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GLOBAL CLASSICAL SOLUTIONS OF SEMILINEAR WAVE EQUATIONS ON R^(3)×T WITH CUBIC NONLINEARITIES
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作者 陶飞 《Acta Mathematica Scientia》 SCIE CSCD 2024年第1期115-128,共14页
In this paper,we establish global classical solutions of semilinear wave equations with small compact supported initial data posed on the product space R^(3)×T.The semilinear nonlinearity is assumed to be of the ... In this paper,we establish global classical solutions of semilinear wave equations with small compact supported initial data posed on the product space R^(3)×T.The semilinear nonlinearity is assumed to be of the cubic form.The main ingredient here is the establishment of the L^(2)-L^(∞)decay estimates and the energy estimates for the linear problem,which are adapted to the wave equation on the product space.The proof is based on the Fourier mode decomposition of the solution with respect to the periodic direction,the scaling technique,and the combination of the decay estimates and the energy estimates. 展开更多
关键词 semilinear wave equation product space decay estimate energy estimate global solution
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Existence for Semilinear Wave Equations with Low Regularity
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作者 YANG Ning YANG Han 《Chinese Quarterly Journal of Mathematics》 CSCD 北大核心 2006年第1期49-56,共8页
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-△u=F(u,Du) u(0,x)=f(x)∈H^s,δtu(0,x)=g(x)∈H^s-1... 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-△u=F(u,Du) u(0,x)=f(x)∈H^s,δtu(0,x)=g(x)∈H^s-1,where F is quadratic in Du with D = (δr, δx1,…, δxn).We proved that the range of s is s ≥n+1/2 + δ, respectively, with δ 〉 1/4 if n = 2, and δ 〉 0 if n = 3, and δ ≥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. 展开更多
关键词 semilinear wave equations local existence low regularity
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Convergence on Finite Difference Solution for Semilinear Wave Equation in One Space Variable
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作者 鲁百年 房少梅 《Chinese Quarterly Journal of Mathematics》 CSCD 1997年第3期35-40, ,共6页
In this paper,the convergence and stability of the ’Leap-frog’ finite difference scheme for the semilinear wave equation are proved by using of the bounded extensive method under more generalized condition for the n... In this paper,the convergence and stability of the ’Leap-frog’ finite difference scheme for the semilinear wave equation are proved by using of the bounded extensive method under more generalized condition for the nonlinear term. The more complex standard a priori estimates are avoided so that the theoretical results are complemented for the scheme which was presented by Perring and Skyrne (1962). 展开更多
关键词 semilinear wave equation Leap-frog finite difference scheme convergence and stability
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The Local Existence for Semilinear Wave Equations
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作者 Yang Han(杨晗) +1 位作者 Lai Shaoyong(赖绍永) 《Journal of Southwest Jiaotong University(English Edition)》 2002年第2期134-138,共5页
The Cauchy problem for the semilinear wave equation has been studied and results show that the problem is locally well-posed in Hs ( Rn ) for s > max [ 0, n/2 - 1]. We extend the results by Lindblad in R3 to R2 and... The Cauchy problem for the semilinear wave equation has been studied and results show that the problem is locally well-posed in Hs ( Rn ) for s > max [ 0, n/2 - 1]. We extend the results by Lindblad in R3 to R2 and R1. The methods used in this paper are different from those of Lindblad and also the methods are more simple. 展开更多
关键词 semilinear wave equations Cauchy problem
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Life-Span of Classical Solutions to a Semilinear Wave Equation with Time-Deppendent Damping
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作者 GUO Fei LIANG Jinling XIAO Changwang 《Journal of Partial Differential Equations》 CSCD 2023年第3期235-261,共27页
This paper is concerned with the Cauchy problem for a semilinear wave equation with a time-dependent damping.In case that the space dimension n=1 and the nonlinear power is bigger than 2,the life-span T(ε)and global ... This paper is concerned with the Cauchy problem for a semilinear wave equation with a time-dependent damping.In case that the space dimension n=1 and the nonlinear power is bigger than 2,the life-span T(ε)and global existence of the classical solution to the problem has been investigated in a unified way.More precisely,with respect to different values of an index K,which depends on the time-dependent damping and the nonlinear term,the life-span T(ε)can be estimated below byε-p/1-k,e^(ε)-p or+∞,where e is the scale of the compact support of the initial data. 展开更多
关键词 semilinear wave equation time-dependent damping life-span global iteration method.
