Discrete and Continuous Dynamical Systems - Series A (DCDS-A)

Qualitative properties of solutions for linear and nonlinear hyperbolic PDE's

Pages: 517 - 542, Volume 10, Issue 1/2, January/February 2004      doi:10.3934/dcds.2004.10.517

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Armen Shirikyan - Laboratoire de Mathématiques, Université de Paris-Sud XI, Bâtiment 425, 91405 Orsay Cedex, France (email)
Leonid Volevich - Keldysh Institute of Applied Mathematics, Russian Academy of Sciences, 4 Miusskaya Square, 125047 Moscow, Russian Federation (email)

Abstract: We present a number of results concerning large-time qualitative behavior of solutions for high-order hyperbolic equations and first-order hyperbolic systems. We discuss the properties of exponential stability and exponential dichotomy, construction of stable, unstable, and center manifolds, Grobman--Hartman type theorems on linearization of the phase portrait, and existence and uniqueness of time-bounded and almost periodic (AP) solutions.

Keywords:  hyperbolic equations and systems; bounded and almost periodic solutions;exponential dichotomy; stable, unstable and center manifolds; linearization.
Mathematics Subject Classification:  35L25, 35L40, 35L60, 35L75, 35B15, 35B40.

Received: July 2001;      Revised: May 2002;      Available Online: October 2003.