R~2上一类非牛顿流体动力学方程组解的最优衰减率

本文利用Fourier分解方法讨论一类二维不可压缩非牛顿流体动力学方程组弱解的衰减性,证明了弱解在L~2范数下衰减率为(1+t)~(-1/2),和线性热传导方程解的衰减率一致,在此意义下本文的结果是最优的。     

偶数维空间带粘性项的非线性波动方程解的衰减估计

基于对线性化方程格林函数的详细分析,研究了偶数维空间带粘性项的非线性波动方程解的大时间状态.得到了解的最佳衰减估计,与惠更斯原理相符.     

Asymptotic behavior of solutions to a class of nonlinear wave equations of sixth order with damping

We investigate the initial value problem for a class of nonlinear wave equations of sixth order with damping. The decay structure of this equation is of the regularity‐loss type, which causes difficulty in high‐frequency region. By using the Fourier splitting frequency technique and energy method in Fourier space, we establish asymptotic profiles of solutions to the linear equation that is given by the convolution of the fundamental solutions of heat and free wave equation. Moreover, the asymptotic profile of solutions shows the decay estimate of solutions to the corresponding linear equation obtained in this paper that is optimal under some conditions. Finally, global existence and optimal decay estimate of solutions to this equation are also established. Copyright © 2016 John Wiley & Sons, Ltd.     

一类非线性高阶Kirchhoff型方程的初边值问题

本文研究了一类具有非线性耗散项的高阶Kirchhoff型方程的初边值问题.通过构造稳定集讨论了此问题整体解的存在性,应用Nakao的差分不等式建立了解能量的衰减估计.在初始能量为正的条件下,证明了解在有限时间内发生blow-up,并且给出了解的生命区间估计.     

Asymptotic stability and decay rate of solutions for a nonlinear fourth order wave equation

This paper is concerned with investigating the global asymptotic behavior of the zero solution of the initial-boundary value problem for a nonlinear fourth order wave equation. Moreover an estimate of the rate of decay of the solutions is obtained.     

Large time behavior to the system of incompressible non-Newtonian fluids in R

In this paper, the authors study the large time behavior for the weak solutions to a class system of the incompressible non-Newtonian fluids in R². It is proved that the weak solutions decay in L² norm at (1+t)^−1/2 and the estimate for the decay rate is sharp in the sense that it coincides with the decay rate of a solution to the heat equation.     

Behaviors of solutions for a singular diffusion equation

This paper is devoted to some behaviors of solutions of the initial-boundary problem for a singular diffusion equation, namely, localization and large time behavior. After given some special explicit solutions it is proved that solutions of the problem possess the localization property. Next, L² decay estimate as t→∞ is proved by a rather standard energy method. Finally, by comparison with a special solution the expected L^∞ decay estimate is derived.     

On the decay of solutions to the 2D Neumann exterior problem for the wave equation

We consider the exterior problem in the plane for the wave equation with a Neumann boundary condition and study the asymptotic behavior of the solution for large times. For possible application we are interested to show a decay estimate which does not involve weighted norms of the initial data. In the paper we prove such an estimate, by a combination of the estimate of the local energy decay and decay estimates for the free space solution.     

LARGE TIME BEHAVIOR OF SOLUTIONS TO THE PERTURBED HASEGAWA-MIMA EQUATION

The large time behavior of solutions to the two-dimensional perturbed HasegawaMima equation with large initial data is studied in this paper. Based on the time-frequency decomposition and the method of Green function,we not only obtain the optimal decay rate but also establish the pointwise estimate of global classical solutions.     

Global existence and asymptotic behavior of solutions for a semi-linear wave equation

In this paper, we investigate the initial value problem for a semi-linear wave equation in n-dimensional space. Based on the decay estimate of solutions to the corresponding linear equation, we define a set of time-weighted Sobolev spaces. Under small condition on the initial value, we prove the global existence and asymptotic behavior of the solution in the corresponding Sobolev spaces by the contraction mapping principle.     

