Funkcialaj Ekvacioj

Asymptotic Behavior of Oscillatory Solutions of First Order Delay Differential Equations of Unstable Type

Tateo Baba

2013 Volume 56 Issue 2 Pages 177-191
Published: 2013
Released on J-STAGE: August 26, 2013

DOIhttps://doi.org/10.1619/fesi.56.177

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In this paper, we study the asymptotic behavior of solutions of the first order delay differential equations of unstable type. A new sufficient condition is given under which every oscillatory solution of the delay equation of unstable type tends to zero asymptotically.

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On the Cauchy Problem of Fractional Schrödinger Equation with Hartree Type Nonlinearity

Yonggeun Cho, Hichem Hajaiej, Gyeongha Hwang, Tohru Ozawa

2013 Volume 56 Issue 2 Pages 193-224
Published: 2013
Released on J-STAGE: August 26, 2013

DOIhttps://doi.org/10.1619/fesi.56.193

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We study the Cauchy problem for the fractional Schrödinger equation i∂_tu = (m²−Δ)^α/2u + F(u) in R¹⁺ⁿ, where n ≥ 1, m ≥ 0, 1 < α < 2, and F stands for the nonlinearity of Hartree type F(u) = λ (ψ (·) |·|^−γ ∗ |u|²)u with λ = ±1, 0 < γ < n, and 0 ≤ ψ ∈ L^∞ (Rⁿ). We prove the existence and uniqueness of local and global solutions for certain α, γ, λ, ψ. We also remark on finite time blowup of solutions when λ = −1.

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Characterization of PDE Reducible to ODE under a Certain Homogeneity and Applications to Singular Cauchy Problems

Takuya Watanabe, Jiichiroh Urabe

2013 Volume 56 Issue 2 Pages 225-247
Published: 2013
Released on J-STAGE: August 26, 2013

DOIhttps://doi.org/10.1619/fesi.56.225

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We give a necessary and sufficient condition for a homogeneous partial differential equation in two variables to be reduced to a homogeneous ordinary one under a certain change of variables. It is described by means of the commutator with a first order partial differential operator which characterizes a homogeneity. Moreover we obtain the explicit representation of the reduced ordinary differential equation. This result is a generalization of such a reduction which had been applied to singular Cauchy problems in our previous works [U, WU1]. This fact suggests that local structures of the solutions to partial differential equations can be described by global structures of those to ordinary ones.

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Nonlinear Neumann Problems with Constraints

Leszek Gasiński, Nikolaos S. Papageorgiou

2013 Volume 56 Issue 2 Pages 249-270
Published: 2013
Released on J-STAGE: August 26, 2013

DOIhttps://doi.org/10.1619/fesi.56.249

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We consider a C¹-functional ψ defined on the "Neumann" Sobolev space W_n^1,p(Ω). If M is a C¹-submanifold, then for ψ|_M we show that any local C_n¹($\overline{\Omega}$)-minimizer is also a local W_n^1,p(Ω)-minimizer. Then we use this general result on local minimizers to show that a nonlinear parametric Neumann problem driven by the p-Laplace differential operator and restricted on a sphere, has at least three distinct smooth solutions.

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Parametric Transformations between the Heun and Gauss Hypergeometric Functions

Raimundas Vidunas, Galina Filipuk

2013 Volume 56 Issue 2 Pages 271-321
Published: 2013
Released on J-STAGE: August 26, 2013

DOIhttps://doi.org/10.1619/fesi.56.271

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The hypergeometric and Heun functions are classical special functions. Transformation formulas between them are commonly induced by pull-back transformations of their differential equations, with respect to some coverings P¹ → P¹. This gives expressions of Heun functions in terms of better understood hypergeometric functions. This article presents the list of hypergeometric-to-Heun pull-back transformations with a free continuous parameter, and illustrates most of them by a Heun-to-hypergeometric reduction formula. In total, 61 parametric transformations exist, of maximal degree 12.

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Uniqueness of Steady Navier-Stokes Flows in Exterior Domains

Tomoyuki Nakatsuka

2013 Volume 56 Issue 2 Pages 323-337
Published: 2013
Released on J-STAGE: August 26, 2013

DOIhttps://doi.org/10.1619/fesi.56.323

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We consider the uniqueness of stationary solutions to the Navier-Stokes equation in 3-dimensional exterior domains within the class u ∈ L_{3, ∞} with ∇u ∈ L_{3/2, ∞}, where L_{3, ∞} and L_{3/2, ∞} are the Lorentz spaces. It is shown that if solutions u and v satisfy the conditions that u is small in L_{3, ∞} and v ∈ L₃ + L_∞, then u = v. The proof relies upon the regularity theory for the perturbed Stokes equation.

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