Weighted Energy Method Research Articles

Stability and uniqueness of traveling waves of a non-local dispersal SIR epidemic model

This paper is mainly concerned with the exponential stability and uniqueness of traveling waves of a delayed nonlocal dispersal SIR epidemic model. We first prove the stability of traveling waves by using the weighted energy method, where the traveling waves are allowed to be non-monotone. Next we establish the exact asymptotic behavior of traveling waves at-8 by using Ikehara's theorem. Then the uniqueness of traveling waves is obtained by the stability result. Finally, we discuss how the nonlocal dispersal affects the stability of traveling waves. The conclusion shows that the nonlocal dispersal slows down the convergence rate of the solution to the traveling waves.

Stability of non-monotone non-critical traveling waves in discrete reaction-diffusion equations with time delay

This paper is concerned with traveling waves for temporally delayed, spatially discrete reaction-diffusion equations without quasi-monotonicity. We first establish the existence of non-critical traveling waves (waves with speeds c>c*, where c* is minimal speed). Then by using the weighted energy method with a suitably selected weight function, we prove that all noncritical traveling waves Φ(x+ct) (monotone or nonmonotone) are time-asymptotically stable, when the initial perturbations around the wavefronts in a certain weighted Sobolev space are small.

Exponential Stability of Traveling Waves for a Reaction Advection Diffusion Equation with Nonlinear-Nonlocal Functional Response

The stability of a reaction advection diffusion equation with nonlinear-nonlocal functional response is concerned. By using the technical weighted energy method and the comparison principle, the exponential stability of all noncritical traveling waves of the equation can be obtained. Moreover, we get the rates of convergence. Our results improve the previous ones. At last, we apply the stability result to some real models, such as an epidemic model and a population dynamic model.

Stationary solutions to the one-dimensional micropolar fluid model in a half line: Existence, stability and convergence rate

In this paper, we study the asymptotic behavior of solutions to the initial boundary value problem for the one-dimensional micropolar fluid model in a half line R+:=(0,∞). Our idea mainly comes from [12] which describes the large time behavior of solutions for non-isentropic Navier–Stokes equations in a half line. Compared with Navier–Stokes equations in the absence of the microrotation velocity, the microrotation velocity brings us some additional troubles. We obtain the convergence rate of global solutions toward corresponding stationary solutions if the initial perturbation belongs to the weighted Sobolev space. The proofs are given by a weighted energy method.

Time-delayed reaction–diffusion equations with boundary effect: (I) convergence to non-critical traveling waves

This paper is concerned with time-delayed reaction–diffusion equations on half space with boundary effect. When the birth rate function is non-monotone, the solution of the delayed equation subjected to appropriate boundary condition is proved to converge time-exponentially to a certain (monotone or non-monotone) traveling wave profile with wave speed , where is the minimum wave speed, when the initial data is a small perturbation around the wave, while the wave is shifted far away from the boundary so that the boundary layer is sufficiently small. The adopted method is the technical weighted-energy method with some new flavors to handle the boundary terms. However, when the birth rate function is monotone under consideration, then, for all traveling waves with , no matter what size of the boundary layers is, these monotone traveling waves are always globally stable. The proof approach is the monotone technique and squeeze theorem but with some new development.

Stability of traveling wavefronts for a discrete diffusive Lotka–Volterra competition system

In this paper, we study a discrete diffusive Lotka–Volterra competition system. It is known that this system has traveling wavefronts. We prove that the traveling wavefronts are exponentially stable, when the initial perturbation around the traveling wavefronts decays exponentially as x→−∞, but can be arbitrarily large in other locations. The approach we use here is the comparison principle and the weighted energy method.

The Vlasov–Poisson–Boltzmann system for the whole range of cutoff soft potentials

The dynamics of dilute electrons can be modeled by the fundamental one-species Vlasov–Poisson–Boltzmann system which describes mutual interactions of the electrons through collisions in the self-consistent electrostatic field. For cutoff intermolecular interactions, although there is some progress on the construction of global smooth solutions to its Cauchy problem near Maxwellians recently, the problem for the case of very soft potentials remains unsolved. By introducing a new time–velocity weighted energy method and based on some new optimal temporal decay estimates on the solution itself and some of its derivatives with respect to both the spatial and the velocity variables, it is shown in this manuscript that the Cauchy problem of the one-species Vlasov–Poisson–Boltzmann system for all cutoff soft potentials does exist a unique global smooth solution for general initial perturbation which is unnecessary to satisfy the neutral condition imposed in [13] for the case of cutoff moderately soft potentials but is assumed to be small in certain weighted Sobolev spaces. Our approach applies also to the case of cutoff hard potentials and thus provides a satisfactory global well-posedness theory to the one-species Vlasov–Poisson–Boltzmann system near Maxwellians for the whole range of cutoff intermolecular interactions in the perturbative framework.

