Global properties of the wave equation on non-globally hyperbolic manifolds.
The author studies the equation \(P u=0\) or \( P u+Vu=0\) where \(P u\) is defined on \(\mathbb{R}^{4} \) by \( (1- C(r,z) / r ^{2}) \partial _{t} ^{2}u - \Delta _{x}u- 2 (C(r,z)/ r ^{2}) \partial _{t} \partial {\varphi }u\). The coordinates in \( {\mathbb{R}}^{4} \) are \(( x ^{1}, x ^{2}, x ^{3},t)\), but on \( {\mathbb{R}}^{3} \) also the cylindrical coordinates \( r = ( | x ^{1}| ^{2}+ | x ^{2} | ^{2}) ^{1/2}\), \( x ^{1} = r \cos \varphi\) , \( x ^{2}= r \sin \varphi\), \(z = x ^{3}\) are used. In particular, \( \Delta _{x}\) is the Laplace operator in \( {\mathbb{R}}^{3} \) and \( P \) corresponds to the wave operator on \( {\mathbb{R}}^{4} \), if the latter is endowed with the Lorentzian metric \(d t ^{2} - ( r ^{2} - C ^{2}( r,z)) d \varphi ^{2} - d r ^{2} - d z ^{2}\). It is assume that \( C \geq 0\) and that \(C\) vanishes if \( (r,z) \, \notin [ r _{-}, r _{+}], \times [ z _{-}, z_{+}]\), which means in particular that the metric is the Minkowski metric outside some compact set. The operator is globally hyperbolic with respect to the surfaces \(t = \) const if \( C(r,z) < r\), but the main interest of the author is precisely to study what happens when this condition does not hold. In particular, when \( \Sigma = \{ C=r\}\) is not void, there exist closed null geodesics, and when \(T= \{C>r\}\) is non-void, then \({\mathbb{R}}^{4} \) is ``totally vicious'' (in the terminology of the paper) in that any two points can be connected by timelike future-pointing curves. In this kind of contexts the author studies a variety of questions: uniqueness of the Cauchy problem, resonances, asymptotically free solutions, meromorphic continuation of the scattering matrix, local energy decay. As a justification for the equation the author mentions that the metric under consideration is a particular case of the so-called Papapetrou metric which appears in the study of the Einstein equations.
scientific article; zbMATH DE number 1985992
scientific article