THE ASYMPTOTIC BEHAVIOR AND OSCILLATION FOR A CLASS OF THIRD-ORDER NONLINEAR DELAY DYNAMIC EQUATIONS

In this paper,we consider a class of third-order nonlinear delay dynamic equations.First,we establish a Kiguradze-type lemma and some useful estimates.Second,we give a sufficient and necessary condition for the existence of eventually positive solutions having upper bounds and tending to zero.Third,we obtain new oscillation criteria by employing the Potzsche chain rule.Then,using the generalized Riccati transformation technique and averaging method,we establish the Philos-type oscillation criteria.Surprisingly,the integral value of the Philos-type oscillation criteria,which guarantees that all unbounded solutions oscillate,is greater than θ_(4)(t_(1),T).The results of Theorem 3.5 and Remark 3.6 are novel.Finally,we offer four examples to illustrate our results.

ASYMPTOTIC BEHAVIOR OF SOLUTION FOR NONLOCAL REACTION-DIFFUSION SYSTEM

This paper deals with reaction-diffusion system with nonlocal source. It is proved that there exists a unique classical solution and the solution either exists globally or blows up in finite time. Furthermore, its blow-up set and asymptotic behavior are obtained provided that the solution blows up in finite time.

COMPLEXITY OF ASYMPTOTIC BEHAVIOR OF SOLUTIONS FOR THE POROUS MEDIUM EQUATION WITH ABSORPTION

In this paper we analyze the large time behavior of nonnegative solutions of the Cauchy problem of the porous medium equation with absorption ut - △um ＋ yup = 0,where γ≥0,m〉 1and P〉m＋2/N We will show that if γ=0 and 0〈μ〈 2N/n（m-1）＋2 or γ 〉 0 and 1/p-1 〈 μ 〈 2N/N（m-1）＋2 then for any nonnegative function φ in a nonnegative countable subset F of the Schwartz space S（RN）, there exists an initial-value u0 ∈ C（RN） with limx→∞ uo（x）= 0 such that φ is an w-limit point of the rescaled solutions tμ/2u（tβ, t）, Where β = 2-μ（m-1）/4.

GLOBAL EXISTENCE AND ASYMPTOTIC BEHAVIOR FOR THE 3D COMPRESSIBLE NON–ISENTROPIC EULER EQUATIONS WITH DAMPING

We investigate the global existence and asymptotic behavior of classical solutions for the 3D compressible non-isentropic damped Euler equations on a periodic domain. The global existence and uniqueness of classical solutions are obtained when the initial data is near an equilibrium. Furthermore, the exponential convergence rates of the pressure and velocity are also proved by delicate energy methods.

ASYMPTOTIC BEHAVIOR OF LINEAR ADVANCED DIFFERENTIAL EQUATIONS

In this article, we consider a general class of linear advanced differential equa- tions, and obtain explicitly sufficient conditions of convergence and exponential convergence to zero. A necessary condition is provided as well.

Asymptotic Behavior of a Stochastic SIRS Model with Non-linear Incidence and Levy Jumps

A stochastic susceptible-infective-recovered-susceptible( SIRS) model with non-linear incidence and Levy jumps was considered. Under certain conditions, the SIRS had a global positive solution. The stochastically ultimate boundedness of the solution of the model was obtained by using the method of Lyapunov function and the generalized Ito's formula. At last,asymptotic behaviors of the solution were discussed according to the value of R0. If R0< 1,the solution of the model oscillates around a steady state, which is the diseases free equilibrium of the corresponding deterministic model,and if R0> 1,it fluctuates around the endemic equilibrium of the deterministic model.

ASYMPTOTIC BEHAVIOR FOR COMMUTATIVE SEMIGROUPS OF ALMOST ASYMPTOTICALLY NONEXPANSIVE TYPE MAPPINGS

This article introduces the concept of commutative semigroups of almost asymptotically nonexpansive-type mappings in a Banach space X which has the Opial property and whose norm is UKK, and establishes the weak convergence theorems for almostorbits of this class of commutative semigroups. The author improves, extends and develops some recent and earlier results.

