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International Journal of Mathematics and Mathematical Sciences
Volume 13, Issue 4, Pages 625-644
http://dx.doi.org/10.1155/S0161171290000886

Boundary value problems in time for wave equations on RN

Department of Mathematics, Iowa State University, Ames 50011, Iowa, USA

Received 5 December 1989

Copyright © 1990 Hindawi Publishing Corporation. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

Let Lλ denote the linear operator associated with the radially symmetric form of the wave operator t2Δ+λ together with the side conditions of decay to zero as r=x+ and T-periodicity in time. Thus Lλω=ωtt(ωrr+N1rωr)+λω, when there are N space variables. For δ,R,T>0 let DT,R=(0,T)×(R,+) and Lδ2(D) denote the weighted L2 space with weight function exp(δr). It is shown that Lλ is a Fredholm operator from dom(Lλ)L2(D) onto Lδ2(D) with non-negative index depending on λ. If [2πj/T]2<λ[2π(j+1)/T]2 then the index is 2j+1. In addition it is shown that Lλ has a bounded partial inverse Kλ:Lδ2(D)Hδ1(D)Lδ(D), with all spaces weighted by the function exp(δr). This provides a key ingredient for the analysis of nonlinear problems via the method of alternative problems.