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

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.