Precise asymptotics of boundary layers for damped simple pendulum equations
The author considers the simple pendulum equation with friction \[ -u''(t)- |u'(t)|+ g(u(t))\equiv\lambda \sin u(t),\quad t\in (-T,T), \] \[ u(t)> 0,\quad t\in (-T, T),\qquad u(\pm T)= 0, \] where \(T> 0\) and \(\lambda> 0\) is a parameter; \(g\in C^m(\mathbb{R})\) \((m\geq 1)\) and \(g(u)> 0\) for \(u> 0\); \(g(0)= g'(0)= 0\); \(g(u)/u\) is strictly increasing for \(0\leq u\leq n\). The author establishes the asymptotic formula for the boundary layers of the solution \(u_\lambda\) for the equation above, and shows that its boundary slope is steeper than that of the solution without damping term \(|u'(t)|\).
scientific article; zbMATH DE number 5800218