Oscillatory behavior near blow-up of the solutions to some second-order nonlinear ODE

Abstract

We study the oscillation properties of solutions to the nonlinear scalar second-order ODE \begin{equation} u''(t)+\vert u(t) \vert ^{\beta}u(t)+g(u'(t))=0,\quad t\leq0, \end{equation} where $\beta$ is a positive constant and $g:\mathbf{R}\rightarrow \mathbf{R}$ is an increasing and locally Lipschitz function behaving globally like $\vert v \vert^{\alpha}v$, $\alpha>0.$