Global Asymptotic Behavior Research Articles

Solving a fully parabolic chemotaxis system with periodic asymptotic behavior using Generalized Finite Difference Method

This work studies a parabolic-parabolic chemotactic PDE's system which describes the evolution of a biological population “U” and a chemical substance “V”, using a Generalized Finite Difference Method, in a two dimensional bounded domain with regular boundary. In a previous paper [12], the authors asserted global classical solvability and periodic asymptotic behavior for the continuous system in 2D. In this continuous work, a rigorous proof of the global classical solvability to the discretization of the model proposed in [12] is presented in two dimensional space. Numerical experiments concerning the convergence in space and in time, and long-time simulations are presented in order to illustrate the accuracy, efficiency and robustness of the developed numerical algorithms.

Global existence and asymptotic behavior for some multidimensional quasilinear hyperbolic systems

For the Cauchy problem of multidimensional quasilinear hyperbolic systems of diagonal form without self-interaction, we show the global existence of classical solutions with small initial data. We also prove that the global solution will converge to a solution of the linearized system as time tends to infinity, and the convergence rate is also determined.

Global Existence and Asymptotic Behavior of Solutions for Compressible Two-Fluid Euler–Maxwell Equation

We study the global existence and asymptotic behavior of the solutions for two-fluid compressible isentropic Euler–Maxwell equations by the Fourier transform and energy method. We discuss the case when the pressure for two fluids is not identical, and we also add friction between the two fluids. In addition, we discuss the rates of decay of Lp−Lq norms for a linear system. Moreover, we use the result for Lp−Lq estimates to prove the decay rates for the nonlinear systems.

Global Behavior of Solutions in a Predator-Prey Cross-Diffusion Model with Cannibalism

The global asymptotic behavior of solutions in a cross-diffusive predator-prey model with cannibalism is studied in this paper. Firstly, the local stability of nonnegative equilibria for the weakly coupled reaction-diffusion model and strongly coupled cross-diffusion model is discussed. It is shown that the equilibria have the same stability properties for the corresponding ODE model and semilinear reaction-diffusion model, but under suitable conditions on reaction coefficients, cross-diffusion-driven Turing instability occurs. Secondly, the uniform boundedness and the global existence of solutions for the model with SKT-type cross-diffusion are investigated when the space dimension is one. Finally, the global stability of the positive equilibrium is established by constructing a Lyapunov function. The result indicates that, under certain conditions on reaction coefficients, the model has no nonconstant positive steady state if the diffusion matrix is positive definite and the self-diffusion coefficients are large enough.

Global threshold dynamics of an infection age-structured SIR epidemic model with diffusion under the Dirichlet boundary condition

In this paper, we are concerned with the global asymptotic behavior of an infection age-structured SIR epidemic model with diffusion in a general n-dimensional bounded spatial domain under the homogeneous Dirichlet boundary condition. By using the method of characteristics, we reformulate the model into a system of a reaction-diffusion equation and a Volterra integral equation. We define the basic reproduction number R0 by the spectral radius of a compact positive linear operator and show that if R0<1, then the disease-free steady state is globally attractive, whereas if R0>1, then a positive endemic steady state exists and the system is uniformly persistent. By numerical simulation for the 2-dimensional case, we show that R0 depends on the shape of the spatial domain. This result is in contrast with the case of the homogeneous Neumann boundary condition, in which R0 is independent of the spatial domain.

Global Existence and Stability of Shock Front Solution to 1-D Piston Problem for Exothermically Reacting Euler Equations

In this paper, we are concerned with the global existence and stability of strong shock front for one-dimensional piston problem. We use the exothermically reacting Euler equations as a mathematical model to describe the gas motion. Under the assumptions that the total variations of initial data and the perturbation of piston velocity are sufficiently small, we establish the global existence and asymptotic behavior of entropy solutions including a strong shock front without restriction on the strength. A modified fractional wave front tracking scheme is developed and a modified Glimm-type functional is carefully designed.

