Funkcialaj Ekvacioj

Equation of Motion for Incompressible Mixed Fluid Driven by Evaporation and Its Application to Online Rankings

Kumiko Hattori, Tetsuya Hattori

2009 Volume 52 Issue 3 Pages 301-319
Published: 2009
Released on J-STAGE: February 04, 2010

DOIhttps://doi.org/10.1619/fesi.52.301

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We consider a system of partial differential equations for the densities of components of one dimensional incompressible fluid mixture whose motion is driven by evaporation. We prove existence and give explicit form of unique global classical non-negative solution to initial value problem for the system. The solution is known to be an infinite particle limit of stochastic ranking processes, which is a simple stochastic model of time evolutions of Amazon Sales Ranks. As a practical application, we collected data from the web and performed statistical fits to our formula. The results suggest that the system of equations and solutions, though very simple, may have applications in the analysis of online rankings.

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On State and Output Stabilization of Linear Parabolic Systems

Takao Nambu

2009 Volume 52 Issue 3 Pages 321-341
Published: 2009
Released on J-STAGE: February 04, 2010

DOIhttps://doi.org/10.1619/fesi.52.321

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In feedback stabilization studies, a control scheme is designed so that the "state" of the system decays with a designated rate as t → ∞. Thus the "output", which is subordinate to the state, also decays at least with the same rate. We study in the paper a class of linear parabolic systems, and construct a specific control scheme such that the output decays faster than the state.

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Structure Theorems for Positive Radial Solutions of the Generalized Scalar Curvature Equation

Matteo Franca

2009 Volume 52 Issue 3 Pages 343-369
Published: 2009
Released on J-STAGE: February 04, 2010

DOIhttps://doi.org/10.1619/fesi.52.343

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In this paper we analyze radial solutions for the generalized scalar curvature equation. In particular we prove the existence of ground states and singular ground states when the curvature K (r) is monotone as r → 0 and as r → ∞. The results are new even when p = 2, that is when we consider the usual Laplacian. The proofs use a new Fowler transform which allow us to consider a 2-dimensional dynamical system thus giving a geometrical point of view on the problem. A key role in the analysis is played by an energy function which is a dynamical interpretation of the Pohozaev function used in [21] and [22].

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Instability of Standing Waves for a System of Nonlinear Schrödinger Equations with Three-Wave Interaction

Mathieu Colin, Thierry Colin, Masahito Ohta

2009 Volume 52 Issue 3 Pages 371-380
Published: 2009
Released on J-STAGE: February 04, 2010

DOIhttps://doi.org/10.1619/fesi.52.371

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We consider a three-component system of nonlinear Schrödinger equations related to the Raman amplification in a plasma. In dimension N ≤ 3, we study the orbital instability of standing wave solution of the form (0,0,e^2iωtψ), where ψ is a ground state of scalar nonlinear Schrödinger equation. Using time derivative instead of space derivatives to estimate nonlinear terms, we improve an instability result in our previous paper [4], and also give a simpler proof.

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Approximate Controllability of Impulsive Stochastic Evolution Equations

Rathinasamy Sakthivel

2009 Volume 52 Issue 3 Pages 381-393
Published: 2009
Released on J-STAGE: February 04, 2010

DOIhttps://doi.org/10.1619/fesi.52.381

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In this paper, we study the approximate controllability of nonlinear impulsive stochastic differential equations in a real separable Hilbert space. We prove the approximate controllability of nonlinear impulsive stochastic control systems under the assumption that the corresponding linear system is approximately controllable. By using the stochastic analysis theory and a fixed point strategy, sufficient conditions are formulated and proved. Moreover, an example is also provided to illustrate the effectiveness of the result.

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Existence of the Solution of an Implicit Integro-Differential Equation of Order One

Jagmohan Tyagi, Venkataramanarao Raghavendra

2009 Volume 52 Issue 3 Pages 395-410
Published: 2009
Released on J-STAGE: February 04, 2010

DOIhttps://doi.org/10.1619/fesi.52.395

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In this study, we establish the existence of the solution of a class of first order implicit integro-differential equation of type F(t, (Kx)(t), x(t), x′(t)) = 0, where F : J × R³ → R, K : C(J) → C(J) defined as (Kx)(t) = ∫₀^tk(t, s)x(s)ds, ∀x ∈ C(J), using the concept of viscosity solution.

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Critical Exponent for Semilinear Wave Equations with Space-Dependent Potential

Ryo Ikehata, Grozdena Todorova, Borislav Yordanov

2009 Volume 52 Issue 3 Pages 411-435
Published: 2009
Released on J-STAGE: February 04, 2010

DOIhttps://doi.org/10.1619/fesi.52.411

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We study the balance between the effect of spatial inhomogeneity of the potential in the dissipative term and the focusing nonlinearity. Sharp critical exponent results will be presented in the case of slow decaying potential.

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Multiple Constant Sign and Nodal Solutions for Superlinear Elliptic Equations

Sophia Th. Kyritsi, Nikolaos S. Papageorgiou

2009 Volume 52 Issue 3 Pages 437-473
Published: 2009
Released on J-STAGE: February 04, 2010

DOIhttps://doi.org/10.1619/fesi.52.437

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We consider semilinear elliptic problems with a superlinear right hand side nonlinearity, which however, need not satisfy the Ambrosetti-Rabinowitz condition. Using a combination of variational methods, with Morse theory (critical groups) and truncation techniques, we prove multiplicity theorems providing precise sign information for the solutions. We show that the problem can have seven and eight solutions, all with prescribed sign.

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