## Differential and Integral Equations

- Differential Integral Equations
- Volume 17, Number 11-12 (2004), 1293-1307.

### On the critical exponent for the Schrödinger equation with a nonlinear boundary condition

#### Abstract

We study the Schrödinger equation: $iu_t+u_{xx}=0,$ $ x\in {\bf R}_+,$ $ t>0$ with a nonlinear boundary condition $-u_x(0,t)=\vert u(0,t)\vert ^{p-1} u(0,t),$ $ t>0$. We show that if $1 <p <3,$ every solution is global in $H^1({\bf R}_+)$, while if $p\ge 3$, then nonglobal solutions exist.

#### Article information

**Source**

Differential Integral Equations, Volume 17, Number 11-12 (2004), 1293-1307.

**Dates**

First available in Project Euclid: 21 December 2012

**Permanent link to this document**

https://projecteuclid.org/euclid.die/1356060247

**Mathematical Reviews number (MathSciNet)**

MR2100028

**Zentralblatt MATH identifier**

1150.35303

**Subjects**

Primary: 35J65: Nonlinear boundary value problems for linear elliptic equations

Secondary: 35B33: Critical exponents 35J05: Laplacian operator, reduced wave equation (Helmholtz equation), Poisson equation [See also 31Axx, 31Bxx]

#### Citation

Ackleh, Azmy S.; Deng, Keng. On the critical exponent for the Schrödinger equation with a nonlinear boundary condition. Differential Integral Equations 17 (2004), no. 11-12, 1293--1307. https://projecteuclid.org/euclid.die/1356060247