Precise Asymptotic Behavior of Solutions to Damped Simple Pendulum Equations

We consider the simple pendulum equation −u′′(t) + f(u′(t)) = λ sinu(t), t ∈ I := (−1, 1), u(t) > 0, t ∈ I, u(±1) = 0, where 0 < ≤ 1, λ > 0, and the friction term is either f(y) = ±|y| or f(y) = −y. Note that when f(y) = −y and = 1, we have well known original damped simple pendulum equation. To understand the dependance of solutions, to the damped simple pendulum equation with λ 1, upon the term f(u′(t)), we present asymptotic formulas for the maximum norm of the solutions. Also we present an asymptotic formula for the time at which maximum occurs, for the case f(u) = −u.