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The Lifespan of Smooth Solutions to Semilinear Wave Equations in Schwarzschild Space-Time
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作者 LOU Qiong LUO Shaoying 《Journal of Partial Differential Equations》 CSCD 2023年第4期404-413,共10页
This paper considers the Cauchy problem of the semilinear wave equations with small initial data in the Schwarzschild space-time,□_(g)u=|ut|^(P),where g denotes the Schwarzschild metric.When 1<p<2 and the initi... This paper considers the Cauchy problem of the semilinear wave equations with small initial data in the Schwarzschild space-time,□_(g)u=|ut|^(P),where g denotes the Schwarzschild metric.When 1<p<2 and the initial data are supported far away from the black hole,we can prove that the lifespan of the spherically symmetric solu-tion obtains the same order as the semilinear wave equation evolving in the Minkowski space-time by introducing an auxiliary function. 展开更多
关键词 semilinear wave equations Schwarzschild spacetime BLOW-UP LIFESPAN
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Blow Up of Solutions to Semilinear Wave Equations with Critical Exponent in High Dimensions 被引量:22
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作者 Yi ZHOU 《Chinese Annals of Mathematics,Series B》 SCIE CSCD 2007年第2期205-212,共8页
In this paper, the author considers equations with critical exponent in n ≥4 space tions on the initial data, it is proved that there small the initial data are. the Cauchy problem for semilinear wave dimensions. Und... In this paper, the author considers equations with critical exponent in n ≥4 space tions on the initial data, it is proved that there small the initial data are. the Cauchy problem for semilinear wave dimensions. Under some positivity condican be no global solutions no matter how 展开更多
关键词 semilinear wave equation Critical exponent Cauchy problem Blow up
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Blow Up of Solutions to One Dimensional Initial-Boundary Value Problems for Semilinear Wave Equations with Variable Coefficients 被引量:5
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作者 HAN Wei 《Journal of Partial Differential Equations》 2013年第2期138-150,共13页
This paper is devoted to studying the following initial-boundary value prob- lem for one-dimensional semilinear wave equations with variable coefficients and with subcritical exponent:We wili establish a blowup resul... This paper is devoted to studying the following initial-boundary value prob- lem for one-dimensional semilinear wave equations with variable coefficients and with subcritical exponent:We wili 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. 展开更多
关键词 semilinear wave equation critical exponent initial-boundary value problem BLOWUP lifespan.
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RAPID EXACT CONTROLLABILITY OFTHE SEMILINEAR WAVE EQUATION 被引量:1
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作者 ZHANG Xu(Institute of Mathematics, Fudan University, Shanghai 200433, China.)(Project supported by the Science Foundation of the Ministry of Education of China and the Youth Science Foundation of Shanghai’s Universities.) 《Chinese Annals of Mathematics,Series B》 SCIE CSCD 1999年第3期377-384,共8页
In this paper the semilinear wave equation with homogeneous Dirichlet boundary condition having a locally distributed controller is considered, and the rapid exact controllability of this system is obtained by changin... In this paper the semilinear wave equation with homogeneous Dirichlet boundary condition having a locally distributed controller is considered, and the rapid exact controllability of this system is obtained by changing the shape and/or the location of the controller under proper conditions. For this purpose, the author derives an (rapid) observability inequality for wave equations with linear time-variant potential by means of the energy estimate. The main difference of the method from the previous ones is that any unique continuation property of the corresponding linear time-variant wave equations is not needed. 展开更多
关键词 Rapid exact controllability semilinear wave equation Changing controller Energy estimate
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THE ASYMPTOTIC THEORY OF INITIAL VALUE PROBLEMS FOR SEMILINEAR PERTURBED WAVE EQUATIONS IN TWO SPACE DIMENSIONS
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作者 赖绍永 胡青龙 《Applied Mathematics and Mechanics(English Edition)》 SCIE EI 2003年第1期82-91,共10页
The asymptotic theory of initial value problems for semilinear wave equations in two space dimensions was dealt with.The well_posedness and vaildity of formal approximations on a long time scale were discussed in the ... The asymptotic theory of initial value problems for semilinear wave equations in two space dimensions was dealt with.The well_posedness and vaildity of formal approximations on a long time scale were discussed in the twice continuous classical space. These results describe the behavior of long time existence for the validity of formal approximations. And an application of the asymptotic theory is given to analyze a special wave equation in two space dimensions. 展开更多
关键词 semilinear wave equation ASYMPTOTICS long time scale application
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THE GLOBAL AND LOCAL C^(2)-SOLUTIONS FOR THE WAVE EQUATION □u = G(u_t, Du) IN THREE SPACE DIMENSIONS 被引量:2
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作者 赖绍永 《Acta Mathematica Scientia》 SCIE CSCD 2000年第4期495-503,共9页
Kunio Hidano[4] has shown that the global and local C2-solutions for semilinear wave equations with spherical symmetry in three space dimensions. This paper studies the global and local C2-solutions for the semilinea... Kunio Hidano[4] has shown that the global and local C2-solutions for semilinear wave equations with spherical symmetry in three space dimensions. This paper studies the global and local C2-solutions for the semilinear wave equations without spherical symmetry in three space dimensions. A problem put forward by Hiroyuki Takamura[2] is partially answered. 展开更多
关键词 semilinear wave equations global and local C^(2)-solutions three space dimensions
