On the growth of high Sobolev norms of the fourth-order Schrödinger equation

Abstract

This paper is concerned with the Sobolev norm growth for the solution to the one-dimensional cubic fourth-order Schrödinger equation. Applying Tao's $[k;Z]$-multiplier method, we gain some bilinear estimates. Then we show the solution satisfies $$ \|u(t)\|_{H^s}\leq \|u(\tau)\|_{H^s}+ C\|u(\tau)\|_{H^s}^{1-\delta}, \qquad\delta^{-1}=(s-2)^+ $$ and then derive a polynomial upper bound of time $t$.