Global behavior of the Solutions to a Class of Nonlinear, Singular Second Order ODE

TL;DR

This paper investigates the initial value problem and global behavior of solutions for a class of nonlinear, singular second order ODEs, establishing existence, uniqueness, decay rates, and oscillatory properties based on parameter conditions.

Contribution

It provides a comprehensive analysis of existence, uniqueness, regularity, decay rates, and oscillatory behavior for solutions of a specific class of nonlinear second order ODEs.

Findings

Solutions exist and are unique under certain conditions.

Optimal decay rates to zero are established.

Oscillatory and non-oscillatory behaviors are characterized.

Abstract

In this paper the initial value problem and global properties of solutions are studied for the scalar second order ODE: $ (|u'|^{l}u')' + c|u'|^{\alpha}u' + d|u|^\beta u=0$, where $\alpha,\beta,l,c, d$ are positive constants. In particular, existence, uniqueness and regularity as well as optimal decay rates of solutions to 0 are obtained depending on the various parameters, and the oscillatory or non-oscillatory behavior is elucidated.