Electron. J. Diff. Eqns., Vol. 2007(2007), No. 64, pp. 1-11.

Asymptotic shape of solutions to the perturbed simple pendulum problems

Tetsutaro Shibata

Abstract:
We consider the positive solution of the perturbed simple pendulum problem
$$ u''(r) + \frac{N-1}{r}u'(r) - g(u(t)) + \lambda \sin u(r) = 0, $$
with 0 less than r less than R, $ u'(0) = u(R) = 0$. To understand well the shape of the solution $u_\lambda$ when $\lambda \gg 1$, we establish the leading and second terms of $\Vert u_\lambda\Vert_q$ ($1 \le q$ less than $\infty$) with the estimate of third term as $\lambda \to \infty$. We also obtain the asymptotic formula for $u_\lambda'(R)$ as $\lambda \to \infty$.

Submitted May 12, 2006. Published May 9, 2007.
Math Subject Classifications: 35J60.
Key Words: Asymptotic formulas; Lq-norm; simple pendulum.

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Tetsutaro Shibata
Department of Applied Mathematics
Graduate School of Engineering
Hiroshima University
Higashi-Hiroshima, 739-8527, Japan
email: shibata@amath.hiroshima-u.ac.jp

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