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Media type:
E-Article
Title:
New singular solutions of Protter′s problem for the 3D wave equation
Contributor:
Grammatikopoulos, M. K.;
Popivanov, N. I.;
Popov, T. P.
Published:
Wiley, 2004
Published in:
Abstract and Applied Analysis, 2004 (2004) 4, Seite 315-335
Language:
English
DOI:
10.1155/s1085337504306111
ISSN:
1085-3375;
1687-0409
Origination:
Footnote:
Description:
In 1952, for the wave equation,Protter formulated some boundaryvalue problems (BVPs), which are multidimensionalanalogues of Darboux problems on the plane. He studied theseproblems in a 3D domain Ω0, bounded by twocharacteristic cones Σ1 and Σ2,0 and aplane region Σ0. What is the situation around theseBVPs now after 50 years? It is well known that, for the infinitenumber of smooth functions in the right‐hand side of theequation, these problems do not have classical solutions.Popivanov and Schneider (1995) discovered the reason of this factfor the cases of Dirichlet′s or Neumann′s conditions on Σ0. In the present paper, we consider the case of third BVP onΣ0 and obtain the existence of many singularsolutions for the wave equation. Especially, for Protter′sproblems in ℝ3, it is shown here that for any n ∈ ℕ there exists a ‐ right‐handside function, for which the corresponding unique generalizedsolution belongs to buthas a strong power‐type singularity of order n at the pointO. This singularity is isolated only at the vertex O of thecharacteristic cone Σ2,0 and does not propagate alongthe cone.