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dc.contributor.authorGrammatikopoulos, Myron K. ( )
dc.contributor.authorHristov, Tzvetan D. ( Orcid Icon 0000-0002-6417-1816 )
dc.contributor.authorPopivanov, Nedyu I. ( Orcid Icon 0000-0003-4439-5211 )
dc.date.accessioned2020-09-10T19:20:25Z
dc.date.available2020-09-10T19:20:25Z
dc.date.issued2003-01-02
dc.identifier.citationGrammatikopoulos, M. K., Hristov, T. D., & Popivanov, N. I. (2003). Singular solutions to Protter's problem for the 3-D wave equation involving lower order terms. Electronic Journal of Differential Equations, 2003(03), pp. 1-31.en_US
dc.identifier.issn1072-6691
dc.identifier.urihttps://digital.library.txstate.edu/handle/10877/12570
dc.description.abstractIn 1952, at a conference in New York, Protter formulated some boundary value problems for the wave equation, which are three-dimensional analogues of the Darboux problems (or Cauchy-Goursat problems) on the plane. Protter studied these problems in a 3-D domain Ω0, bounded by two characteristic cones Σ1 and Σ2,0, and by a plane region Σ0. It is well known that, for an infinite number of smooth functions in the right-hand side, these problems do not have classical solutions. Popivanov and Schneider (1995) discovered the reason of this fact for the case of Dirichlet's and Neumann's conditions on Σ0: the strong power-type singularity appears in the generalized solution on the characteristic cone Σ2,0. In the present paper we consider the case of third boundary-value problem on Σ0 and obtain the existence of many singular solutions for the wave equation involving lower order terms. Especifica ally, for Protter's problems in ℝ3 it is shown here that for any n ∈ N there exists a Cn(ˉΩ0)-function, for which the corresponding unique generalized solution belongs to Cn(ˉΩ0\O) and has a strong power type singularity at the point O. This singularity is isolated at the vertex O of the characteristic cone Σ2,0 and does not propagate along the cone. For the wave equation without lower order terms, we presented the exact behavior of the singular solutions at the point O.en_US
dc.formatText
dc.format.extent31 pages
dc.format.medium1 file (.pdf)
dc.language.isoenen_US
dc.publisherSouthwest Texas State University, Department of Mathematicsen_US
dc.sourceElectronic Journal of Differential Equations, 2003, San Marcos, Texas: Southwest Texas State University and University of North Texas.
dc.subjectWave equationen_US
dc.subjectBoundary value problemsen_US
dc.subjectGeneralized solutionsen_US
dc.subjectSingular solutionsen_US
dc.subjectPropagation of singularitiesen_US
dc.titleSingular Solutions to Protter's Problem for the 3-D Wave Equation Involving Lower Order Termsen_US
dc.typepublishedVersion
txstate.documenttypeArticle
dc.rights.licenseCreative Commons License
This work is licensed under a Creative Commons Attribution 4.0 International License.


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