Dynamics of some Kirchhoff-type PDEs

In this thesis we study the long-time dynamics of a class of Kirchhoff-type partial differen- tial equations, which provide fundamental examples of quasilinear Hamiltonian PDEs with nonlocal nonlinearities. In the first part of the thesis we focus on a special Kirchhoff equation introduced by Po- hozaev, for which global-in-time existence is known to hold for sufficiently regular initial data. Our goal is to understand this phenomenon from a dynamical systems perspective. To this end, we perform a quasilinear normal form reduction of the associated vector field and analyze the structure of the resonant terms at low orders. We show that all resonant interac- tions up to a relatively high degree are ineffective in producing growth of Sobolev norms. As a consequence, we obtain long-time stability results for small-amplitude solutions and improved lower bounds on their lifespan. In order to further investigate the dynamical properties of the equation, we study the formal Birkhoff normal form of the corresponding Hamiltonian. By means of an algebraic analysis based on formal power series, we prove that, despite the absence of effective resonances at low orders, the system fails to be integrable at higher orders due to the presence of nontrivial resonant terms. In the second part of the thesis we consider a semilinear Kirchhoff-type equation and address the existence of small-amplitude almost-periodic solutions. Adopting a dynamical sys- tems viewpoint, the equation can be interpreted as an infinite chain of coupled harmonic oscillators. We construct invariant tori supporting almost-periodic solutions by means of a KAM scheme. Although the linear dispersion relation leads to a delicate small divisor prob- lem, the special structure of the Kirchhoff nonlinearity allows us to overcome these difficulties. We prove that, for almost every choice of the external potential, the equation admits infinitely many weak almost-periodic solutions, among which infinitely many are classical and infinitely many are non-classical.