In this paper,we apply Fokas unified method to study the initial boundary value(IBV)problems for nonlinear integrable equation with 3×3 Lax pair on the finite interval[0,L].The solution can be expressed by the so...In this paper,we apply Fokas unified method to study the initial boundary value(IBV)problems for nonlinear integrable equation with 3×3 Lax pair on the finite interval[0,L].The solution can be expressed by the solution of a 3×3 Riemann-Hilbert(RH)problem.The relevant jump matrices are written in terms of matrix-value spectral functions s(k),S(k),S_(l)(k),which are determined by initial data at t=0,boundary values at x=0 and boundary values at x=L,respectively.What's more,since the eigenvalues of 3×3 coefficient matrix of k spectral parameter in Lax pair are three different values,search for the path of analytic functions in RH problem becomes a very interesting thing.展开更多
To show some theorems on the existence of singular initial value problem with n Laplacian operator, topology method and methods of analysis are employed. Some existence theorems for initial value problems...To show some theorems on the existence of singular initial value problem with n Laplacian operator, topology method and methods of analysis are employed. Some existence theorems for initial value problems with n Laplacian operators are established in three singular cases.展开更多
The multisplitting algorithm for solving large systems of ordinary differential equations on parallel computers was introduced by Jeltsch and Pohl in [1]. On fixed time intervals conver gence results could be derived ...The multisplitting algorithm for solving large systems of ordinary differential equations on parallel computers was introduced by Jeltsch and Pohl in [1]. On fixed time intervals conver gence results could be derived if the subsystems are solving exactly.Firstly,in theis paper,we deal with an extension of the waveform relaxation algorithm by us ing multisplittin AOR method based on an overlapping block decomposition. We restricted our selves to equidistant timepoints and dealed with the case that an implicit integration method was used to solve the subsystems numerically in parallel. Then we have proved convergence of multi splitting AOR waveform relaxation algorithm on a fixed window containing a finite number of timepoints.展开更多
The nonlinear interactions between the monochromatic wave have been considered by K. Matsunchi, who also proposed one class of the nonlinear Schrdinger equation system with wave operator. We also obtain the same type ...The nonlinear interactions between the monochromatic wave have been considered by K. Matsunchi, who also proposed one class of the nonlinear Schrdinger equation system with wave operator. We also obtain the same type of equations, which are satisfied by transverse velocity of higher frequency electron, as we study soliton in plasma physics. In this paper, under some condition we study the existence and nonexistence for this equations in the cases possessing different signs in nonlinear term.展开更多
In this paper, we study the asymptotic behavior of the solutions to the initial boundary value problem for unipolar drift diffusion equations for semiconductors. Under the proper assumptions on doping profile and init...In this paper, we study the asymptotic behavior of the solutions to the initial boundary value problem for unipolar drift diffusion equations for semiconductors. Under the proper assumptions on doping profile and initial value, we prove that the smooth solutions to these evolutionary problems tend to the unique stationary solution exponentially as time tends to infinity.展开更多
In this paper, a new set of sufficient conditions related to an initial value problem and global homeomorphism is obtained in discussing the existence and uniqueness of 2π-periodic solution for 2kth order differentia...In this paper, a new set of sufficient conditions related to an initial value problem and global homeomorphism is obtained in discussing the existence and uniqueness of 2π-periodic solution for 2kth order differential equations with resonance. The key role is played by nonnegative auxiliary scalar coercive function. The result of this paper generalizes some existed theorems.展开更多
In this paper, the global existence of weak s olutions to the initial boundary value problem for Boltzmann-Poisson system is proved. The proof is based on the regularization and the stability of the veloci ty averages...In this paper, the global existence of weak s olutions to the initial boundary value problem for Boltzmann-Poisson system is proved. The proof is based on the regularization and the stability of the veloci ty averages and the compactness results on L 1-theory.展开更多
The paper deal with the asymptotic behavior of the solutions to the initial boundary value problem for unipolar drift diffusion equations for semiconductors. Under the proper assumptions on doping profile and initial ...The paper deal with the asymptotic behavior of the solutions to the initial boundary value problem for unipolar drift diffusion equations for semiconductors. Under the proper assumptions on doping profile and initial value, we prove that the smooth solutions to these evolutionary problems tend to the unique stationary solution exponentially as time tends to infinity.展开更多
In this paper, the existence and uniqueness of the local generalized solution of the initial boundary value problem for a nonlinear hyperbolic equation are proved by the contraction mapping principle and the sufficien...In this paper, the existence and uniqueness of the local generalized solution of the initial boundary value problem for a nonlinear hyperbolic equation are proved by the contraction mapping principle and the sufficient conditions of blow_up of the solution in finite time are given.展开更多
This paper is concerned with the generalzed global solution and its asymptotic properties for the initial value problem of the partial differential equationu t+u x 3 =F(u).
