On Multistep-galerkin Discretizations of Semilinear Hyperbolic and Parabolic Equations*

Henceforth, it will be assumed that (1.1) has a unique solution U. Precise hypotheses on the required smoothness of the solution will be made when needed. For s a nonnegative integer, let H” = H”(Q) be the Sobolev space W,“(R) of real-valued functions on R and let 11. /Is denote its usual norm. Let fi’ = ii’(Q) = {u E H’ :ulm = 0). Th e inner product on L* = L’(Q) is denoted by (. , .) and the associated norm by 11. /I. Let also 11. II ,.,’ d enote the norm on L” = L”(R). If (X, 11 I( ,y) is a Banach space and 1 d p d r;o, E(0, t; X) will be the Banach space of (classes of) strongly measurable functions I;: (0, t) -+ X such that