Differential and Integral Equations

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

Azmy S. Ackleh and Keng Deng

Full-text: Open access

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


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