caa a22 4500 388057386 CHVBK 20180307125111.0 cr unu---uuuuu 161130e199801 xx s 000 0 eng 10.1017/S096249290000283X doi S096249290000283X pii (NATIONALLICENCE)cambridge-10.1017/S096249290000283X Stability for time-dependent differential equations [Elektronische Daten] In this paper we review results on asymptotic stability of stationary states of PDEs. After scaling, our normal form is ut = Pu + ε f(u, ux, ) + 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 → ∞, and εf(u, ...) is a nonlinear perturbation. We will discuss conditions that ensure u → 0 for t → ∞ when |ε| 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 Lyapunov technique will also be given. The emphasis on the resolvent technique is motivated by the recent interest in pseudospectra. Copyright © Cambridge University Press 1998 Kreiss Heinz-Otto Department of Mathematics, UCLA, Los Angeles, CA 90095, USA E-mail: kreiss@math.ucla.edu Lorenz Jens Department of Mathematics and Statistics, UNM, Albuquerque, NM 87131, USA E-mail: lorenz@math.unm.edu Acta Numerica Cambridge University Press 7(1998-01), 203-285 0962-4929 7<203 1998 7 ANU https://doi.org/10.1017/S096249290000283X text/html Onlinezugriff via DOI 1 research-article jats NATIONALLICENCE 856 40 https://doi.org/10.1017/S096249290000283X text/html Onlinezugriff via DOI NATIONALLICENCE 700 1- Kreiss Heinz-Otto Department of Mathematics, UCLA, Los Angeles, CA 90095, USA E-mail: kreiss@math.ucla.edu NATIONALLICENCE 700 1- Lorenz Jens Department of Mathematics and Statistics, UNM, Albuquerque, NM 87131, USA E-mail: lorenz@math.unm.edu NATIONALLICENCE 773 0- Acta Numerica Cambridge University Press 7(1998-01), 203-285 0962-4929 7<203 1998 7 ANU CC0 http://creativecommons.org/publicdomain/zero/1.0 nationallicence BK010053 XK010053 XK010000 NATIONALLICENCE NATIONALLICENCE NL-cambridge