On the asymptotic behavior of solutions of the cauchy problem for a hyperbolic equation with periodic coefficients [Articol]

Abstract

The paper establishes the asymptotic behavior as t → ∞and the principle of limiting amplitude of solutions to the Cauchy problem for a second-order hyperbolic equation with periodic coefficients for large values of the time parameter t, with zero initial data, where the right side of the equation is the function f(x) exp{−i ω t}, ω > 0. To obtain an asymptotic expansion as t → ∞, the basic methods of the spectral theory of differential operators are used, as well as the properties of the spectrum of the Hill operator with periodic coefficients.