New Trends in the Theory of Hyperbolic Equations

The present volume is dedicated to modern topics of the theory of hyperbolic equations such as evolution equations, multiple characteristics, propagation phenomena, global existence, influence of nonlinearities.
It is addressed to beginners as well as specialists in these fields. The contributions are to a large extent self-contained.
Key topics include:

• low regularity solutions to the local Cauchy problem associated with wave maps; local well-posedness, non-uniqueness and ill-posedness results are proved
• coupled systems of wave equations with different speeds of propagation; here pointwise decay estimates for solutions in spaces with hyperbolic weights come in
• damped wave equations in exterior domains; the energy method is combined with the geometry of the exterior domain; for the critical part of the boundary a restricted localized effective dissipation is employed
• the phenomenon of parametric resonance for wave map type equations; the influence of time-dependent oscillations on the existence of global small data solutions is studied - a unified approach to attack degenerate hyperbolic problems as weakly hyperbolic ones and Cauchy problems for strictly hyperbolic equations with non-Lipschitz coefficients
• weakly hyperbolic Cauchy problems with finite time degeneracy; the precise loss of regularity depending on the spatial variables is determined; the main step is to find the correct class of pseudodifferential symbols and to establish a calculus which contains a symmetrizer.

Résumé
This book presents several recent developments in the theory of hyperbolic equations. The carefully selected invited and peer-reviewed contributions deal with questions of low regularity, critical growth, ill-posedness, decay estimates for solutions of different non-linear hyperbolic models, and introduce new approaches based on microlocal methods.

Contenu
Wave Maps and Ill-posedness of their Cauchy Problem.- On the Global Behavior of Classical Solutions to Coupled Systems of Semilinear Wave Equations.- Decay and Global Existence for Nonlinear Wave Equations with Localized Dissipations in General Exterior Domains.- Global Existence in the Cauchy Problem for Nonlinear Wave Equations with Variable Speed of Propagation.- On the Nonlinear Cauchy Problem.- Sharp Energy Estimates for a Class of Weakly Hyperbolic Operators.