In this paper,the dynamical behavior of a reaction-diffusion system with quiescence in a closed environment is investigated.The global existence of the solution is obtained by the upper and lower solution method,and t...In this paper,the dynamical behavior of a reaction-diffusion system with quiescence in a closed environment is investigated.The global existence of the solution is obtained by the upper and lower solution method,and the dissipative structure of the system is derived by constructing Lyapunov functions.展开更多
This paper theoretically analyses and studies stationary patterns in diffusively coupled bistable elements. Since these stationary patterns consist of two types of stationary mode structure: kink and pulse, a mode an...This paper theoretically analyses and studies stationary patterns in diffusively coupled bistable elements. Since these stationary patterns consist of two types of stationary mode structure: kink and pulse, a mode analysis method is proposed to approximate the solutions of these localized basic modes and to analyse their stabilities. Using this method, it reconstructs the whole stationary patterns. The cellular mode structures (kink and pulse) in bistable media fundamentally differ from stationary patterns in monostable media showing spatial periodicity induced by a diffusive Taring bifurcation.展开更多
In this paper,we study a semi-linear reaction-diffusion system with a weighted nonlocal source,subject to the null Dirichlet boundary condition.Under certain conditions,we prove that the classical solution exists glob...In this paper,we study a semi-linear reaction-diffusion system with a weighted nonlocal source,subject to the null Dirichlet boundary condition.Under certain conditions,we prove that the classical solution exists globally and blows up in finite time respectively,and then obtain the uniform blow-up rate in the interior.展开更多
The complex Ginzburg-Landau equation (CGLE) has been used to describe the travelling wave behaviour in reaction-diffusion (RD) systems. We argue that this description is valid only when the RD system is close to t...The complex Ginzburg-Landau equation (CGLE) has been used to describe the travelling wave behaviour in reaction-diffusion (RD) systems. We argue that this description is valid only when the RD system is close to the Hopf bifurcation, and is not valid when a RD system is away from the onset. To test this, we study spirals and anti-spirals in the chlorite-iodide-malonic acid (CIMA) reaction and the corresponding OGLE. Numerical simulations confirm that the OGLE can only be applied to the CIMA reaction when it is very near the Hopf onset.展开更多
By the degree theory on positive cone together with the technique of a priori estimate, the nontrivial equilibrium solutions of a strong nonlinearity and weak coupling reaction diffusion system and the structure of t...By the degree theory on positive cone together with the technique of a priori estimate, the nontrivial equilibrium solutions of a strong nonlinearity and weak coupling reaction diffusion system and the structure of the equilibrium solutions are discussed.展开更多
In this paper, the problem of initial boundary value for nonlinear coupled reaction-diffusion systems arising in biochemistry, engineering and combustion_theory is considered.
This paper is concerned with an Initial Boundary Value Problem (IBVP) for a strongly coupled semilinear reaction-diffusion system. By using the upper and lower solutions method and Leray-Schauder fixed point theorem a...This paper is concerned with an Initial Boundary Value Problem (IBVP) for a strongly coupled semilinear reaction-diffusion system. By using the upper and lower solutions method and Leray-Schauder fixed point theorem and so on, the authors prove the global existence and uniqueness of a. smooth. solution for this IBVP under some appropriate conditions.展开更多
In this paper, we study properties of solutions to doubly nonlinear reaction-diffusion systems with variable density and source. We demonstrate the possibilities of the self-similar approach to studying the qualitativ...In this paper, we study properties of solutions to doubly nonlinear reaction-diffusion systems with variable density and source. We demonstrate the possibilities of the self-similar approach to studying the qualitative properties of solutions of such reaction-diffusion systems. We also study the finite speed of propagation (FSP) properties of solutions, an asymptotic behavior of the compactly supported solutions and free boundary asymptotic solutions in quick diffusive and critical cases.展开更多
Differential method and homotopy analysis method are used for solving the two-dimensional reaction-diffusion model. And the structure of the solutions is analyzed. Finally, the homotopy series solutions are simulated ...Differential method and homotopy analysis method are used for solving the two-dimensional reaction-diffusion model. And the structure of the solutions is analyzed. Finally, the homotopy series solutions are simulated with the mathematical software Matlab, so the Turing patterns will be produced. Overall analysis and experimental simulation of the model show that the different parameters lead to different Turing pattern structures. As time goes on, the structure of Turing patterns changes, and the final solutions tend to stationary state.展开更多
