A direct method for the boundary stabilization of the wave equation (Q1098814)

We consider the wave equation \(y''-\Delta y=0\) in a bounded domain \(\Omega\subset {\mathbb{R}}^ n \)with smooth boundary \(\Gamma\), subject to mixed boundary conditions \(y=0\) on \(\Gamma_ 1\) and \(\partial y/\partial v=F(x,y')\) on \(\Gamma_ 0\), \((\Gamma_ 0,\Gamma_ 1)\) being a partition of \(\Gamma\). We study the boundary stabilizability of the solutions i.e. the existence of a partition \((\Gamma_ 0,\Gamma_ 1)\) and of a boundary feedback F(\(\cdot,\cdot)\) such that every solution decays exponentially in the energy space as \(t\to \infty\). We prove in this paper the stabilizability of the system without geometrical hypotheses. The proof is based on the use of a feedback \(F(x,y')=-b(x)y'\) with b(x)\(\geq 0\) and \(b(x)=0\) on the interface points \(x\in cl(\Gamma_ 0)\cap cl(\Gamma_ 1)\), and on the construction of energy functionals, well adapted to the system. This method is rather general and can be adapted to other evolution systems (e.g. models of plates, elasticity systems) as well. The semilinear wave equation \(y''-\Delta y+f(y)=0\) is also treated.