Hostname: page-component-586b7cd67f-dlnhk Total loading time: 0 Render date: 2024-11-26T21:42:05.418Z Has data issue: false hasContentIssue false
  • English
  • Français

Uniqueness of solutions to certain hyperbolic boundary value problems in a semi-infinite strip

Part of: General higher-order equations and systems

Published online by Cambridge University Press:  26 February 2010

C. M. Khalique
C. M. Khalique
Affiliation:
Department of Mathematical Sciences, University of North-West, Private Bag X2046, Mmabatho 2735, South Africa.
Get access

Abstract

It is well known that boundary value problems for hyperbolic equations are in general “not well posed” problems. This paper is concerned with the uniqueness of solutions to boundary value problems for the hyperbolic equation uxx − Qu = utt. Here Q is a function of the variable x alone, and satisfies the following conditions:

(a) Q:[0, ∞) → ℝ;

(b) Q is Lebesgue integrable on any compact subinterval of [0, ∞);

(c) Q(x)→ ∞ as x → ∞.

MSC classification

Secondary: 35G15: Boundary value problems for linear higher-order equations
Type
Research Article
Information
Mathematika , Volume 46 , Issue 1 , June 1999 , pp. 155 - 163
Copyright
Copyright © University College London 1999

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

1.Bourgin, D. G. and Duffin, R.. The Dirichlet problem for the vibrating string equation, Bull. Amer. Math. Soc, 45, (1939), 851–858. MR1#12O.CrossRefGoogle Scholar
2.Diaz, J. B. and Young, E. C.. Uniqueness of solutions of certain boundary value problems for ultrahyperbolic equations. Proc. Amer. Math. Soc. 29 (1971), 569–574. MR44#626.CrossRefGoogle Scholar
3.Dunninger, D. R. and Zachmanoglou, E. C.. The condition for uniqueness of solutions of the Dirichlet problem for the wave equation in coordinate rectangles. J. Math. Anal. Appl., 20 (1967), 17–21. MR37#1807.CrossRefGoogle Scholar
4.Dunninger, D. R. and Zachmanoglou, E. C.. The condition for uniqueness of the Dirichlet problem for hyperbolic equations in cylindrical domains. J. Math. Mech. 18 (1969), 763–766. MR#4817.Google Scholar
5.Khalique, C. M.. Some existence and uniqueness theorems for the Dirichlet and Neumann problems for hyperbolic differential equations. Ph.D. thesis, University of Dundee, Scotland, 1979.Google Scholar
6.Titchmarsh, E. C.. Theory of Functions (Oxford University Press, 1939).Google Scholar
7.Titchmarsh, E. C.. Eigenfunction expansions associated with second order differential equations. Part I (2nd edn.) (Oxford University Press, 1962).CrossRefGoogle Scholar
8.Travis, C. C.. On the uniqueness of solutions to hyperbolic boundary value problems. Trans. Amer. Math. Soc, 216 (1976) 327–336.Google Scholar
9.Travis, C. C. and Young, E. C.. Uniqueness of solutions to singular boundary value problems. SIAM. J. Math. Anal., 8, (1977), 111–117.CrossRefGoogle Scholar
10.Young, E. C.. Uniqueness of solutions of the Dirichlet and Neumann problems for hyperbolic equations. Trans. Amer. Math. Soc, 160 (1971), 403–409. MR43#7777.CrossRefGoogle Scholar
11.Young, E. C.. Uniqueness of solutions of the Dirichlet problem for singular ultrahyperbolic equations. Proc. Amer. Math. Soc. 36 (1972), 130–136.CrossRefGoogle Scholar