Energy decay estimates for the dissipative wave equation with space–time dependent potential

We study the asymptotic behavior of solutions of dissipative wave equations with space–time‐dependent potential. When the potential is only time‐dependent, Fourier analysis is a useful tool to derive sharp decay estimates for solutions. When the potential is only space‐dependent, a powerful technique has been developed by Todorova and Yordanov to capture the exact decay of solutions. The presence of a space–time‐dependent potential, as in our case, requires modifications of this technique. We find the energy decay and decay of the L² norm of solutions in the case of space–time‐dependent potential. Copyright © 2010 John Wiley & Sons, Ltd.     

三维空间中半线性波动方程组的整体解和自相似解

本文证明了一类半线性波动方程组Cauchy问题整体解的存在唯一性．特别地,证明了自相似解的存在唯一性．同时还得到了渐近自相似解．     

The influence of oscillations on global existence for a class of semi‐linear wave equations

The goal of this paper is to study the global existence of small data solutions to the Cauchy problem for the nonlinear wave equation In particular we are interested in statements for the 1D case. We will explain how the interplay between the increasing and oscillating behavior of the coefficient will influence global existence of small data solutions. Copyright © 2011 John Wiley & Sons, Ltd.     

含奇异项的半线性抛物方程组的Cauchy问题

本文考察含奇异项的半线性抛物方程组的Ｃａｕｃｈｙ问题,算出了该问题的猝灭临界指标和猝灭临界维数．     

Large time behavior of solutions to semilinear systems of wave equations

This paper is concerned with the initial value problem for semilinear systems of wave equations. First we show a global existence result for small amplitude solutions to the systems. Then we study asymptotic behavior of the global solution. We underline that ``modified' free profiles are obtained for all global solutions to the systems even in the case where the free profile might not exist. Moreover, we prove non–existence of any free profiles for the global solution in some cases where the effect of the nonlinearity is strong enough. The first author was partially supported by Grant-in-Aid for Science Research (14740114), JSPS.     

Existence for Semilinear Wave Equations with Low Regularity

In this paper, we study how much regularity of initial data is needed to ensure existence of a local solution to the following semilinear wave equations utt-Δu=F(u, Du), u(0, x)=f(x)∈HS,(?)tu(0, x)=g(x)∈HS-1, where F is quadratic in Du with D = ((?)t,(?)x1,…,(?)xn). We proved that the range of s is s≥n 1/2 δ, respectively, withδ>1/4 if n = 2, andδ>0 if n = 3, andδ≥0 if n≥4. Which is consistent with Lindblad's counterexamples [3] for n = 3, and the main ingredient is the use of the Strichartz estimates and the refinement of these.     

Semilinear structural damped waves

We study the Cauchy problem for the semilinear structural damped wave equation with source term with σ ∈ (0,1] in space dimension n ≥ 2 and with a positive constant μ. We are interested in the influence of σ on the critical exponent p_crit in | f(u) | ≈ | u | ^p. This critical exponent is the threshold between global existence in time of small data solutions and blow‐up behavior for some suitable range of p. Our results are optimal for σ = 1 ∕ 2. Copyright © 2013 John Wiley & Sons, Ltd.     

Periodic problem for the nonlinear damped wave equation with convective nonlinearity

We study the nonlinear damped wave equation with a linear pumping and a convective nonlinearity. We consider the solutions, which satisfy the periodic boundary conditions. Our aim is to prove global existence of solutions to the periodic problem for the nonlinear damped wave equation by applying the energy-type estimates and estimates for the Green operator. Moreover, we study the asymptotic profile of global solutions.     

Global attractors for the viscous hyperelastic‐rod wave equation

We consider the viscous hyperelastic‐rod wave equation subject to an external force, where the viscous term is given by second order differential operator in divergence form. Under some mild assumptions on the viscous term, first, we establish the global well‐posedness in both the periodic case and the case of the whole line, afterwards, we show the existence of global attractors for the two cases, respectively. Copyright © 2012 John Wiley & Sons, Ltd.     

一类非线性Schr(o)dinger方程的整体小解

本文研究带有非线性项|u|~pu的高阶非线性Schr(?)dinger方程的Cauchy问题．对于p的某一取值范围。我们证明了此问题整体解的存在唯一性,并得到了解关于初值的连续依赖性及解具有较强的衰减估计。     

Cauchy problem for quasi-linear wave equations with nonlinear damping and source terms

The paper studies the existence and non-existence of global weak solutions to the Cauchy problem for a class of quasi-linear wave equations with nonlinear damping and source terms. It proves that when α?max{m,p}, where m+1, α+1 and p+1 are, respectively, the growth orders of the nonlinear strain terms, the nonlinear damping term and the source term, under rather mild conditions on initial data, the Cauchy problem admits a global weak solution. Especially in the case of space dimension N=1, the weak solutions are regularized and so generalized and classical solution both prove to be unique. On the other hand, if 0?α<1, and the initial energy is negative, then under certain opposite conditions, any weak solution of the Cauchy problem blows up in finite time. And an example is shown.     

Long time decay for 2D Klein-Gordon equation

We obtain a dispersive long-time decay in weighted energy norms for solutions of the 2D Klein-Gordon equations. The decay extends the results obtained by Jensen, Kato and Murata for the equations of Schrödinger's type by the spectral approach. For the proof we modify the approach to make it applicable to relativistic equations.     

Cauchy problem for quasi-linear wave equations with viscous damping

The paper studies the global existence and asymptotic behavior of weak solutions to the Cauchy problem for quasi-linear wave equations with viscous damping. It proves that when pmax{m,α}, where m+1, α+1 and p+1 are, respectively, the growth orders of the nonlinear strain terms, the nonlinear damping term and the source term, the Cauchy problem admits a global weak solution, which decays to zero according to the rate of polynomial as t→∞, as long as the initial data are taken in a certain potential well and the initial energy satisfies a bounded condition. Especially in the case of space dimension N=1, the solutions are regularized and so generalized and classical solution both prove to be unique. Comparison of the results with previous ones shows that there exist clear boundaries similar to thresholds among the growth orders of the nonlinear terms, the states of the initial energy and the existence, asymptotic behavior and nonexistence of global solutions of the Cauchy problem.