The universal bound property for a class of second order ODEs

Abstract

We consider the scalar second order ODE $u^{\prime \prime }+|u^{\prime }{|}^{\alpha }{u}^{\prime }+|u{|}^{\beta }u=0$, where $\alpha ,\beta$ are two positive numbers and the non-linear semi-group $S(t)$ generated on ${\mathbb{R}}^2$ by the system in $(u,u^{\prime })$. We prove that $S(t){\mathbb{R}}^2$ is bounded for all $t>0$ whenever $0<\alpha <\beta$ and moreover there is a constant $C$ independent of the initial data such that