Global Stability Of Solutions Research Articles

Optimal decay for the 2D anisotropic Navier–Stokes equations with mixed partial dissipation

This paper studies the stability and large time behavior to the 2D anisotropic Navier–Stokes equations with mixed partial dissipation. We establish the uniform upper bounds and the global stability of solutions, and obtain the optimal decay properties to these global solutions and their higher order derivatives without any small assumptions on the initial data.

Global stability of solutions with large initial conditions for the Navier–Stokes equations with damping

Global stability of solutions with large initial conditions for the Navier–Stokes equations with damping

The Long-Time Stability of Solutions for Intermittently Controlled Viscoelastic Wave Equations with Memory Terms

In this paper, we investigate the asymptotic stability of global solutions to the following intermittently controlled semilinear viscoelastic equation with memory term $$\begin{aligned}u(x,t)\in \varOmega \times [0,\infty ), \end{aligned}$$ under the null Dirichlet boundary condition and $$\tau \in \{t,\infty \}$$ . By virtue of appropriate new Lyapunov functional and Łojasiewicz–Simon inequality, we show that any global bounded solution converges to a steady state and get the rate of convergence as well, when damping coefficients are integral positive and positive-negative, respectively. Moreover, under the assumptions of on–off or sign-changing dampings, we derive the asymptotic stability of solutions.

Well-Posedness and Stability for the Generalized Incompressible Magneto-Hydrodynamic Equations in Critical Fourier-Besov-Morrey Spaces

This paper concerns the Cauchy problem of the 3D generalized incompressible magneto-hydrodynamic (GMHD) equations. By using the Fourier localization argument and the Littlewood-Paley theory, we get local well-posedness results of the GMHD equations with large initial data (u0, b0) belonging to the critical Fourier-Besov-Morrey spaces $${\cal F}\dot {\cal N}_{p,{\rm{\lambda }},q}^{1 - 2\alpha + {3 \over {p\prime }} + {{\rm{\lambda }} \over p}}\left( {{\mathbb{R}^3}} \right)$$ . Moreover, stability of global solutions is also discussed.

Global asymptotic stability for a SEI reaction–diffusion model of infectious diseases with immigration

This paper studies the local and global stability of solutions for a spatially spread SEI epidemic model with immigration of individuals using a Lyapunov functional. It is shown that in the presence of diffusion, the unique steady state remains globally stable. Numerical results obtained through Matlab simulations are presented to confirm the findings of this study.

Combined ZK-mZK equation for Rossby solitary waves with complete Coriolis force and its conservation laws as well as exact solutions

In the present paper, a new partial differential equation has been obtained to describe the Rossby solitary waves with complete Coriolis force by employing multi-scale analysis and perturbation method, we call it combined ZK-mZK equation. The equation can reflect the propagation of Rossby waves on the plane and is more appropriate for the real ocean and atmosphere than the (1+1) dimensional models (such as KdV and mKdV), which can only represent the propagation of Rossby solitary waves in a line. Furthermore, by adopting the multiplier method, we construct conservation laws of the combined ZK-mZK equation, which is meaningful for researching the global stability of solutions. Finally, we deduce the exact solutions of the combined ZK-mZK equation via the semi-inverse variational principle. By applying these exact solutions, some propagation features of Rossby solitary waves are analyzed.

Global stability of solutions in a reaction-diffusion system of predator-prey model

In this article, we investigate the global asymptotic stability of a reaction-diffusion system of predator-prey model. By applying the comparison principle and iteration method, we prove the global asymptotic stability of the unique positive equilibrium solution of (1.1).

On the stability of global solutions to the three-dimensional Navier-Stokes equations

We prove a weak stability result for the three-dimensional homogeneous incom-pressible Navier-Stokes system. More precisely, we investigate the following problem : if a sequence (u_{0,n})_{ n∈\in N} of initial data, bounded in some scaling invariant space, converges weakly to an initial data u0 which generates a global smooth solution, does u0,n generate a global smooth solution ? A positive answer in general to this question would imply global regularity for any data, through the following examples u_{0,n} = nϕ0(n·) or u_{0,n}) = ϕ0(· − x_n) with |x_n| → ∞. We therefore introduce a new concept of weak convergence (rescaled weak convergence) under which we are able to give a positive answer. The proof relies on profile decompositions in anisotropic spaces and their propagation by the Navier-Stokes equations.

Global well-posedness and long time decay of the 3D Boussinesq equations

The incompressible 3D Boussinesq equations is considered in this paper. The global well-posedness of mild solutions is proved. By using Fourier analysis and standard techniques, the global solutions u,θ will decay to zero in sense of the norm of the space χ−1 as time goes to infinity and the stability of global solutions is proved.

