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 a continuation of recent work by Guo-Xiang-Zheng[10].We deduce the sharp Morrey regularity theory for weak solutions to the fourth order nonhomogeneous Lamm-Rivière equation △^{2}u=△(V▽u)+div(w▽...This paper is a continuation of recent work by Guo-Xiang-Zheng[10].We deduce the sharp Morrey regularity theory for weak solutions to the fourth order nonhomogeneous Lamm-Rivière equation △^{2}u=△(V▽u)+div(w▽u)+(▽ω+F)·▽u+f in B^(4),under the smallest regularity assumptions of V,ω,ω,F,where f belongs to some Morrey spaces.This work was motivated by many geometrical problems such as the flow of biharmonic mappings.Our results deepens the Lp type regularity theory of[10],and generalizes the work of Du,Kang and Wang[4]on a second order problem to our fourth order problems.展开更多
In this paper, under the hypothesis that y is upper bounded, we construct a Lyapunov functional for the multidimensional isentropic compressible magnetohydrodynamic equations and show that the weak solutions decay exp...In this paper, under the hypothesis that y is upper bounded, we construct a Lyapunov functional for the multidimensional isentropic compressible magnetohydrodynamic equations and show that the weak solutions decay exponentially to the equilibrium state in L2 norm. Our result verifies that the method of Daoyuan Fang, Ruizhao Zi and Ting Zhang I1] can be adapted to magnetohydrodynamic equations.展开更多
L^p- L^q decay estimate of solution to Cauchy problem of a linear thermoviscoelastic system is studied. By using a diagonalization argument of frequency analysis, the coupled system will be decoupled micrologically. T...L^p- L^q decay estimate of solution to Cauchy problem of a linear thermoviscoelastic system is studied. By using a diagonalization argument of frequency analysis, the coupled system will be decoupled micrologically. Then with the help of the information of characteristic roots for the coefficient matrix of the system, L^p- L^q decay estimate of parabolic type of solution to the Cauchy problem is obtained.展开更多
This paper investigates the spatial behavior of the solutions of the Stokes equations in a semi-infinite cylinder.We consider four kinds of semi-infinite cylinders with boundary conditions of Dirichlet type.For each t...This paper investigates the spatial behavior of the solutions of the Stokes equations in a semi-infinite cylinder.We consider four kinds of semi-infinite cylinders with boundary conditions of Dirichlet type.For each type of cylinder we obtain the spatial decay estimates for the solutions.To make the attenuation meaningful,we derive the explicit bound for the total energy in terms of the initial boundary data.展开更多
In this paper,we study the decay estimate and scattering theory for the Klein-Gordon-Hartree equation with radial data in space dimension d≥3.By means of a compactness strategy and two Morawetz-type estimates which c...In this paper,we study the decay estimate and scattering theory for the Klein-Gordon-Hartree equation with radial data in space dimension d≥3.By means of a compactness strategy and two Morawetz-type estimates which come from the linear and nonlinear parts of the equation,respectively,we obtain the corresponding theory for energy subcritical and critical cases.The exponent range of the decay estimates is extended to 0〈γ≤4 and γ〈d with Hartree potential V(x) =|x|-γ.展开更多
In this paper, Saint-Venant's principle for anisotropic material in an end-loaded. semi-infinite elastic strip is established. Energy method is used to establish the lower bounds of the decay estimate of stress ef...In this paper, Saint-Venant's principle for anisotropic material in an end-loaded. semi-infinite elastic strip is established. Energy method is used to establish the lower bounds of the decay estimate of stress effect. An explicit estimate formula in terms of the elastic constants of the anisotropic materials is presented. Finally, a numerical example for an end-loaded, off-axis, graphite-epoxy strip is given to illustrate the results.展开更多
We are concerned with the following quasilinear wave equation involving variable sources and supercritical damping:■Generally speaking,when one tries to use the classical multiplier method to analyze tRhe asymptotic ...We are concerned with the following quasilinear wave equation involving variable sources and supercritical damping:■Generally speaking,when one tries to use the classical multiplier method to analyze tRhe asymptotic behavior of solutions,an inevitable step is to deal with the integralΩ|ut|^(m−2)utudx.A usual technique is to apply Young’s inequality and Sobolev embedding inequality to use the energy function and its derivative to control this integral for the subcritical or critical damping.However,for the supercritical case,the failure of the Sobolev embedding inequality makes the classical method be impossible.To do this,our strategy is to prove the rate of the integral RΩ|u|^(m)dx grows polynomially as a positive power of time variable t and apply the modified multiplier method to obtain the energy functional decays logarithmically.These results improve and extend our previous work[12].Finally,some numerical examples are also given to authenticate our results.展开更多