ASYMPTOTIC BEHAVIOR OF SOLUTIONS TO THE HEAT EQUATIONS WITH NONLINEAR BOUNDARY CONDITIONS

The purpose of this paper is to investigate the stability and asymptotic behavior of the time-dependent solutions to a linear parabolic equation with nonlinear boundary condition in relation to their corresponding steady state solutions. Then, the above results are extended to a semilinear parabolic equation with nonlinear boundary condition by analyzing the corresponding eigenvalue problem and using the method of upper and lower solutions.

EXISTENCE AND ASYMPTOTIC BEHAVIOR OF SOLUTIONS TO A SINGULAR PARABOLIC EQUATION

In this paper,we study the initial-boundary value problem for a class of singular parabolic equations.Under some conditions,we obtain the existence and asymptotic behavior of solutions to the problem by parabolic regularization method and the sub-super solutions method.As a byproduct,we prove the existence of solutions to some problems with gradient terms,which blow up on the boundary.

EXISTENCE AND ASYMPTOTIC BEHAVIOR OF THE SOLUTIONS FOR A NONLINEAR ELLIPTIC EQUATION ARISING IN ASTROPHYSICS

The author first analyzes the existence of ground state solutions and cylindrically symmetric solutions and then the asymptotic behavior of the ground state solution of the equation -△u =φ（r）u^p-1, u 〉 0 in R^N, u ∈ D^1,2（R^N）, where N ≥ 3, x = （x^1,z） ∈ R^K×R^N-K,2 ≤ K ≤ N,r = ｜x′｜. It is proved that for 2（N-s）/（N-2） 〈 p 〈 2^＊ = 2N/（N - 2）, 0 〈 s 〈 2, the above equation has a ground state solution and a cylindrically symmetric solution. For p = 2^＊, the above equation does not have a ground state solution but a cylindrically symmetric.solution, and when p close to 2^＊, the ground state solutions are not cylindrically symmetric. On the other hand, it is proved that as p close to 2＊, the ground state solution Up has a unique maximum point xp = （x′p, Zp） and as p → 2^＊, ｜x′p｜ → r0 which attains the maximum of φ on R^N. The asymptotic behavior of ground state solution Up is also given, which also deduces that the ground state solution is not cylindrically symmetric as p goes to 2^＊.

THE ASYMPTOTIC BEHAVIOR AND SYMMETRY OF POSITIVE SOLUTIONS TO p-LAPLACIAN EQUATIONS IN A HALF-SPACE

We study a nonlinear equation in the half-space with a Hardy potential,specifically,−Δ_(p)u=λu^(p−1)x_(1)^(p)−x_(1)^(θ)f(u)in T,where Δp stands for the p-Laplacian operator defined by Δ_(p)u=div(∣Δu∣^(p−2)Δu),p>1,θ>−p,and T is a half-space{x_(1)>0}.When λ>Θ(where Θ is the Hardy constant),we show that under suitable conditions on f andθ,the equation has a unique positive solution.Moreover,the exact behavior of the unique positive solution as x_(1)→0^(+),and the symmetric property of the positive solution are obtained.

Existence and asymptotic behavior for systems of nonlinear hyperbolic equations

The initial-boundary value problem for a class of nonlinear hyperbolic equations system in bounded domain is studied. The existence of global solutions for this problem is proved by constructing a stable set, and obtain the asymptotic stability of global solutions by means of a difference inequality.

ASYMPTOTIC BEHAVIOR TOWARD THE RAREFACTION WAVE FOR SOLUTIONS OF RATE-TYPE VISCOELASTIC SYSTEM WITH BOUNDARY EFFECT

The initial boundary value problems for the system of rate-type viscoelasticityis considered in the present paper.It is shown that if the initial data are a small perturbationof a forward smooth rarefaction wave, then there is a global solutions to the system, whichtends to the rarefaction wave time-asymptotically.

