# The coexistence of quasi-periodic and blow-up solutions in a superlinear Duffing equation

## TL;DR

This paper constructs a specific superlinear Duffing equation with a positive periodic coefficient that admits both finite-time blow-up solutions and infinitely many subharmonic and quasi-periodic solutions, revealing complex coexistence phenomena.

## Contribution

It demonstrates the coexistence of blow-up and quasi-periodic solutions in a superlinear Duffing equation with a specially designed periodic coefficient.

## Key findings

- Existence of solutions that blow up in finite time.
- Presence of infinitely many subharmonic and quasi-periodic solutions.
- Construction of a periodic coefficient function enabling this coexistence.

## Abstract

In this paper we will construct a continuous positive periodic function $p(t)$ such that the corresponding superlinear Duffing equation $$ x"+a(x)\,x^{2n+1}+p(t)\,x^{2m+1}=0,\ \ \ \ n+2\leq 2m+1<2n+1 $$ possesses a solution which escapes to infinity in some finite time, and also has infinitely many subharmonic and quasi-periodic solutions, where the coefficient $a(x)$ is an arbitrary positive smooth periodic function defined in the whole real axis.