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.     

ASYMPTOTIC PROPERTY FOR THE SOLUTION TO THE GENERALIZED KORTEWEG－DE VRIES EQUATION

ＡＳＹＭＰＴＯＴＩＣＰＲＯＰＥＲＴＹＦＯＲＴＨＥＳＯＬＵＴＩＯＮＴＯＴＨＥＧＥＮＥＲＡＬＩＺＥＤＫＯＲＴＥＷＥＧ－ＤＥＶＲＩＥＳＥＱＵＡＴＩＯＮＺＨＡＮＧＬＩＮＧＨＡＩ（张领海）（ＤｅｐｏｒｔｍｅｎｔｏｆＭａｔｈｅｍａｔｉｃｓ，ｔｈｅＯｈｉｏＳｔａｔ...     

奇摄动分数阶微分方程的渐近解

本文研究了一类奇摄动非线性分数阶微分方程初值问题.利用伸长变量构造出解的形式展开式,并利用微分不等式理论,证明了解的一致有效的渐近式.所得的结果具有较好精度的近似解.     

On the asymptotic behaviour of solution for the generalized double dispersion equation

In this article, we investigate the Cauchy problem for the generalized double dispersion equation in n-dimensional space. We establish the decay estimates of solution to the corresponding linear equation. Under smallness condition on the initial data, we prove the global existence and asymptotic behaviour of the small amplitude solution in the time-weighted Sobolev space by the contraction mapping principle.     

Stable and unstable sets for the Cauchy problem for a nonlinear wave equation with nonlinear damping and source terms

For the Cauchy problem for the nonlinear wave equation with nonlinear damping and source terms we define stable and unstable sets for the initial data. We prove that, if during the evolution the solution enters into the stable set, the solution is global and we are able to estimate the decay rate of the energy. If during the evolution the solution enters into the unstable set, the solution blows up in finite time.     

Convergence rates toward the travelling waves for a model system of the radiating gas

The present paper is concerned with an asymptotics of a solution to the model system of radiating gas. The previous researches have shown that the solution converges to a travelling wave with a rate (1 + t)^?1/4 as time t tends to infinity provided that an initial data is given by a small perturbation from the travelling wave in the suitable Sobolev space and the perturbation is integrable. In this paper, we make more elaborate analysis under suitable assumptions on initial data in order to obtain shaper convergence rates than previous researches. The first result is that if the initial data decays at the spatial asymptotic point with a certain algebraic rate, then this rate reflects the time asymptotic convergence rate. Precisely, this convergence rate is completely same as the spatial convergence rate of the initial perturbation. The second result is that if the initial data is given by the Riemann data, an admissible weak solution, which has a discontinuity, converges to the travelling wave exponentially fast. Both of two results are proved by obtaining decay estimates in time through energy methods with suitably chosen weight functions. Copyright © 2006 John Wiley & Sons, Ltd.     

Asymptotic profile of solutions for the damped wave equation with a nonlinear convection term

This paper is concerned with the large time behavior of solutions to the initial value problem for the damped wave equations with nonlinear convection in one‐dimensional whole space. In 2007, Ueda and Kawashima showed that the solution tends to a self similar solution of the Burgers equation. However, they did not mention that their decay estimate is optimal or not. Under this situation, the aim of this paper was to find out the sharp decay estimate by studying the second asymptotic profile of solutions. The explicit representation formula and the decay estimates of the solution for the linearized equation including the lower order term play crucial roles in our analysis.     

Asymptotic stability of the exact boundary controllability of nodal profile for 1D quasi-linear wave equations

In this paper, we consider the asymptotic stability of the exact boundary controllability of nodal profile for 1D quasi-linear wave equations. First, for 1D quasi-linear hyperbolic systems with zero eigenvalues, we establish the existence and uniqueness of semiglobal classical solution to the one-sided mixed initial-boundary value problem on a semibounded initial axis and discuss the asymptotic behavior of the corresponding solutions under different hypotheses on the initial data. Based on these results, we obtain the asymptotic stability of the exact boundary controllability of nodal profile for 1D quasi-linear wave equations on a semibounded time interval.     

Global existence of solutions for damped wave equations with nonlinear memory

This article studies the Cauchy problem for the damped wave equation with nonlinear memory. For a noncompactly supported initial data with small energy, global existence and asymptotic behaviour of solutions are obtained when 1?≤?n?≤?3. This result generalized the previous result by Fino [Critical exponent for damped wave equations with nonlinear memory, Nonlinear Anal. 74 (2011), pp. 5495–5505], which dealt with the solution with compactly supported initial data.     

