apply the results to the rapid solution of parabolic problems which march from t=0 to T>0 with N degrees of freedom in O(N|log N|^a) work.