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Existence and Stability of Standing Waves of Fourth Order Nonlinear Schrödinger Type Equation Related to Vortex Filament

Masaya Maeda, Jun-ichi Segata

2011 年 54 巻 1 号 p. 1-14
発行日: 2011年
公開日: 2011/04/21

DOIhttps://doi.org/10.1619/fesi.54.1

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In this paper, we study the fourth order nonlinear Schrödinger type equation (4NLS) which is a generalization of the Fukumoto-Moffatt [5] model that arising in the context of the motion of a vortex filament. Firstly, we mention the existence of standing wave solution and the conserved quantities. We next investigate the case that the equation is completely integrable and show that the standing wave obtained in [20] is orbitally stable in Sobolev spaces H^m with m ∈ N. Further, we show that the completely integrable equation is ill-posed in H^s with s ∈ (-1/2,1/2) by following Kenig-Ponce-Vega [13].

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Moment Estimates and Higher-Order Asymptotic Expansions of Solutions to a Parabolic System of Chemotaxis in the Whole Space

Tetsuya Yamada

2011 年 54 巻 1 号 p. 15-51
発行日: 2011年
公開日: 2011/04/21

DOIhttps://doi.org/10.1619/fesi.54.15

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We study the asymptotic profiles of bounded solutions to the initial value problem for a parabolic system of chemotaxis. Very recently, it was shown in Yamada [18] that under suitable initial conditions, Theorem 2.1 of Yamada [17] also holds true for bounded solutions by deriving the moment estimate of n-th order for the solution. In this paper, we prove that if the l-th moment of initial data is finite for some l ≥ 1, then the bounded solution admits the moment estimate of l-th order. Moreover, as an application, we give the asymptotic profiles of bounded solutions by space-time higher-order asymptotic expansions under suitable moment conditions on the initial data.

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Existence Results for a Functional Boundary Value Problem on an Infinite Interval

Kyriakos G. Mavridis, Panagiotis Ch. Tsamatos

2011 年 54 巻 1 号 p. 53-68
発行日: 2011年
公開日: 2011/04/21

DOIhttps://doi.org/10.1619/fesi.54.53

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Based on a fixed point theorem due to Avery and Henderson, we prove that a second order functional boundary value problem has at least two positive solutions.

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Multiscale Asymptotic Behavior of the Schrödinger Equation

Daoyuan Fang, Jian Xie, Thierry Cazenave

2011 年 54 巻 1 号 p. 69-94
発行日: 2011年
公開日: 2011/04/21

DOIhttps://doi.org/10.1619/fesi.54.69

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In this paper, we construct solutions e^itΔφ of the Schrödinger equation on R^N which have nontrivial asymptotic properties simultaneously on different time and space scales. More precisely, given μ ∈ (0,N) and β ≥ 1/2 we consider the set ω^β_μ,r(φ) of limit points in L^r(R^N) as t → ∞ of t^μ/2[e^itΔφ](·t^β). We show in particular that, given 0 < ν < N and an arbitrary countable set S ⊂ (ν,N), there exists an initial value ϕ such that ω^β_μ,r(ϕ) = L^r(R^N) for all μ ∈ (0,N) and β ≥ 1/2 such that μ/2β ∈ S, and all sufficiently large r. We also establish a result of a similar nature for a nonlinear Schrödinger equation.

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Nonlinear Gauge Invariant Evolution of Superposed Plane Waves

Kazuyuki Doi

2011 年 54 巻 1 号 p. 95-118
発行日: 2011年
公開日: 2011/04/21

DOIhttps://doi.org/10.1619/fesi.54.95

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We consider nonlinear gauge invariant evolution of superposed plane waves. We treat not only the case where the plane wave has one wave number vector, but also it has two wave number vectors including the case they are not Q-linearly independent (see (1.7) below). Assuming that the evolution is of dispersive type, we find a solution expressed by a special form. Moreover, we study the global behavior of the solution from its representation.

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Global Well-Posedness for Dissipative Korteweg-de Vries Equations

Stéphane Vento

2011 年 54 巻 1 号 p. 119-138
発行日: 2011年
公開日: 2011/04/21

DOIhttps://doi.org/10.1619/fesi.54.119

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This paper is devoted to the well-posedness for dissipative KdV equations u_t + u_xxx + |D_x|^2αu + uu_x = 0, 0 < α ≤ 1. An optimal bilinear estimate is obtained in Bourgain's type spaces, which provides global well-posedness in H^s(R), s > -3/4 for α ≤ 1/2 and s > -3/(5-2α) for α > 1/2.

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Global Attractor for Some Partial Functional Differential Equations with Infinite Delay

Hassane Bouzahir, Honglian You, Rong Yuan

2011 年 54 巻 1 号 p. 139-156
発行日: 2011年
公開日: 2011/04/21

DOIhttps://doi.org/10.1619/fesi.54.139

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This paper deals with a class of partial functional differential equations with infinite delay. Supposing that the linear part is a Hille-Yosida operator but not necessarily densely defined, and employing the integrated semigroups and dissipative dynamics theory, we present some appropriate conditions to guarantee existence of a global attractor.

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Singularities of Solutions to Schrödinger Equations with Constant Magnetic Fields

Shikuan Mao

2011 年 54 巻 1 号 p. 157-171
発行日: 2011年
公開日: 2011/04/21

DOIhttps://doi.org/10.1619/fesi.54.157

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In this paper we study the singularities of solutions to Schrödinger equations with magnetic fields. We characterize the microlocal singularities of the solutions to Schrödinger equations in the sense of Sobolev spaces.

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