On the Spectral Properties and Stabilization of Acoustic Flow
Бо Лю, Walter Littman · SIAM Journal on Applied Mathematics · 1998
In this paper we use perturbation theory to study the spectral properties and energy decay of two-dimensional acoustic flow (cf. [J.T. Beale, Indiana Univ. Math. J., 25 (1976), pp.895--917], [P.M. Morse and K.U. Ingard, Theoretical Acoustics, McGraw-Hill, New York, 1968]):$\phi_{tt}-c^2\Delta \phi=0$ in $\Omega\times(0,\infty)$, $m\delta_{tt}+d\delta_t+k\delta=-\rho\phi_t$ and $\phi_x=\delta_t$ on $\Gamma_0\times(0,\infty)$, $\frac{\partial\phi}{\partial u}=0$ on $\Gamma_1\times(0,\infty)$ with initial data $\phi(0)=\phi_0,\ \phi_t(0)=\phi_1$ in $\Omega$ and $\delta(0)=\delta_0,\ \delta_t(0)=\delta_1$ on $\Gamma_0$, where $\Omega=(0,1)\times (0,1)$, $\Gamma_0=\{(1,y); \0 -1 (even like $t^{-\beta}$) if initial data satisfy certain smoothness are proved.