This paper addresses a nonlinear partial differential control system arising in population dynamics.The system consist of three diffusion equations describing the evolutions of three biological species:prey,predator,a...This paper addresses a nonlinear partial differential control system arising in population dynamics.The system consist of three diffusion equations describing the evolutions of three biological species:prey,predator,and food for the prey or vegetation.The equation for the food density incorporates a hysteresis operator of generalized stop type accounting for underlying hysteresis effects occurring in the dynamical process.We study the problem of minimization of a given integral cost functional over solutions of the above system.The set-valued mapping defining the control constraint is state-dependent and its values are nonconvex as is the cost integrand as a function of the control variable.Some relaxationtype results for the minimization problem are obtained and the existence of a nearly optimal solution is established.展开更多
In this work, we apply the bifurcation method of dynamical systems to investigate the underlying complex dynamics of traveling wave solutions to a highly nonlinear Fujimoto–Watanabe equation. We identify all bifurcat...In this work, we apply the bifurcation method of dynamical systems to investigate the underlying complex dynamics of traveling wave solutions to a highly nonlinear Fujimoto–Watanabe equation. We identify all bifurcation conditions and phase portraits of the system in different regions of the three-dimensional parametric space, from which we present the sufficient conditions to guarantee the existence of traveling wave solutions including solitary wave solutions, periodic wave solutions, kink-like(antikink-like) wave solutions, and compactons. Furthermore, we obtain their exact expressions and simulations, which can help us understand the underlying physical behaviors of traveling wave solutions to the equation.展开更多
We study the heat flow of equation of H-surface with non-zero Dirichlet boundary in the present article. Introducing the "stable set" M2 and "unstable set" M1, we show that there exists a unique gl...We study the heat flow of equation of H-surface with non-zero Dirichlet boundary in the present article. Introducing the "stable set" M2 and "unstable set" M1, we show that there exists a unique global solution provided the initial data belong to M2 and the global solution converges to zero in H^1 exponentially as time goes to infinity. Moreover, we also prove that the local regular solution must blow up at finite time provided the initial data belong to M1.展开更多
We prove the global existence and exponential decay of strong solutions to the three-dimensional nonhomogeneous asymmetric fluid equations with nonnegative density provided that the initial total energy is suitably sm...We prove the global existence and exponential decay of strong solutions to the three-dimensional nonhomogeneous asymmetric fluid equations with nonnegative density provided that the initial total energy is suitably small.Note that although the system degenerates near vacuum,there is no need to require compatibility conditions for the initial data via time-weighted techniques.展开更多
A conformal multi-symplectic method has been proposed for the damped Korteweg–de Vries(DKdV) equation, which is based on the conformal multi-symplectic structure. By using the Strang-splitting method and the Preissma...A conformal multi-symplectic method has been proposed for the damped Korteweg–de Vries(DKdV) equation, which is based on the conformal multi-symplectic structure. By using the Strang-splitting method and the Preissmann box scheme,we obtain a conformal multi-symplectic scheme for multi-symplectic partial differential equations(PDEs) with added dissipation. Applying it to the DKdV equation, we construct a conformal multi-symplectic algorithm for it, which is of second order accuracy in time. Numerical experiments demonstrate that the proposed method not only preserves the dissipation rate of mass exactly with periodic boundary conditions, but also has excellent long-time numerical behavior.展开更多
In this paper,we derive a residual based a posteriori error estimator for a modified weak Galerkin formulation of second order elliptic problems.We prove that the error estimator used for interior penalty discontinuou...In this paper,we derive a residual based a posteriori error estimator for a modified weak Galerkin formulation of second order elliptic problems.We prove that the error estimator used for interior penalty discontinuous Galerkin methods still gives both upper and lower bounds for the modified weak Galerkin method,though they have essentially different bilinear forms.More precisely,we prove its reliability and efficiency for the actual error measured in the standard DG norm.We further provide an improved a priori error estimate under minimal regularity assumptions on the exact solution.Numerical results are presented to verify the theoretical analysis.展开更多
