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

Xianfa Song,Xiaowei An,Desheng Li

doi:10.1155/2013/238410

Abstract

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.

Highlights

In this paper, 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, (1)u (x, 0) = u0 (x) ∈ Σ, x ∈ RN, where V(x) and W(x) are real-valued potentials, V(x) ≥ 0 and W(x) is even, f(x, |u|2) is measurable in x and continuous in |u|2,(W (x) ⋆ |u|2) u (x) = (∫ RN W (x − y)󵄨󵄨󵄨󵄨u (y)󵄨󵄨󵄨󵄨2dy)
We will prove Theorems 3 and 4, which give some sufficient conditions on global existence and blowup of the solution to (1)
As a corollary of Theorem 5, we obtain the sharp threshold for global existence and blowup of the solution of (8) as follows

Summary

Introduction

We obtain some sufficient conditions and establish two sharp thresholds for the blowup and global existence of the solution to the problem. In [2], Oh obtained the local well-posedness and global existence results of (9) under some conditions. In [3, 5], Gan et al and Zhang, respectively, established the sharp thresholds for the global existence and blowup of the solutions to (9) under some conditions.

Results

Conclusion