An unconditional estimate for solutions of a wave equation (Q1312735)

Under some conditions on the coefficients of the equation \(c(x)u_{tt} - \Delta u + a(x)u = 0\), \((x,t) \in Q = \Omega \times (0,T)\), it is proved that there are constants \(T > 0\), \(C > 0\), such that the energy \(Eu(t)\) of any solution in \(Q\) is bounded by a constant \(C\) times a norm of its boundary data, i.e. \[ -{1\over 2} \int_ \Omega [Cu^ 2_ t + | \nabla u|^ 2 + 2au^ 2]dx \leq C \int_{\Sigma} (u^ 2_ t + | \nabla u|^ 2 + u^ 2)ds. \] For the proof the multiplier method is applied.