In this paper,we consider a planar piecewise smooth differential system consisting of a linear system and a quadratic Hamiltonian system.The quadratic system has some folds on the discontinuity line.The linear system ...In this paper,we consider a planar piecewise smooth differential system consisting of a linear system and a quadratic Hamiltonian system.The quadratic system has some folds on the discontinuity line.The linear system may have a focus,saddle or node.Our results show that this piecewise smooth differential system will have two limit cycles and a sliding cycle.Moreover,this piecewise smooth system will undergo pseudo-homoclinic bifurcation,Hopf bifurcation and critical crossing bifurcation CC.Some examples are given to illustrate our results.展开更多
Combining symplectic algorithm,splitting technique and compact method,a compact splitting symplectic scheme is proposed to solve the fourth-order dispersive Schr¨odinger equation with cubic-quintic nonlinear term...Combining symplectic algorithm,splitting technique and compact method,a compact splitting symplectic scheme is proposed to solve the fourth-order dispersive Schr¨odinger equation with cubic-quintic nonlinear term.The scheme has fourth-order accuracy in space and second-order accuracy in time.The discrete charge conservation law and stability of the scheme are analyzed.Numerical examples are given to confirm the theoretical results.展开更多
基金supported by National Natural Science Foundation of China(12071165 and 62076104)Natural Science Foundation of Fujian Province(2020J01072)+2 种基金Program for Innovative Research Team in Science and Technology in Fujian Province University,Quanzhou High-Level Talents Support Plan(2017ZT012)Scientific Research Funds of Huaqiao University(605-50Y 19017,605-50Y14040)supported by Ministry of Science and Higher Education of Russian Federation(075-15-2020-787,large scientific project"Fundamentals,methods and technologies for digital monitoring and forecasting of the environmental situation on the Baikal natural territory")。
文摘This paper addresses a nonlinear partial differential control system arising in population dynamics.The system consist of three diffusion equations describing the evolutions of three biological species:prey,predator,and food for the prey or vegetation.The equation for the food density incorporates a hysteresis operator of generalized stop type accounting for underlying hysteresis effects occurring in the dynamical process.We study the problem of minimization of a given integral cost functional over solutions of the above system.The set-valued mapping defining the control constraint is state-dependent and its values are nonconvex as is the cost integrand as a function of the control variable.Some relaxationtype results for the minimization problem are obtained and the existence of a nearly optimal solution is established.
基金Project supported by the National Natural Science Foundation of China(Grant Nos.11701191 and 11871232)the Program for Innovative Research Team in Science and Technology in University of Fujian Province,Quanzhou High-Level Talents Support Plan(Grant No.2017ZT012)the Subsidized Project for Cultivating Postgraduates’ Innovative Ability in Scientific Research of Huaqiao University
文摘In this work, we apply the bifurcation method of dynamical systems to investigate the underlying complex dynamics of traveling wave solutions to a highly nonlinear Fujimoto–Watanabe equation. We identify all bifurcation conditions and phase portraits of the system in different regions of the three-dimensional parametric space, from which we present the sufficient conditions to guarantee the existence of traveling wave solutions including solitary wave solutions, periodic wave solutions, kink-like(antikink-like) wave solutions, and compactons. Furthermore, we obtain their exact expressions and simulations, which can help us understand the underlying physical behaviors of traveling wave solutions to the equation.
基金supported by National Natural Science Foundation of China(11701193,11671086)Natural Science Foundation of Fujian Province(2018J05005)+3 种基金Program for Innovative Research Team in Science and Technology in Fujian Province University Quanzhou High-Level Talents Support Plan(2017ZT012)part supported by National Natural Science Foundation of China(11271305,11531010)Jiankai Xu’s research was in part supported by National Natural Science Foundation(11671086,11871208)Natural Science Foundation of Hunan Province(2018JJ2159)
文摘We study the heat flow of equation of H-surface with non-zero Dirichlet boundary in the present article. Introducing the "stable set" M2 and "unstable set" M1, we show that there exists a unique global solution provided the initial data belong to M2 and the global solution converges to zero in H^1 exponentially as time goes to infinity. Moreover, we also prove that the local regular solution must blow up at finite time provided the initial data belong to M1.
