Boundedness of solutions for Duffing equation with low regularity in time

Abstract

It is shown that all solutions are bounded for Duffing equation $\ddot{x}+ x^{2n+1}+\sum_{j=0}^{2n}P_{j}(t)x^{j}=0,$ provided that for each $n+1\le j\le 2n$, $P_j(t)\in C^{\gamma}(\mathbb T)$ with $\gamma>1-\frac1n$ and for each $0\le j\le n$, $P_j\in L(\mathbb T^1)$.