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
Rayleigh waves in the two-dimensional half-plane linear elasticity were investigated. First, the solutions of the equations of motion of linear elasticity were generalized, which has been studied by Lord Rayleigh. The...Rayleigh waves in the two-dimensional half-plane linear elasticity were investigated. First, the solutions of the equations of motion of linear elasticity were generalized, which has been studied by Lord Rayleigh. Then the explicit formula with different decay rates was also obtained. Secondly, by the free boundary conditions, the secular equation is derived. It is shown that some Rayleigh waves with different decay rates does exist.展开更多
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
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
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
The authors study the Cauchy problem for the semi-linear damped wave equation utt-△u＋b（t）ut=f（u）,u（0,χ）=u0（χ）,ut（0,χ）=u1（χ） in any space dimension n ≥ 1. It is assumed that the time-dependent dampin...The authors study the Cauchy problem for the semi-linear damped wave equation utt-△u＋b（t）ut=f（u）,u（0,χ）=u0（χ）,ut（0,χ）=u1（χ） in any space dimension n ≥ 1. It is assumed that the time-dependent damping term b（t）〉 0 is effective, and in particular tb（t） →∞ as t →∞. The global existence of small energy data solutions for｜f（u）｜≈｜u｜^p in the supercritical case of p 〉 1＋ 2/n and p ≤n/n-2 for n ≥ 3 is proved.展开更多
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
The authors prove the global exact boundary controllability for the cubic semilinear wave equation in three space dimensions,subject to Dirichlet,Neumann,or any other kind of boundary controls which result in the well...The authors prove the global exact boundary controllability for the cubic semilinear wave equation in three space dimensions,subject to Dirichlet,Neumann,or any other kind of boundary controls which result in the well-posedness of the corresponding initial-boundary value problem.The exponential decay of energy is first established for the cubic semi-linear wave equation with some boundary condition by the multiplier method,which reduces the global exact boundary controllability problem to a local one.The proof is carried out in line with [2,15].Then a constructive method that has been developed in [13] is used to study the local problem.Especially when the region is star-complemented,it is obtained that the control function only need to be applied on a relatively open subset of the boundary.For the cubic Klein-Gordon equation,similar results of the global exact boundary controllability are proved by such an idea.展开更多
In this paper, we consider the exterior problem of the critical semilinear wave equation in three space dimensions with variable coefficients and prove the global existence of smooth solutions. As in the constant coef...In this paper, we consider the exterior problem of the critical semilinear wave equation in three space dimensions with variable coefficients and prove the global existence of smooth solutions. As in the constant coefficients case, we show that the energy cannot concentrate at any point (t, x) ∈ (0, ∞) ×Ω. For that purpose, following Ibrahim and Majdoub's paper in 2003, we use a geometric multiplier similar to the well-known Morawetz multiplier used in the constant coefficients case. We then use the comparison theorem from Riemannian geometry to estimate the error terms. Finally, using the Strichartz inequality as in Smith and Sogge's paper in 1995, we confirm the global existence.展开更多
This paper concerns the global existence of solutions to the semi-linear wave equation utt-△u = G(u) in five space dimensions, where G(u) -|u|p with p > 3+17^(1/17). We used the classical iteration method and tech...This paper concerns the global existence of solutions to the semi-linear wave equation utt-△u = G(u) in five space dimensions, where G(u) -|u|p with p > 3+17^(1/17). We used the classical iteration method and technique estimates to show that a classical global solution exists for the radially symmetric equations with small and compact supported initial data.展开更多
文摘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).
基金National Natural Science Foundation of China(No. 10371073)
文摘Rayleigh waves in the two-dimensional half-plane linear elasticity were investigated. First, the solutions of the equations of motion of linear elasticity were generalized, which has been studied by Lord Rayleigh. Then the explicit formula with different decay rates was also obtained. Secondly, by the free boundary conditions, the secular equation is derived. It is shown that some Rayleigh waves with different decay rates does exist.
基金Project supported by the National Natural Science Foundation of China (No. 10225102)the 973 Project of the Ministry of Science and Technology of China.
文摘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
基金Project supported by the 973 Project of the National Natural Science Foundation of China,the Key Teachers Program and the Doctoral Program Foundation ofthe Miistry of Education of China.
文摘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.
基金Project supported by the National Natural Science Foundation of China (No.10271108).
文摘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.
基金Project supported by a grant of DFG (Deutsche Forschungsgemeinschaft) for the research project "Influence of time-dependent coefficients on semi-linear wave models" (No. RE 961/17-1)
文摘The authors study the Cauchy problem for the semi-linear damped wave equation utt-△u＋b（t）ut=f（u）,u（0,χ）=u0（χ）,ut（0,χ）=u1（χ） in any space dimension n ≥ 1. It is assumed that the time-dependent damping term b（t）〉 0 is effective, and in particular tb（t） →∞ as t →∞. The global existence of small energy data solutions for｜f（u）｜≈｜u｜^p in the supercritical case of p 〉 1＋ 2/n and p ≤n/n-2 for n ≥ 3 is proved.
基金Project supported by Grant-in-Aid for Science Research (No.12740105, No.14204011), JSPS.
文摘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.
基金supported by the National Natural Science Foundation of China (No. 10728101)the 973 Project ofthe Ministry of Science and Technology of China+1 种基金the Doctoral Program Foundation of the Ministry of Ed-ucation of Chinathe "111" Project and the Postdoctoral Science Foundation of China (No. 20070410160)
文摘The authors prove the global exact boundary controllability for the cubic semilinear wave equation in three space dimensions,subject to Dirichlet,Neumann,or any other kind of boundary controls which result in the well-posedness of the corresponding initial-boundary value problem.The exponential decay of energy is first established for the cubic semi-linear wave equation with some boundary condition by the multiplier method,which reduces the global exact boundary controllability problem to a local one.The proof is carried out in line with [2,15].Then a constructive method that has been developed in [13] is used to study the local problem.Especially when the region is star-complemented,it is obtained that the control function only need to be applied on a relatively open subset of the boundary.For the cubic Klein-Gordon equation,similar results of the global exact boundary controllability are proved by such an idea.
基金supported by National Natural Science Foundation of China (Grant No. 10728101)National Basic Research Program of China+3 种基金Doctoral Program Foundation of the Ministry of Education of Chinathe "111" projectSGST 09DZ2272900supported by the Outstanding Doctoral Science Foundation Program of Hehai University
文摘In this paper, we consider the exterior problem of the critical semilinear wave equation in three space dimensions with variable coefficients and prove the global existence of smooth solutions. As in the constant coefficients case, we show that the energy cannot concentrate at any point (t, x) ∈ (0, ∞) ×Ω. For that purpose, following Ibrahim and Majdoub's paper in 2003, we use a geometric multiplier similar to the well-known Morawetz multiplier used in the constant coefficients case. We then use the comparison theorem from Riemannian geometry to estimate the error terms. Finally, using the Strichartz inequality as in Smith and Sogge's paper in 1995, we confirm the global existence.
文摘This paper concerns the global existence of solutions to the semi-linear wave equation utt-△u = G(u) in five space dimensions, where G(u) -|u|p with p > 3+17^(1/17). We used the classical iteration method and technique estimates to show that a classical global solution exists for the radially symmetric equations with small and compact supported initial data.