In this paper the singularly perturbed initial boundary value problems for the nonlocal reaction diffusion system are considered. Unsing the iteration method and the comparison theorem, the existence, uniqueness and i...In this paper the singularly perturbed initial boundary value problems for the nonlocal reaction diffusion system are considered. Unsing the iteration method and the comparison theorem, the existence, uniqueness and its asymptotic behavior of solution for the problem are studied.展开更多
An FDTD system associated with uniaxial perfectly matched layer（UPML） for an electromagnetic scattering problem in two-dimensional space in polar coordinates is considered.Particularly the FDTD system of an initial-...An FDTD system associated with uniaxial perfectly matched layer（UPML） for an electromagnetic scattering problem in two-dimensional space in polar coordinates is considered.Particularly the FDTD system of an initial-boundary value problems of the transverse magnetic（TM） mode to Maxwell＇s equations is obtained by Yee＇s algorithm,and the open domain of the scattering problem is truncated by a circle with a UPML.Besides,an artificial boundary condition is imposed on the outer boundary of the UPML.Afterwards,stability of the FDTD system on the truncated domain is established through energy estimates by the Gronwall inequality.Numerical experiments are designed to approve the theoretical analysis.展开更多
We consider a wave equation with nonlocal nonlinear damping and source terms.We prove a general energy decay property for solutions by constructing a stable set and using the multiplier technique.The main difficult is...We consider a wave equation with nonlocal nonlinear damping and source terms.We prove a general energy decay property for solutions by constructing a stable set and using the multiplier technique.The main difficult is how to handle with the nonlocal nonlinear damping term.Our result extends and improves the result in the literature such as the work by Jorge Silva and Narciso(Evolution Equation and Control Theory,2017(6):437-470)and Narciso(Evolution Equations and Control Theory,2020,9(2):487-508).展开更多
The periodic initial value problem of a fifth-order shallow water equation t u 2 x t u ＋ 3 x u 5 x u ＋ 3u x u 2 x u 2 x u u 3 x u = 0 is shown to be globally well-posed in Sobolev spaces˙ H s （T） for s 〉 2/3 by ...The periodic initial value problem of a fifth-order shallow water equation t u 2 x t u ＋ 3 x u 5 x u ＋ 3u x u 2 x u 2 x u u 3 x u = 0 is shown to be globally well-posed in Sobolev spaces˙ H s （T） for s 〉 2/3 by I-method. For this equation lacks scaling invariance, we first reconsider the local result and pay special attention to the relationship between the lifespan of the local solution and the initial data, and then prove the almost conservation law, and finally obtain the global well-posedness by an iteration process.展开更多
Recently, the notion of an S-metric space is defined and extensively studied as a generalization of a metric space. In this paper, we define the notion of the S∞-space and prove its completeness. We obtain a new gene...Recently, the notion of an S-metric space is defined and extensively studied as a generalization of a metric space. In this paper, we define the notion of the S∞-space and prove its completeness. We obtain a new generalization of the classical ＂Picard Theorem＂.展开更多
In this article, we study the initial boundary value problem of generalized Pochhammer-Chree equation utt-uxx-uxxt-uxxtt=f（u）xx,x∈Ω,t〉0,u（x,0）=u0（x）,ut（x,0）=u1（x）,x∈Ω,u（0,t）=u（1,t）=0,t≥0, where Ω...In this article, we study the initial boundary value problem of generalized Pochhammer-Chree equation utt-uxx-uxxt-uxxtt=f（u）xx,x∈Ω,t〉0,u（x,0）=u0（x）,ut（x,0）=u1（x）,x∈Ω,u（0,t）=u（1,t）=0,t≥0, where Ω = （0, 1）. First, we obtain the existence of local Wkp solutions. Then, we prove that, if f（s） ∈ Ck＋1（R） is nondecreasing, f（0） = 0 and ｜f（u）｜ ≤ C1｜u｜∫0uf（s）ds ＋ C2, u0（x）,u1（x） ∈ WkP（Ω） ∩ W01,P（Ω）, k≥ 1, 1 〈 p≤ ∞, then for any T 〉 0 the problem admits a unique solution u（x, t） ∈ W2,∞（0, T; WkP（Ω） ∩ W01,P（Ω））. Finally, the finite time blow-up of solutions and global Wkp solution of generalized IMBo equations are discussed.展开更多
