  We use the local orthogonal decomposition technique to derive a generalized
finite element method for linear and semilinear parabolic equations with
spatial multiscale diffusion coefficient. We consider nonsmooth initial data
and a backward Euler scheme for the temporal discretization. Optimal order
convergence rate, depending only on the contrast, but not on the variations in
the diffusion coefficient, is proven in the $L_\infty(L_2)$-norm. We present
numerical examples, which confirm our theoretical findings.
