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May/June 2011 Well-posedness for the fourth-order Schrödinger equations with quadratic nonlinearity
Jiqiang Zheng
Adv. Differential Equations 16(5/6): 467-486 (May/June 2011).

Abstract

This paper is concerned with one-dimensional quadratic semilinear fourth-order Schrödinger equations. Motivated by the quadratic Schrödinger equations in the pioneering work of Kenig-Ponce-Vega [12], three bilinearities $uv,$ $\overline{uv},$ and $\overline{u}v $ for functions $u,v: \mathbb{R}\times[0,T]\mapsto \mathbb{C} $ are sharply estimated in function spaces $X_{s,b}$ associated to the fourth-order Schrödinger operator $i\partial_t+\Delta^2-\varepsilon\Delta$. These bilinear estimates imply local wellposedness results for fourth-order Schrödinger equations with quadratic nonlinearity. To establish these bilinear estimates, we derive a fundamental estimate on dyadic blocks for the fourth-order Schrödinger from the $[k,Z]$-multiplier norm argument of Tao [20].

Citation

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Jiqiang Zheng. "Well-posedness for the fourth-order Schrödinger equations with quadratic nonlinearity." Adv. Differential Equations 16 (5/6) 467 - 486, May/June 2011.

Information

Published: May/June 2011
First available in Project Euclid: 17 December 2012

zbMATH: 1232.35160
MathSciNet: MR2816113

Subjects:
Primary: 35Q40, 35Q55, 47J35

Rights: Copyright © 2011 Khayyam Publishing, Inc.

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Vol.16 • No. 5/6 • May/June 2011
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