Scattering for Semilinear Wave Equation with Small Data in High Space Dimensions

In this paper we study the scattering theory for the semilincar wave equation u_{tt} - Δu = F(u(t, x), Du(t, x)) in R^n (n ≥ 4) with smooth and small data. We show that the scattering operator exists for the nonlinear term F = F(λ) = O(|λ|^{1, α}), where α is an integer and satisfies α ≥ 2, n = 4; α ≥ I, n ≥ 5.     

半线性波方程的一个反问题

该文利用Banach不动点原理讨论了一个半线性波方程的反问题,文中给出了该问题解的存在性、唯一性和稳定性．     

具有阻尼的半线性波动方程解的衰减估计

研究具有阻尼的半线性波动方程的初边值问题u_(tt)-△u+βu_t=|u|~(p-1)u,x∈Ω,t>0u(x,0)=u_0(x),u_t(x,0)=u_1(x),x∈Ωu|_((?)Ω)=0,t≥0其中γ为正常数,Ω■R~n为有界域,当n≥3时,1相似文献   

渐近欧氏流形上带有阻尼和位势项的波动方程的生命跨度估*

本文研究了渐近欧氏流形上带有阻尼和位势的半线性波动方程的有限时间破裂以及解的生命跨度上界估计,其半线性项是形如c₁ |u_t|^p + c₂ |u|^p的混合项. 该问题与Strauss猜测和Glassey猜测紧密相关.     

半线性波动方程的自相似解

研究了具有非线性项|u|~αu的半线性波动方程的Cauclly问题,利用仿积分解及交换子估计等技术,证明了当α为一般的实数且满足一定的限制时,Cauchy问题自相似解的存在性。本文的结果回答了Planchon在其工作中所遗留的问题。     

Interaction of Three Conormal Waves for Semilinear Wave Equations

In this paper, we consider the interaction of triple of conormal waves with different singularities for semilinear wave equations. We will show that if three characteristic hyperplanes carrying different conormal singularities intersect transversally at the origin, then the solution will be conormal with respect to the three hyperplanes, and a new singularity will be produced on the surface of the light cone at later times. We can also prove here that the strength of the new singularity will be dependent only on the weakest one and strongest one in the three hyperplanes.     

Radiation Fields on Asymptotically Euclidean Manifolds

Abstract

F.G. Friedlander introduced the notion of radiation fields for asymptotically Euclidean manifolds. Here we answer some of the questions he proposed and apply the results to give a unitary translation representation of the wave group, and to obtain the scattering matrix for such manifolds. We also obtain a support theorem for the radiation fields.     

The Einstein-Scalar Field Constraints on Asymptotically Euclidean Manifolds

By using the conformal method, solutions of the Einstein-scalar field gravitational constraint equations are obtained. Handling scalar fields is a bit more challenging than handling matter fields such as fluids, Maxwell fields or Yang-Mills fields, because the scalar field introduces three extra terms into the Lichnerowicz equation, rather than just one. The proofs are constructive and allow for arbitrary dimension (> 2) as well as low regularity initial data.     

Blow Up of Solutions to Semilinear Wave Equations with Critical Exponent in High Dimensions

In this paper, the author considers the Cauchy problem for semilinear wave equations with critical exponent in n≥4 space dimensions. Under some positivity conditions on the initial data, it is proved that there can be no global solutions no matter how small the initial data are.     