Global existence and asymptotic behaviour of solution for a damped nonlinear hyperbolic equation

In this paper we study the initial–boundary value problem for a damped nonlinear hyperbolic equation utt+Δ2u+αΔ2ut+Δf(Δu)=0,x∈Ω,t>0,u(x,0)=u0(x),ut(x,0)=u1(x),x∈Ω,u=0Δu=0,x∈∂Ω,t≥0, where Ω⊂Rn, n≥1 and α is a positive constant. Under some assumptions on f(s) and initial data, we prove the global existence of solution. Furthermore, we prove that the solution decays to zero exponentially.

A new result for asymptotic stability in a two-species chemotaxis model with signal-dependent sensitivity

This paper deals with a two-species chemotaxis model with signal-dependent sensitivity ut=Δu−∇⋅(uχ1(w)∇w)+μ1u(1−u),x∈Ω,t>0,vt=Δv−∇⋅(vχ2(w)∇w)+μ2v(1−v),x∈Ω,t>0,wt=dΔw+h(u,v,w),x∈Ω,t>0,under homogeneous Neumann boundary conditions in a bounded domain Ω⊂Rn (n≥1), where the parameters μ1>0, μ2>0 and d>0, and the chemotactic function χi(w) (i=1,2) are smooth and satisfy some conditions. The global existence and asymptotic behavior of solutions to this system have been studied under some conditions (Mizukami and Yokota, 2016; Mizukami, 2016). The purpose of this work is to improve the conditions assumed in (Mizukami and Yokota, 2016; Mizukami, 2016) for asymptotic behavior.

Global existence and blow up of solutions for pseudo-parabolic equation with singular potential

We consider the initial boundary value problem of pseudo-parabolic equation with singular potential. We obtain global existence, asymptotic behavior and blowup of solutions with initial energy J(u0)≤d. Moreover, we estimate the upper bound of the blowup time for J(u0)<0 and 0<J(u0)<d respectively. Finally, we prove the finite time blowup and estimate the upper bound of the blowup time for the high energy level, i.e., J(u0)>0.

Dirichlet-boundary value problem for one dimensional nonlinear Schrödinger equations with large initial and boundary data

We consider the inhomogeneous Dirichlet-boundary value problem with large initial and boundary data for nonlinear Schrodinger equations in one space dimension. Global existence and asymptotic behavior in time of solutions to the problem are obtained by using the classical energy method and factorization techniques.

Optimal global and boundary asymptotic behavior of large solutions to the Monge-Ampère equation

This paper is mainly concerned with optimal global and boundary asymptotic behavior of strict convex large solutions to the Monge-Ampère equation detD2u=b(x)f(u), x∈Ω, where Ω is a strict convex and bounded smooth domain in Rn with n≥2, f∈C1[0,∞) (or f∈C1(R)), which is increasing in [0,∞) (or R) and satisfies the Keller-Osserman type condition, b∈C∞(Ω) is positive in Ω, but may vanish or blow up on the boundary properly. We find new structure conditions on f which play a crucial role in both global and boundary behavior of such solutions. Moreover, we reveal asymptotic behavior of such solutions when the parameters on b tend to the corresponding critical values. In addition, when f does not satisfy the Keller-Osserman type condition and Ω is a ball, we supply a necessary and sufficient condition on b for the existence of an infinitude of strict convex radially symmetric positive solutions to such problem.

Global existence and asymptotic behavior of solutions to the Euler equations with time-dependent damping

ABSTRACT We study the isentropic Euler equations with time-dependent damping, given by . Here, λ and μ are two non-negative constants to describe the decay rate of damping with respect to time. We will investigate the global existence and asymptotic behavior of small data solutions to the Euler equations when in multi-dimensions . Our strategy of proving the global existence is to convert the Euler system to a time-dependent damped wave equation and use a kind of weighted energy estimate. Investigation to the asymptotic behavior of the solution is based on the detailed analysis to the fundamental solutions of the corresponding linear damped wave equation and it coincides with that of standard results if λ deduces to zero.

A Deterministic Model for Q Fever Transmission Dynamics within Dairy Cattle Herds: Using Sensitivity Analysis and Optimal Controls.

