Diffusive stability of rolls in the two-dimensional real and complex Swift-Hohenberg equation

We show the nonlinear stability of small bifurcating stationary rolls u(eps ilon,kappa) for the Swift-Hohenberg-equation on the domain R-2. In Bloch wa ve representation the linearization around a marginal stable roll u(epsilon ,kappa) has continuous spectrum up to 0 with a locally parabolic shape at t he critical Bloch vector 0. Using an abstract renormalization theorem we sh ow that small spatially localized integrable perturbations decay diffusivel y to zero. Moreover we estimate the size of the domain of attraction of a r oll u(epsilon,kappa) in terms of its modulus and Fourier wavenumber. To exp lain the method we also treat the nonlinear stability of stationary rolls f or the complex Swift-Hohenberg equation on R-2.