Stability for time-dependent differential equations

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.