一阶双曲型方程组解的衰减率

本文在研究波动方程时引入的整体Sobolev不等式推广到双曲组的情形．得到了一阶双曲组Cauchy问题解的几个衰减估计．特别是当初始资料给在指定的带权Sobolev空间中时,定理1．5的结果提供了最佳的衰减率．在定理的证明中我们将双曲组化到相应的一阶拟微分方程的情形,进而利用微局部分析建立所需要的估计．     

Characterization of polynomial decay rate for the solution of linear evolution equation

In this paper, we study the decay rate of solutions to strongly stable, but not exponentially stable linear evolution equations. It is known that the resolvent operator of such an equation must be unbounded on the imaginary axis. Our main result is an estimate of the decay rate when the unboundedness is of polynomial order. We then apply our main theorem to three strongly stable but not exponentially stable systems to obtain the decay rate, which is not available in the literature.Received: July 21, 2003     

BBM-Burgers方程解得衰减估计

本文研究了一维空间的BBM-Burgers方程.利用时频分解和能量估计等工具,在解整体存在的前提下,得到了本方程柯西问题解的某些衰减估计.     

Decay rate of solutions to hyperbolic system of first order

In this paper we generalize the global Sobolev inequality introduced by Klainerman in studying wave equation to the hyperbolic system case. We obtain several decay estimates of solutions of a hyperbolic system of first order by different norms of initial data. In particular, the result mentioned in Theorem 1.5 offers an optimal decay rate of solutions, if the initial data belongs to the assigned weighted Sobolev space. In the proof of the theorem we reduce the estimate of solutions of a hyperbolic system to the corresponding case for a scalar pseudodifferential equation of the first order, and then establish the required estimate by using microlocal analysis. This work is partly supported by NNSF of China and Doctoral Programme Foundation of IHEC     

一类退化非线性抛物型方程整体解的衰减估计

J.L.L ions用紧致性方法证明了一类退化非线性抛物型方程初边值问题整体解的存在唯一性,但解的衰减性很少有人考虑.应用M.N akao建立的差分不等式研究了整体解的衰减估计.     

On the asymptotic behavior of solution for the generalized IBq equation with Stokes damped term

In this paper, we investigate the Cauchy problem for the generalized improved Boussinesq equation with Stokes damped term in n-dimensional space. We observe that the dissipative structure of the linearized equation is of the regularity-loss type. This means that we have the optimal decay estimates of solutions under the additional regularity assumption on the initial data. Based on the decay estimates of solutions to the corresponding linear equation and smallness condition on the initial data, we prove the global existence and asymptotic of the small amplitude solution in the time-weighted Sobolev space by the contraction mapping principle.     

Large time behavior of solutions to a nonlinear hyperbolic relaxation system with slowly decaying data

We consider the large time asymptotic behavior of the global solutions to the initial value problem for the nonlinear damped wave equation with slowly decaying initial data. When the initial data decay fast enough, it is known that the solution to this problem converges to the self-similar solution to the Burgers equation called a nonlinear diffusion wave, and its optimal asymptotic rate is obtained. In this paper, we focus on the case that the initial data decay more slowly than previous works and derive the corresponding asymptotic profile. Moreover, we investigate how the change of the decay rate of the initial values affect its asymptotic rate.     

Beam evolution equation with variable coefficients

We investigate a initial‐boundary value problem for the nonlinear beam equation with variable coefficients on the action of a linear internal damping. We show the existence of a unique global weak solution and that the energy associated with this solution has a rate decay estimate. Besides, we prove the existence and uniqueness of non‐local strong solutions. Copyright © 2004 John Wiley & Sons, Ltd.     

一类双退化非线性抛物型方程

In this paper we study the decay estimate of global solutions to the initialboundary value problem for double degenerate nonlinear parabolic equation by using a difference inequality.     

GENERAL DECAY OF A TRANSMISSION PROBLEM FOR KIRCHHOFF TYPE WAVE EQUATIONS WITH BOUNDARY MEMORY CONDITION

In this paper, we investigate the influence of boundary dissipation on the de-cay property of solutions for a transmission problem of Kirchhoff type wave equation with boundary memory condition. By introducing suitable energy and Lyapunov functionals, we establish a general decay estimate for the energy, which depends on the behavior of relaxation function.