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Nonlinear oscillations of Hamiltonian PDEs. (English) Zbl 1146.35002

Progress in Nonlinear Differential Equations and Their Applications 74. Basel: Birkhäuser (ISBN 978-0-8176-4680-6/hbk). xiv, 180 p. (2007).
This well written monograph is an introduction to recent research mainly due to the author and his collaborators on periodic solutions of infinite-dimensional Hamiltonian systems. As a model problem treated throughout the book the author discusses in detail the nonlinear wave equation on an interval with Dirichlet boundary conditions.
Chapter 1 contains the classical results of Lyapunov, Weinstein, Moser, Fadell-Rabinowitz on periodic solutions of finite-dimensional Hamiltonian systems near an equilibrium. Proofs are given in detail for simplified versions. They are based on bifurcation techniques, in particular the variational Lyapunov-Schmidt reduction, and critical point theory. These techniques will also be used later in the book for infinite-dimensional Hamiltonian systems.
Chapter 2 deals with the autonomous nonlinear wave equation
\[ \left \{ \begin{aligned} & u_{tt}-u_{xx}=a(x)u^p+O(u^{p+1}), \quad t\in\mathbb{R} ,\;x\in(0,\pi)\\ {}&u(t,0)=u(t,\pi)=0, \end{aligned}\right.\tag{1} \]
in particular with the completely resonant case \(p\geq 2\) (\(p\in\mathbb{N}\)). The “small divisors” problem is explained and the existence of \(2\pi/\omega\)-periodic solutions near \(0\) is proved under strong nonresonance conditions on the frequency \(\omega\).
Chapter 3 contains a tutorial in Nash-Moser theory. The Nash-Moser implicit function theorem is then applied in Chapter 4 in order to find periodic solutions near \(0\) of (1) for \(p\geq 2\) for sets of frequencies \(\omega\) close to \(1\) with asymptotically full measure.
Chapter 5 deals with forced vibrations, more precisely with the problem
\[ \left\{ \begin{aligned} {}&u_{tt}-u_{xx}=\varepsilon f(t,x,u), \quad t\in\mathbb{R},\;x\in(0,\pi)\\ {}&u(t,0)=u(t,\pi)=0 \end{aligned} \right.\tag{2} \]
where \(f\) is \(2\pi\)-periodic in \(t\). Using purely variational methods existence theorems for \(2\pi\)-periodic solutions are obtained provided \(\varepsilon\) is small.
The book concludes with several appendices. The first contains Hamiltonian formulations for some PDEs, the second an introduction to critical point theory, in particular the mountain pass theorem. This is used in the third appendix to prove a result about the existence of one periodic solution of \(u_{tt}-u_{xx}=| u| ^{p-2}u\), \(p>2\), which is not close to \(0\). The fourth appendix collects some elementary results in number theory related to the nonresonance conditions.

MSC:

35-02 Research exposition (monographs, survey articles) pertaining to partial differential equations
37-02 Research exposition (monographs, survey articles) pertaining to dynamical systems and ergodic theory
37K50 Bifurcation problems for infinite-dimensional Hamiltonian and Lagrangian systems
35L70 Second-order nonlinear hyperbolic equations
58E05 Abstract critical point theory (Morse theory, Lyusternik-Shnirel’man theory, etc.) in infinite-dimensional spaces
37K55 Perturbations, KAM theory for infinite-dimensional Hamiltonian and Lagrangian systems
35B10 Periodic solutions to PDEs
35B34 Resonance in context of PDEs
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