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 $t$.
Citation
Qionglei Chen. Mingming Deng. "On the growth of high Sobolev norms of the fourth-order Schrödinger equation." Differential Integral Equations 36 (7/8) 661 - 678, July/August 2023. https://doi.org/10.57262/die036-0708-661
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