Turing demonstrated that spatially heterogeneous patterns can be self-organized, when the two substances interact locally and diffuse randomly. Turing systems have been applied not only to explain patterns observed wi...Turing demonstrated that spatially heterogeneous patterns can be self-organized, when the two substances interact locally and diffuse randomly. Turing systems have been applied not only to explain patterns observed within the biological and chemical fields, but also to develop image information processing tools. In a twin study, to evaluate the V-shaped bundle of the inner ear outer hair, we developed a method that utilizes a reaction-diffusion system with anisotropic diffusion that exhibited triangular patterns with the introduction of a certain anisotropy strength. In this study, we explored the parameter range over which these periodic triangular patterns were obtained. First, we defined an index for triangular clearness, TC. Triangular patterns can be obtained by introducing a large anisotropy δ, but the range of δ depends on the diffusion coefficient. We found an explanatory variable that can explain the change in TC based on a heuristic argument of the relative distance of the pitchfork bifurcation point between the maximum and minimum anisotropic diffusion function values. Clear periodic triangular patterns were obtained when the distance between the minimum anisotropic function value and pitchfork bifurcation point was over 2.5 times the distance to the anisotropic diffusion function maximum value. By changing the diffusion coefficients or the reaction terms, we further confirmed the accuracy of this condition using computer simulation. Its relevance to diffusion instability has also been discussed.展开更多
This paper focuses on the analytical and numerical asymptotical stability of neutral reaction-diffusion equations with piecewise continuous arguments.First,for the analytical solutions of the equations,we derive their...This paper focuses on the analytical and numerical asymptotical stability of neutral reaction-diffusion equations with piecewise continuous arguments.First,for the analytical solutions of the equations,we derive their expressions and asymptotical stability criteria.Second,for the semi-discrete and one-parameter fully-discrete finite element methods solving the above equations,we work out the sufficient conditions for assuring that the finite element solutions are asymptotically stable.Finally,with a typical example with numerical experiments,we illustrate the applicability of the obtained theoretical results.展开更多
In this paper, the existence and nonexistence of finite travelling waves (FTWs) for a semilinear degenerate reaction-diffusion system (u<sub>i</sub><sup>αi</sup>t=u<sub>ixx</sub>...In this paper, the existence and nonexistence of finite travelling waves (FTWs) for a semilinear degenerate reaction-diffusion system (u<sub>i</sub><sup>αi</sup>t=u<sub>ixx</sub>-multiply from j=1 to N u<sub>j</sub><sup>mij</sup>, x∈R, t】0,i=1,. . . ,N (Ⅰ) is studied. where 0【a<sub>i</sub>【1. mij≥0 and sum from j=1 to N mij】0, i, j=1, . . . ,N .Necessary and sufficient conditions on existence and large time behaviours of FTWs of (Ⅰ) are obtained by using the matrix theory. Schauder’s fixed point theorem, and upper and lower solutious method.展开更多
Boundary control for a class of partial integro-differential systems with space and time dependent coefficients is consid- ered. A control law is derived via the partial differential equation (PDE) backstepping. The...Boundary control for a class of partial integro-differential systems with space and time dependent coefficients is consid- ered. A control law is derived via the partial differential equation (PDE) backstepping. The existence of kernel equations is proved. Exponential stability of the closed-loop system is achieved. Simulation results are presented through figures.展开更多
This paper is concerned with the spreading speeds of time dependent partially degenerate reaction-diffusion systems with monostable nonlinearity.By using the principal Lyapunov exponent theory,the author first proves ...This paper is concerned with the spreading speeds of time dependent partially degenerate reaction-diffusion systems with monostable nonlinearity.By using the principal Lyapunov exponent theory,the author first proves the existence,uniqueness and stability of spatially homogeneous entire positive solution for time dependent partially degenerate reaction-diffusion system.Then the author shows that such system has a finite spreading speed interval in any direction and there is a spreading speed for the partially degenerate system under certain conditions.The author also applies these results to a time dependent partially degenerate epidemic model.展开更多
In this paper, the nonlocal nonlinear reaction-diffusion singularly perturbed problems with two parameters are studied. Using a singular perturbation method, the structure of the solutions to the problem is discussed ...In this paper, the nonlocal nonlinear reaction-diffusion singularly perturbed problems with two parameters are studied. Using a singular perturbation method, the structure of the solutions to the problem is discussed in relation to two small parameters. The asymptotic solutions of the problem are given.展开更多
This work is concerned with the numerical simulations for two reactiondiffusion systems,i.e.,the Brusselator model and the Gray-Scott model.The numerical algorithm is based upon a moving finite element method which he...This work is concerned with the numerical simulations for two reactiondiffusion systems,i.e.,the Brusselator model and the Gray-Scott model.The numerical algorithm is based upon a moving finite element method which helps to resolve large solution gradients.High quality meshes are obtained for both the spot replication and the moving wave along boundaries by using proper monitor functions.Unlike[33],this work finds out the importance of the boundary grid redistribution which is particularly important for a class of problems for the Brusselator model.Several ways for verifying the quality of the numerical solutions are also proposed,which may be of important use for comparisons.展开更多
