Nonlinear Hyperbolic Equations, Spectral Theory, and Wavelet Transformations A Volume of Advances in Partial Differential Equations

This volume focuses on recent developments in non-linear and hyperbolic equations. In the first contribution, the singularities of the solutions of several classes of non-linear partial differential equations are investigated. Applications concern the Monge-Ampère equation, quasi-linear systems arising in fluid mechanics as well as integro-differential equations for media with memory. There follows an article on L_p-L_q decay estimates for Klein-Gordon equations with time-dependent coefficients, explaining, in particular, the influence of the relation between the mass term and the wave propagation speed. The next paper addresses questions of local existence of solutions, blow-up criteria, and C^8 regularity for quasilinear weakly hyperbolic equations. Spectral theory of semibounded selfadjoint operators is the topic of a further contribution, providing upper and lower bounds for the bottom eigenvalue as well as an upper bound for the second eigenvalue in terms of capacitary estimates. TOC:Contributions: Nonlinear PDE. Singularities, Propagation, Applications (P.R. Popivanov).- From Wave to Klein-Gordon Type Decay Rates (F. Hirosawa and M. Reissig).- Local Solutions to Quasilinear Qeakly Hyperbolic Differential Equations (M. Dreher).- S(M,g)-pseudo-differential Calculus of Manifolds (F. Baldus).- Domain Perturbations and Capacity in General Hilbert Spaces and Applications to Spectral Theory (A. Noll).- An Interpolation Family between Gabor and Wavelet Transformations (B. Nazaret and M. Holschneider).- Formes de Torsion Analytique et Fibrations Singulières (Xiaonan Ma).- Regularisation of Secondary Characteristic Classes and Unusual Index Formulas for Operator-Valued Symbols (G. Rozenblum).