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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

 

Diophantine approximation, Bessel functions and radially symmetric periodic solutions of semilinear wave equations in a ball


Authors: J. Berkovits and J. Mawhin
Journal: Trans. Amer. Math. Soc. 353 (2001), 5041-5055
MSC (1991): Primary 35B10, 35L05, 35L20, 11J70, 47H10
Published electronically: July 13, 2001
MathSciNet review: 1852093
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Abstract:

The aim of this paper is to consider the radially-symmetric periodic-Dirichlet problem on $[0,T] \times B^n[a]$ for the equation

\begin{displaymath}u_{tt} - \Delta u = f(t,\vert x\vert,u),\end{displaymath}

where $\Delta$ is the classical Laplacian operator, and $B^n[a]$ denotes the open ball of center $0$ and radius $a$ in ${\mathbb R}^n.$ When $\alpha = a/T$ is a sufficiently large irrational with bounded partial quotients, we combine some number theory techniques with the asymptotic properties of the Bessel functions to show that $0$ is not an accumulation point of the spectrum of the linear part. This result is used to obtain existence conditions for the nonlinear problem.


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Additional Information

J. Berkovits
Affiliation: Department of Mathematics, University of Oulu, Oulu, Finland
Email: jberkovi@sun3.oulu.fi

J. Mawhin
Affiliation: Université Catholique de Louvain, Institut Mathématique, B-1348 Louvain-la-Neuve, Belgium
Email: mawhin@amm.ucl.ac.be

DOI: http://dx.doi.org/10.1090/S0002-9947-01-02875-6
PII: S 0002-9947(01)02875-6
Keywords: Diophantine approximations, Bessel functions, wave equation, periodic solutions
Received by editor(s): December 28, 1999
Published electronically: July 13, 2001
Article copyright: © Copyright 2001 American Mathematical Society