Asymptotic behavior as \(t\to \infty\) of solutions of exterior mixed problems periodic with respect to t (Q920279)

The paper refers to a mixed exterior problem associated with \(Lu=0\), where \(L=L(t,x,\partial_ t,\partial_ x)\) is described by a matrix of differential operators, \((t,x)\in D\), with D is the exterior of a curvilinear cylinder from \({\mathbb{R}}_{t,x}^{n+1}\). The author studies the asymptotic behavior for \(t\to \infty\) of the solutions of the exterior mixed problems that are periodic with respect to t. The main result focuses on the study of the asymptotic behavior of the solutions of problems defined by five special conditions. These conditions refer to correctness, homogeneity, translations, Sobolev spaces, Schwarz kernel, weak Huygens principle. Let us remark that the correctness condition is fulfilled for Cauchy problems for general hyperbolic systems. The proof of the main theorem makes use of many functional, analytic and operator techniques, several of them described by the author in previous publicactions [cf. ``Asymptotic methods in equations of mathematical physics'' (1982; Zbl 0518.35002)].