In this paper, we study the threshold result for the initial boundary value problem of non-homogeneous semilinear parabolic equations {μt-△μ=g（μ）＋λf（x）,（x,t）∈Ω×（0,T）,μ=0,（x,t）∈ Ω×[0,T）...In this paper, we study the threshold result for the initial boundary value problem of non-homogeneous semilinear parabolic equations {μt-△μ=g（μ）＋λf（x）,（x,t）∈Ω×（0,T）,μ=0,（x,t）∈ Ω×[0,T）,μ（x,0）=μ0（x）≥0,x∈Ω.By combining a priori estimate of global solution with property of stationary solution set of problem （P）, we prove that the minimal stationary solution Uλ（x） of problem （P） is stable, whereas, any other stationary solution is an initial datum threshold for the existence and nonexistence of global solution to problem （P）.展开更多
Is this Paper, the global existence of smooth solutions to the Antial value problem for the fourth order nonlinear Schrodinger equation in the Lax hierarchy of the nonlinear fSchrodinger equation(NLS equation) is esta...Is this Paper, the global existence of smooth solutions to the Antial value problem for the fourth order nonlinear Schrodinger equation in the Lax hierarchy of the nonlinear fSchrodinger equation(NLS equation) is established by using the so-called continuation method and delicate a priori estimate. In addition, the asylnptotic properties of the solutions as|×|+∞ are discussed.展开更多
The initial boundary value problems for the system of rate-type viscoelasticityis considered in the present paper.It is shown that if the initial data are a small perturbationof a forward smooth rarefaction wave, then...The initial boundary value problems for the system of rate-type viscoelasticityis considered in the present paper.It is shown that if the initial data are a small perturbationof a forward smooth rarefaction wave, then there is a global solutions to the system, whichtends to the rarefaction wave time-asymptotically.展开更多
Fractional differential equations and systems are gradually becoming an essential ap-proach to real world applications in science and engineering technology.The boundary value prob-lems of the fractional differential ...Fractional differential equations and systems are gradually becoming an essential ap-proach to real world applications in science and engineering technology.The boundary value prob-lems of the fractional differential equations and systems were investigated by using several different methods.Recently,much attention has been paid to investigate fractional differential equations with nonlocal conditions by using the fixed point method.In this paper,by using the Banach fixed point theorem and Krasnoselkii fixed point theorem,the existence and uniqueness of the solutions to the initial value problem of the nonlinear fractional differential equations with nonlocal condi-tions are investigated,and some sufficient conditions are obtained.We extend some results that already exist.Finally,an example is given to show the usefulness of the theoretical results.展开更多
基金supported by the National Natural Science Foundation of China(11901167,11971313 and 51879045)Key scientific research projects of higher education institutions in Henan,China(18B110008).
文摘In this paper,we apply Fokas unified method to study the initial boundary value(IBV)problems for nonlinear integrable equation with 3×3 Lax pair on the finite interval[0,L].The solution can be expressed by the solution of a 3×3 Riemann-Hilbert(RH)problem.The relevant jump matrices are written in terms of matrix-value spectral functions s(k),S(k),S_(l)(k),which are determined by initial data at t=0,boundary values at x=0 and boundary values at x=L,respectively.What's more,since the eigenvalues of 3×3 coefficient matrix of k spectral parameter in Lax pair are three different values,search for the path of analytic functions in RH problem becomes a very interesting thing.
文摘To show some theorems on the existence of singular initial value problem with n Laplacian operator, topology method and methods of analysis are employed. Some existence theorems for initial value problems with n Laplacian operators are established in three singular cases.
