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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| Language: | English |
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Wiley
1990-01-01
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| Series: | International Journal of Mathematics and Mathematical Sciences |
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| Online Access: | http://dx.doi.org/10.1155/S0161171290000886 |
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| _version_ | 1850214465802338304 |
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| author | M. W. Smiley A. M. Fink |
| author_facet | M. W. Smiley A. M. Fink |
| author_sort | M. W. Smiley |
| collection | DOAJ |
| description | 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. |
| format | Article |
| id | doaj-art-85ffd6604e2446ebbd162965cd688c69 |
| institution | OA Journals |
| issn | 0161-1712 1687-0425 |
| language | English |
| publishDate | 1990-01-01 |
| publisher | Wiley |
| record_format | Article |
| series | International Journal of Mathematics and Mathematical Sciences |
| spelling | doaj-art-85ffd6604e2446ebbd162965cd688c692025-08-20T02:08:54ZengWileyInternational Journal of Mathematics and Mathematical Sciences0161-17121687-04251990-01-0113462564410.1155/S0161171290000886Boundary value problems in time for wave equations on RNM. W. Smiley0A. M. Fink1Department of Mathematics, Iowa State University, Ames 50011, Iowa, USADepartment of Mathematics, Iowa State University, Ames 50011, Iowa, USALet 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.http://dx.doi.org/10.1155/S0161171290000886wave equationradial symmetryboundary value problemeigenvalue problemHilbert spaceweighted Sobolev spaceFredholm operatorLaplace transformBessel functions. |
| spellingShingle | M. W. Smiley A. M. Fink Boundary value problems in time for wave equations on RN International Journal of Mathematics and Mathematical Sciences wave equation radial symmetry boundary value problem eigenvalue problem Hilbert space weighted Sobolev space Fredholm operator Laplace transform Bessel functions. |
| title | Boundary value problems in time for wave equations on RN |
| title_full | Boundary value problems in time for wave equations on RN |
| title_fullStr | Boundary value problems in time for wave equations on RN |
| title_full_unstemmed | Boundary value problems in time for wave equations on RN |
| title_short | Boundary value problems in time for wave equations on RN |
| title_sort | boundary value problems in time for wave equations on rn |
| topic | wave equation radial symmetry boundary value problem eigenvalue problem Hilbert space weighted Sobolev space Fredholm operator Laplace transform Bessel functions. |
| url | http://dx.doi.org/10.1155/S0161171290000886 |
| work_keys_str_mv | AT mwsmiley boundaryvalueproblemsintimeforwaveequationsonrn AT amfink boundaryvalueproblemsintimeforwaveequationsonrn |