Oscillatory behavior near blow-up of the solutions to some second-order nonlinear ODE. (English) Zbl 1265.34124

The author investigates the equation \[ u''(t)+| u(t)| ^{\beta }u(t)+g(u'(t))=0, \] where \(\beta >0\) and \(g\) is a locally Lipschitz continuous function which satisfies: \[ \exists c>0\;\exists \eta >0: \eta | v| ^{\alpha +2}\leq g(v)v\leq c| v| ^{\alpha +1} \] for some \(\alpha >0\). The behaviour and the oscillatory properties of solutions of the above equation are determined by the constants \(\alpha \) and \(\beta \). The article generalizes results of A. Haraux [Anal. Appl., Singap. 9, No. 1, 49–69 (2011; Zbl 1227.34052)] which concern the case of \(g(u')=b| u'| ^{\alpha }u'\) and \(t\geq 0\), and the results of P. Souplet [Differ. Integral Equ. 11, No. 1, 147–167 (1998; Zbl 1015.34038)] and M. Balabane et al. [Discrete Contin. Dyn. Syst. 9, No. 3, 577–584 (2003; Zbl 1048.34074)] who studied the equation \( u''+| u| ^{\beta }u=b| u'| ^{\alpha }u' \) for \(t\geq 0\).


34C10 Oscillation theory, zeros, disconjugacy and comparison theory for ordinary differential equations
34D05 Asymptotic properties of solutions to ordinary differential equations
34C20 Transformation and reduction of ordinary differential equations and systems, normal forms