The authors are concerned with the linear hyperbolic equation \[ \begin{split} D_t^2u(x,t) - p(x,t)\Delta u(x,t) - \sum_{k=1}^n\,q_k(x,t)D_{x_k}u(x,t)\\ - q_{n+1}(x,t)D_tu(x,t) - q_0(x,t)u(x,t)=0,\quad (x,t)\in \Omega\times (-T,T), \end{split}\tag{1} \] where \(\Omega\subset {\mathbb R^n}\) is a bounded domain with a boundary of class \(C^2\), while the coefficients \(p\) and \(q_j\) satisfy the properties: (i) \(p\in C^1({\overline \Omega}\times {\mathbb R})\); (ii) \(p(x,t)>0\) for all \((x,t)\in {\overline \Omega}\times {\mathbb R}\); (iii) \(q_j\in L^\infty(\Omega\times {\mathbb R})\), \(j=0,\ldots,n+1\).
The authors show a local (conditioned) stability result related to a point \(x_0\in \partial \Omega\) such that: (iv) \(D_\nu p(x_0,0)\leq 0\), \(\nu\) denoting the exterior normal derivative; (v) \(B(x_0,\delta)\cap \Omega\) is convex for some \(\delta>0\). Under the a priori given estimate \(\| u\| _{H^1(\Omega\times (-T,T))}\leq M\) it can be proved that there exist \(y\in {\mathbb R}^n\), \(\beta>0\), \(r_1>0\), \(\gamma>0\), \(C>0\) and \(\kappa\in (0,1)\) such that \(x_0\in Q_\gamma(y)=\{(x,t)\in \Omega \times {\mathbb R}: | x| \leq r_1,\;| x-y| ^2-\beta t^2>\gamma\}\) and \[ \| u\| _{Q_\gamma(y)}\leq C{\mathcal E}^\kappa({\mathcal E}^{1-\kappa}+M^{1-\kappa}), \] where \[ \begin{aligned} {\mathcal E} = &\| u\| _{H^{3/2}((\partial \Omega\cap B(x_0,\delta))\times (-T,T))} + \| u\| _{H^{2}((-T,T);L^2(\partial \Omega\cap B(x_0,\delta)))}\\ +&\| D_\nu u\| _{H^{2}((-T,T);L^2(\partial \Omega\cap B(x_0,\delta)))} +\| D_\nu u\| _{L^{2}((-T,T);H^{1/2}(\partial \Omega\cap B(x_0,\delta)))}.\end{aligned} \] Such a property immediately implies a unique continuation result, i.e., if \(u=| \nabla u| =0\) on \(\Gamma \times (-T,T)\), \(\Gamma\) being an open set in \(\partial \Omega\), then there exists a neighborhood \(U\) of \((x_0,0)\) such that \(u=0\) in \(U\).