On the Global Existence and Blowup Phenomena of Schrödinger Equations with Multiple Nonlinearities

2010/09/02 by Xianfa Song, Song, Xianfa
Mathematics · #Analysis of PDEs (math.AP) #FOS: Mathematics #Primary 35Q55 #math.AP #msc:35Q55

paper · pdf · doi:10.48550/arxiv.1009.0328

24 pages

arxiv created 2010/09/02 · arxiv updated 2010/09/03

Abstract

In this paper, we consider the global existence and blowup phenomena of the following Cauchy problem \· -i u_t=Δu-V(x)u+f(x,|u|²)u+(W⋆|u|²)u, x∈ℝ^N, t>0, · u(x,0)=u₀(x), x∈ℝ^N, . where V(x) and W(x) are real-valued potentials with V(x)≥ 0 and W is even, f(x,|u|²) is measurable in x and continuous in |u|², and u₀(x) is a complex-valued function of x. We obtain some sufficient conditions and establish two sharp thresholds for the blowup and global existence of the solution to the problem. These results can be looked as the supplement to Chapter 6 of \citeCazenave2. In addition, our results extend those of \citeZhang and improve some of \citeTao2.

Related

• Global existence, blowup phenomena, and asymptotic behavior for quasilinear Schrödinger equations
• Global existence and blow-up for the variable coefficient Schrödinger equations with a linear potential
• Blow-up solutions to a class of nonlinear coupled Schrödinger systems with power-type-growth nonlinearities
• Global existence results for nonlinear Schrodinger equations with quadratic potentials
• On nonlinear Schrödinger equations with repulsive inverse-power potentials