文摘The multisplitting algorithm for solving large systems of ordinary differential equations on parallel computers was introduced by Jeltsch and Pohl in [1]. On fixed time intervals conver gence results could be derived if the subsystems are solving exactly.Firstly,in theis paper,we deal with an extension of the waveform relaxation algorithm by us ing multisplittin AOR method based on an overlapping block decomposition. We restricted our selves to equidistant timepoints and dealed with the case that an implicit integration method was used to solve the subsystems numerically in parallel. Then we have proved convergence of multi splitting AOR waveform relaxation algorithm on a fixed window containing a finite number of timepoints.
文摘The nonlinear interactions between the monochromatic wave have been considered by K. Matsunchi, who also proposed one class of the nonlinear Schrdinger equation system with wave operator. We also obtain the same type of equations, which are satisfied by transverse velocity of higher frequency electron, as we study soliton in plasma physics. In this paper, under some condition we study the existence and nonexistence for this equations in the cases possessing different signs in nonlinear term.
文摘In this paper, we study the asymptotic behavior of the solutions to the initial boundary value problem for unipolar drift diffusion equations for semiconductors. Under the proper assumptions on doping profile and initial value, we prove that the smooth solutions to these evolutionary problems tend to the unique stationary solution exponentially as time tends to infinity.
基金Foundation item： Supported by the Natural Science Foundation of Changzhou Instituty of Technology（YN09090） Supported by the Natural Science Foundation of Jiangsu Province（13KJD110001）
文摘In this paper, a new set of sufficient conditions related to an initial value problem and global homeomorphism is obtained in discussing the existence and uniqueness of 2π-periodic solution for 2kth order differential equations with resonance. The key role is played by nonnegative auxiliary scalar coercive function. The result of this paper generalizes some existed theorems.
文摘In this paper, the global existence of weak s olutions to the initial boundary value problem for Boltzmann-Poisson system is proved. The proof is based on the regularization and the stability of the veloci ty averages and the compactness results on L 1-theory.
文摘The paper deal with the asymptotic behavior of the solutions to the initial boundary value problem for unipolar drift diffusion equations for semiconductors. Under the proper assumptions on doping profile and initial value, we prove that the smooth solutions to these evolutionary problems tend to the unique stationary solution exponentially as time tends to infinity.
文摘In this paper, the existence and uniqueness of the local generalized solution of the initial boundary value problem for a nonlinear hyperbolic equation are proved by the contraction mapping principle and the sufficient conditions of blow_up of the solution in finite time are given.
文摘This paper is concerned with the generalzed global solution and its asymptotic properties for the initial value problem of the partial differential equationu t+u x 3 =F(u).
基金The project is supported by The National Natural Science Foundation of China(10071048)"Hundred People Project" of Chinese Academy of Sciences
文摘In this paper the singularly perturbed initial boundary value problems for the nonlocal reaction diffusion system are considered. Unsing the iteration method and the comparison theorem, the existence, uniqueness and its asymptotic behavior of solution for the problem are studied.
文摘An FDTD system associated with uniaxial perfectly matched layer（UPML） for an electromagnetic scattering problem in two-dimensional space in polar coordinates is considered.Particularly the FDTD system of an initial-boundary value problems of the transverse magnetic（TM） mode to Maxwell＇s equations is obtained by Yee＇s algorithm,and the open domain of the scattering problem is truncated by a circle with a UPML.Besides,an artificial boundary condition is imposed on the outer boundary of the UPML.Afterwards,stability of the FDTD system on the truncated domain is established through energy estimates by the Gronwall inequality.Numerical experiments are designed to approve the theoretical analysis.
基金Supported by National Natural Science Foundation of China(11601122,11801145)。
文摘We consider a wave equation with nonlocal nonlinear damping and source terms.We prove a general energy decay property for solutions by constructing a stable set and using the multiplier technique.The main difficult is how to handle with the nonlocal nonlinear damping term.Our result extends and improves the result in the literature such as the work by Jorge Silva and Narciso(Evolution Equation and Control Theory,2017(6):437-470)and Narciso(Evolution Equations and Control Theory,2020,9(2):487-508).
