## Advances in Differential Equations

- Adv. Differential Equations
- Volume 16, Number 5/6 (2011), 467-486.

### Well-posedness for the fourth-order Schrödinger equations with quadratic nonlinearity

#### 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].

#### Article information

**Source**

Adv. Differential Equations Volume 16, Number 5/6 (2011), 467-486.

**Dates**

First available in Project Euclid: 17 December 2012

**Permanent link to this document**

https://projecteuclid.org/euclid.ade/1355703297

**Mathematical Reviews number (MathSciNet)**

MR2816113

**Zentralblatt MATH identifier**

1232.35160

**Subjects**

Primary: 35Q40: PDEs in connection with quantum mechanics 35Q55: NLS-like equations (nonlinear Schrödinger) [See also 37K10] 47J35: Nonlinear evolution equations [See also 34G20, 35K90, 35L90, 35Qxx, 35R20, 37Kxx, 37Lxx, 47H20, 58D25]

#### Citation

Zheng, Jiqiang. Well-posedness for the fourth-order Schrödinger equations with quadratic nonlinearity. Adv. Differential Equations 16 (2011), no. 5/6, 467--486. https://projecteuclid.org/euclid.ade/1355703297