Outwards pointing hysteresis operators and asymptotic behaviour of evolution equations

The paper deals with the long-time behaviour of evolution systems described by ODEs and PDEs with hysteresis operators. The analysis is based on two concepts. The first one is the outward pointing property of the involved hysteresis operators which implies uniform a priori bounds for solutions, the second one is related to the hysteresis modelling itself and consists in introducing a class of thermodynamically consistent generalized Prandtl–Ishlinskii operators as a model for a nonlinear elastoplastic material law. A stability result for solutions in one-dimensional visco-elasto-plasticity is derived as an illustration of the theory.     

一个具有Hysteresis的双曲方程

讨论了一个具有Hysteresis的双曲方程,并给出这个方程一维初值问题古典解和多维初边值问题强解的存在唯一性.     

Continuity of solutions of a quasilinear hyperbolic equation with hysteresis

This paper is devoted to the investigation of quasilinear hyperbolic equations of first order with convex and nonconvex hysteresis operator. It is shown that in the nonconvex case the equation, whose nonlinearity is caused by the hysteresis term, has properties analogous to the quasilinear hyperbolic equation of first order. Hysteresis is represented by a functional describing adsorption and desorption on the particles of the substance. An existence result is achieved by using an approximation of implicit time discretization scheme, a priori estimates and passage to the limit; in the convex case it implies the existence of a continuous solution.     

Asymptotic behaviour of solutions to some pseudoparabolic equations

The aim of this paper is to investigate the behaviour as t→∞ of solutions to the Cauchy problem u_t−△u_t−v△u−(b,∇u)=∇⋅F(u),u(x,0)=u₀(x), where v>0 is a fixed constant, t≥0, x∈Ω, Ω is a bounded domain in Rⁿ. We will first establish an a priori estimate. Then, we establish the global existence, uniqueness and continuous dependence of the weak solution for the Sobolev-Galpern type equation with the Dirichlet boundary.     

Asymptotic behaviour of global classical solutions of diagonalizable quasilinear hyperbolic systems

This paper is concerned with the asymptotic behaviour of global classical solutions of diagonalizable quasilinear hyperbolic systems with linearly degenerate characteristic fields. Based on the existence results of global classical solutions, we prove that when t tends to infinity, the solution approaches a combination of C¹ travelling wave solutions, provided that L¹ ∩ L^∞ norm of the initial data as well as its derivative are bounded. Application is given for the time‐like extremal surface in Minkowski space. Copyright © 2006 John Wiley & Sons, Ltd.     

Some bv properties of the preisach hysteresis operator

We discuss mapping and Lipschitz continuity properties of the Preisach model for hysteresis in the spaces BV[0,1] and W^1,1[0,1]     

Asymptotic behaviour of some nonlocal diffusion problems

We deal with a parabolic equation having a diffusion coefficient depending on a nonlocal quantity. We investigate the convergence of the solution towards a steady state, extending previous results obtained by [M. Chipot and B. Lovat (1999). On the asympotic behaviour of some nonlocal problems. Positivity, 3, 65-81]. Using the dynamical systems point of view, we are able to treat the case of a continuum of steady states     

Asymptotic behaviour of certain zero-balanced hypergeometric series

In this paper an attempt has been made to give a very simple method of extending certain results of Ramanujan, Evans and Stanton on obtaining the asymptotic behaviour of a class of zero-balanced hypergeometric series. A more recent result of Saigo and Srivastava has also been used to obtain a Ramanujan type of result for a partial sum of a zero-balanced₄F₃ (1) and similar other partial series of higher order.     

Asymptotic behaviour and bifurcation in competitive Lotka-Volterra Systems

A conjecture about global attraction in autonomous competitive Lotka-Volterra systems is clarified by investigating a special system with a circular matrix. Under suitable assumptions, this system meets the condition of the conjecture but Hopf bifurcation occurs in a particular instance. This shows that the conjecture is not true in general and the condition of the conjecture is too weak to guarantee global attraction of an equilibrium. Sufficient conditions for global attraction are also obtained for this system.     

Asymptotic behaviour of the stochastic Lotka-Volterra model

This paper examines the asymptotic behaviour of the stochastic extension of a fundamentally important population process, namely the Lotka-Volterra model. The stochastic version of this process appears to have far more intriguing properties than its deterministic counterpart. Indeed, the fact that a potential deterministic population explosion can be prevented by the presence of even a tiny amount of environmental noise shows the high level of difference which exists between these two representations.     

Asymptotic behaviour of solutions to cable stayed bridge equations

Asymptotic behaviour of solutions to the nonlinear variational equations of the two basic types of cable stayed bridges is studied. The equations describe vertical and torsional oscillations of the center span, which is the part of road bed between pylons. Due to damping, amplitude and velocity of oscillations remain bounded if wind force is bounded as well. Moreover, the solution to the nonlinear equations converges to the solution of the steady state problem if wind drops. If the center span has certain aerodynamical properties and the initial state is near to the solution of the steady state problem, then the solution converges to the steady state solution even if the forces representing wind do not vanish.     

Asymptotic behaviour for the 3-D Schrödinger-Poisson System in the attractive case with positive energy

In this paper, we study the asymptotic behaviour of solutions to the three-dimensional Schrödinger-Poisson system in the attractive case with positive energy. In this case, it is proved that for a real initial condition, the solutions expand unboundedly as time goes to infinity. The proof of this result is based on the derivation of a dispersive equation relating density and linear momentum, as well as on optimal bounds for the kinetic energy.     

Inverse coefficient problem for a quasilinear hyperbolic equation with final overdetermination

The inverse problem of recovering a solution-dependent coefficient multiplying the lowest derivative in a hyperbolic equation is investigated. As overdetermination is required in the inverse problem, an additional condition is imposed on the solution to the equation with a fixed value of the timelike variable. Global uniqueness and local existence theorems are proved for the solution to the inverse problem. An iterative method is proposed for solving the inverse problem.     

Asymptotic behaviour of the stochastic Gilpin-Ayala competition models

In this paper, we investigate a stochastic Gilpin-Ayala competition system, which is more general and more realistic than the classical Lotka-Volterra competition system.We discuss the asymptotic behaviour in detail of the stochastic Gilpin-Ayala competition system, and comparing the classical Lotka-Volterra with Gilpin-Ayala competition system, we find that the latter has better properties.     

A semi-discrete convergent scheme for a quasilinear hyperbolic equation

We establish here the convergence (thereby proving the existence) of a semi-discrete scheme for the quasilinear hyperbolic equation where x ∈ Rⁿ, t ∈ [0,T], and ? ∈ L^∞ (Rⁿ). It is well known that the above problem does not necessarily have global classical solutions and the usual concepts of weak solution. do not lead to a unique solution The existence of a unique solution to the above problem in a suitable sense was established in [3], where a parabolic problem obtained by introducing the term ??Δu was studied and then the behavior as ? → 0 was discussed. A difference scheme approach to a problem of the above type where ?_i does not depend on x and t and Ψ does not depend on u was also studied in [2]. The aim of this paper is to present a proof for the case when ? depends on x, Ψ depends on u, and the technical complications in this case are nontrivial. The discussions in this paper my be considered as continuation of the ideas in the above papers.