On the Cauchy problem for second order hyperbolic equation with time dependent coefficients

Abstract: In this talk we introduce a recent approach for the consideration of stability of the solutions to some second order hyperbolic equations with time depending coefficients taking account of their singular behavior; differentiability, degeneration, oscillation, etc. In consequence, we introduce new Levi conditions of $C^infty$ type for second order hyperbolic Cauchy problems due to the special influence of coupled oscillating coefficients.