Ada Numerica (1998), pp. 203-285 © Cambridge University Press, 1998 Stability for time-dependent differential equations Heinz-Otto Kreiss* Department of Mathematics, UCLA, Los Angeles, CA 90095, USA E-mail: kreiss@math.ucla.edu Jens Lorenz^ Department of Mathematics and Statistics, UNM, Albuquerque, NM 87131, USA E-mail: lorenzQmath. imm. edu In this paper we review results on asymptotic stability of stationary states of PDEs. After scaling, our normal form is u t = Pu + ef(u, u x ,...) + F(x, t), where the (vector-valued) function u(x, t) depends on the space variable x and time t. The differential operator P is linear, F(x, t) is a smooth forcing, which decays to zero for t —> oo, and ef(u,...) is a nonlinear perturbation. We will discuss conditions that ensure u —> 0 for t —> oo when |e| is sufficiently small. If this holds, we call the problem asymptotically stable. While there are many approaches to show asymptotic stability, we mainly concentrate on the resolvent technique. However, comparisons with the Lya- punov technique will also be given. The emphasis on the resolvent technique is motivated by the recent interest in pseudospectra. * Supported by Office of Naval Research N 00014 92 J 1890. + Supported by NSF Grant DMS-9404124 and DOE Grant DE-FG03-95ER25235. www.DownloadPaper.ir www.DownloadPaper.ir