IDEAS home Printed from https://ideas.repec.org/a/wly/jnlaaa/v2012y2012i1n278542.html

Exact Asymptotic Expansion of Singular Solutions for the (2 + 1)‐D Protter Problem

Author

Listed:
  • Lubomir Dechevski
  • Nedyu Popivanov
  • Todor Popov

Abstract

We study three‐dimensional boundary value problems for the nonhomogeneous wave equation, which are analogues of the Darboux problems in ℝ2. In contrast to the planar Darboux problem the three‐dimensional version is not well posed, since its homogeneous adjoint problem has an infinite number of classical solutions. On the other hand, it is known that for smooth right‐hand side functions there is a uniquely determined generalized solution that may have a strong power‐type singularity at one boundary point. This singularity is isolated at the vertex of the characteristic light cone and does not propagate along the cone. The present paper describes asymptotic expansion of the generalized solutions in negative powers of the distance to this singular point. We derive necessary and sufficient conditions for existence of solutions with a fixed order of singularity and give a priori estimates for the singular solutions.

Suggested Citation

  • Lubomir Dechevski & Nedyu Popivanov & Todor Popov, 2012. "Exact Asymptotic Expansion of Singular Solutions for the (2 + 1)‐D Protter Problem," Abstract and Applied Analysis, John Wiley & Sons, vol. 2012(1).
    • Handle: RePEc:wly:jnlaaa:v:2012:y:2012:i:1:n:278542
    DOI: 10.1155/2012/278542
    as

    Download full text from publisher

    File URL: https://doi.org/10.1155/2012/278542
    Download Restriction: no
    File URL: https://libkey.io/10.1155/2012/278542?utm_source=ideas
    LibKey link: if access is restricted and if your library uses this service, LibKey will redirect you to where you can use your library subscription to access this item
    ---><---

    References listed on IDEAS

    as
    1. M. K. Grammatikopoulos & N. I. Popivanov & T. P. Popov, 2004. "New singular solutions of Protter′s problem for the 3D wave equation," Abstract and Applied Analysis, John Wiley & Sons, vol. 2004(4), pages 315-335.
    2. M. K. Grammatikopoulos & N. I. Popivanov & T. P. Popov, 2004. "New singular solutions of Protter's problem for the 3 D wave equation," Abstract and Applied Analysis, Hindawi, vol. 2004, pages 1-21, January.
    Full references (including those not matched with items on IDEAS)

    Citations

    Citations are extracted by the CitEc Project, subscribe to its RSS feed for this item.
    as


    Cited by:

    1. Nedyu Popivanov & Todor Popov & Allen Tesdall, 2014. "Semi‐Fredholm Solvability in the Framework of Singular Solutions for the (3+1)‐D Protter‐Morawetz Problem," Abstract and Applied Analysis, John Wiley & Sons, vol. 2014(1).

    Most related items

    These are the items that most often cite the same works as this one and are cited by the same works as this one.
    1. Nedyu Popivanov & Todor Popov & Allen Tesdall, 2014. "Semi‐Fredholm Solvability in the Framework of Singular Solutions for the (3+1)‐D Protter‐Morawetz Problem," Abstract and Applied Analysis, John Wiley & Sons, vol. 2014(1).

    More about this item

    Statistics

    Access and download statistics

    Corrections

    All material on this site has been provided by the respective publishers and authors. You can help correct errors and omissions. When requesting a correction, please mention this item's handle: RePEc:wly:jnlaaa:v:2012:y:2012:i:1:n:278542. See general information about how to correct material in RePEc.

    If you have authored this item and are not yet registered with RePEc, we encourage you to do it here. This allows to link your profile to this item. It also allows you to accept potential citations to this item that we are uncertain about.

    If CitEc recognized a bibliographic reference but did not link an item in RePEc to it, you can help with this form .

    If you know of missing items citing this one, you can help us creating those links by adding the relevant references in the same way as above, for each refering item. If you are a registered author of this item, you may also want to check the "citations" tab in your RePEc Author Service profile, as there may be some citations waiting for confirmation.

    For technical questions regarding this item, or to correct its authors, title, abstract, bibliographic or download information, contact: Wiley Content Delivery (email available below). General contact details of provider: https://onlinelibrary.wiley.com/journal/4058 .

    Please note that corrections may take a couple of weeks to filter through the various RePEc services.

    IDEAS is a RePEc service. RePEc uses bibliographic data supplied by the respective publishers.