The existence and the nonexistence,the uniqueness and the energy decay estimate of solution for the fourth-order nonlinear wave equation utt+αΔ2 u-bΔut-βΔu+ut|ut|^r+g(u)=0 in Ω×(0,∞) are studied w...The existence and the nonexistence,the uniqueness and the energy decay estimate of solution for the fourth-order nonlinear wave equation utt+αΔ2 u-bΔut-βΔu+ut|ut|^r+g(u)=0 in Ω×(0,∞) are studied with the boundary condition u=(u)/(υ)=0 onΩ and the initial condition u(x,0)=u0(x),ut(x,0)=u1(x,0) in bounded domain ΩR^n ,n≥1.The energy decay rate of the global solution is estimated by the multiplier method.The blow-up result of the solution in finite time is established by the ideal of a potential well theory,and the existence of the solution is gotten by the Galekin approximation method.展开更多
In this article, first, the authors prove that there exists a unique global smooth solution for the Cauthy problem to the hyperbolic conservation laws systems with relaxation; second, in the large time station, they p...In this article, first, the authors prove that there exists a unique global smooth solution for the Cauthy problem to the hyperbolic conservation laws systems with relaxation; second, in the large time station, they prove that the global smooth solutions of the hyperbolic conservation laws systems with relaxation converge to rarefaction waves solution at a determined L^P(p ≥ 2) decay rate.展开更多
In this paper,we investigate the initial boundary value problem for a plate equation with nonlocal source term.The local,global existence and exponential decay result are established under certain conditions.Moreover,...In this paper,we investigate the initial boundary value problem for a plate equation with nonlocal source term.The local,global existence and exponential decay result are established under certain conditions.Moreover,we also prove the blow-up in finite time and the lifespan of solution under certain conditions.展开更多
In the present paper,we study partial collapsing degeneration of Hamiltonian-perturbed Floer trajectories for an adiabatic ε-family and its reversal adiabatic gluing,as the prototype of the partial collapsing degener...In the present paper,we study partial collapsing degeneration of Hamiltonian-perturbed Floer trajectories for an adiabatic ε-family and its reversal adiabatic gluing,as the prototype of the partial collapsing degeneration of 2-dimensional(perturbed)J-holomorphic maps to 1-dimensional gradient segments.We consider the case when the Floer equations are S^(1)-invariant on parts of their domains whose adiabatic limit has positive length as ε→0,which we call thimble-flow-thimble configurations.The main gluing theorem we prove also applies to the case with Lagrangian boundaries such as in the problem of recovering holomorphic disks out of pearly configuration.In particular,our gluing theorem gives rise to a new direct proof of the chain isomorphism property between the Morse-Bott version of Lagrangian intersection Floer complex of L by Fukaya-Oh-Ohta-Ono and the pearly complex of L Lalonde and Biran-Cornea.It also provides another proof of the present authors’earlier proof of the isomorphism property of the PSS map without involving the target rescaling and the scale-dependent gluing.展开更多
We study the large time behavior of a 3-D MHD(magneto-hydrodynamical)-type system without magnetic diffusion introduced by Lin and Zhang(2014). By using the elementary energy method and interpolation technique, we pro...We study the large time behavior of a 3-D MHD(magneto-hydrodynamical)-type system without magnetic diffusion introduced by Lin and Zhang(2014). By using the elementary energy method and interpolation technique, we prove the global existence and decay estimate of smooth solution near the equilibrium state(x3, 0).展开更多
In this paper we consider the initial boundary value problem for a class of logarithmic wave equation. By constructing an appropriate Lyapunov function, we obtain the decay estimates of energy for the logarithmic wave...In this paper we consider the initial boundary value problem for a class of logarithmic wave equation. By constructing an appropriate Lyapunov function, we obtain the decay estimates of energy for the logarithmic wave equation with linear damping and some suitable initial data. The results extend the early results.展开更多
