Semilinear hyperbolic equations
The aim of the lecture note is to explain how to derive sharp a priori estimates which enable one to prove a global in time existence of solutions to semilinear wave equations and nonlinear Klein-Gordon equations. The first chapters contain some basic facts of functional analysis, interpolation theorems, Fourier transform on hyperboloid, weighted Sobolev spaces of fractional order and pseudodifferential operators. The core of the work is devoted to Fourier transform on manifolds of constant negative curvature and sharp \(L^p\)-weighted a priori estimates of solutions to wave and Klein-Gordon equations. Global existence and asymptotic behaviour of the solutions of Cauchy problems for semilinear wave and Klein-Gordon equations with small initial data are proved in the last chapter. The lecture note is useful for graduate students as well as for specialists in the field of partial differential equations.
weighted Sobolev space of fractional order
manifolds of constant negative curvature
scientific article; zbMATH DE number 1565319