Time-periodic scattering of symmetric hyperbolic systems
The authors consider symmetric hyperbolic systems which have several nonvanishing wave speeds and present additional examples to their previous work on the subject [Indiana Univ. Math. J. 34, 33-83 (1985; Zbl 0582.47011)]. First, they consider systems of the form \[ u_ t=\sum^{n}_{j=1}A_ j(x,t)u_{x_ j}+c(x,t)u,\text{ n is odd, }n\geq 3, \] \(A_ j\) are \(m\times m\) hermitian matrices. Outside a ball each \(A_ j\) is a constant matrix \(A^ 0_ j\) and C vanishes there. Furthermore, they assume that \(A_ j\) and C are at least continuously differentiable and periodic functions (in time) with period T. Secondly, the authors consider systems of the form \(u_ t=\sum^{n}_{j=1}A^ 0_ j u_{x_ j}\) outside a moving body. The exterior region is assumed to be with smooth time-like boundary, the obstacle changes its shape in a periodic way but it is always contained in a fixed ball. In this case they assume dissipative boundary conditions. In both cases they show that conditions P1) and P2) of the above mentioned paper hold. In particular, the expansion \[ u(x,t)\sim \sum^{\infty}_{j=1}a_ je^{i\sigma_ jt} P_ j(x,t)\quad as\quad t\to +\infty \] is valid in either case (in the sense of \(L^ 2_{loc}({\mathbb{R}}^ n))\), where \(\sigma_ j\) are the scattering frequencies.
scientific article; zbMATH DE number 4030089
scientific article