We study the global existence of solution to one di- mensional convection-diffusion equation. Through constructing a Cauchy sequence in a Banach space, we get the local existence of solution to the equation, t3ased on...We study the global existence of solution to one di- mensional convection-diffusion equation. Through constructing a Cauchy sequence in a Banach space, we get the local existence of solution to the equation, t3ased on the global bounds of the solu- tion, we extend the local one to a global one that decays in Hl space.展开更多
For quantum fluids governed by the compressible quantum Navier-Stokes equations in R;with viscosity and heat conduction, we prove the optimal L;- L;decay rates for the classical solutions near constant states. The pro...For quantum fluids governed by the compressible quantum Navier-Stokes equations in R;with viscosity and heat conduction, we prove the optimal L;- L;decay rates for the classical solutions near constant states. The proof is based on the detailed linearized decay estimates by Fourier analysis of the operators, which is drastically different from the case when quantum effects are absent.展开更多
For a sparse non-singular matrix A, generally A- 1 is a dense matrix. However, for a class of matrices, A-1 can be a matrix with off-diagonal decay properties, i.e., |Aij^-1| decays fast to 0 with respect to the inc...For a sparse non-singular matrix A, generally A- 1 is a dense matrix. However, for a class of matrices, A-1 can be a matrix with off-diagonal decay properties, i.e., |Aij^-1| decays fast to 0 with respect to the increase of a properly defined distance between i and j. Here we consider the off-diagonal decay properties of discretized Green's functions for SchrSdinger type operators. We provide decay estimates for discretized Green's functions obtained from the finite difference discretization, and from a variant of the pseudo-spectral discretization. The asymptotic decay rate in our estimate is independent of the domain size and of the discretization parameter. We verify the decay estimate with numerical results for one-dimensional Schr6dinger type operators.展开更多
We establish the optimal rates of decay estimates of global solutions of some abstract differential equations, which include many partial differential equations. We provide a general treatment so that any future probl...We establish the optimal rates of decay estimates of global solutions of some abstract differential equations, which include many partial differential equations. We provide a general treatment so that any future problem will enjoy the decay estimates displayed here as long as the general hypotheses are satisfied. The main hypotheses are the existence of global solutions of the equations and some growth control of the Fourier transform of the solutions. We establish the optimal rates of decay of the solutions for initial data in different spaces. The main ingredients and technical tools are the Fourier splitting method, the iteration skill and the energy estimates.展开更多
This work is concerned with the proof of Lp-Lq decay estimates for solutions of the Cauchy problem for utt-λ2(t)b2(t) △ u =0. The coefficient consists of an increasing smooth function λ and an oscillating smoot...This work is concerned with the proof of Lp-Lq decay estimates for solutions of the Cauchy problem for utt-λ2(t)b2(t) △ u =0. The coefficient consists of an increasing smooth function λ and an oscillating smooth and bounded function b which are uniformly separated from zero. The authors’ main interest is devoted to the critical case where one has an interesting interplay between the growing and the oscillating part.展开更多
Consider the Cauchy problem for the n-dimensional incompressible NavierStokes equations:??tu-α△u+(u·?)u+?p = f(x, t), with the initial condition u(x, 0) = u_0(x) and with the incompressible conditions ? · ...Consider the Cauchy problem for the n-dimensional incompressible NavierStokes equations:??tu-α△u+(u·?)u+?p = f(x, t), with the initial condition u(x, 0) = u_0(x) and with the incompressible conditions ? · u = 0, ? · f = 0 and ? · u_0= 0. The spatial dimension n ≥ 2.Suppose that the initial function u_0∈ L1(Rn) ∩ L^2(Rn) and the external force f ∈ L^1(Rn× R+) ∩ L^1(R+, L^2(Rn)). It is well known that there holds the decay estimate with sharp rate:(1 + t)1+n/2∫Rn|u(x, t)|2 dx ≤ C, for all time t > 0, where the dimension n ≥ 2, C > 0 is a positive constant, independent of u and(x, t).The main purpose of this paper is to provide two independent proofs of the decay estimate with sharp rate, both are complete, systematic, simplified proofs, under a weaker condition on the external force. The ideas and methods introduced in this paper may have strong influence on the decay estimates with sharp rates of the global weak solutions or the global smooth solutions of similar equations, such as the n-dimensional magnetohydrodynamics equations, where the dimension n ≥ 2.展开更多
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.
