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

Berkovits, J.; Mawhin, J.

doi:10.1090/S0002-9947-01-02875-6

Diophantine approximation, Bessel functions and radially symmetric periodic solutions of semilinear wave equations in a ball
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by J. Berkovits and J. Mawhin PDF

Trans. Amer. Math. Soc. 353 (2001), 5041-5055 Request permission

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 \[ u_{tt} - \Delta u = f(t,|x|,u),\] 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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