基金supported by NSFC (10771074)NSFC-NSAF(10976026)+1 种基金Yang was partially supported by NSFC (10801055 10901057)
文摘The periodic initial value problem of a fifth-order shallow water equation t u 2 x t u ＋ 3 x u 5 x u ＋ 3u x u 2 x u 2 x u u 3 x u = 0 is shown to be globally well-posed in Sobolev spaces˙ H s （T） for s 〉 2/3 by I-method. For this equation lacks scaling invariance, we first reconsider the local result and pay special attention to the relationship between the lifespan of the local solution and the initial data, and then prove the almost conservation law, and finally obtain the global well-posedness by an iteration process.
文摘Recently, the notion of an S-metric space is defined and extensively studied as a generalization of a metric space. In this paper, we define the notion of the S∞-space and prove its completeness. We obtain a new generalization of the classical ＂Picard Theorem＂.
基金supported by National Natural Science Foundation of China(10871055,10926149)Natural Science Foundation of Heilongjiang Province (A2007-02+2 种基金A200810)Science and Technology Foundation of Education Office of Heilongjiang Province(11541276)Foundational Science Founda-tion of Harbin Engineering University
文摘In this article, we study the initial boundary value problem of generalized Pochhammer-Chree equation utt-uxx-uxxt-uxxtt=f（u）xx,x∈Ω,t〉0,u（x,0）=u0（x）,ut（x,0）=u1（x）,x∈Ω,u（0,t）=u（1,t）=0,t≥0, where Ω = （0, 1）. First, we obtain the existence of local Wkp solutions. Then, we prove that, if f（s） ∈ Ck＋1（R） is nondecreasing, f（0） = 0 and ｜f（u）｜ ≤ C1｜u｜∫0uf（s）ds ＋ C2, u0（x）,u1（x） ∈ WkP（Ω） ∩ W01,P（Ω）, k≥ 1, 1 〈 p≤ ∞, then for any T 〉 0 the problem admits a unique solution u（x, t） ∈ W2,∞（0, T; WkP（Ω） ∩ W01,P（Ω））. Finally, the finite time blow-up of solutions and global Wkp solution of generalized IMBo equations are discussed.
基金supported by Natural Science Foundation of China(10971061)Hunan Provincial Innovation Foundation For Postgraduate(CX2010B209)
文摘In this paper, we study the threshold result for the initial boundary value problem of non-homogeneous semilinear parabolic equations {μt-△μ=g（μ）＋λf（x）,（x,t）∈Ω×（0,T）,μ=0,（x,t）∈ Ω×[0,T）,μ（x,0）=μ0（x）≥0,x∈Ω.By combining a priori estimate of global solution with property of stationary solution set of problem （P）, we prove that the minimal stationary solution Uλ（x） of problem （P） is stable, whereas, any other stationary solution is an initial datum threshold for the existence and nonexistence of global solution to problem （P）.
文摘Is this Paper, the global existence of smooth solutions to the Antial value problem for the fourth order nonlinear Schrodinger equation in the Lax hierarchy of the nonlinear fSchrodinger equation(NLS equation) is established by using the so-called continuation method and delicate a priori estimate. In addition, the asylnptotic properties of the solutions as|×|+∞ are discussed.
文摘The initial boundary value problems for the system of rate-type viscoelasticityis considered in the present paper.It is shown that if the initial data are a small perturbationof a forward smooth rarefaction wave, then there is a global solutions to the system, whichtends to the rarefaction wave time-asymptotically.
基金supported by the Natural Science Foundation of Shanxi Province(No.201801D121024)the Scientific and Tech-nological Innovation Programs of Higher Education Institu-tions in Shanxi(STIP)(No.2019L0903).
文摘Fractional differential equations and systems are gradually becoming an essential ap-proach to real world applications in science and engineering technology.The boundary value prob-lems of the fractional differential equations and systems were investigated by using several different methods.Recently,much attention has been paid to investigate fractional differential equations with nonlocal conditions by using the fixed point method.In this paper,by using the Banach fixed point theorem and Krasnoselkii fixed point theorem,the existence and uniqueness of the solutions to the initial value problem of the nonlinear fractional differential equations with nonlocal condi-tions are investigated,and some sufficient conditions are obtained.We extend some results that already exist.Finally,an example is given to show the usefulness of the theoretical results.