Stability of non-constant steady-state solutions for bipolar non-isentropic Euler–Maxwell equations with damping terms

In this article, we consider the periodic problem for bipolar non-isentropic Euler–Maxwell equations with damping terms in plasmas. By means of an induction argument on the order of the time-space derivatives of solutions in energy estimates, the global smooth solution with small amplitude was established close to a non-constant steady-state solution with asymptotic stability property. Furthermore, we obtain the global stability of solutions with exponential decay in time near the non-constant steady-states for bipolar non-isentropic Euler–Poisson equations. This phenomenon on the charge transport shows the essential relation and difference between the bipolar non-isentropic and the bipolar isentropic Euler–Maxwell/Poisson equations.

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 decay of solutions to 3D MHD equations

In the very recent work Wang and Wang (2014) proved that the initial value problem for the three dimension incompressible MHD equations has a global solutions (u,B)∈C([0,∞);χ−1), provided that the norms of the initial data are bounded exactly by the minimal value of the viscosity coefficients. In this paper, we prove that global solutions u,B that was obtained in Wang and Wang (2014) is asymptotic to zero in sense of the norm of the space χ−1 as time goes to infinity. Moreover, stability of global solutions is also discussed.

Analytical and Numerical Methods for Solving Partial Differential Equations and Integral Equations Arising in Physical Models 2014

Analytical and Numerical Methods for Solving Partial Differential Equations and Integral Equations Arising in Physical Models 2014

Asymptotic Behavior of Solution for Higher-Order Wave Equation

In this paper the asymptotic stability of global solutions to the initial-boundary value problem for some nonlinear wave equation with nonlinear damping and source terms is studied by using a difference ineauality.

Exponential Decay of Global Solutions for some Quasilinear Higher-Order Wave Equation

In this paper we prove the uniform stabilization of global solutions for some quasilinear higher-order wave equation with linear damping term and source term by applying a lemma due to V.Komornik.

The Role of Viral Infection on a Bacterial Ecosystem with Nutrient Enrichment

A mathematical analysis of a virus–marine bacteria interaction model is proposed to study the role of virus infection on a bacterial community in the presence of a nutrient. Our model includes four state variables, namely, nutrient concentration, susceptible bacteria biomass, infected bacteria biomass and phage virus concentration. The phage virus infection induces viral lysis, which releases into the marine environment on average \(b\) (viral replication factor) viruses per cell. The dynamical behavior of the system is studied by deriving the existence and bounded conditions and criteria for local and global stability of solutions. The local stability for the interior positive equilibrium is studied by using Routh–Hurwitz conditions. The general bifurcation analysis is shown. The qualitative behaviour of the system is verified through numerical simulation. It is observed that inclusion of nutrients in the system changes the nature of stability of different equilibria.

Existence and Decay Estimate of Global Solutions to Systems of Nonlinear Wave Equations with Damping and Source Terms

The initial-boundary value problem for a class of nonlinear wave 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 through the use of a difference inequality.

Global stability of solutions of nonstationary monotone differential equations with impulsive action in the pseudolinear form

We consider a generalization of the principle of comparison for pseudolinear differential equations with impulsive perturbations in a Banach space. On the basis of these results, we establish the conditions for the global stability in a cone of the trivial solution for the analyzed class of equations. The obtained results are used for the investigation of stability in the Takagi–Sugeno models.

Non-linear spring equations and stability

We discuss the boundary in the Poincaré phase plane for boundedness of solutions to spring model equations of the form and and report the results of a systematic numerical investigation on the global stability of solutions to initial value problems as the initial conditions are allowed to vary. We demonstrate that the unforced spring models explain some of the unbounded behaviour observed with the forced models and thus give at least an idea of where to expect bounded behaviour to occur. We also discuss the behaviour when viscous damping is added to the unforced models.

Stability of Global Solution for the Relativistic Enskog Equation near Vacuum

The Cauchy problem of the relativistic Enskog equation with near-vacuum data is considered in this paper. Under the same assumption as that in Jiang (J. Stat. Phys. 127:805–812, 2007) for the relativistic Enskog equation, we obtain the uniform L∞-stability of the solution. What’s more important, is that for two new types of the scattering cross section σ, we give the global existence and L1(x,v)-stability for mild solution when the initial data lies in the space L1(x,v). As a corollary, we have a BV-type estimate. It is worth mentioning that the stability results in this paper can be applied to the case in Jiang (J. Stat. Phys. 127:805–812, 2007).