The motion of the self-gravitational gaseous stars can be described by the Euler-Poisson equations. The main purpose of this paper is concerned with the existence of stationary solutions of Euler-Poisson equations for...The motion of the self-gravitational gaseous stars can be described by the Euler-Poisson equations. The main purpose of this paper is concerned with the existence of stationary solutions of Euler-Poisson equations for some velocity fields and entropy functions that solve the conservation of mass and energy. Under different restriction to the strength of velocity field, we get the existence and multiplicity of the stationary solutions of Euler-Poisson system.展开更多
This paper is concerned with the system of Euler-Poisson equations as a model to describe the motion of the self-induced gravitational gaseous stars. When ~ 〈 7 〈 2, under the weak smoothness of entropy function, we...This paper is concerned with the system of Euler-Poisson equations as a model to describe the motion of the self-induced gravitational gaseous stars. When ~ 〈 7 〈 2, under the weak smoothness of entropy function, we find a sufficient condition to guarantee the existence of stationary solutions for some velocity fields and entropy function that solve the conservation of mass and energy.展开更多
This paper investigates the dynamical properties of nonstationary solutions in one-dimensional two-component Bose-Einstein condensates. It gives three kinds of stationary solutions to this model and develops a general...This paper investigates the dynamical properties of nonstationary solutions in one-dimensional two-component Bose-Einstein condensates. It gives three kinds of stationary solutions to this model and develops a general method of constructing nonstationary solutions. It obtains the unique features about general evolution and soliton evolution of nonstationary solutions in this model.展开更多
In this paper, the uniqueness of stationary solutions with vacuum of Euler-Poisson equations is considered. Through a nonlinear transformation which is a function of density and entropy, the corresponding problem can ...In this paper, the uniqueness of stationary solutions with vacuum of Euler-Poisson equations is considered. Through a nonlinear transformation which is a function of density and entropy, the corresponding problem can be reduced to a semilinear elliptic equation with a nonlinear source term consisting of a power function, for which the classical theory of the elliptic equations leads the authors to the uniqueness result under some assumptions on the entropy function S(x). As an example, the authors get the uniqueness of stationary solutions with vacuum of Euler-Poisson equations for S(x) =|x|θandθ∈{0}∪[2(N-2),+∞).展开更多
In this paper,we consider an inflow problem for the non-isentropic Navier-StokesPoisson system in a half line(0,∞).For the general gas including ideal polytropic gas,we first give some results for the existence of th...In this paper,we consider an inflow problem for the non-isentropic Navier-StokesPoisson system in a half line(0,∞).For the general gas including ideal polytropic gas,we first give some results for the existence of the stationary solution with the aid of center manifold theory on a 4×4 system of autonomous ordinary differential equations.We also show the time asymptotic stability of the stationary solutions with small strength under smallness assumptions on the initial perturbations by using an elementary energy method.展开更多
In this work, we are mainly concerned with the existence of stationary solutions for a generalized Kadomtsev-Petviashvili equation in bounded domain of Rn. We utilize variational method and critical point theory to es...In this work, we are mainly concerned with the existence of stationary solutions for a generalized Kadomtsev-Petviashvili equation in bounded domain of Rn. We utilize variational method and critical point theory to establish our main results.展开更多
The outflow problem for the viscous two-phase flow model in a half line is investigated in the present paper.The existence and uniqueness of the stationary solution is shown for both supersonic state and sonic state a...The outflow problem for the viscous two-phase flow model in a half line is investigated in the present paper.The existence and uniqueness of the stationary solution is shown for both supersonic state and sonic state at spatial far field,and the nonlinear time stability of the stationary solution is also established in the weighted Sobolev space with either the exponential time decay rate for supersonic flow or the algebraic time decay rate for sonic flow.展开更多
For the Boltzmann equation with an external force in the form of the gradient of a potential function in space variable, the stability of its stationary solutions as local Maxwellians was studied by S. Ukai et al. （2...For the Boltzmann equation with an external force in the form of the gradient of a potential function in space variable, the stability of its stationary solutions as local Maxwellians was studied by S. Ukai et al. （2005） through the energy method. Based on this stability analysis and some techniques on analyzing the convergence rates to stationary solutions for the compressible Navier-Stokes equations, in this paper, we study the convergence rate to the above stationary solutions for the Boltzmann equation which is a fundamental equation in statistical physics for non-equilibrium rarefied gas. By combining the dissipation from the viscosity and heat conductivity on the fluid components and the dissipation on the non-fluid component through the celebrated H-theorem, a convergence rate of the same order as the one for the compressible Navier-Stokes is obtained by constructing some energy functionals.展开更多