一类半线性波动方程的Sobolev指数

Gaustavo Ponce与Thomas C.Sideris猜测：对一些具有特殊非线性项的半线性波动方程,如utt-△u=u^k(Du)^αx∈R^n,k∈Z^ ,ρ＝│α│≥2,其中Sobloev指数会在[n/2,n/2 1]中,他们在x∈R^3时回答了这一问题,本文在R^n(n≥4）中得到了半线性波动方程utt-△u=u^k(Du)^α（x∈R^n,k∈R^n,k∈Z^ ,p=│α│≥2）的Sobolev指数为max{n/2,(n/2-1)1-3/l-1 2},此数确实在区间[n/2,n/2 1]中,特别当ρ≤n-1时,我们得到了此半线性波动方程的Sobolev指数为n/2。     

Blow Up of Solutions to One Dimensional Initial-boundary Value Problems for Semilinear Wave Equations with Variable Coefficients

This paper is devoted to studying the following initial-boundary value problemfor one-dimensional semilinearwave equationswith variable coefficients andwith subcritical exponent: $u_{tt}-∂_x(a(x)∂_xu)=|u|^p, x > 0, t > 0, n=1,$ where $u=u(x,t)$ is a real-valued scalar unknown function in $[0,+∞)×[0,+∞)$, here a(x) is a smooth real-valued function of the variable $x∈(0,+∞)$. The exponents p satisfies $1 < p < +∞$ in (0.1). It is well-known that the number $p_c(1)=+∞$ is the critical exponent of the semilinear wave equation (0.1) in one space dimension (see for e.g., [1]). We will establish a blowup result for the above initial-boundary value problem, it is proved that there can be no global solutions no matter how small the initial data are, and also we give the lifespan estimate of solutions for above problem.     

半线性抛物方程支配系统的最优性条件

本文讨论了控制变量含于高阶导数项系数中的抛物型偏微分方程支配系统的最优控制问题,给出了最优控制的必要条件．     

Asymptotic Properties for a Semilinear Edge-Degenerate Parabolic Equation

The author deals with a semi-linear edge-degenerate parabolic equation, and proves that the solution increases exponentially under the initial energy J(u₀) ≤ d, where d is the mountain-pass level. Moreover, the author estimates the blow-up time and the blow-up rate for the solution under J(u₀) < 0.     

Convergence on Finite Difference Solution for Semilinear Wave Equation in One Space Variable

61.IntroductionThesemilinearwaveequationareoftenappearinthestudyofphysical,chemical,mechani-cal,biologicalandotherproblems,onespecialexampleisthewell-knownKlein-Gordonequa-tlonutt-u..=f(u)(l)whichtakesanimportantrolesinthestudyofSoliton.Choicedifferentnonlineartermf,thevariousstandardequations['jaregotsuchastakingf(u)=sin(u),then(l)asaSine-Gordonequationandtakingf(u)=u-u3lthen(1)asagi-Wequationwhichisanimportantmodell.insolidphysicsandhighenergyphysics['j.Moreover,takingf(u)=f[sin(u) isin(7)…     

二维空间中半线性波动方程渐近理论的一个新结果

研究二维空间中一类半线性波动方程初值问题解的渐近理论 ,在 C2 (J× R2 ) (J ={ t|0≤ t≤ O(|ε|-1 2 -k(p-1 ) ) ,ε→ 0 ,0 <2 - k(p - 1 ) <1 ,0 相似文献   

关于黎曼流形上的一个扩散方程的一些梯度估计

我们给出关于黎曼流形上的扩散方程θtu=Δu-▽φ·▽u（这里φ是一个C^2函数）的一些梯度估计。这推广了R.Hamilton和Qi S.Zhang关于热方程的一些梯度估计。     

Global Existence and Blow-up for Semilinear Wave Equations with Variable Coefficients

The authors consider the critical exponent problem for the variable coefficients wave equation with a space dependent potential and source term. For sufficiently small data with compact support, if the power of nonlinearity is larger than the expected exponent, it is proved that there exists a global solution. Furthermore, the precise decay estimates for the energy, L₂ and L^p+1 norms of solutions are also established. In addition, the blow-up of the solutions is proved for arbitrary initial data with compact support when the power of nonlinearity is less than some constant.     

浅水波方程和Klein-Gordon方程新的精确行波解

采用了一种新的方法来求解浅水波方程和Klein-Gordon的行波解.在该方法下,Klein-Gordon方程和浅水波方程都得到了其精确的周期孤立波解,从而该方法的有效性得到了验证.