Existence and stability of stationary solution to compressible Navier–Stokes–Poisson equations in half line

In this paper, we investigate the asymptotic stability of the stationary solution to the outflow problem for the compressible Navier–Stokes–Poisson system in a half line. We show the existence of the stationary solution with the aid of the stable manifold theory. The time asymptotic stability of the stationary solution is obtained by the elementary energy method. Furthermore, for the supersonic flow at spatial infinity, we also obtain an algebraic and an exponential decay rate, when the initial perturbation belongs to the corresponding weighted Sobolev space. The proof is based on a time and space weighted energy method.

Nonlinear Stability of Traveling Wavefronts for Delayed Reaction-diffusion Equation with Nonlocal Diffusion

In this paper, we consider a class of nonlocal dispersal equation with nonlocal time-delayed reaction. We prove that all noncritical wavefronts are globally exponentially stable by the weighted energy method and comparison principle. However, for the critical wavefronts, we prove that they are globally asymptotically stable.

The stability of stationary solution for outflow problem on the navier-stokes-poisson system

In this article, we are concerned with the stability of stationary solution for outflow problem on the Navier-Stokes-Poisson system. We obtain the unique existence and the asymptotic stability of stationary solution. Moreover, the convergence rate of solution towards stationary solution is obtained. Precisely, if an initial perturbation decays with the algebraic or the exponential rate in space, the solution converges to the corresponding stationary solution as time tends to infinity with the algebraic or the exponential rate in time. The proof is based on the weighted energy method by taking into account the effect of the self-consistent electric field on the viscous compressible fluid.

Long-time behavior of solution for the compressible nematic liquid crystal flows in [formula omitted

In this paper, we investigate the global existence and long-time behavior of classical solution for the compressible nematic liquid crystal flows in three-dimensional whole space. First of all, the global existence of classical solution is established under the condition that the initial data are close to the constant equilibrium state in HN(R3) (N≥3)-framework. Then, one establishes algebraic time decay for the classical solution by weighted energy method. Finally, the algebraic decay rate of classical solution in Lp(R3)-norm with 2≤p≤∞ and optimal decay rate of their spatial derivative in L2(R3)-norm are obtained if the initial perturbation belong to L1(R3) additionally.

The $$r^{p}$$ r p -Weighted Energy Method of Dafermos and Rodnianski in General Asymptotically Flat Spacetimes and Applications

In [11], Dafermos and Rodnianski presented a novel approach to establish uniform decay rates for solutions $${\upvarphi }$$ to the scalar wave equation $$\square _{g}{\upvarphi }=0$$ on Minkowski, Schwarzschild and other asymptotically flat backgrounds. This paper generalises the methods and results of [11] to a broad class of asymptotically flat spacetimes $$(\mathcal {M},g)$$ , including Kerr spacetimes in the full subextremal range $$|a|<M$$ , but also radiating spacetimes with no exact symmetries in general dimension $$d+1$$ , $$d\ge 3$$ . As a soft corollary, it is shown that the Friedlander radiation field for $${\upvarphi }$$ is well defined on future null infinity. Moreover, polynomial decay rates are established for $${\upvarphi }$$ , provided that an integrated local energy decay statement (possibly with a finite loss of derivatives) holds and the near region of $$(\mathcal {M},g)$$ satisfies some mild geometric conditions. The latter conditions allow for $$(\mathcal {M},g)$$ to be the exterior of a black hole spacetime with a non-degenerate event horizon (having possibly complicated topology) or the exterior of a compact moving obstacle in an ambient globally hyperbolic spacetime satisfying suitable geometric conditions.

The relativistic Vlasov-Maxwell-Boltzmann system for short range interaction

We are concerned with the Cauchy problem of the relativistic Vlasov-Maxwell-Boltzmann system for short range interaction. For perturbative initial data with suitable regularity and integrability, we prove the large time stability of solutions to the relativistic Vlasov-Maxwell-Boltzmann system, and also obtain as a byproduct the convergence rates of solutions. Our proof is based on a new time-velocity weighted energy method and some optimal temporal decay estimates on the solution itself. The results also extend the case of ``hard ball model considered by Guo and Strain [Comm. Math. Phys. 310: 49--673 (2012)] to the short range interactions.

Stability of traveling waves of a diffusive susceptible-infective-removed (SIR) epidemic model

This paper is concerned with the stability and uniqueness of traveling waves of a delayed diffusive susceptible-infective-removed (SIR) epidemic model. We first prove the exponential stability of traveling waves by using the weighted energy method, where the traveling waves are allowed to be non-monotone. Then we establish the exact asymptotic behavior of traveling waves at −∞ by using Ikehara’s theorem. Finally, the uniqueness of traveling waves is proved by the stability result of traveling waves.

