We study the linearized water-wave problem in a bounded domain ( a finite pond of water) of ${ℝ}^3$, having a cuspidal boundary irregularity created by a submerged body. In earlier publications the authors discovered that in this situation the spectrum of the problem may contain a continuous component in spite of the boundedness of the domain. Here, we proceed to impose and study radiation conditions at a point $𝒪$ of the water surface, where a submerged body touches the surface (see...

Here, we prove the uniform observability of a two-grid method for the semi-discretization of the -wave equation for a time $T>2\sqrt{2}$; this time, if the observation is made in $(-T/2,T/2)$, is optimal and this result improves an earlier work of Negreanu and Zuazua [ (2004) 413–418]. Our proof follows an Ingham type approach.

Let be a possibly unbounded positive operator on the Hilbert space , which is boundedly invertible. Let be a bounded operator from $𝒟\left(A_0^{\frac{1}{2}}\right)$ to another Hilbert space . We prove that the system of equations $$\ddot{z}\left(t\right)+A_{0}z\left(t\right)+\frac{1}{2}{C}_0^{*}{C}_0\dot{z}\left(t\right)=C_0^{*}u\left(t\right)$$ $$y\left(t\right)=-C_0\dot{z}\left(t\right)+u\left(t\right),$$ determines a well-posed linear system with input and output . The state of this system is $$x\left(t\right)=\left[\begin{array}{c}z\left(t\right)\\ \dot{z}\left(t\right)\end{array}\right]\in 𝒟\left(A_0^{\frac{1}{2}}\right)\times H=X,$$ where is the state space. Moreover, we have the energy identity $${\parallel x\left(t\right)\parallel }_X^2-{\parallel x\left(0\right)\parallel }_X^2={\int }_0^T{\parallel u\left(t\right)\parallel }_U^2\mathrm{d}t-{\int }_0^T{\parallel y\left(t\right)\parallel }_U^2\mathrm{d}t.$$ We show that the system described above is isomorphic to its dual, so that...