Non-Linear Hyperbolic Equations in Domains with Conical Points: Existence and Regularity of Solutions (Mathematical Research) - Softcover

Witt, Ingo

9783055016912: Non-Linear Hyperbolic Equations in Domains with Conical Points: Existence and Regularity of Solutions (Mathematical Research)

Softcover

ISBN 10: 3055016912 ISBN 13: 9783055016912

Publisher: Wiley-VCH, 1995

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These notes build to a proof of a local-in-time existence result for quasilinear hyperbolic evolution equations of second order in domains with conical points. In the first part, the existence of solutions to the corresponding linear equations is addressed, including the asymptotics of solutions near conical points. Using this information, the quasilinear equations are then solved by the standard iteration procedure. The exposition is based on Kato's semigroup-theoretic approach for solving abstract linear hyperbolic equations and Schulze's theory of pseudo-differential operators on manifolds with conical singularities. The former method provides the general framework, whereas the latter is the basic tool in treating the specific difficulties of the nonsmooth situation. Significantly, Schulze's theory admits a parameter-dependent version, which allows the description of the branching behaviour in time of discrete asymptotics of solutions near conical points. The calculus is presented in a form in which the operators are permitted to have symbols with limited smoothness, as arises in nonlinear problems. In an appendix, the applicability of energy methods is briefly discussed.

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From the Publisher

The aim of these notes is to prove a local-in-time existence result for quasilinear hyperbolic evolution equations of second order in domains with conical points. The exposition is based on Kato's semigroup-theoretic approach for solving abstract linear hyperbolic equations and Schulze's theory of pseudo-differential operators on manifolds with conical singularities.

About the Author

Author's affiliation:
Ingo Witt, Dr. rer. nat., studied at the Humboldt University of Berlin and the Moscow State University, doctorate at the University of Bonn, now at the University of Potsdam

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