This paper presents a differential equation model which describes a possible transmission route for Q fever dynamics in cattle herds. The model seeks to ascertain epidemiological and theoretical inferences in understanding how to avert an outbreak of Q fever in dairy cattle herds (livestock). To prove the stability of the model's equilibria, we use a matrix-theoretic method and a Lyapunov function which establishes the local and global asymptotic behaviour of the model. We introduce time-dependent vaccination, environmental hygiene, and culling and then solve for optimal strategies. The optimal control strategies are necessary management practices that may increase animal health in a Coxiella burnetii-induced environment and may also reduce the transmission of the disease from livestock into the human population. The sensitivity analysis presents the relative importance of the various generic parameters in the model. We hope that the description of the results and the optimality trajectories provides some guidelines for animal owners and veterinary officers on how to effectively minimize the bacteria and control cost before/during an outbreak.

Global well-posedness, asymptotic behavior and blow-up of solutions for a class of degenerate parabolic equations

The initial–boundary value problem for a class of degenerate parabolic equations is studied. Some new results on global existence of solutions are established by introducing a family of potential wells. In addition, asymptotic behavior and finite time blow-up of solutions are obtained in the case of subcritical initial energy and critical initial energy, respectively.

Global existence and asymptotic behavior in a two-species chemotaxis system with logistic source

In this work we consider the two-species chemotaxis system with logistic source. We present the global existence of generalized solutions under appropriate regularity assumptions on the initial data. In addition, the asymptotic behavior of the solutions is studied, and our results generalize and improve some well-known results in the literature.

Global dynamics of the diffusive Lotka–Volterra competition model with stage structure

The global asymptotic behavior of the classical diffusive Lotka–Volterra competition model with stage structure is studied. A complete classification of the global dynamics is given for the weak competition case. It is shown that under otherwise same conditions, the species with shorter maturation time prevails. The method is also applied to the global dynamics of another competition models with time delays.

Asymptotic behavior and finite time blow up for damped fourth order nonlinear evolution equation

This paper investigates the initial boundary value problem for a class of fourth order nonlinear damped wave equations modeling longitudinal motion of an elasto-plastic bar. By applying a suitable potential well-convexity method, we derive the global existence, asymptotic behavior and finite time blow up for the considered problem with more generalized nonlinear functions at subcritical initial energy level. Further for arbitrarily positive initial energy we give some sufficient conditions ensuring finite time blow up.

Global existence, asymptotic behavior and uniform attractor for a non-autonomous Timoshenko system of type III with weak damping

In this paper, we investigate one-dimensional thermoelastic system of Timoshenko type III with double memory dampings. At first we give the global existence of solutions by using semigroup theory. Then we can prove the energy decay of solutions by constructing a series of Lyapunov functionals and obtain the existence of absorbing ball. Finally, we prove the asymptotic compactness by using uniform contractive functions and obtain the existence of uniform attractor.

Global existence and asymptotic behavior for the 3D compressible magneto-micropolar fluids in a bounded domain

This paper analyzes the global existence and asymptotic behavior of the initial boundary value problem for the 3D compressible magnetomicropolar fluids. Based on the delicate energy estimates over the region away from the boundary and near the boundary, we obtain the global existence, uniqueness and exponential convergence rates of strong solutions in a bounded domain.

Global existence and asymptotic behavior of solutions to a chemotaxis system with chemicals and prey-predator terms

This paper is concerned with a general asymptotic stabilization of arbitrary global positive bounded solutions for the Lotka Volterra reaction diffusion systems, with an additional chemotactic influence and constant coefficients. We consider the dynamics of a mathematical model involving two biological species, both of which move according to random diffusion and are attracted/ repulsed by chemical stimulus produced by the other. The biological species present the ability to orientate their movement towards the concentration of the chemical secreted by the other species. The nonlinear system consists of two parabolic equations with Lotka-Volterra-type kinetic terms coupled with chemotactic cross-diffusion, along with two elliptic equations describing the behavior of the chemicals. We prove that the solution to the corresponding Neumann initial boundary value problem is global and bounded for regular and positive initial data. Moreover, for different ranges of parameters, we show that any positive and bounded solution converges to a spatially constant homogeneous state.