IDEAS home Printed from https://ideas.repec.org/a/wly/jnlaaa/v2004y2004i4p315-335.html
   My bibliography  Save this article

New singular solutions of Protter′s problem for the 3D wave equation

Author

Listed:
  • M. K. Grammatikopoulos
  • N. I. Popivanov
  • T. P. Popov

Abstract

In 1952, for the wave equation,Protter formulated some boundary value problems (BVPs), which are multidimensional analogues of Darboux problems on the plane. He studied these problems in a 3D domain Ω0, bounded by two characteristic cones Σ1 and Σ2,0 and a plane region Σ0. What is the situation around these BVPs now after 50 years? It is well known that, for the infinite number of smooth functions in the right‐hand side of the equation, these problems do not have classical solutions. Popivanov and Schneider (1995) discovered the reason of this fact for the cases of Dirichlet′s or Neumann′s conditions on Σ0. In the present paper, we consider the case of third BVP on Σ0 and obtain the existence of many singular solutions for the wave equation. Especially, for Protter′s problems in ℝ3, it is shown here that for any n ∈ ℕ there exists a Cn(Ω¯0) ‐ right‐hand side function, for which the corresponding unique generalized solution belongs to Cn(Ω¯0\O), but has a strong power‐type singularity of order n at the point O. This singularity is isolated only at the vertex O of the characteristic cone Σ2,0 and does not propagate along the cone.

Suggested Citation

Handle: RePEc:wly:jnlaaa:v:2004:y:2004:i:4:p:315-335
DOI: 10.1155/S1085337504306111
as

Download full text from publisher

File URL: https://doi.org/10.1155/S1085337504306111
Download Restriction: no
File URL: https://libkey.io/10.1155/S1085337504306111?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
---><---

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:2004:y:2004:i:4:p:315-335. 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.

We have no bibliographic references for this item. You can help adding them by using 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://doi.org/10.1155/4058 .

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.