Lp-Lq decay estimates for the solutions of strictly hyperbolic equations of second order with increasing in time coefficients

One of the most important questions in the theory of nonlinear wave equations is that for global existence of solutions. An essential tool is the Strichartz inequality for special solutions of the wave equation. In the last time different results were proved generalizing the classical one of STRICHARTZ. In the present paper L-p - L-q estimates are proved for the solutions of strictly hyperbolic equations of second order with time dependent coefficients where these are unbounded at infinity. In the first step the WKB method is applied to the construction of a fundamental system of solutions for ordinary differential equations depending on a parameter. In a second step the method of stationary phase yields the asymptotical behaviour of Fourier multipliers with nonstandard phase functions depending on a parameter.