Hyperbolic partial di?erential equations describe phenomena of material or wave transport in the applied sciences. Despite of considerable progress in the past decades,the mathematical theory still faces fundamental questions concerningthe in?uenceofnonlinearitiesormultiple characteristicsofthe hyperbolicoperatorsor geometric properties of the domain in which the evolution process is considered. The current volume is dedicated to modern topics of the theory of hyperbolic equations such as evolution equations - multiple characteristics - propagation phenomena - global existence - in?uence of nonlinearities. It is addressed both to specialists and to beginners in these ?elds. The c- tributions are to a large extent self-contained. The ?rst contribution is written by Piero D'Ancona and Vladimir Georgiev. Piero D'Ancona graduated in 1982 from Scuola Normale Superiore of Pisa. Since 1997he isfull professorat the Universityof Rome1. Vladimir Georgievgraduated in1981fromtheUniversityofSo?a.Since2000heisfullprofessorattheUniversity of Pisa. The ?rst part of the paper treats the existence of low regularity solutions to the local Cauchy problem associated with wave maps.This introductory part f- lows the classical approach developed by Bourgain, Klainerman, Machedon which yields local well-posedness results for supercritical regularity of the initial data. The nonuniqueness results are establishedbytheauthors under the assumption that the regularity of the initial data is subcritical. The approach is based on the use of self-similar solutions. The third part treats the ill-posedness results of the Cauchy problem for the critical Sobolev regularity. The approach is based on the e?ective application of the properties of a special family of solutions associated with geodesics on the target manifold.
