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.

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.

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.

Global Existence, Asymptotic Behavior and Uniform Attractors for Damped Timoshenko Systems

In this paper, we consider the Timoshenko system as an initial-boundary value problem in a one-dimensional bounded domain. And then we establish the global existence and asymptotic behavior of solution by using the semigroup method and multiplicative techniques, then further prove the existence of a uniform attractor by using the method of uniform contractive function. The main advantage of this method is that we need only to verify compactness condition with the same type of energy estimate as that for establishing absorbing sets.

Asymptotic Behavior for A Class of Non-autonomous Nonclassical Parabolic Equations with Delay on Unbounded Domain

In this article,we investigate the longtime behavior for the following nonautonomous nonclassical parabolic equations on unbounded domain ut−∆ut−∆u+λu=f(x,u(x,t−ρ(t)))+g(x,t).Under some suitable conditions on the delay term f and the non-autonomous forcing term g,we prove the existence of uniform attractors in Banach space CH1(RN)for the multivalued process generated by non-autonomous nonclassical parabolic equations with delays in unbounded domain.

DISCUSSION ON″THE BOUNDEDNESS AND ASYMPTOTIC BEHAVIOR OR SOLUTION DIFFERENTIAL SYSTEM OF SECOND-ORDER WITH VARIABLE COEFFICIENT" (App1ied Mathematics and Mechanics,Vo1.3,No.4,1982)

After reading the article 'The Boundedness and Asymptotic Behavior of Solution of Differential System of Second-Order with Variable Coefficient' in 'Applied Mathematics and Mechanics', Vol. 3, No. 4, 1982, we would like to put forward a few points to discuss with the author and the readers. Our opinions are presented as follows:

On the Asymptotic Behavior of Second Order Quasilinear Difference Equations

In this paper, we investigate the asymptotic behavior of the following quasilinear difference equations (E) where , . We classified the solutions into six types by means of their asymptotic behavior. We establish the necessary and/or sufficient conditions for such equations to possess a solution of each of these six types.

Some Criteria for the Asymptotic Behavior of a Certain Second Order Nonlinear Perturbed Differential Equation

In this paper we give sufficient conditions so that for every nonoscillatory u(t) solution of (r(t)ψ(u)u')'+Q(t,u,u'), we have lim inf|u(t)|=0. Our results contain the some known results in the literature as particular cases.

Asymptotic Behavior of a Bi-Dimensional Hybrid System

We study the asymptotic behavior of the solutions of a Hybrid System wrapping an elliptic operator.

Asymptotic Behavior of Solutions of Retarded Differential Equations

A sufficient condition is obtained for every solution of the nonlinear retarded differential equationx'(t) +f(t,x(t-τ)) =0to tend to zero as t→∞ , which extends and improves the corresponding results obtained by Ladas, Sficas and Gopalsamy.

Asymptotic Behavior of Solutions to a Class of Semilinear Parabolic Equations with Boundary Degeneracy

This paper concerns the asymptotic behavior of solutions to one-dimensional semilinear parabolic equations with boundary degeneracy both in bounded and unbounded intervals.For the problem in a bounded inter-val,it is shown that there exist both nontrivial global solutions for small initial data and blowing-up solutions for large one if the degeneracy is not strong.Whereas in the case that the degeneracy is strong enough,the nontrivial solu-tion must blow up in a finite time.For the problem in an unbounded interval,blowing-up theorems of Fujita type are established.It is shown that the critical Fujita exponent depends on the degeneracy of the equation and the asymptotic behavior of the diffusion coefficient at infinity,and it may be equal to one or infinity.Furthermore,the critical case is proved to belong to the blowing-up case.

Complicated Asymptotic Behavior of Solutions for the Cauchy Problem of Doubly Nonlinear Diffusion Equation

In this paper,we analyze the large time behavior of nonnegative solutions to the doubly nonlinear diffusion equation u_(t)−div(|∇u^(m)|^(p−2)∇u^(m))=0 in R^(N)with p>1,m>0 and m(p−1)−1>0.By using the finite propagation property and the L^(1)−L^(∞)smoothing effect,we find that the complicated asymptotic behavior of the rescaled solutions t^(μ/2)u(t^(β)⋅,t)for 0<μ<2 N/(N[m(p−1)−1]+p)andβ>(2−μ[m(p−1)−1])/(2 p)can take place.

Asymptotic behavior of the principal eigenvalue and the basic reproduction ratio for periodic patch models

This paper is devoted to the study of the asymptotic behavior of the principal eigenvalue and the basic reproduction ratio associated with periodic population models in a patchy environment for small and large dispersal rates.We first deal with the eigenspace corresponding to the zero eigenvalue of the connectivity matrix.Then we investigate the limiting profile of the principal eigenvalue of an associated periodic eigenvalue problem as the dispersal rate goes to zero and infinity,respectively.We further establish the asymptotic behavior of the basic reproduction ratio in the case of small and large dispersal rates.Finally,we apply these results to a periodic Ross-Macdonald patch model.

On the Asymptotic Behavior of Nearest Neighbor Estimator of Conditional Median

This paper deals with the estimation in nonparametrio regression model.Sincethe conditional mean is sensitive to the tail behavior of the conditional distributionof the model,instead conditional median is considered.For estimation of theconditional median,the sequence of the nearest neighbor estimators is shown to beasymptotio normal and consistent.

