Long-time behavior of solutions of superlinear systems of differential equations

TL;DR

This paper analyzes the long-term asymptotic behavior of solutions to nonlinear differential systems with superlinear terms, showing solutions decay like a specific power of time with an explicit decay rate.

Contribution

It provides a precise description of the asymptotic decay of solutions for nonlinear systems with positively homogeneous lowest order terms, extending understanding beyond linear approximations.

Findings

Solutions decay as $\xi t^{-p}$ for large t

Decay rate p is explicitly determined

Behavior characterized by a nonzero vector \xi

Abstract

This paper establishes the precise asymptotic behavior, as time $t$ tends to infinity, for nontrivial, decaying solutions of genuinely nonlinear systems of ordinary differential equations. The lowest order term in these systems, when the spatial variables are small, is not linear, but rather positively homogeneous of a degree greater than one. We prove that the solution behaves like $\xi t^{-p}$, as $t\to\infty$, for a nonzero vector $\xi$ and an explicit number $p>0$.