基金supported by the National Natural Science Foundation of China(12271296,12271195).
文摘This paper is a continuation of recent work by Guo-Xiang-Zheng[10].We deduce the sharp Morrey regularity theory for weak solutions to the fourth order nonhomogeneous Lamm-Rivière equation △^{2}u=△(V▽u)+div(w▽u)+(▽ω+F)·▽u+f in B^(4),under the smallest regularity assumptions of V,ω,ω,F,where f belongs to some Morrey spaces.This work was motivated by many geometrical problems such as the flow of biharmonic mappings.Our results deepens the Lp type regularity theory of[10],and generalizes the work of Du,Kang and Wang[4]on a second order problem to our fourth order problems.
基金Supported by the National Natural Science Foundation of China(10976026)the Fujian Provincial Department of Science and Technology(JK2009045)
文摘In this paper, under the hypothesis that y is upper bounded, we construct a Lyapunov functional for the multidimensional isentropic compressible magnetohydrodynamic equations and show that the weak solutions decay exponentially to the equilibrium state in L2 norm. Our result verifies that the method of Daoyuan Fang, Ruizhao Zi and Ting Zhang I1] can be adapted to magnetohydrodynamic equations.
基金supported by the National Natural Science Foundation of China (10771055)HNSF(07JJ3007)
文摘L^p- L^q decay estimate of solution to Cauchy problem of a linear thermoviscoelastic system is studied. By using a diagonalization argument of frequency analysis, the coupled system will be decoupled micrologically. Then with the help of the information of characteristic roots for the coefficient matrix of the system, L^p- L^q decay estimate of parabolic type of solution to the Cauchy problem is obtained.
基金Supported by the Key Projects of Universities in Guangdong Province(NATURAL SCIENCE)(Grant No.2019KZDXM042)Research Team Project of Guangzhou Huashang College(Grant No.2021HSKT01).
文摘This paper investigates the spatial behavior of the solutions of the Stokes equations in a semi-infinite cylinder.We consider four kinds of semi-infinite cylinders with boundary conditions of Dirichlet type.For each type of cylinder we obtain the spatial decay estimates for the solutions.To make the attenuation meaningful,we derive the explicit bound for the total energy in terms of the initial boundary data.
基金H.G.Wu was supported by the National Science Foundation of China (11071057,10801015)China Postdoctoral Science Foundation (20100470570)+1 种基金the Guozhi Xu Posdoctoral Research FoundationDoctoral Foundation of Henan Polytechnic University
文摘In this paper,we study the decay estimate and scattering theory for the Klein-Gordon-Hartree equation with radial data in space dimension d≥3.By means of a compactness strategy and two Morawetz-type estimates which come from the linear and nonlinear parts of the equation,respectively,we obtain the corresponding theory for energy subcritical and critical cases.The exponent range of the decay estimates is extended to 0〈γ≤4 and γ〈d with Hartree potential V(x) =|x|-γ.
文摘In this paper, Saint-Venant's principle for anisotropic material in an end-loaded. semi-infinite elastic strip is established. Energy method is used to establish the lower bounds of the decay estimate of stress effect. An explicit estimate formula in terms of the elastic constants of the anisotropic materials is presented. Finally, a numerical example for an end-loaded, off-axis, graphite-epoxy strip is given to illustrate the results.