The exact solutions for stationary responses of one class of the second order and three classes of higher order nonlinear systems to parametric and/or external while noise excitations are constructed by using Fokkcr-P...The exact solutions for stationary responses of one class of the second order and three classes of higher order nonlinear systems to parametric and/or external while noise excitations are constructed by using Fokkcr-Planck-Kolmogorov et/ualion approach. The conditions for the existence and uniqueness and the behavior of the solutions are discussed. All the systems under consideration are characterized by the dependence ofnonconservative fqrces on the first integrals of the corresponding conservative systems and arc catted generalized-energy-dependent f G.E.D.) systems. It is shown taht for each of the four classes of G.E.D. nonlinear stochastic systems there is a family of non-G.E.D. systems which are equivalent to the G.E.D. system in the sense of having identical stationary solution. The way to find the equivalent stochastic systems for a given G.E.D. system is indicated and. as an example, the equivalent stochastic systems for the second order G.E. D. nonlinear stochastic system are given. It is pointed out and illustrated with example that the exact stationary solutions for many non-G.E.D. nonlinear stochastic systems may he found by searching the equivalent G.E.D. systems.展开更多
In this paper,we study the asymptotic dynamics of a single-species model with resource-dependent dispersal in one dimension.To overcome the analytical difficulties brought by the resource-dependent dispersal,we use th...In this paper,we study the asymptotic dynamics of a single-species model with resource-dependent dispersal in one dimension.To overcome the analytical difficulties brought by the resource-dependent dispersal,we use the idea of changing variables to transform the model into a uniform dispersal one.Then the existence and uniqueness of positive stationary solution to the model can be verified by the squeezing argument,where the solution plays a crucial role in later analyses.Moreover,the asymptotic behavior of solutions to the model is obtained by the upper-lower solutions method.The result indicates that the solutions of the model converge to the corresponding positive stationary solution locally uniformly in one dimension as time goes to infinity.展开更多
In this paper, the nonlinear Schrodinger (NLS) equation was analytically solved. Firstly, the stationary solutions of NLS equation were explicitly given by the elliptic functions. Then a family of exact solutions ...In this paper, the nonlinear Schrodinger (NLS) equation was analytically solved. Firstly, the stationary solutions of NLS equation were explicitly given by the elliptic functions. Then a family of exact solutions of NLS equation were obtained from these stationary solutions by a method for finding new exact solutions from the stationary solutions of integrable evolution equations.展开更多
In this article, we are concerned with the stability of stationary solution for outflow problem on the Navier-Stokes-Poisson system. We obtain the unique existence and the asymptotic stability of stationary solution. ...In this article, we are concerned with the stability of stationary solution for outflow problem on the Navier-Stokes-Poisson system. We obtain the unique existence and the asymptotic stability of stationary solution. Moreover, the convergence rate of solution towards stationary solution is obtained. Precisely, if an initial perturbation decays with the algebraic or the exponential rate in space, the solution converges to the corresponding stationary solution as time tends to infinity with the algebraic or the exponential rate in time. The proof is based on the weighted energy method by taking into account the effect of the self-consistent electric field on the viscous compressible fluid.展开更多
This paper considers a stochastic Lienard equation with Markovian switching. The Feller continuity of its solution is proved by the coupling method and a truncation argument. The existence of a stationary solution for...This paper considers a stochastic Lienard equation with Markovian switching. The Feller continuity of its solution is proved by the coupling method and a truncation argument. The existence of a stationary solution for the equation is also proved under the Foster-Lyapunov drift condition.展开更多
This paper studies the stationary probability density function（PDF） solution of a nonlinear business cycle model subjected to random shocks of Gaussian white-noise type. The PDF solution is controlled by a Fokker–P...This paper studies the stationary probability density function（PDF） solution of a nonlinear business cycle model subjected to random shocks of Gaussian white-noise type. The PDF solution is controlled by a Fokker–Planck– Kolmogorov（FPK） equation, and we use exponential polynomial closure（EPC） method to derive an approximate solution for the FPK equation. Numerical results obtained from EPC method, better than those from Gaussian closure method, show good agreement with the probability distribution obtained with Monte Carlo simulation including the tail regions.展开更多
