On the Spectral Properties and Stabilization of Acoustic Flow

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 $\Gamma_0\times(0,\infty)$, $\frac{\partial\phi}{\partial\nu}=0$ 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 $\Omega=(0,1)\times (0,1)$, $\Gamma_0=\{(1,y); \0<y <1\}$, $\Gamma_1=\partial\Omega\setminus\Gamma_0$, and $\nu$ is the external normal direction on the boundary. Locations of eigenvalues of the infinitesimal generator of semigroup associated with the above system are estimated. A certain "Fourier" expansion is obtained. That the energy decays to zero and like t-1 (even like $t^{-\beta}...