Precise Asymptotics of Boundary Layers for Damped Simple Pendulum Equations

We consider the simple pendulum equation with friction: -u ''(t) - |u'(t)| + g(u(t)) = lambda sin u(t), t is an element of I := (-T, T), u(t) > 0, t is an element of I, u(+/- T) = 0, where T > 0 is a constant and lambda > 0 is a parameter. The case without friction is known as the simple pendulum equation with self interaction, and the asymptotic shape of the solution as lambda -> infinity is well understood. In this paper, we establish the asymptotic formula for the boundary layers of the solution u(lambda) for the equation above, and show that its boundary slope is steeper than that of the solution without damping term |u'(t)|.