Stabilization of the wave equations with potential and indefinite damping
In this paper the authors study the energy \({\mathcal E}(t)\) of the wave equation \[ \begin{cases} y_{tt}-\Delta y+\bigl[a(x) +b(x)\bigr]y_t +c(x)y=0, \quad & \text{in }(0,\infty) \times\Omega,\\ y=0,\quad &\text{on }(0,\infty)\times \Gamma,\\ y(0)=y_0,\;y_t(0)= y_1,\quad &\text{in }\Omega, \end{cases} \] where \(\Omega\subset \mathbb{R}^n\) is a bounded domain with \(C^2\) boundary \(\Gamma\), \(a(\cdot) \geq 0\), \(b(\cdot)\leq 0\) and \(c(\cdot)\geq 0\). It is assumed that the damping \(a(\cdot)+ b(\cdot)>0\) in a subdomain \(G\) of \(\Omega\), which is a neighborhood of the ``star-shaped'' part of \(\Gamma\), and \(|b |_{L^\infty (\Omega)}\) is small compared to \(|a|_{L^\infty (\Omega)}\). Using a global Carleman-type inequality and a classical perturbation result, an explicit exponential decay rate of \({\mathcal E}(t)\) is obtained. Also, in the same context, an upper bound estimate of the negative damping \(b(\cdot)\) is given.
upper bound estimate of the negative damping
scientific article; zbMATH DE number 1797507