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

In this paper, we construct a continuous positive periodic function p(t) such that the corresponding superlinear Duffing equation

$$x'' + a\left( x \right){x^{2n + 1}} + p\left( t \right){x^{2m + 1}} = 0,n + 2 \leqslant 2m + 1 < 2n + 1$$

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.