Stability of implicit and implicit–explicit multistep methods for nonlinear parabolic equations in Hilbert spaces

We consider the discretization of a class of nonlinear parabolic equations in Hilbert spaces by both implicit and implicit–explicit multistep methods, and establish localstability under best possible and best possible linear stability conditions, respectively. Our approach is based on a spectral and Fourier stability technique and uses a suitable quantification of the non-self-adjointness of linear elliptic operators as well as a discrete perturbation argument.