Remarks on global uniqueness theorems for partial differential equations (Q2708684)

The author describes a method that transforms local results into global results. In this paper he proved the following results. Let NEWLINE\[NEWLINEPu=u_{tt}-\sum^n_{i,j=1} a^{ij}u_{x_ix_j}+ b^iu_{x_i} +cuNEWLINE\]NEWLINE be defined in \(Q^*=\Omega^* = \Omega^* \times(0,T)\), where \(\Omega^*\) is an open set in \(\mathbb{R}^n\) containing the colosure \(\overline\Omega\) of the bounded domain \(\Omega\). Assume that the elliptic principal part is uniformly elliptic in \(\Omega^*\) with \(C^1\) coefficients independent of \(t\), while the other coefficients are analytic in \(t\) with values in \(L^\infty\). Let \(ds^2=\sum a_{ij} dx_idx_j\) be the Riemannian metric associated with \(P\), i.e. \((a_{ij})=(a^{ij})^{-1}\) and \(\Gamma\) be an open nonempty subset of \(\partial\Omega\). Assume \(Pu=0\) in \(Q\), that \(u\in H^1 (Q)\), where \(Q=\Omega\times(0,T)\), that \(u\) vanishes on \(\partial\Omega \times (0,T)\) and that \(u\) can be continued by zero as a solution to \(Pu=0\) across \(\Gamma \times (0,T)\). Then \(u\equiv 0\) in \(Q\) provided \(T>T_0=2 \max_{x\in \Omega} \{\text{dist} (x,\Gamma)\}\).NEWLINENEWLINEFor the entire collection see [Zbl 0956.00042].