Asymptotic behavior of solutions of the Cauchy problem for a hyperbolic equation with periodic coefficients. the principle of the limiting amplitude
Abstract
This paper establishes the principle of limiting amplitude and the asymptotic behavior as $ t\to\infty $ 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, and the right side of the equation is the function $ f(x)\, \exp\{-i\, \omega\, t\}, \quad \omega>0 $. To obtain an asymptotic expansion as $ t\to\infty $, the basic methods of the spectral theory of differential operators were used, as well as the properties of the spectrum of the Hill operator with periodic coefficients.
Fields of Study & Subjects
Differential Equations and Boundary Problems
Spectral Theory in Mathematical Physics
Advanced Mathematical Physics Problems
Publication details
- Journal / source
- Evolution equations and control theory
- Volume
- 19
- Issue
- 0
- Pages
- 271–299
- Publication date
- 1 January 2026