Weighted Energy Method Research Articles

Asymptotic stability of non-monotone traveling waves for time-delayed nonlocal dispersion equations

This paper is concerned with the stability of non-monotone traveling waves to a nonlocal dispersion equation with time-delay, a time-delayed integro-differential equation. When the equation is crossing-monostable, the equation and the traveling waves both loss their monotonicity, and the traveling waves are oscillating as the time-delay is big. In this paper, we prove that all non-critical traveling waves (the wave speed is greater than the minimum speed), including those oscillatory waves, are time-exponentially stable, when the initial perturbations around the waves are small. The adopted approach is still the technical weighted-energy method but with a new development. Numerical simulations in different cases are also carried out, which further confirm our theoretical result. Finally, as a corollary of our stability result, we immediately obtain the uniqueness of the traveling waves for the non-monotone integro-differential equation, which was open so far as we know.

Stability of the relativistic Vlasov–Maxwell–Boltzmann system for short range interaction

The Cauchy problem of the relativistic Vlasov–Maxwell–Boltzmann system for short range interaction is investigated. 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. For the proof, a new interactive instant energy functional is introduced to capture the the macroscopic dissipation and the very weak electro-magnetic dissipation of the linearized system. A refined time–velocity weighted energy method is also applied to compensate the weaker dissipation of the linearized collision operator in the case of non-hard potential models. The results also extend the case of “hard ball” model considered by Guo and Strain (2012) to the short range interactions.

Stability of non-monotone critical traveling waves for reaction–diffusion equations with time-delay

This paper is concerned with the stability of critical traveling waves for a kind of non-monotone time-delayed reaction–diffusion equations including Nicholson's blowflies equation which models the population dynamics of a single species with maturation delay. Such delayed reaction–diffusion equations possess monotone or oscillatory traveling waves. The latter occurs when the birth rate function is non-monotone and the time-delay is big. It has been shown that such traveling waves ϕ(x+ct) exist for all c≥c⁎ and are exponentially stable for all wave speed c>c⁎[13], where c⁎ is called the critical wave speed. In this paper, we prove that the critical traveling waves ϕ(x+c⁎t) (monotone or oscillatory) are also time-asymptotically stable, when the initial perturbations are small in a certain weighted Sobolev norm. The adopted method is the technical weighted-energy method with some new flavors to handle the critical oscillatory waves. Finally, numerical simulations for various cases are carried out to support our theoretical results.

Stability of travelling wavefronts in discrete reaction–diffusion equations with nonlocal delay effects

This paper deals with travelling wavefronts for temporally delayed, spatially discrete reaction–diffusion equations. Using a combination of the weighted energy method and the Green function technique, we prove that all noncritical wavefronts are globally exponentially stable, and critical wavefronts are globally algebraically stable when the initial perturbations around the wavefront decay to zero exponentially near minus infinity regardless of the magnitude of time delay.

Convergence rate of solutions toward stationary solutions to a viscous liquid-gas two-phase flow model in a half line

In this paper we study an asymptotic behavior of a solution to the initial boundary value problem for a viscous liquid-gas two-phase flow model in a half line $R_+:=(0,\infty).$ Our ideamainly comes from [23] and [29] which describe anisothermal Navier-Stokes equation in a half line. We obtain theconvergence rate of the time global solution towards corresponding stationary solution in Eulerian coordinates. 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. These theorems are proved by a weighted energy method.

Asymptotic Stability of Plasma Boundary Layers to the Euler--Poisson Equations with Fluid-Boundary Interaction

In the present paper, we discuss the behavior of a boundary layer, called a sheath, arising in the plasma physics. We define the sheath by a monotone stationary solution to the Euler--Poisson equations under a condition known as the Bohm criterion and consider a situation in which charged particles accumulate on the boundary due to the flux from the inner region. Under this fluid-boundary interactive setting, we prove the asymptotic stability of the sheath for both the nondegenerate and degenerate cases based on a weighted energy method. At the same time, we obtain convergence rates toward the stationary solution subject to the spatial decay rates of the initial perturbation.

Critical exponent for the Cauchy problem to the weakly coupled damped wave system

In this paper, we consider a system of weakly coupled semilinear damped wave equations. We determine the critical exponent for any space dimensions. Our proof of the global existence of solutions for supercritical nonlinearities is based on a weighted energy method, whose multiplier is appropriately modified in the case where one of the exponent of the nonlinear term is less than the so called Fujita’s critical exponent. We also give estimates of the lifespan of solutions from above for subcritical nonlinearities.

