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ISSN 1088-6850(e) ISSN 0002-9947(p)

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

Author(s): J. Berkovits; J. Mawhin
Journal: Trans. Amer. Math. Soc. 353 (2001), 5041-5055.
MSC (1991): Primary 35B10, 35L05, 35L20, 11J70, 47H10
Posted: 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

denotes the open ball of center

and radius

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

is not an accumulation point of the spectrum of the linear part. This result is used to obtain existence conditions for the nonlinear problem.

References:

1.: M. Abramowitz and I. Stegun (eds), Handbook of Mathematical Functions, Dover, New York, 1966. MR 34:8606
2.: A. Baker, Continued fractions of transcendental numbers, Matematika 9 (1962), 1-8. MR 26:2394
3.: A.K. Ben-Naoum and J. Berkovits, On the existence of periodic solutions for semilinear wave equations on ball in ${\mathbb R}^n$ with the space dimension odd, Nonlinear Anal. TMA 24 (1995), 241-250. MR 95k:35016
4.: A.K. Ben-Naoum and J. Mawhin, The periodic-Dirichlet problem for some semilinear wave equations, J. Differential Equations 96 (1992), 340-354. MR 93h:35009
5.: A.K. Ben-Naoum and J. Mawhin, Periodic solutions of some semilinear wave equations on balls and on spheres, Topological Methods in Nonlinear Anal. 1 (1993), 113-137. MR 94c:35019
6.: F. Bernstein, Ueber eine Anwendung der Mengenlehre auf ein aus der Theorie der säkularen Störungen herrührendes Problem, Math. Ann. 71 (1912), 417-439.
7.: E. Borel, Sur les équations aux dérivées partielles à coefficients constants et les fonctions non analytiques, C.R. Acad. Sci. Paris 121 (1895), 933-935.
8.: E. Borel, Les probabilités dénombrables et leurs applications arithmétiques, Rend. Circ. Mat. Palermo 27 (1909), 247-271.
9.: E. Borel, Sur un problème de probabilités relatifs aux fractions continues, Math. Ann. 72 (1912), 578-584.
10.: F.E. Browder, Nonlinear operators and nonlinear equations of evolution in Banach spaces, Proc. Symp. Pure Math. No. 18-2, Amer. Math. Soc., Providence, 1976. MR 53:8982
11.: G.H. Hardy and E.M. Wright, An Introduction to the Theory of Numbers, 5th ed., Oxford Univ. Press, 1979. MR 81i:10002
12.: V. Jarník, Zur metrischen Theorie der diophantischen Approximationen, Prace Mat.-Fiz. 36 (1928), 91-106.
13.: S. Lang, Introduction to Diophantine Approximations, New Enlarged Edition, Springer, New York, 1995. MR 96h:11067
14.: J. Mawhin, Solutions périodiques d'équations aux dérivées partielles hyperboliques non linéaires, Mélanges Théodore Vogel, Rybak, Janssens, Jessel ed., Presses de l'ULB, Bruxelles, 1978, 301-315. MR 87d:00007
15.: J. Mawhin, Semilinear equations of gradient type in Hilbert spaces and applications to differential equations, Nonlinear Differential Equations, Invariance, Stability and Bifurcation, Academic Press, New York, 1981, 269-282. MR 82i:47095
16.: J. Mawhin, Periodic solutions of some semilinear wave equations and systems : a survey, Chaos, Solitons and Fractals 5 (1995), 1651-1669. MR 96m:34074
17.: P.J. McKenna, On solutions of a nonlinear wave equation when the ratio of the period to the length of the interval is irrational, Proc. Amer. Math. Soc. 93 (1985), 59-64. MR 86f:35017
18.: I. Niven and H.S. Zuckerman, The Theory of Numbers, 4th ed., Wiley, New York, 1980. MR 81g:10001
19.: A.W. Rockett and P. Szüsz, Continued Fractions, World Scientific, Singapore, 1992. MR 93m:11060
20.: G. Sell, The prodigal integral, Amer. Math. Monthly 84 (1977), 162-167. MR 55:587
21.: J. Shallit, Real numbers with bounded partial quotients : a survey, Enseignement math. 38 (1992), 151-187. MR 93g:11011
22.: M.W. Smiley, Time periodic solutions of nonlinear wave equations in balls, Oscillations, Bifurcation and Chaos, Toronto 1986, Canad. Math. Soc. Confer. Proc., 1987, 287-297. MR 89c:35106
23.: G. Watson, A Treatise on the Theory of Bessel Functions, University Press, Cambridge, 1922.

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