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.     

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.     

Global Existence for Some 4-D Quasilinear Wave Equations with Low Regularity

In this paper, we prove the global existence for some 4-D quasilinear wave equations with small, radial data in H~3× H~2. The main idea is to exploit local energy estimates with variable coefficients, together with the trace estimates.     

Strichartz Estimates for Wave Equations with Coefficients of Sobolev Regularity

Wave packet techniques provide an effective method for proving Strichartz estimates on solutions to wave equations whose coefficients are not smooth. We use such methods to show that the existing results for C ^{1, 1} and C ^{1, α} coefficients can be improved when the coefficients of the wave operator lie in a Sobolev space of sufficiently high order.     

外区域上半线性波动方程解的整体存在性

该文研究带耗散项的线性和半线性波动方程外问题. 首先利用一个Sobolev型不等式得到了线性耗散波动方程在外区域上的整体能量衰减估计, 此结果用来证明非线性项为|u|^p (2+) 的半线性波动方程解的整体存在性. 为此, 该文主要研究N维(3≤ N≤7)外区域上球对称解的情形.     

五维空间中半线性波动方程整体解的存在性定理

本文研究五维空间中半线性波动方程u_tt-△u=G（u）整体解的存在性,其中G（u）～|u|^p并且p>（3+(17)^1/2)/4.利用经典的迭代方法证明了:如果初始值很小并且紧支的,径向对称方程有一个经典整体解.     

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.     

The Semilinear Wave Equation on Asymptotically Euclidean Manifolds

We consider the quadratically semilinear wave equation on (? ^d , 𝔤), d ≥ 3. The metric 𝔤 is non-trapping and approaches the Euclidean metric like ?x?^?ρ. Using Mourre estimates and the Kato theory of smoothness, we obtain, for ρ > 0, a Keel–Smith–Sogge type inequality for the linear equation. Thanks to this estimate, we prove long time existence for the nonlinear problem with small initial data for ρ ≥ 1. Long time existence means that, for all n > 0, the life time of the solution is a least δ^?n , where δ is the size of the initial data in some appropriate Sobolev space. Moreover, for d ≥ 4 and ρ > 1, we obtain global existence for small 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.     

Strichartz estimates in the hyperbolic space and global existence for the semilinear wave equation

The aim of this article is twofold. First we consider the wave equation in the hyperbolic space and obtain the counterparts of the Strichartz type estimates in this context. Next we examine the relationship between semilinear hyperbolic equations in the Minkowski space and in the hyperbolic space. This leads to a simple proof of the recent result of Georgiev, Lindblad and Sogge on global existence for solutions to semilinear hyperbolic problems with small data. Shifting the space-time Strichartz estimates from the hyperbolic space to the Minkowski space yields weighted Strichartz estimates in which extend the ones of Georgiev, Lindblad, and Sogge.

     

Nontrivial Solutions for Some Semilinear Elliptic Equations with Critical Sobolev Exponents

Let Ω be a bounded domain in R^4(n ≥ 4) with smooth boundary ∂Ω. We discuss the existence of nontrivial solutions of the Dirichlet problem {- Δu = a(x) |u|^{4/(a-2)}u + λu + g(x, u), \quad x ∈ Ω u = 0, \quad x ∈ ∂Ω where a(x) is a smooth function which is nonnegative on \overline{Ω} and positive somewhere, λ> 0 and λ ∉ σ(-Δ). We weaken the conditions on a(x) that are generally assumed in other papers dealing with this problem.     

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.     

一类半线性热方程整体解的存在性与非存在性

本文研究半线性热方程的初值问题u_t－△u＝u~γ＋cu,（γ＞1）;u（x,0）=（x）非负整体L~P解的存在性与非存在性．首先证明,若C＞0,则不存在非负整体解．而后,对C＜0情形给出了解的整体存在与非存在的充分条件,特别证明了,若P＞（γ一1）或,则当。充分小时存在非负整体L~P解.最后,对系数C和初值（x）得到无穷多个门槛结果．     

一类非线性四阶波动方程整体解的存在性

The initial boundary value problem for the fourth-order wave equation utt △2u u=|t|p-1u is considered.The existence and uniqueness of global weak solutions is obtained by using the Galerkin method and the concept of stable set due to Sattinger.     

关于半线性椭圆型方程爆破解的存在性

文中得到半线性椭圆型方程的爆破问题解的存在性,其中或者是Rn中的有界区域,C3,C4,C5,C6是正常数,并且C5,C3（0,1）．     

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.     

具临界Sobolev-Hardy指数的半线性椭圆方程的非平凡解

本文研究了如下问题：-div（｜x｜β△u）=｜x｜^a｜u｜^2（α,β）-2u＋λ｜x｜σ｜u｜^q-2,x∈Ω,u=0,x∈δΩ,这里Ω∪→R^N是有界光滑区域且0∈Ω，2（α，β）=2（N＋α）/N＋β-2，运用Sobolev-Hardy不等式和山路几何，证明了在一定的条件下方程至少存在一个非平凡解。     

Global Existence for the n-Dimensional Semilinear Tricomi-Type Equations

In this article we investigate the issue of global existence of the solutions of the Cauchy problem for semilinear Tricomi-type equations in ? ⁿ⁺¹, n > 1. We give some sufficient conditions for existence of the global weak solutions. These conditions tie together nonlinearity with the speed of propagation and with the dimension n. We also prove necessity of these (or close) conditions. In fact, we extend these necessity results to the nonlocal semilinear equations.     

Stability and Unconditional Uniqueness of Solutions for Energy Critical Wave Equations in High Dimensions

In this paper we establish a complete local theory for the energy-critical nonlinear wave equation (NLW) in high dimensions ? × ? ^d with d ≥ 6. We prove the stability of solutions under the weak condition that the perturbation of the linear flow is small in certain space-time norms. As a by-product of our stability analysis, we also prove local well-posedness of solutions for which we only assume the smallness of the linear evolution. These results provide essential technical tools that can be applied towards obtaining the extension to high dimensions of the analysis of Kenig and Merle [17 Kenig , C.E. , Merle , F. ( 2008 ). Global well-posedness, scattering and blow-up for the energy critical focusing non-linear wave equation . Acta Math. 201 : 147 – 212 .[Crossref], [Web of Science ®] , [Google Scholar]] of the dynamics of the focusing (NLW) below the energy threshold. By employing refined paraproduct estimates we also prove unconditional uniqueness of solutions for d ≥ 6 in the natural energy class. This extends an earlier result by Planchon [26 Planchon , F. ( 2003 ). On uniqueness for semilinear wave equations . Math. Z. 244 : 587 – 599 .[Web of Science ®] , [Google Scholar]].