基金supported by the Scientific and Technological Project of jilin Province's Education Department in Thirteenth Five-Year(JKH20180111KI)supported by NSFC(11301211).
文摘We are concerned with the following quasilinear wave equation involving variable sources and supercritical damping:■Generally speaking,when one tries to use the classical multiplier method to analyze tRhe asymptotic behavior of solutions,an inevitable step is to deal with the integralΩ|ut|^(m−2)utudx.A usual technique is to apply Young’s inequality and Sobolev embedding inequality to use the energy function and its derivative to control this integral for the subcritical or critical damping.However,for the supercritical case,the failure of the Sobolev embedding inequality makes the classical method be impossible.To do this,our strategy is to prove the rate of the integral RΩ|u|^(m)dx grows polynomially as a positive power of time variable t and apply the modified multiplier method to obtain the energy functional decays logarithmically.These results improve and extend our previous work[12].Finally,some numerical examples are also given to authenticate our results.
文摘The existence and the nonexistence,the uniqueness and the energy decay estimate of solution for the fourth-order nonlinear wave equation utt+αΔ2 u-bΔut-βΔu+ut|ut|^r+g(u)=0 in Ω×(0,∞) are studied with the boundary condition u=(u)/(υ)=0 onΩ and the initial condition u(x,0)=u0(x),ut(x,0)=u1(x,0) in bounded domain ΩR^n ,n≥1.The energy decay rate of the global solution is estimated by the multiplier method.The blow-up result of the solution in finite time is established by the ideal of a potential well theory,and the existence of the solution is gotten by the Galekin approximation method.
基金This research is supported by "Foundation of office of overseas Chinese affair under the state council: 03QZR09"
文摘In this article, first, the authors prove that there exists a unique global smooth solution for the Cauthy problem to the hyperbolic conservation laws systems with relaxation; second, in the large time station, they prove that the global smooth solutions of the hyperbolic conservation laws systems with relaxation converge to rarefaction waves solution at a determined L^P(p ≥ 2) decay rate.
基金Supported by National Natural Science Foundation of China(11801145)Key Scientific Research Foundation of the Higher Education Institutions of Henan Province,China(Grant No.19A110004)and(2018GGJS068)。
文摘In this paper,we investigate the initial boundary value problem for a plate equation with nonlocal source term.The local,global existence and exponential decay result are established under certain conditions.Moreover,we also prove the blow-up in finite time and the lifespan of solution under certain conditions.
文摘In the present paper,we study partial collapsing degeneration of Hamiltonian-perturbed Floer trajectories for an adiabatic ε-family and its reversal adiabatic gluing,as the prototype of the partial collapsing degeneration of 2-dimensional(perturbed)J-holomorphic maps to 1-dimensional gradient segments.We consider the case when the Floer equations are S^(1)-invariant on parts of their domains whose adiabatic limit has positive length as ε→0,which we call thimble-flow-thimble configurations.The main gluing theorem we prove also applies to the case with Lagrangian boundaries such as in the problem of recovering holomorphic disks out of pearly configuration.In particular,our gluing theorem gives rise to a new direct proof of the chain isomorphism property between the Morse-Bott version of Lagrangian intersection Floer complex of L by Fukaya-Oh-Ohta-Ono and the pearly complex of L Lalonde and Biran-Cornea.It also provides another proof of the present authors’earlier proof of the isomorphism property of the PSS map without involving the target rescaling and the scale-dependent gluing.
基金supported by National Natural Science Foundation of China (Grant Nos. 11371039 and 11425103)
文摘We study the large time behavior of a 3-D MHD(magneto-hydrodynamical)-type system without magnetic diffusion introduced by Lin and Zhang(2014). By using the elementary energy method and interpolation technique, we prove the global existence and decay estimate of smooth solution near the equilibrium state(x3, 0).
文摘In this paper we consider the initial boundary value problem for a class of logarithmic wave equation. By constructing an appropriate Lyapunov function, we obtain the decay estimates of energy for the logarithmic wave equation with linear damping and some suitable initial data. The results extend the early results.