In this paper we are interested in the large time behavior of the nonlinear diffusion equationWe consider functions which allow the equation to possess traveling wave solutions. We first present an existence and uniqu...In this paper we are interested in the large time behavior of the nonlinear diffusion equationWe consider functions which allow the equation to possess traveling wave solutions. We first present an existence and uniqueness as well as some comparison principle result of generalized solutions to the Cauchy problem. Then we give for some threshold results, from which we can see that u=a is stable, while u= 0 or u=1 is unstable under some assumptions, etc.展开更多
The existence of classical solutions to a stationary simplified quantum energytransport model for semiconductor devices in 1-dimensional space is proved.The model consists of a nonlinear elliptic third-order equation ...The existence of classical solutions to a stationary simplified quantum energytransport model for semiconductor devices in 1-dimensional space is proved.The model consists of a nonlinear elliptic third-order equation for the electron density,including a temperature derivative,an elliptic nonlinear heat equation for the electron temperature,and the Poisson equation for the electric potential.The proof is based on an exponential variable transformation and the Leray-Schauder fixed-point theorem.展开更多
The classical Lotka-Volterra （LV） model is a well-known mathematical model for prey-predator ecosystems. In the present paper, the pulse-type version of stochastic LV model, in which the effect of a random natural e...The classical Lotka-Volterra （LV） model is a well-known mathematical model for prey-predator ecosystems. In the present paper, the pulse-type version of stochastic LV model, in which the effect of a random natural environment has been modeled as Poisson white noise, is in- vestigated by using the stochastic averaging method. The averaged generalized It6 stochastic differential equation and Fokkerlanck-Kolmogorov （FPK） equation are derived for prey-predator ecosystem driven by Poisson white noise. Approximate stationary solution for the averaged generalized FPK equation is obtained by using the perturbation method. The effect of prey self-competition parameter e2s on ecosystem behavior is evaluated. The analytical result is confirmed by corresponding Monte Carlo （MC） simulation.展开更多
We give the definition of the weak solution for the equilibrium systems of ferro-magnetic chain and get the existence result of the Dirichlet problems for this systems. We get the regularity result for the weakly stat...We give the definition of the weak solution for the equilibrium systems of ferro-magnetic chain and get the existence result of the Dirichlet problems for this systems. We get the regularity result for the weakly stationary solution.展开更多
The basic equations of free capillary_gravity surface_waves in a circular cylindrical basin were derived from Luke's principle. Taking Galerkin's expansion of the velocity potential and the free surface elevat...The basic equations of free capillary_gravity surface_waves in a circular cylindrical basin were derived from Luke's principle. Taking Galerkin's expansion of the velocity potential and the free surface elevation, the second_order perturbation equations were derived by use of expansion of multiple scale. The nonlinear interactions with the second order internal resonance of three free surface_waves were discussed based on the above. The results include:derivation of the couple equations of resonant interactions among three waves and the conservation laws; analysis of the positions of equilibrium points in phase plane; study of the resonant parameters and the non_resonant parameters respectively in all kinds of circumstances; derivation of the stationary solutions of the second_order interaction equations corresponding to different parameters and analysis of the stability property of the solutions; discussion of the effective solutions only in the limited time range. The analysis makes it clear that the energy transformation mode among three waves differs because of the different initial conditions under nontrivial circumstance. The energy may either exchange among three waves periodically or damp or increase in single waves.展开更多
The existence and stability of stationary solutions for a reaction-diffusion-ODE system are investigated in this paper.We first show that there exist both continuous and discontinuous stationary solutions.Then a good ...The existence and stability of stationary solutions for a reaction-diffusion-ODE system are investigated in this paper.We first show that there exist both continuous and discontinuous stationary solutions.Then a good understanding of the stability of discontinuous stationary solutions is gained under an appropriate condition.In addition,we demonstrate the influences of the diffusion coefficient on stationary solutions.The results we obtained are based on the super-/sub-solution method and the generalized mountain pass theorem.Finally,some numerical simulations are given to illustrate the theoretical results.展开更多
基金supported by NSFC (10631030, 11071094)the fund of CCNU for Ph.D students (2009021)
文摘The motion of the self-gravitational gaseous stars can be described by the Euler-Poisson equations. The main purpose of this paper is concerned with the existence of stationary solutions of Euler-Poisson equations for some velocity fields and entropy functions that solve the conservation of mass and energy. Under different restriction to the strength of velocity field, we get the existence and multiplicity of the stationary solutions of Euler-Poisson system.
