Boundary value problems in time for wave equations on RN
Let Lλ denote the linear operator associated with the radially symmetric form of the wave operator ∂t2−Δ+λ together with the side conditions of decay to zero as r=‖x‖→+∞ and T-periodicity in time. Thus Lλω=ωtt−(ωrr+N−1rωr)+λω, when there are N space variables. For δ,R,T>0 let DT,R=(0,T)×(R,+∞) an...
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| Main Authors: | , |
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| Format: | Article |
| Language: | English |
| Published: |
Wiley
1990-01-01
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| Series: | International Journal of Mathematics and Mathematical Sciences |
| Subjects: | |
| Online Access: | http://dx.doi.org/10.1155/S0161171290000886 |
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| Summary: | Let Lλ denote the linear operator associated with the radially symmetric form of the wave operator ∂t2−Δ+λ together with the side conditions of decay to zero as r=‖x‖→+∞ and T-periodicity in time. Thus Lλω=ωtt−(ωrr+N−1rωr)+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. |
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| ISSN: | 0161-1712 1687-0425 |