  A variable stepsize exponential multistep integrator, with contour integral
approximation of the operator-valued exponential functions, is proposed for
solving semilinear parabolic equations with nonsmooth initial data. By this
approach, the exponential k-step method would have $k$th-order convergence in
approximating a mild solution, possibly nonsmooth at the initial time. In
consistency with the theoretical analysis, a numerical example shows that the
method can achieve high-order convergence in the maximum norm for semilinear
parabolic equations with discontinuous initial data.