基金supported by the Fundamental Research Funds for the Central Universities(2011-1a-021)
文摘This paper is concerned with the system of Euler-Poisson equations as a model to describe the motion of the self-induced gravitational gaseous stars. When ~ 〈 7 〈 2, under the weak smoothness of entropy function, we find a sufficient condition to guarantee the existence of stationary solutions for some velocity fields and entropy function that solve the conservation of mass and energy.
基金supported by the National Natural Science Foundation of China (Grant No. 1057411)the Foundation for Researching Group by Beijing Normal Universitythe Foundation for Outstanding Doctoral Dissertation by Beijing Normal University
文摘This paper investigates the dynamical properties of nonstationary solutions in one-dimensional two-component Bose-Einstein condensates. It gives three kinds of stationary solutions to this model and develops a general method of constructing nonstationary solutions. It obtains the unique features about general evolution and soliton evolution of nonstationary solutions in this model.
基金the Natural Science Foundation of China and the Excellent Teachers Foundation of Ministry of Education of China.
文摘In this paper, the uniqueness of stationary solutions with vacuum of Euler-Poisson equations is considered. Through a nonlinear transformation which is a function of density and entropy, the corresponding problem can be reduced to a semilinear elliptic equation with a nonlinear source term consisting of a power function, for which the classical theory of the elliptic equations leads the authors to the uniqueness result under some assumptions on the entropy function S(x). As an example, the authors get the uniqueness of stationary solutions with vacuum of Euler-Poisson equations for S(x) =|x|θandθ∈{0}∪[2(N-2),+∞).
文摘In this paper,we consider an inflow problem for the non-isentropic Navier-StokesPoisson system in a half line(0,∞).For the general gas including ideal polytropic gas,we first give some results for the existence of the stationary solution with the aid of center manifold theory on a 4×4 system of autonomous ordinary differential equations.We also show the time asymptotic stability of the stationary solutions with small strength under smallness assumptions on the initial perturbations by using an elementary energy method.
基金The NSF(10971046 and 11371117) of Chinathe Shandong Provincial Natural Science Foundation(ZR2013AM009)+2 种基金GIIFSDU(yzc12063)IIFSDU(2012TS020)the Project of Shandong Province Higher Educational Science and Technology Program(J09LA55)
文摘In this work, we are mainly concerned with the existence of stationary solutions for a generalized Kadomtsev-Petviashvili equation in bounded domain of Rn. We utilize variational method and critical point theory to establish our main results.
基金the paper is supported by the National Natural Science Foundation of China(Nos.11871047,11671384,11931010)the key research project of Academy for Multidisciplinary Studies,Capital Normal University,and by the Capacity Building for Sci-Tech Innovation-Fundamental Scientific Research Funds(No.007/20530290068).
文摘The outflow problem for the viscous two-phase flow model in a half line is investigated in the present paper.The existence and uniqueness of the stationary solution is shown for both supersonic state and sonic state at spatial far field,and the nonlinear time stability of the stationary solution is also established in the weighted Sobolev space with either the exponential time decay rate for supersonic flow or the algebraic time decay rate for sonic flow.
