一类(QL)型随机微分方程解的轨道唯一性判别及其应用

考虑了一类拟左连续(QL)型随机微分方程(S.D.E.)解的轨道唯一性,应用随机分析方法获得了唯一性成立的一般判别定理,并在方程系数满足局部(或非)Lipschitz条件下给出了一些应用实例.     

An initial boundary value problem for a pseudoparabolic equation with a nonlinear boundary condition

An initial boundary value problem for a quasilinear equation of pseudoparabolic type with a nonlinear boundary condition of the Neumann–Dirichlet type is investigated in this work. From a physical point of view, the initial boundary value problem considered here is a mathematical model of quasistationary processes in semiconductors and magnets, which takes into account a wide variety of physical factors. Many approximate methods are suitable for finding eigenvalues and eigenfunctions in problems where the boundary conditions are linear with respect to the desired function and its derivatives. Among these methods, the Galerkin method leads to the simplest calculations. On the basis of a priori estimates, we prove a local existence theorem and uniqueness for a weak generalized solution of the initial boundary value problem for the quasilinear pseudoparabolic equation. A special place in the theory of nonlinear equations is occupied by the study of unbounded solutions, or, as they are called in another way, blow-up regimes. Nonlinear evolutionary problems admitting unbounded solutions are globally unsolvable. In the article, sufficient conditions for the blow-up of a solution in a finite time in a limited area with a nonlinear Neumann–Dirichlet boundary condition are obtained.     

一类带积分边界条件的三阶微分方程边值问题的解

运用Banach压缩映射原理以及Leray-Schauder连续性原理,在非线性项为L1-Caratheodory函数的条件下,研究了一类带积分边界条件的三阶微分方程边值问题解的唯一性、存在性以及解集的紧性.     

Burgers equation with random boundary conditions

We prove an existence and uniqueness theorem for stationary solutions of the inviscid Burgers equation on a segment with random boundary conditions. We also prove exponential convergence to the stationary distribution.

     

Fractional stochastic differential equation with discontinuous diffusion

In this article, we study a class of stochastic differential equations driven by a fractional Brownian motion with H > 1/2 and a discontinuous coefficient in the diffusion. We prove existence and uniqueness for the solution of these equations. This is a first step to define a fractional version of the skew Brownian motion.     

Small sets for Neumann boundary conditions

For an open set D ? ?ⁿ and a relatively closed subset E ? D of Lebesgue measure zero, we investigate conditions for the property that Brownian motion with reflexion at the boundary on D and D \ E are the same. (© 2008 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Existence and exponential decay for a nonlinear wave equation with nonlocal boundary conditions of 2N‐point type

This paper is devoted to the study of a nonlinear wave equation with initial conditions and nonlocal boundary conditions of 2N‐point type, which connect the values of an unknown function u(x,t) at x = 1, x = 0, x = η_i(t) , and x = θ_i(t), where $0<{\eta }_1(t)<{\eta }_2(t)<\dots <{\eta }_{{}_{N?1}}(t)<1,$ $0<{\theta }_1(t)<{\theta }_2(t)<\dots <{\theta }_{{}_{N?1}}(t)<1,$ for all t ≥ 0. First, we prove local existence of a unique weak solution by using density arguments and applying the Banach's contraction principle. Next, under the suitable conditions, we show that the problem considered has a unique global solution u(t) with energy decaying exponentially as t → +∞ . Finally, we present numerical results.     

On an evolution equation with acoustic boundary conditions

In this paper, we analyze from the mathematical point of view a model for small vertical vibrations of an elastic string with weak internal damping and quadratic term, coupled with mixed boundary conditions of Dirichlet type and acoustic type. Our goal is to extend some of the results of Frota‐Goldstein work in the sense of considering a weaker internal damping and one more quadratic nonlinearity in the elastic string equation. Copyright © 2011 John Wiley & Sons, Ltd.     

Free boundary problem for the equation of one-dimensional motion of compressible gas with density-dependent viscosity

We consider a free boundary problem for the equation of the one-dimensional isentropic motion with density-dependent viscosity μ =b _ϱ ^β, whereb and β are positive constants. We prove that there exists an unique weak solution globally in time, provided that β<1/3.

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Sunto Si considera un problema di frontiera libera per l’equazione del moto unidimensionale isoentropico con viscosità dipendente dalla densità secondo la legge μ =b _ϱ ^β, doveb e β sono costanti positive. Si dimostra che esiste un’unica soluzione debole globale nel tempo, purché β<1/3.

