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Diophantine Approximation, Bessel Functions and Radially Symmetric Periodic Solutions of Semilinear Wave Equations in a Ball

J. Berkovits and J. Mawhin
Transactions of the American Mathematical Society
Vol. 353, No. 12 (Dec., 2001), pp. 5041-5055
Stable URL: http://www.jstor.org/stable/2693916
Page Count: 15
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Since scans are not currently available to screen readers, please contact JSTOR User Support for access. We'll provide a PDF copy for your screen reader.
Diophantine Approximation, Bessel Functions and Radially Symmetric Periodic Solutions of Semilinear Wave Equations in a Ball
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Abstract

The aim of this paper is to consider the radially-symmetric periodic-Dirichlet problem on [0, T] × Bn[a] for the equation utt - Δ u = f(t, |x|, u), where Δ is the classical Laplacian operator, and Bn[a] denotes the open ball of center 0 and radius a in Rn. When α = 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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