基金Project supported by the Grant-in-Aid for Scientific Research (C) (No. 136470207)the Japan Society for the Promotion of Science (JSPS)+1 种基金the Strategic Research Grant of City University of Hong Kong (No.7001608)the National Natural Science Foundation of China (No.10431060, No.10329101).
文摘For the Boltzmann equation with an external force in the form of the gradient of a potential function in space variable, the stability of its stationary solutions as local Maxwellians was studied by S. Ukai et al. （2005） through the energy method. Based on this stability analysis and some techniques on analyzing the convergence rates to stationary solutions for the compressible Navier-Stokes equations, in this paper, we study the convergence rate to the above stationary solutions for the Boltzmann equation which is a fundamental equation in statistical physics for non-equilibrium rarefied gas. By combining the dissipation from the viscosity and heat conductivity on the fluid components and the dissipation on the non-fluid component through the celebrated H-theorem, a convergence rate of the same order as the one for the compressible Navier-Stokes is obtained by constructing some energy functionals.
基金Project Supported by The National Natural Science Foundation of China
文摘The exact solutions for stationary responses of one class of the second order and three classes of higher order nonlinear systems to parametric and/or external while noise excitations are constructed by using Fokkcr-Planck-Kolmogorov et/ualion approach. The conditions for the existence and uniqueness and the behavior of the solutions are discussed. All the systems under consideration are characterized by the dependence ofnonconservative fqrces on the first integrals of the corresponding conservative systems and arc catted generalized-energy-dependent f G.E.D.) systems. It is shown taht for each of the four classes of G.E.D. nonlinear stochastic systems there is a family of non-G.E.D. systems which are equivalent to the G.E.D. system in the sense of having identical stationary solution. The way to find the equivalent stochastic systems for a given G.E.D. system is indicated and. as an example, the equivalent stochastic systems for the second order G.E. D. nonlinear stochastic system are given. It is pointed out and illustrated with example that the exact stationary solutions for many non-G.E.D. nonlinear stochastic systems may he found by searching the equivalent G.E.D. systems.
基金supported by the National Natural Science Foundation of China (Nos.12301101,12101121)the Guangdong Basic and Applied Basic Research Foundation (Nos.2022A1515110019,2020A1515110585)。
文摘In this paper,we study the asymptotic dynamics of a single-species model with resource-dependent dispersal in one dimension.To overcome the analytical difficulties brought by the resource-dependent dispersal,we use the idea of changing variables to transform the model into a uniform dispersal one.Then the existence and uniqueness of positive stationary solution to the model can be verified by the squeezing argument,where the solution plays a crucial role in later analyses.Moreover,the asymptotic behavior of solutions to the model is obtained by the upper-lower solutions method.The result indicates that the solutions of the model converge to the corresponding positive stationary solution locally uniformly in one dimension as time goes to infinity.
文摘In this paper, the nonlinear Schrodinger (NLS) equation was analytically solved. Firstly, the stationary solutions of NLS equation were explicitly given by the elliptic functions. Then a family of exact solutions of NLS equation were obtained from these stationary solutions by a method for finding new exact solutions from the stationary solutions of integrable evolution equations.
基金supported by the National Natural Science Foundation of China(11331005,11471134)the Program for Changjiang Scholars and Innovative Research Team in University(IRT13066)the Scientific Research Funds of Huaqiao University(15BS201,15BS309)
文摘In this article, we are concerned with the stability of stationary solution for outflow problem on the Navier-Stokes-Poisson system. We obtain the unique existence and the asymptotic stability of stationary solution. Moreover, the convergence rate of solution towards stationary solution is obtained. Precisely, if an initial perturbation decays with the algebraic or the exponential rate in space, the solution converges to the corresponding stationary solution as time tends to infinity with the algebraic or the exponential rate in time. The proof is based on the weighted energy method by taking into account the effect of the self-consistent electric field on the viscous compressible fluid.
