Electron. J. Differential Equations, Vol. 2018 (2018), No. 55, pp. 1-52.

Global well-posedness of semilinear hyperbolic equations, parabolic equations and Schrodinger equations

Runzhang Xu, Yuxuan Chen, Yanbing Yang, Shaohua Chen, Jihong Shen, Tao Yu, Zhengsheng Xu

Abstract:
This article studies the existence and nonexistence of global solutions to the initial boundary value problems for semilinear wave and heat equation, and for Cauchy problem of nonlinear Schrodinger equation. This is done under three possible initial energy levels, except the NLS as it does not have comparison principle. The most important feature in this article is a new hypothesis on the nonlinear source terms which can include at least eight important and popular power-type nonlinearities as special cases. This article also finds some kinds of divisions for the initial data to guarantee the global existence or finite time blowup of the solution of the above three problems.

Submitted December 5, 2017. Published February 23, 2018.
Math Subject Classifications: 35L05, 35K05, 35Q55, 35A15.
Key Words: Semilinear hyperbolic equation; semilinear parabolic equation; nonlinear Schrodinger equation; global solution; potential well.

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Runzhang Xu
College of Science
Harbin Engineering University, China
email: xurunzh@163.com
Yuxuan Chen
College of Automation
Harbin Engineering University, China
email: chenyuxuan07@126.com}
Yanbing Yang
College of Science
Harbin Engineering University, China
email: yangyanbheu@163.com
Shaohua Chen
Department of Mathematics
Cape Breton University
Sydney, NS, Canada
email: george_chen@cbu.ca
Jihong Shen
College of Science
Harbin Engineering University, China
email: shenjihong@hrbeu.edu.cn
Tao Yu
College of Science
Harbin Engineering University, China
email: yutao@hrbeu.edu.cn
Zhengsheng Xu
College of Science
Harbin Engineering University, China
email: xuzhengsheng1@163.com

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