     

Analysis of a stochastic predator-prey system with foraging arena scheme

ABSTRACT

This paper focuses on a predator-prey system with foraging arena scheme incorporating stochastic noises. This SDE model is generated from a deterministic framework by the stochastic parameter perturbation. We then study how the correlations of the environmental noises affect the long-time behaviours of the SDE model. Later on the existence of a stationary distribution is pointed out under certain parametric restrictions. Numerical simulations are carried out to substantiate the analytical results.     

A variational approach to stochastic nonlinear diffusion problems with dynamical boundary conditions

We study a nonlinear partial differential equation of the calculus of variation in a bounded domain, perturbed by noise; we allow stochastic boundary conditions that depend on the time derivative of the solution on the boundary. This work provides the existence and uniqueness of the solution and it shows the existence of an ergodic invariant measure for the corresponding transition semigroup; furthermore, under suitable additional assumptions, uniqueness and strong asymptotic stability of the invariant measure are proved.     

The second initial boundary value problem for the gravitation-gyroscopic wave equation in exterior domains

We study the existence and uniqueness of the solution to the second initial boundary value problem for the gravitation-gyroscopic wave equation in an exterior multiply connected domain with various types of conditions at infinity. Translated fromMatematicheskie Zametki, Vol. 60, No. 1, pp. 40–57, July, 1996. This research was partially supported by the Russian Foundation for Basic Research under grant No. 96-01-01411.     

Polynomial stabiization of the wave equation with Ventcel's boundary conditions

We consider the wave equation on the unit square of the plane with Ventcel boundary conditions on a part of the boundary. It was shown by A. Heminna [8] that this problem is not exponentially stable. Here using a Fourier analysis and a careful analysis of the 1‐d problem with respect to the Fourier parameter l, we show a polynomial stability of this system (© 2010 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Well-posedness for a variable-coefficient wave equation with nonlinear damped acoustic boundary conditions

We consider a variable-coefficient wave equation with nonlinear damped acoustic boundary conditions. Well-posedness in the Hadamard sense for strong and weak solutions is proved by using the theory of nonlinear semigroups.     

Absorbing boundary conditions for reaction-diffusion equations

We treat here of the question of absorbing boundary conditionsfor nonlinear diffusion equations. We use the conditions designedfor the linear equation, we prove them to be well posed forthe nonlinear problem, and through numerical experiments thatthey are well suited for reaction–diffusion equations.     

On uniform decay for transmission problem of Kirchhoff type viscoelastic wave equation

Abstract In this paper we consider the large time behavior of solutions to an n-dimensional transmission problem for two Kirchhoff type viscoelastic wave equations, that is, the wave propagation over bodies consisting of two physically different types of materials. One component is a simple elastic part while the other is a viscoelastic component endowed with a long range memory. We show that the dissipation produced by the viscoelastic part is strong enough to produce exponential or polynomial decay of the solution     

The unique solution for a fractional q-difference equation with three-point boundary conditions

The aim of this paper is to investigate the existence and uniqueness of solutions for nonlinear fractional $q$-difference equations with three-point boundary conditions. Our approach relies on a new fixed point theorem of increasing $\psi ?(h,r)?$concave operators defined on ordered sets. Further, we can construct a monotone explicit iterative scheme to approximate the unique solution. Finally, the main results are illustrated with the aid of two interesting examples.     

Heat equation with dynamical boundary conditions of reactive-diffusive type

This paper deals with the heat equation posed in a bounded regular domain Ω of RN (N?2) coupled with a dynamical boundary condition of reactive-diffusive type. In particular we study the problem

     

Nonclassical problem with integral boundary conditions for elliptic system

In the present paper, a mixed nonclassical problem for multidimensional second-order elliptic system with Dirichlet and nonlocal integral boundary conditions is considered. Since Lax-Milgram theorem cannot be applied straightforwardly for such a nonlocal problem, we consider the problem in the spaces of vector-valued distributions with respect to one space variable with values in the spaces of functions with respect to the other space variables. We introduce special multipliers and applying them we obtain suitable new a priori estimates, and under minimal conditions on the coefficients of the elliptic operator we prove the existence and uniqueness of the solution in appropriate spaces of vector-valued distributions with values in Sobolev spaces.