Local decay of solutions of conservative first order hyperbolic systems in odd dimensional space

Abstract

This paper deals with symmetric hyperbolic systems, ∂ u / ∂ t = L u \partial u/\partial t = Lu , where L is equal to the homogeneous, constant coefficient operator L 0 {L_0} for | x | > R |x| > R . Under the hypothesis that L has simple null bicharacteristics and these propagate to infinity, local decay of solutions and completeness of the wave operators relating solutions of ∂ u / ∂ t = L u \partial u/\partial t = Lu and solutions of ∂ u / ∂ t = L 0 u \partial u/\partial t = {L_0}u are established. Results of this type for elliptic L are due to Lax and Phillips. The proof here is based, in part, on a new estimate of the regularity of the L 2 {L^2} -solutions of the equation L u + ( i λ + ε ) u = g Lu + (i\lambda + \varepsilon )u = g for smooth g with support in | x | ≤ R |x| \leq R .