文摘This paper considers a stochastic Lienard equation with Markovian switching. The Feller continuity of its solution is proved by the coupling method and a truncation argument. The existence of a stationary solution for the equation is also proved under the Foster-Lyapunov drift condition.
基金supported by the National Natural Science Foundation of China(11302157)Fundamental Research Funds for the Central Universities(K5051370008)Chinese-Serbian Science&Technology Cooperation(2-14)
文摘This paper studies the stationary probability density function（PDF） solution of a nonlinear business cycle model subjected to random shocks of Gaussian white-noise type. The PDF solution is controlled by a Fokker–Planck– Kolmogorov（FPK） equation, and we use exponential polynomial closure（EPC） method to derive an approximate solution for the FPK equation. Numerical results obtained from EPC method, better than those from Gaussian closure method, show good agreement with the probability distribution obtained with Monte Carlo simulation including the tail regions.
文摘In this paper we are interested in the large time behavior of the nonlinear diffusion equationWe consider functions which allow the equation to possess traveling wave solutions. We first present an existence and uniqueness as well as some comparison principle result of generalized solutions to the Cauchy problem. Then we give for some threshold results, from which we can see that u=a is stable, while u= 0 or u=1 is unstable under some assumptions, etc.
基金the Vital Science Research Foundation of Henan Province Education Department(No.12A110024)
文摘The existence of classical solutions to a stationary simplified quantum energytransport model for semiconductor devices in 1-dimensional space is proved.The model consists of a nonlinear elliptic third-order equation for the electron density,including a temperature derivative,an elliptic nonlinear heat equation for the electron temperature,and the Poisson equation for the electric potential.The proof is based on an exponential variable transformation and the Leray-Schauder fixed-point theorem.
基金supported by the National Natural Science Foundation of China(10932009,11072212,11272279,and 11321202)
文摘The classical Lotka-Volterra （LV） model is a well-known mathematical model for prey-predator ecosystems. In the present paper, the pulse-type version of stochastic LV model, in which the effect of a random natural environment has been modeled as Poisson white noise, is in- vestigated by using the stochastic averaging method. The averaged generalized It6 stochastic differential equation and Fokkerlanck-Kolmogorov （FPK） equation are derived for prey-predator ecosystem driven by Poisson white noise. Approximate stationary solution for the averaged generalized FPK equation is obtained by using the perturbation method. The effect of prey self-competition parameter e2s on ecosystem behavior is evaluated. The analytical result is confirmed by corresponding Monte Carlo （MC） simulation.
文摘We give the definition of the weak solution for the equilibrium systems of ferro-magnetic chain and get the existence result of the Dirichlet problems for this systems. We get the regularity result for the weakly stationary solution.
文摘The basic equations of free capillary_gravity surface_waves in a circular cylindrical basin were derived from Luke's principle. Taking Galerkin's expansion of the velocity potential and the free surface elevation, the second_order perturbation equations were derived by use of expansion of multiple scale. The nonlinear interactions with the second order internal resonance of three free surface_waves were discussed based on the above. The results include:derivation of the couple equations of resonant interactions among three waves and the conservation laws; analysis of the positions of equilibrium points in phase plane; study of the resonant parameters and the non_resonant parameters respectively in all kinds of circumstances; derivation of the stationary solutions of the second_order interaction equations corresponding to different parameters and analysis of the stability property of the solutions; discussion of the effective solutions only in the limited time range. The analysis makes it clear that the energy transformation mode among three waves differs because of the different initial conditions under nontrivial circumstance. The energy may either exchange among three waves periodically or damp or increase in single waves.
基金supported by National Natural Science Foundation of China(Grant No.11790273,52276028).
文摘The existence and stability of stationary solutions for a reaction-diffusion-ODE system are investigated in this paper.We first show that there exist both continuous and discontinuous stationary solutions.Then a good understanding of the stability of discontinuous stationary solutions is gained under an appropriate condition.In addition,we demonstrate the influences of the diffusion coefficient on stationary solutions.The results we obtained are based on the super-/sub-solution method and the generalized mountain pass theorem.Finally,some numerical simulations are given to illustrate the theoretical results.