基金Supported by the National Natural Science Foundation of China(11101121)
文摘We study the global existence of solution to one di- mensional convection-diffusion equation. Through constructing a Cauchy sequence in a Banach space, we get the local existence of solution to the equation, t3ased on the global bounds of the solu- tion, we extend the local one to a global one that decays in Hl space.
基金supported in part by NSFC(11471057)Natural Science Foundation Project of CQ CSTC(cstc2014jcyjA50020)the Fundamental Research Funds for the Central Universities(Project No.106112016CDJZR105501)
文摘For quantum fluids governed by the compressible quantum Navier-Stokes equations in R;with viscosity and heat conduction, we prove the optimal L;- L;decay rates for the classical solutions near constant states. The proof is based on the detailed linearized decay estimates by Fourier analysis of the operators, which is drastically different from the case when quantum effects are absent.
基金supported by Laboratory Directed Research and Development Funding from Berkeley Labprovided by the Director,Office of Science,of the US Department of Energy(Grant No.DE-AC02-05CH11231)+3 种基金the Alfred P Sloan Foundationthe DOE Scientific Discovery through the Advanced Computing Programthe DOE Center for Applied Mathematics for Energy Research Applications Programthe National Science Foundation of USA(Grant Nos.DMS-1312659 and DMS-1454939)
文摘For a sparse non-singular matrix A, generally A- 1 is a dense matrix. However, for a class of matrices, A-1 can be a matrix with off-diagonal decay properties, i.e., |Aij^-1| decays fast to 0 with respect to the increase of a properly defined distance between i and j. Here we consider the off-diagonal decay properties of discretized Green's functions for SchrSdinger type operators. We provide decay estimates for discretized Green's functions obtained from the finite difference discretization, and from a variant of the pseudo-spectral discretization. The asymptotic decay rate in our estimate is independent of the domain size and of the discretization parameter. We verify the decay estimate with numerical results for one-dimensional Schr6dinger type operators.
文摘We establish the optimal rates of decay estimates of global solutions of some abstract differential equations, which include many partial differential equations. We provide a general treatment so that any future problem will enjoy the decay estimates displayed here as long as the general hypotheses are satisfied. The main hypotheses are the existence of global solutions of the equations and some growth control of the Fourier transform of the solutions. We establish the optimal rates of decay of the solutions for initial data in different spaces. The main ingredients and technical tools are the Fourier splitting method, the iteration skill and the energy estimates.
文摘This work is concerned with the proof of Lp-Lq decay estimates for solutions of the Cauchy problem for utt-λ2(t)b2(t) △ u =0. The coefficient consists of an increasing smooth function λ and an oscillating smooth and bounded function b which are uniformly separated from zero. The authors’ main interest is devoted to the critical case where one has an interesting interplay between the growing and the oscillating part.
文摘Consider the Cauchy problem for the n-dimensional incompressible NavierStokes equations:??tu-α△u+(u·?)u+?p = f(x, t), with the initial condition u(x, 0) = u_0(x) and with the incompressible conditions ? · u = 0, ? · f = 0 and ? · u_0= 0. The spatial dimension n ≥ 2.Suppose that the initial function u_0∈ L1(Rn) ∩ L^2(Rn) and the external force f ∈ L^1(Rn× R+) ∩ L^1(R+, L^2(Rn)). It is well known that there holds the decay estimate with sharp rate:(1 + t)1+n/2∫Rn|u(x, t)|2 dx ≤ C, for all time t > 0, where the dimension n ≥ 2, C > 0 is a positive constant, independent of u and(x, t).The main purpose of this paper is to provide two independent proofs of the decay estimate with sharp rate, both are complete, systematic, simplified proofs, under a weaker condition on the external force. The ideas and methods introduced in this paper may have strong influence on the decay estimates with sharp rates of the global weak solutions or the global smooth solutions of similar equations, such as the n-dimensional magnetohydrodynamics equations, where the dimension n ≥ 2.