Small data blow-up of solutions to nonlinear Schrodinger equations without gauge invariance in L2
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Date
2021-03-31
Authors
Ren, Yuanyuan
Li, Yongsheng
Journal Title
Journal ISSN
Volume Title
Publisher
Texas State University, Department of Mathematics
Abstract
In this article we study the Cauchy problem of the nonlinear Schrödinger equations without gauge invariance
i∂tu + Δu = λ(|u|p1 + |v|p2, (t, x) ∈ [0, T) x ℝn,
i∂tv + Δv = λ(|u|p2 + |v|p1, (t, x) ∈ [0, T) x ℝn,
where 1 < p1, p2 < 1 + 4/n and λ ∈ ℂ\{0}. We first prove the existence of a local solution with initial data in L2(ℝn). Then under a suitable condition on the initial data, we show that the L2-norm of the solution must blow up in finite time although the initial data are arbitrarily small. As a by-product, we also obtain an upper bound of the maximal existence time of the solution.
Description
Keywords
Nonlinear Schrödinger equations, Weak solution, Blow up of solutions
Citation
Ren, Y., & Li, Y. (2021). Small data blow-up of solutions to nonlinear Schrodinger equations without gauge invariance in L2. <i>Electronic Journal of Differential Equations, 2021</i>(24), pp. 1-13.
Rights
Attribution 4.0 International