Electronic Journal of Differential Equations,
Vol. 2007(2007), No. 64, pp. 1-11.
Title: Asymptotic shape of solutions to
the perturbed simple pendulum problems
Author: Tetsutaro Shibata (Hiroshima Univ., Higashi-Hiroshima, Japan)
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 $0 < r < R$, $ 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 $\Vert u_\lambda\Vert_q$
($1 \le q < \infty$) 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; $L^q$-norm; simple pendulum.