Abstract
In this paper, we investigate the stability for the nonlinear Hartree equation with time-dependent coefficients \t\t\ti∂tu+Δu+α(t)1|x|u+β(t)(W∗|u|2)u=0.\\documentclass[12pt]{minimal}\t\t\t\t\\usepackage{amsmath}\t\t\t\t\\usepackage{wasysym}\t\t\t\t\\usepackage{amsfonts}\t\t\t\t\\usepackage{amssymb}\t\t\t\t\\usepackage{amsbsy}\t\t\t\t\\usepackage{mathrsfs}\t\t\t\t\\usepackage{upgreek}\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\t\t\t\t\\begin{document}$$i\\partial_{t}u+\\Delta u+ \\alpha(t)\\frac{1}{ \\vert x \\vert }u+\\beta(t) \\bigl(W \\ast \\vert u \\vert ^{2}\\bigr)u=0. $$\\end{document} We first obtain the Lipschitz continuity of the solution u=u(alpha ,beta) with respect to coefficients α and β, and then prove that this equation is stable under the perturbation of coefficients. Our results improve some recent results.
Highlights
We investigate the stability for the nonlinear Hartree equation with time-dependent coefficients i∂tu +
We first obtain the Lipschitz continuity of the solution u = u(α, β) with respect to coefficients α and β, and prove that this equation is stable under the perturbation of coefficients
4 Conclusions In this paper, we consider the stability for the nonlinear Schrödinger equation ( . ) of Hartree type under the perturbation of coefficients
Summary
Motivated by the nonlinearity management and dispersion management [ , ] in the experimental work in Bose-Einstein condensates and optics, nonlinear Schrödinger equations have attracted more and more attention in both the physics and the mathematics fields; see [ – ] and the references therein. We consider the stability for the following nonlinear Schrödinger equation of Hartree type under the perturbation of coefficients:. Let uj be the H solution of the following perturbed Hartree equation: i∂tuj =. Our results hold for more general Hartree nonlinearities (W ∗ |u| )u, where W ∈ Lp(R ) + L∞(R ) for some p ≥ We extend this result to the time-dependent coefficients α(t) and β(t). We prove the locally Lipschitz continuity of the solution u(α, β) with respect to the coefficients α and β. Our results follow if the potential V : RN → R is a real-valued function, satisfying:. In Section , we firstly obtain the Lipschitz continuity of the solution u = u(α, β) with respect to coefficients α and β, and prove Theorem.
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