Asymptotic Behavior of Solutions for a Periodic Transport Equation with Delay

This paper analyzes a class of first order partial differential equations with delay (a model for the blood production system). The asymptotic behavior of solutions are studied.

Asymptotic Behavior of the Finite Difference and the Finite Element Methods for Parabolic Equations

The asymptotic convergence of the solution of the parabolic equation is proved. By the eigenvalues estimation, we obtain that the approximate solutions by the finite difference method and the finite element method are asymptotically convergent. Both methods are considered in continnous time.

ASYMPTOTIC BEHAVIOR OF NON-AUTONOMOUS DISSIPATIVE SYSTEMS IN HILBERT SPACES

This paper dwells upon the asymptotic behavior, as t→∞, of the integral solution u(t) to the nonhaear evolution equation u'(t) ∈A(t)u(f) + g(t),t≥s,u(s) =xo∈D(A(a)),where {A(t)}t≥s is a family m-ipative operator in a Hlilbert space H,and g∈L(loc)(0,∞;H). We prove that converges weakly, as t→∞, uniforluly in h≥0, which applies that u(t) is weak convergence if and only if u(t) is weakly asymptotically regular i.e., u(t + h) -u(f) →0 for h≥0.

Asymptotic Behavior of the Solutions to the One-Dimensional Nonisentropic Hydrodynamic Model for Semiconductors

In this paper, the asymptotic behavior of the global smooth solutions to the Cauchy problem for the one-dimensional nonisentropic Euler-Poisson （or full hydrodynamic） model for semiconductors, where the energy equation with non-zero thermal conductivity coefficient are contained, is discussed. The global existence of smooth solutions for the Cauchy problem with small perturbed initial data is proved. In particular, that the solutions converge to the corresponding stationary solutions exponentially fast as t → ∞ is showed.

THE ASYMPTOTIC BEHAVIOR OF SOLUTION FOR THE NONLINEAR HEAT-CONDUCTION EQUATION AND ITS APPLICATION

In this paper the nonlinear heat-conduction equations rhoc partial derivativew/partial derivativet = partial derivative/partial derivativex (k partial derivativew/partial derivativex) with Dirichlet boundary condition and the nonlinear boundary condition are studied. The asymptotic behavior of the global of solution are analyzed by using Lyapuunov function. As its application, the approximate solutions are constructed.

ASYMPTOTIC BEHAVIOR OF SOLUTIONS FOR THE CHAFEE-INFANTE EQUATION

In higher dimension,there are many interesting and challenging problems about the dynamics of non-autonomous Chafee-Infante equation.This article is concerned with the asymptotic behavior of solutions for the non-autonomous Chafee-Infante equation∂u∂t−Δu=λ(t)(u−u3)in higher dimension,whereλ(t)∈C1[0,T]andλ(t)is a positive,periodic function.We denoteλ1 as the first eigenvalue of−Δφ=λφ,x∈Ω;φ=0,x∈∂Ω.For any spatial dimension N≥1,we prove that ifλ(t)≤λ1,then the nontrivial solutions converge to zero,namely,limt→+∞u(x,t)=0,x∈Ω;ifλ(t)>λ1 as t→+∞,then the positive solutions are"attracted"by positive periodic solutions.Specially,ifλ(t)is independent of t,then the positive solutions converge to positive solutions of−ΔU=λ(U−U^3).Furthermore,numerical simulations are presented to verify our results.

ASYMPTOTIC BEHAVIOR OF SOLUTION FOR A CLASS OF REACTION DIFFUSION EQUATIONS

A class of initial boundary value problems for the reaction diffusion equations are considered.The asymptotic behavior of solution for the problem is obtained using the theory of differential inequality.