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Stability for time-dependent differential equations. (English) Zbl 0909.35017
Acta Numerica 7, 203-285 (1998).
Summary: We review results on asymptotic stability of stationary states of PDEs. After scaling, our normal form is \[ u_t= Pu+\varepsilon f(u,u_x,\dots)+ F(x,t), \] 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 \(t\to\infty\), and \(\varepsilon f(u,\dots)\) is a nonlinear perturbation. We discuss conditions that ensure \(u\to 0\) for \(t\to\infty\) when \(|\varepsilon|\) 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 are also given. The emphasis on the resolvent technique is motivated by the recent interest in pseudospectra.
For the entire collection see [Zbl 0894.00025].

MSC:
35B35 Stability in context of PDEs
37C75 Stability theory for smooth dynamical systems
35G25 Initial value problems for nonlinear higher-order PDEs
47H20 Semigroups of nonlinear operators
65M12 Stability and convergence of numerical methods for initial value and initial-boundary value problems involving PDEs
65M70 Spectral, collocation and related methods for initial value and initial-boundary value problems involving PDEs
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