On the asymptotic behavior, for large values of the time, of solutions of exterior boundary value problems for the wave equation with two space variables
Abstract:
Waves are constructed which characterize the behavior, for large values of time $t$, of the Green's functions of the basic exterior boundary value problems for the wave equation with two space variables (behind the wave front). Representations of the Green's functions (and the solutions) are obtained in the form of series, asymptotic in $t$ as $t\to\infty$. The principle of limiting amplitude is proved, i.e., the existence of the limit $\lim_{t\to\infty}u(t,x)e^{i\omega t}=v(x,\omega)$ is established for solutions of the basic exterior boundary value problems for the wave equation in the case of a time-periodic driving force ($u_{tt}=\Delta u-f(x)e^{-i\omega t}$), and a representation is obtained for the difference $u(t,x)-v(x,\omega)e^{-i\omega t}$ in the form of a series asymptotic in $t$ as $t\to\infty$; it is shown that the rate of emergence of a solution $u(t,x)$ to a periodic regime $v(x,\omega)e^{-i\omega t}$ cannot be greater than a power of $t$.
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