A small initial data criterion of global existence for the damped nonlinear Schrödinger equation

2009 ◽  
Vol 42 (5) ◽  
pp. 055205 ◽  
Author(s):  
Guanggan Chen ◽  
Jian Zhang ◽  
Yunyun Wei
Keyword(s):  
Initial Data ◽  
Global Existence ◽  
Schrödinger Equation ◽  
Nonlinear Schrödinger Equation ◽  
Schrodinger Equation ◽  
Nonlinear Schrodinger Equation ◽  
Small Initial Data ◽  
Nonlinear Schrödinger ◽  
Data Criterion

scholarly journals Phenomena of Blowup and Global Existence of the Solution to a Nonlinear Schrödinger Equation

2013 ◽  
Vol 2013 ◽  
pp. 1-14
Author(s):  
Xiaowei An ◽  
Desheng Li ◽  
Xianfa Song
Keyword(s):  
Cauchy Problem ◽  
Global Existence ◽  
Schrödinger Equation ◽  
Nonlinear Schrödinger Equation ◽  
Schrodinger Equation ◽  
Sufficient Conditions ◽  
Nonlinear Schrodinger Equation ◽  
Nonlinear Schrödinger ◽  
Complex Valued ◽  
Sharp Thresholds

We consider the following Cauchy problem:-iut=Δu-V(x)u+f(x,|u|2)u+(W(x)⋆|u|2)u,x∈ℝN,t>0,u(x,0)=u0(x),x∈ℝN,whereV(x)andW(x)are real-valued potentials andV(x)≥0andW(x)is even,f(x,|u|2)is measurable inxand continuous in|u|2, andu0(x)is a complex-valued function ofx. We obtain some sufficient conditions and establish two sharp thresholds for the blowup and global existence of the solution to the problem.

scholarly journals Finite speed of disturbance for the nonlinear Schrödinger equation

2018 ◽  
Vol 149 (6) ◽  
pp. 1405-1419
Author(s):  
Simão Correia
Keyword(s):  
Initial Data ◽  
Schrödinger Equation ◽  
Nonlinear Schrödinger Equation ◽  
Schrodinger Equation ◽  
Nonlinear Schrodinger Equation ◽  
Finite Speed Of Propagation ◽  
Finite Speed ◽  
Nonlinear Schrödinger ◽  
The Cauchy Problem ◽  
Whole Space

AbstractWe consider the Cauchy problem for the nonlinear Schrödinger equation on the whole space. After introducing a weaker concept of finite speed of propagation, we show that the concatenation of initial data gives rise to solutions whose time of existence increases as one translates one of the initial data. Moreover, we show that, given global decaying solutions with initial data u0, v0, if |y| is large, then the concatenated initial data u0 + v0(· − y) gives rise to globally decaying solutions.

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