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New Trends in the Theory of Hyperbolic Equations (Hardcover, 2005 ed.)
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New Trends in the Theory of Hyperbolic Equations (Hardcover, 2005 ed.)
Series: Operator Theory: Advances and Applications, 159
Expected to ship within 10 - 15 working days
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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.
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