一类非线性发展方程整体解的存在性、渐近性与解的爆破

研究一类非线性发展方程初边值问题整体弱解的存在性，渐近性和解的爆破问题，证明在关于非线性项的不同条件下，上述初边值问题分别在大初值和小初始能量的情况下存在整体弱解，并且讨论了弱解的渐近性。还证明：在相反的条件下，上述弱解在有限时刻爆破，并且给出了一个实例。     

Asymptotic profile of solutions to a non‐linear dissipative evolution system with conservation

We investigate the asymptotic profile to the Cauchy problem for a non‐linear dissipative evolution system with conservational form (1) provided that the initial data are small, where constants α, ν are positive satisfying ν²<4α(1 ? α), α<1. In (J. Phys. A 2005; 38 :10955–10969), the global existence and optimal decay rates of the solution to this problem have been obtained. The aim of this paper is to apply the heat kernel to examine more precise behaviour of the solution by finding out the asymptotic profile. Precisely speaking, we show that, when time t → ∞ the solution and solution in the L^p sense, where G(t, x) denotes the heat kernel and is determined by the initial data and the solution to a reformulated problem obtained in Section 3, β is related to ?₊ and ?_? which are determined by (41) in Section 4. The numerical simulation is presented in the end. The motivation of this work thanks to Nishihara (Asymptotic profile of solutions to nonlinear dissipative evolution system with ellipticity. Z. Angew Math Phys 2006; 57 : 604–614). Copyright © 2007 John Wiley & Sons, Ltd.     

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.     

具阻尼项的非线性波动方程的稳定集与不稳定集

本文利用势井理论讨论一类非线性波动方程的初边值问题 .我们构造其稳定集 W和不稳定集 V,证明了当初值属于 W时 ,对 β∈ R整体弱解存在并且利用乘子法得到当 β>0解的指数衰减估计 ;当初值属于 V时 ,而 β<0时 ,解将爆破     

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.     

Asymptotic behavior for the filtration equation in domains with noncompact boundary

We consider the initial value boundary problem with zero Neumann data for an equation modeled after the porous media equation, with variable coefficients. The spatial domain is unbounded and shaped like a (general) paraboloid, and the solution u is integrable in space and nonnegative. We show that the asymptotic profile for large times of u is one dimensional and given by an explicit function, which can be regarded as the fundamental solution of a one-dimensional differential equation with weights. In the case when the domain is a cone or the whole space (Cauchy problem), we obtain a genuine multidimensional profile given by the well-known Barenblatt solution.     

Long-time asymptotic for the Hirota equation via nonlinear steepest descent method

We present the Riemann–Hilbert problem formalism for the initial value problem for the Hirota equation on the line. We show that the solution of this initial value problem can be obtained from that of associated Riemann–Hilbert problem, which allows us to use nonlinear steepest descent method/Deift–Zhou method to analyze the long-time asymptotic for the Hirota equation.     

Nonlinear Cusped Caustics for Dispersive Waves

A multiple-scale perturbation analysis for slowly varying weakly nonlinear dispersive waves predicts that the wave number breaks or folds and becomes triple-valued. This theory has some difficulties, since the wave amplitude becomes infinite. Energy first focuses along a cusped caustic (an envelope of the rays or characteristics). The method of matched asymptotic expansions shows that a thin focusing region with relatively large wave amplitudes, valid near the cusped caustic, is described by the nonlinear Schrödinger equation (NSE). Solutions of the NSE are obtained from an asymptotic expansion of an equivalent linear singular integral equation related to a Riemann-Hilbert problem. In this way connection formulas before and after focusing are derived. We show that a slowly varying nearly monochromatic wave train evolves into a triple-phased slowly varying similarity solution of the NSE. Three weakly nonlinear waves are simultaneously superimposed after focusing, giving meaning to a triple-valued wave number. Nonlinear phase shifts are obtained which reduce to the linear phase shifts previously described by the asymptotic expansion of a Pearcey integral.     

Long-time asymptotic expansion for the damped semilinear wave equation

The first initial-boundary value problem is considered for the damped semilinear wave equation with the quadratic nonlinearity. For small initial data and homogeneous boundary conditions its solution is constructed in the form of a series in the eigenfunctions of the Laplace operator. The long-time asymptotic expansion is obtained which shows the nonlinear effects of amplitude and frequency multiplication. The same results hold for the admissible initial data that are not small.     

On the stability of contact discontinuity for Cauchy problem of compress Navier-Stokes equations with general initial data

In this paper,we study the stability of solutions of the Cauchy problem for 1-D compressible NarvierStokes equations with general initial data.The asymptotic limit of solution is found,under some conditions.The results in this paper imply the case that the limit function of solution as t →∞ is a viscous contact wave in the sense,which approximates the contact discontinuity on any finite-time interval as the heat conduction coefficients toward zero.As a by-product,the decay rates of the solution for the fast diffusion equations are also obtained.The proofs are based on the elementary energy method and the study of asymptotic behavior of the solution to the fast diffusion equation.     

一类非线性广义Euler-Poisson-Darboux方程整体解的渐近性质

J. L Lions and W. A. Stranss [1] have proved the existence of a global solution of the initial boundary value problem for nonlinear generalized Euler-Possion-Darboux equation. In this paper we are going to investigate the asymptotic behavior of the global solution by a difference inequality.