Asymptotic behavior of radially symmetric solutions for the Burgers equation in several space dimensions

We study large-time behavior of the radially symmetric solution for the Burgers equation on the exterior of a ball in a multi dimensional space, where boundary data at the far field are prescribed. Liu et al. (1998) considered the asymptotic behavior of the solution of scalar viscous conservation law for the case where the corresponding Riemann problem for the hyperbolic part admits a rarefaction wave. In the present paper, it is proved that for a radially symmetric solution to the Burgers equation on a multidimensional space, the asymptotic behaviors are the same as in Liu et al. (1998). Furthermore, we also derive the time convergence rate. The proof is given by a standard L2 energy method and a time weighted energy method.

Exponential Stability of Nonmonotone Traveling Waves for Nicholson's Blowflies Equation

This paper is concerned with Nicholson's blowflies equation, a kind of time-delayed reaction-diffusion equation. It is known that when the ratio of birth rate coefficient and death rate coefficient satisfies $1 e$, the equation losses its monotonicity, and its traveling waves are oscillatory when the time-delay $r$ or the wave speed $c$ is large, which causes the study of stability of these nonmonotone traveling waves to be challenging. In this paper, we use the technical weighted energy method to prove that when $e c_*>0$ are exponentially stable, where $c_*>0$ is the minimum wave speed. Here, we allow the traveling wave to be either monotone or nonmonotone with any speed $c>c_*$ and any size of the time-delay $r>0$; however, when $\frac{p}{d}> e^2$ with a smal...

Global existence and decay of solutions to the Fokker-Planck-Boltzmann equation

The Cauchy problem to the Fokker-Planck-Boltzmann equation under Grad's angular cut-off assumption is investigated. When the initial data is a small perturbation of an equilibrium state, global existence and optimal temporal decay estimates of classical solutions are established. Our analysis is based on the coercivity of the Fokker-Planck operator and an elementary weighted energy method.

Convergence rate to strong boundary layer solutions for generalized BBM-Burgers equations with non-convex flux

This paper is concerned with the initial-boundary value problem forthe generalized Benjamin-Bona-Mahony-Burgers equation in the halfspace $R_+$\begin{eqnarray}u_t-u_{txx}-u_{xx}+f(u)_{x}=0,\quad t>0, x\in R_+,\\u(0,x)=u_0(x)\to u_+, \quad as \ \ x\to +\infty,\\u(t,0)=u_b.\end{eqnarray}Here $u(t,x)$ is an unknown function of $t>0$ and $x\in R_+$,$u_+\not=u_b$ are two given constant states and the nonlinearfunction $f(u)$ is assumed to be a non-convex function which has one or finitely many inflection points. In this paper, we consider $u_b<u_+$. For the non-degenerate case $f'(u_+)<0$, we show the existence andthe global stability of strong boundary layer solution $\phi(x)$ with the above non-convex function $f(u)$ satisfying (1.3). In this case, the corresponding algebraic convergence rate is also obtained. Our analysis is based on the space-time weighted energy method (cf. [4, 11]) combined with some delicate estimates.

Spreading speeds and traveling waves for a nonlocal dispersal equation with convolution-type crossing-monostable nonlinearity

This paper is concerned with the traveling wave solutions and the spreading speeds for a nonlocal dispersal equation with convolution-type crossing-monostable nonlinearity, which is motivated by an age-structured population model with time delay. We first prove the existence of traveling wave solution with critical wave speed c = c*. By introducing two auxiliary monotone birth functions and using a fluctuation method, we further show that the number c = c* is also the spreading speed of the corresponding initial value problem with compact support. Then, the nonexistence of traveling wave solutions for c < c* is established. Finally, by means of the (technical) weighted energy method, we prove that the traveling wave with large speed is exponentially stable, when the initial perturbation around the wave is relatively small in a weighted norm.

NONLINEAR STABILITY OF PLANAR STATIONARY WAVES FOR GENERALIZED BENJAMIN–BONA–MAHONY–BURGERS EQUATIONS IN THE HALF-PLANE

This paper is concerned with the nonlinear stability of planar stationary waves of generalized Benjamin–Bona–Mahony–Burgers equations in half-plane. The planar stationary waves are shown to be globally nonlinear stable and then, by employing the space–time weighted energy method developed by Kawashima and Matsumura in 1995 and a generalized Hardy type inequality with the best possible constant introduced by Kawashima and Kurata in 2009, the corresponding convergence rates (both algebraic and exponential) of the global solutions to the stationary waves are also obtained for both the nondegenerate case and the degenerate case.