Optical theorem,crossing property,and derivative dispersion relations:implications on the asymptotic behavior of σ_(tot)(s) and ρ(s)

In this paper,we present some results on the behavior of the total cross section and p-parameter at asymptotic energies in proton-proton(pp) and antiproton-proton(pp) collisions.Hence,we consider three of the main theoretical results in high energy physics:the crossing property,derivative dispersion relation,and optical theorem.The use of such machinery facilitates the derivation of analytic formulas for a wide set of the measured global scattering parameters and some important relations between them.The suggested parameterizations approximate the energy dependence for the total cross section and ρ-parameter for pp and pp with a statistically acceptable quality in the multi-TeV region.Additionally,the qualitative description is obtained for important interrelations,namely difference,sum,and ratio of the antiparticle-particle and particle-particle total cross sections.Despite the reduced number of experimental data for the total cross section and p-parameter at the TeV-scale,which complicates any prediction for the beginning of the asymptotic domain,the fitting procedures indicates that asymptotia occur in the energy range 25.5-130 TeV.Moreover,in the asymptotic regime,we obtain α_(P)=1.A detailed quantitative study of the energy behavior of the measured scattering parameters and their combinations in the ultra-high energy domain indicates that the scenario with the generalized formulation of the Pomeranchuk theorem is more favorable with respect to the original formulation of this theorem.

Asymptotic Behavior of Solutions for the One-Dimensional Drift-Diffusion Model in the Quarter Plane

In this paper,we study the classical drift-diffusion model arising from the semiconductor device simulation,which is the simplest macroscopic model describing the dynamics of the electron and the hole.We prove the global existence of strong solutions for the initial boundary value problem in the quarter plane.In particular,we show that in large time,these solutions tend to the nonlinear diffusion wave which is different from the steady state,at an algebraic time-decay rate.As far as we know,this is the first result about the nonlinear diffusion wave phenomena of the solutions for the one-dimensional drift-diffusion model in the quarter plane.

Asymptotic Behavior in a Quasilinear Fully Parabolic Chemotaxis System with Indirect Signal Production and Logistic Source

In this paper,we study the asymptotic behavior of solutions to a quasilinear fully parabolic chemotaxis system with indirect signal production and logistic source{ut=■·(D(u)■u)-■·(S(u)■v)+b-μur,x∈Ω,t>0,vt=Δv+a1v+b1w,x∈Ω,t>0,wt=Δw-a2w+b2u,x∈Ω,t>0 under homogeneous Neumann boundary conditions in a smooth bounded domainΩ■R^(N)(N≥1),where b≥0,γ≥1,ai≥1,μ,bi>0(i=1,2,D,S∈C^(2)([0,∞))fulfilling D(S)≥a0(s+1)^(-a),0≤S(s)≤b0(s+1)^(β)for all s≥0,where a0,b0>0 and a,β≤R are constants.The pur purpose of this paper is to prove that if b≥0 andμ>0 sufficiently large,the globally bounded solution(u,v,w)with nonnegative initial data(u0,v0,w0)satisfies||u(·,t)-(b/μ)^(1/γ)L∞(Ω)+||V(·,t)+b1b2/a1a2(b/μ)^(1/γ)||L∞(Ω)+||w(·,t)+b2/a2(b/μ)^(1/γ)||L∞(Ω)→0ast→∞.

Blowup and Asymptotic Behavior of a Free Boundary Problem with a Nonlinear Memory

In this paper,we investigate a reaction-diffusion equation ut-duxx=au+∫t0up(x,τ)dT+k(x)with double free boundaries.We study blowup phenomena in finite time and asymptotic behavior of time-global solutions.Our results show if∫h0-hok(x)φ1dx is large enough,then the blowup occurs.Meanwhile we also prove when T*<+oo,the solution must blow up in finite time.On the other hand,we prove that the solution decays at an exponential rate and the two free boundaries converge to a finite limit provided the initial datum is small sufficiently.

Existence and Asymptotic Behavior of Positive Solutions for Variable Exponent Elliptic Systems

In this paper,our main purpose is to establish the existence of positive solution of the following system{−△ p(x)u=F(x,u,v),x∈W,−D q(x)v=H(x,u,v),x∈W,u=v=0,x∈∂W,where W=B(0,r)⊂RN or W=B(0,r2)\B(0,r1)⊂RN,0<r,0<r1<r2 are constants.F(x,u,v)=λp(x)[g(x)a(u)+f(v)],H(x,u,v)=θq(x)[g1(x)b(v)+h(u)],λ,θ>0 are parameters,p(x),q(x)are radial symmetric functions,−D p(x)=−div(|∇u|p(x)−2∇u)is called p(x)-Laplacian.We give the existence results and consider the asymptotic behavior of the solutions.In particular,we do not assume any symmetric condition,and we do not assume any sign condition on F(x,0,0)and H(x,0,0)either.

On the Asymptotic Behavior of Nonlinear Schrodinger Equations with Magnetic Effect

The purpose of this paper is to study the long time asymptotic behavior for anonlinear Schrdinger equations with magnetic effect. Under certain conditions, we prove theexistence and nonexistence of the non-trivial free asymptotic solutions. In addition, the decayestimates of the solutions are also obtained.