基金supported by National Natural Science Foundation of China(11701193,11671086)Natural Science Foundation of Fujian Province(2018J05005,2017J01562)+3 种基金Program for Innovative Research Team in Science and Technology in Fujian Province University Quanzhou High-Level Talents Support Plan(2017ZT012)supported by National Natural Science Foundation of China(11901474)the Chongqing Talent Plan for Young Topnotch Talents(CQYC202005074)the Innovation Support Program for Chongqing Overseas Returnees(cx2020082).
文摘We prove the global existence and exponential decay of strong solutions to the three-dimensional nonhomogeneous asymmetric fluid equations with nonnegative density provided that the initial total energy is suitably small.Note that although the system degenerates near vacuum,there is no need to require compatibility conditions for the initial data via time-weighted techniques.
基金Project supported by the Program for Innovative Research Team in Science and Technology in Fujian Province University,China,the Quanzhou High Level Talents Support Plan,China(Grant No.2017ZT012)the Promotion Program for Young and Middle-Aged Teacher in Science and Technology Research of Huaqiao University,China(Grant No.ZQN-YX502)
文摘A conformal multi-symplectic method has been proposed for the damped Korteweg–de Vries(DKdV) equation, which is based on the conformal multi-symplectic structure. By using the Strang-splitting method and the Preissmann box scheme,we obtain a conformal multi-symplectic scheme for multi-symplectic partial differential equations(PDEs) with added dissipation. Applying it to the DKdV equation, we construct a conformal multi-symplectic algorithm for it, which is of second order accuracy in time. Numerical experiments demonstrate that the proposed method not only preserves the dissipation rate of mass exactly with periodic boundary conditions, but also has excellent long-time numerical behavior.
基金The first author was supported by Guangdong Basic and Applied Basic Research Foundation(Grant Nos.2018A030307024 and 2020A1515011032)by National Natural Science Foundation of China(Grant No.11526097)+2 种基金The second author was supported by National Natural Science Foundation of China(Grant Nos.11871272 and 11871281)The third author was supported by National Natural Science Foundation of China(Grant No.11701197)The fourth author was supported by Guangdong Basic and Applied Basic Research Foundation(Grant No.2018A0303100016).
文摘In this paper,we derive a residual based a posteriori error estimator for a modified weak Galerkin formulation of second order elliptic problems.We prove that the error estimator used for interior penalty discontinuous Galerkin methods still gives both upper and lower bounds for the modified weak Galerkin method,though they have essentially different bilinear forms.More precisely,we prove its reliability and efficiency for the actual error measured in the standard DG norm.We further provide an improved a priori error estimate under minimal regularity assumptions on the exact solution.Numerical results are presented to verify the theoretical analysis.
基金partially supported by Postgraduate research and innovation ability cultivation plan of Huaqiao Universitypartially supported by NNSF of China grant11671040+3 种基金Cultivation Program for Outstanding Young Scientific talents of Fujian Province in 2017Program for Innovative Research Team in Science and Technology in Fujian Province UniversityQuanzhou High-Level Talents Support Plan under Grant2017ZT012Promotion Program for Young and Middle-aged Teacher in Science and Technology Research of Huaqiao University(ZQN-YX401)
文摘In this paper,we consider a planar piecewise smooth differential system consisting of a linear system and a quadratic Hamiltonian system.The quadratic system has some folds on the discontinuity line.The linear system may have a focus,saddle or node.Our results show that this piecewise smooth differential system will have two limit cycles and a sliding cycle.Moreover,this piecewise smooth system will undergo pseudo-homoclinic bifurcation,Hopf bifurcation and critical crossing bifurcation CC.Some examples are given to illustrate our results.
基金This work is partially supported by grants from the National Natural Science Foun-dation of China(Nos.11701196,11701197 and 11501224)the Promotion Program for Young and Middle-aged Teacher in Science and Technology Research of Huaqiao University(No.ZQN-YX502)the Key Laboratory of Applied Mathematics of Fujian Province University(Putian University)(No.SX201802).
文摘Combining symplectic algorithm,splitting technique and compact method,a compact splitting symplectic scheme is proposed to solve the fourth-order dispersive Schr¨odinger equation with cubic-quintic nonlinear term.The scheme has fourth-order accuracy in space and second-order accuracy in time.The discrete charge conservation law and stability of the scheme are analyzed.Numerical examples are given to confirm the theoretical results.