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Global Existence and Blow-up for Semilinear Wave Equations with Variable Coefficients 被引量:1
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作者 Qian LEI Han YANG 《Chinese Annals of Mathematics,Series B》 SCIE CSCD 2018年第4期643-664,共22页
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 nonlin... 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, L^2 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. 展开更多
关键词 semilinear wave equations Global existence Energy decay L^2 and L^p+1 estimates Blow up
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Oscillatory solution to semilinear dissipative wave equations
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《Chinese Science Bulletin》 SCIE CAS 1998年第9期787-788,共2页
关键词 wave MATH Oscillatory solution to semilinear dissipative wave equations
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Galerkin Finite Element Approximation for Semilinear Stochastic Time-Tempered Fractional Wave Equations with Multiplicative Gaussian Noise and Additive Fractional Gaussian Noise
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作者 Yajing Li Yejuan Wang +1 位作者 Weihua Deng Daxin Nie 《Numerical Mathematics(Theory,Methods and Applications)》 SCIE CSCD 2022年第4期1063-1098,共36页
To model wave propagation in inhomogeneous media with frequency dependent power-law attenuation,it is needed to use the fractional powers of symmetric coercive elliptic operators in space and the Caputo tempered fract... To model wave propagation in inhomogeneous media with frequency dependent power-law attenuation,it is needed to use the fractional powers of symmetric coercive elliptic operators in space and the Caputo tempered fractional derivative in time.The model studied in this paper is semilinear stochastic space-time fractional wave equations driven by infinite dimensional multiplicative Gaussian noise and additive fractional Gaussian noise,because of the potential fluctuations of the external sources.The purpose of this work is to discuss the Galerkin finite element approximation for the semilinear stochastic fractional wave equation.First,the space-time multiplicative Gaussian noise and additive fractional Gaussian noise are discretized,which results in a regularized stochastic fractional wave equation while introducing a modeling error in the mean-square sense.We further present a complete regularity theory for the regularized equation.A standard finite element approximation is used for the spatial operator,and a mean-square priori estimates for the modeling error and the approximation error to the solution of the regularized problem are established.Finally,numerical experiments are performed to confirm the theoretical analysis. 展开更多
关键词 Galerkin finite element method semilinear stochastic time-tempered fractional wave equation fractional Laplacian multiplicative Gaussian noise additive fractional Gaussian noise
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ON THE EQUATION□Φ=|▽Φ|~2 IN FOUR SPACE DIMENSIONS 被引量:2
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作者 ZHOU YI Institute of Mathematics, Fudan University, Shanghai 200433, China. 《Chinese Annals of Mathematics,Series B》 SCIE CSCD 2003年第3期293-302,共10页
This paper considers the following Cauchy problem for semilinear wave equations in n space dimensionswhere A is the wave operator, F is quadratic in (?) with (?) = ( ).The minimal value of s is determined such that th... This paper considers the following Cauchy problem for semilinear wave equations in n space dimensionswhere A is the wave operator, F is quadratic in (?) with (?) = ( ).The minimal value of s is determined such that the above Cauchy problem is locally well-posed in H8. It turns out that for the general equation s must satisfyThis is due to Ponce and Sideris (when n = 3) and Tataru (when n≥5). The purpose of this paper is to supplement with a proof in the case n = 2,4. 展开更多
关键词 semilinear wave equation Cauchy problem Low regularity solution
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LOCAL WELL-POSEDNESS AND ILL-POSEDNESS ON THE EQUATION OF TYPE □u= u^k(u)~α 被引量:1
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作者 FANGDAOYUAN WANGCHENGBO 《Chinese Annals of Mathematics,Series B》 SCIE CSCD 2005年第3期361-378,共18页
This paper undertakes a systematic treatment of the low regularity local well-posedness and ill-posedness theory in H3 and Hs for semilinear wave equations with polynomial nonlinearity in u and (?)u. This ill-posed re... This paper undertakes a systematic treatment of the low regularity local well-posedness and ill-posedness theory in H3 and Hs for semilinear wave equations with polynomial nonlinearity in u and (?)u. This ill-posed result concerns the focusing type equations with nonlinearity on u and (?)tu. 展开更多
关键词 semilinear wave equation Low regularity Local well-posedness ILL-POSEDNESS
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Existence and Asymptotic Behavior of Radially Symmetric Solutions to a Semilinear Hyperbolic System in Odd Space Dimensions
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作者 Hideo KUBO Kji KUBOTA 《Chinese Annals of Mathematics,Series B》 SCIE CSCD 2006年第5期507-538,共32页
This paper is concerned with a class of semilinear hyperbolic systems in odd space dimensions. Our main aim is to prove the existence of a small amplitude solution which is asymptotic to the free solution as t →-∞ i... This paper is concerned with a class of semilinear hyperbolic systems in odd space dimensions. Our main aim is to prove the existence of a small amplitude solution which is asymptotic to the free solution as t →-∞ in the energy norm, and to show it has a free profile as t →+∞. Our approach is based on the work of [11]. Namely we use a weighted L^∞ norm to get suitable a priori estimates. This can be done by restricting our attention to radially symmetric solutions. Corresponding initial value problem is also considered in an analogous framework. Besides, we give an extended result of [14] for three space dimensional case in Section 5, which is prepared independently of the other parts of the paper. 展开更多
关键词 semilinear wave equations Asymptotic behavior Radially symmetric solution
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