Local dynamics for high-order semilinear hyperbolic equations

L. R. Volevich ; A. R. Shirikyan

Izvestiya. Mathematics , Tome 64 (2000) no. 3, pp. 439-485

Résumé

This paper is devoted to studying high-order semilinear hyperbolic equations. It is assumed that the equation is a small perturbation of an equation with real constant coefficients and that the roots of the full symbol of the unperturbed equation with respect to the variable $\tau$ dual to time are either separated from the imaginary axis or lie outside the domain $\nu|{\operatorname{Re}\tau}|\delta$, where $\delta>\nu\geqslant 0$. In the first case, it is proved that the phase diagram of the perturbed equation can be linearized in the neighbourhood of zero using a time-preserving family of homeomorphisms and that the constructed homeomorphisms and their inverses are Holder continuous. In the other case, it is proved that the neighbourhood of zero in the phase space of the equation contains a locally invariant smooth manifold $\mathcal M$ which includes all solutions uniformly bounded on the entire time axis and exponentially attracts the solutions bounded on the half-axis. The manifold $\mathcal M$ can be represented as the graph of a non-linear operator that acts on the phase space and is a small perturbation of a pseudo-differential operator whose symbol can be written explicitly. In this case, the dynamics on the invariant manifold $\mathcal M$ is described by a hyperbolic equation whose order coincides with the number of roots of the full symbol that lie in the strip $|{\operatorname{Re}\tau}|\leqslant\nu$.