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
In this paper,we study the blow-up of solutions to a semilinear double-wave equation with nonlinearity of derivative type.By using the iteration method and the differential inequality techniques,we can get the estimat...In this paper,we study the blow-up of solutions to a semilinear double-wave equation with nonlinearity of derivative type.By using the iteration method and the differential inequality techniques,we can get the estimates of the lifespan and the blow-up of solutions in the subcritical case under some assumptions.展开更多
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
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
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
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
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
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
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Supported by the Natural Science Foundation of China(Grant No.61907010)Innovation Team Project in Colleges and Universities of Guangdong Province(Grant No.2020WCXTD008)Science Foundation of Huashang College Guangdong University of Finance&Economics(Grant No.2020HSDS01)。
文摘In this paper,we study the blow-up of solutions to a semilinear double-wave equation with nonlinearity of derivative type.By using the iteration method and the differential inequality techniques,we can get the estimates of the lifespan and the blow-up of solutions in the subcritical case under some assumptions.
文摘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.
文摘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.
基金supported by the NSF of China(11731007)the Priority Academic Program Development of Jiangsu Higher Education Institutions,and the NSF of Jiangsu Province(BK20181381,BK20221320).
文摘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.
文摘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.
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.
文摘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.