Convergence rate of solutions toward stationary solutions to the bipolar Navier-Stokes-Poisson equations in a half line

In this paper, we show the convergence rate of a solution toward the stationary solution to the initial boundary value problem for the one-dimensional bipolar compressible Navier-Stokes-Poisson equations. For the supersonic flow at spatial infinity, if an initial perturbation decays with the algebraic or the exponential rate in the spatial asymptotic point, the solution converges to the corresponding stationary solution with the same rate in time as time tends to infinity. For the transonic flow at spatial infinity, the solution converges to the stationary solution in time with the lower rate than that of the initial perturbation in the spatial. These results are proved by the weighted energy method. MSC: 35M31; 35Q35

Multidimensional stability of traveling fronts in monostable reaction-diffusion equations with complex perturbations

This paper studies the multidimensional stability of traveling fronts in monostable reaction-diffusion equations, including Ginzburg-Landau equations and Fisher-KPP equations. Eckmann andWayne (1994) showed a one-dimensional stability result of traveling fronts with speeds c ⩽ c* (the critical speed) under complex perturbations. In the present work, we prove that these traveling fronts are also asymptotically stable subject to complex perturbations in multiple space dimensions (n = 2, 3), employing weighted energy methods.

Pointwise decaying rate of large perturbation around viscous shock for scalar viscous conservation law

In this paper, we consider the large perturbation around the viscous shock of the scalar conservation law with viscosity in one dimension case. We divide the time region into t ⩽ T0 and t > T0 for a fixed constant T0 when applying energy method. Since T0 is fixed, the case t ⩽ T0 is easy to deal with and when t > T0, from the decaying property of the solution, there is a priori estimate for the solution. Thus we can succeed to control the nonlinear term and get the pointwise estimate for the perturbation by the weighted energy method.

Nonlinear stability of traveling wavefronts in an age-structured population model with nonlocal dispersal and delay

This paper is concerned with the nonlinear stability of traveling wavefronts for a single species population model with nonlocal dispersal and age structure. By using the weighted energy method together with the comparison principle, we prove that the traveling wavefront is exponentially stable, when the initial perturbation around the wavefronts decays exponentially at –∞, but it can be arbitrarily large in other locations. In particular, our result implies that the time delay is harmless for stability of traveling wavefronts of the model.

Weighted energy method and long wave short wave decomposition on the linearized compressible Navier-Stokes equation

The purpose of this paper is to study asymptotic behaviors of the Green function of the linearized compressible Navier-Stokes equation. Liu, T.-P. and Zeng, Y. obtained a point-wise estimate for the Green function of the linearized compressible Navier-Stokes equation in [Comm. Pure Appl. Math. 47, 1053--1082 (1994)] and [Mem. Amer. Math. Soc. 125 (1997), no. 599]. In this paper, we propose a new methodology to investigate point-wise behavior of the Green function of the compressible Navier-Stokes equation. This methodology consists of complex analysis method and weighted energy estimate which was originally proposed by Liu, T.-P. and Yu, S.-H. in [Comm. Pure Appl. Math., 57, 1543--1608 (2004)] for the Boltzmann equation. We will apply this methodology to get a point-wise estimate of the Green function for large $t>0$.

Stability of the nonrelativistic Vlasov-Maxwell-Boltzmann system for angular non-cutoff potentials

Although there recently have been extensive studies on theperturbation theory of the angular non-cutoff Boltzmann equation(cf. [4] and [17]), it remains mathematicallyunknown when there is a self-consistent Lorentz force coupled withthe Maxwell equations in the nonrelativistic approximation. In thepaper, for perturbative initial data with suitable regularity andintegrability, we establish the large time stability of solutionsto the Cauchy problem of the Vlasov-Maxwell-Boltzmann system withphysical angular non-cutoff intermolecular collisions including theinverse power law potentials, and also obtain as a byproduct theconvergence rates of solutions. The proof is based on a newtime-velocity weighted energy method with two key technical parts:one is to introduce the exponentially weighted estimates into thenon-cutoff Boltzmann operator and the other to design a delicatetemporal energy $X(t)$-norm to obtain its uniform bound. The resultalso extends the case of the hard sphere model considered by Guo[Invent. Math. 153(3): 593--630 (2003)] to the general collisionpotentials.

Stability of traveling waves in a monostable delayed system without quasi-monotonicity

This paper is concerned with traveling waves of a monostable reaction–diffusion system with delay and without quasi-monotonicity. When the initial perturbation around the traveling wave is suitably small in a weighted norm, the exponential stability of all traveling wave solutions for the system with delay is proved by the weighted energy method.