Most of the standard theorems of global in time existence for solutions of the nonlinear evolution equations in mathematical physics depend heavily upon estimates for the solution's total energy. Typically, to prove the global existence of a smooth solution, one argues that a certain amount of energy would necessarily be dissipated in the development of a singularity, which is limited by virtue of small data assumptions so far, except for some semilinear evolution equations with good sign.

Under the small data assumption, the main observation is devoted to the investigation of the dissipative mechanism of linearized equations, which is described by the decay estimate of solutions mathematically. V. Georgiev is one of the most excellent mathematicians who created outstanding a priori estimates about hyperbolic equations in mathematical physics, which yield solutions of the corresponding nonlinear hyperbolic equations under small data assumption.

The aim of this lecture note is to explain how to derive sharp a priori estimates which enable us to prove a global in time existence of solutions to semilinear wave equation and non-linear Klein-Gordon equation.

The core of the lecture note is Section 8, which is devoted to Fourier transform on manifolds with constant negative curvature. Combining this with the interpolation method and psudodifferential operator approach enables us to obtain better $L^p$ weighted a priori estimates.