Rate of convergence of solutions to traveling fronts for some quasi-linear relaxation systems with viscous

This paper is concerned with the exponential convergence rates of solutions to non-degenerate traveling fronts in some exponentially weighted spaces and the algebraic convergence rates of solutions to non-degenerate and degenerate traveling fronts in some algebraically weighted spaces for quasi-linear relaxation systems with small viscous using weighted energy method, where the small initial perturbations of the traveling fronts are in some appropriate weighted spaces and the wave strengths are small.

Degenerate boundary layers for a system of viscous conservation laws

This paper is concerned with existence and asymptotic stability of a boundary layer solution which is a smooth stationary wave for a system of viscous conservation laws in one-dimensional half space. With the aid of the center manifold theory, it is shown that the degenerate boundary layer solution exists under the situation that one characteristic is zero and the other characteristics are negative. Asymptotic stability of the degenerate boundary layer solution is also proved in an algebraically weighted Sobolev space provided that the weight exponent [Formula: see text] satisfies [Formula: see text]. The stability analysis is based on deriving the a priori estimate by using the weighted energy method combined with the Hardy type inequality with the best possible constant.

Stability of traveling waves in a population dynamics model with spatio-temporal delay

In this paper, we are concerned with the stability of all traveling waves for a population dynamics model with spatio-temporal delay. By the weighted-energy method combining comparison principle, the global exponential stability of all traveling waves for the model is established, even including the slower waves whose wave speed is close to the critical speed.

The Vlasov–Maxwell–Boltzmann system with a uniform ionic background near Maxwellians

The two-species Vlasov–Maxwell–Boltzmann system is an important model for plasma physics describing the time evolution of dilute charged particles consisting of electrons and ions under the influence of the self-consistent internally generated Lorentz forces. In physical situations the ion mass is usually much larger than the electron mass so that the electrons move much faster than the ions. Thus, the ions are often described by a fixed ion background and only the electrons move. For such a simple case, the two-species Vlasov–Maxwell–Boltzmann system is reduced to the one-species Vlasov–Maxwell–Boltzmann system.Although the one-species Vlasov–Maxwell–Boltzmann system is a simplified model of the two-species Vlasov–Maxwell–Boltzmann system, its global well-posedness theory near a given global Maxwellian in the perturbative framework is more difficult than the two-species case, which is partially due to the slow-decay of the electromagnetic field and up to now, the problem on the construction of global in time solutions near a given global Maxwellian in the perturbative framework for the Cauchy problem of the one-species Vlasov–Maxwell–Boltzmann system with cutoff non-hard sphere intermolecular collisions remains unsolved. It is shown in this paper that the Cauchy problem of the one-species Vlasov–Maxwell–Boltzmann system with cutoff non-hard sphere intermolecular collisions including the cutoff inverse power law potentials is globally well-posed provided that the perturbative initial data satisfies certain regularity, smallness, and integrability conditions. Our analysis is based on a new time-velocity weighted energy method with two key technical parts: one is to introduce the exponentially weighted estimates into the cutoff Boltzmann operator and the other is to design a delicate temporal energy X(t)-norm to obtain its uniform bound. As a by-product of our analysis, we can also deduce certain temporal decay estimates on the global solutions constructed above for the same range of intermolecular collisions. Notice that even for the hard sphere model, although the global solutions have been successfully constructed in [8], its large time behaviors are unknown, cf. [8].

Stability of traveling fronts in a population model with nonlocal delay and advection

In this paper, we are concerned with the stability of traveling fronts in a population model with nonlocal delay and advection under the large initial perturbation (i.e. the initial perturbation around the traveling wave decays exponentially as $x \rightarrow-\infty$, but it can be arbitrarily large in other locations). The globally exponential stability of traveling fronts is established by the weighted-energy method combining with comparison principle, including even the slower waves whose wave speed are close to the critical speed.

Uniqueness and stability of traveling waves for cellular neural networks with multiple delays

In this paper, we investigate the properties of traveling waves to a class of lattice differential equations for cellular neural networks with multiple delays. Following the previous study [38] on the existence of the traveling waves, here we focus on the uniqueness and the stability of these traveling waves. First of all, by establishing the a priori asymptotic behavior of traveling waves and applying Ikehara's theorem, we prove the uniqueness (up to translation) of traveling waves ϕ(n−ct) with c≤c⁎ for the cellular neural networks with multiple delays, where c⁎<0 is the critical wave speed. Then, by the weighted energy method together with the squeezing technique, we further show the global stability of all non-critical traveling waves for this model, that is, for all monotone waves with the speed c<c⁎, the original lattice solutions converge time-exponentially to the corresponding traveling waves, when the initial perturbations around the monotone traveling waves decay exponentially at far fields, but can be arbitrarily large in other locations.