In this article, a two-species predator-prey reaction-diffusion system with Holling type-IV functional response and subject to the homogeneous Neumann boundary condition is regarded. In the absence of the spatial diff...In this article, a two-species predator-prey reaction-diffusion system with Holling type-IV functional response and subject to the homogeneous Neumann boundary condition is regarded. In the absence of the spatial diffusion, the local asymptotic stability, the instability and the existence of Hopf bifurcation of the positive equilibria of the corresponding local system are analyzed in detail by means of the basic theory for dynamical systems. As well, the effect of the spatial diffusion on the stability of the positive equilibria is considered by using the linearized method and analyzing in detail the distribution of roots in the complex plane of the associated eigenvalue problem. In order to verify the obtained theoretical predictions, some examples and numerical simulations are also included by applying the numerical methods to solve the ordinary and partial differential equations.展开更多
Singular perturbation reaction-diffusion problem with Dirichlet boundary condition is considered. It is a multi-scale problem. Presence of small parameter leads to boundary layer phenomena in both sides of the region....Singular perturbation reaction-diffusion problem with Dirichlet boundary condition is considered. It is a multi-scale problem. Presence of small parameter leads to boundary layer phenomena in both sides of the region. A non-equidistant finite difference method is presented according to the property of boundary layer. The region is divided into an inner boundary layer region and an outer boundary layer region according to transition point of Shishkin. The steps sizes are equidistant in the outer boundary layer region. The step sizes are gradually increased in the inner boundary layer region such that half of the step sizes are different from each other. Truncation error is estimated. The proposed method is stable and uniformly convergent with the order higher than 2. Numerical results are given, which are in agreement with the theoretical result.展开更多
The stochastic cracking and healing behaviors of reaction-diffusion growth of thin filmswere studied by means of Markov processes analysis. We chose the thermal growth ofoxide scales on metals as an example of reactio...The stochastic cracking and healing behaviors of reaction-diffusion growth of thin filmswere studied by means of Markov processes analysis. We chose the thermal growth ofoxide scales on metals as an example of reaction-diffusion growth. The thermal growthof oxide films follows power law when no cracking occurs. Our results showed that thegrowth kinetics under stochastic cracking and healing conditions was different fromthat without cracking. It might be altered to either pseudo-linear or pseudo-power lawsdependent upon the intensity and frequency of the cracking of the films. When thehoping items dominated, the growth followed pseudo-linear law; when the diffusionalitems dominated, it followed pseudo-power law with the exponentials lower than theintrinsical values. The numerical results were in good agreement with the meassuredkinetics of isothermal and cyclic oxidation of NiAl-0.1 Y (at. %) alloys in air at 1273K.展开更多
This paper is concerned with the stability of traveling wavefronts for a population dynamics model with time delay. Combining the weighted energy method and the comparison principle, the global exponential stability o...This paper is concerned with the stability of traveling wavefronts for a population dynamics model with time delay. Combining the weighted energy method and the comparison principle, the global exponential stability of noncritical traveling wavefronts (waves with speeds c 〉 c*, where c=c* is the minimal speed) is established, when the initial perturbations around the wavefront decays to zero exponentially in space as x → -∞, but it can be allowed arbitrary large in other locations, which improves the results in[9, 18, 21].展开更多
文摘In this paper,the dynamical behavior of a reaction-diffusion system with quiescence in a closed environment is investigated.The global existence of the solution is obtained by the upper and lower solution method,and the dissipative structure of the system is derived by constructing Lyapunov functions.
基金Project partially supported by the Outstanding Oversea Scholar Foundation of the Chinese Academy of Sciences (Bairenjihua)the Scientific Research Foundation for the Returned Overseas Chinese Scholars,State Education Ministry
文摘This paper theoretically analyses and studies stationary patterns in diffusively coupled bistable elements. Since these stationary patterns consist of two types of stationary mode structure: kink and pulse, a mode analysis method is proposed to approximate the solutions of these localized basic modes and to analyse their stabilities. Using this method, it reconstructs the whole stationary patterns. The cellular mode structures (kink and pulse) in bistable media fundamentally differ from stationary patterns in monostable media showing spatial periodicity induced by a diffusive Taring bifurcation.
基金Project supported by the Research Program of Natural Science of Universities in Jiangsu Province (Grant No.09KJD110008)the Natural Science Foundation of Nanjing Xiaozhuang University (Grant No.005NXY11)
文摘In this paper,we study a semi-linear reaction-diffusion system with a weighted nonlocal source,subject to the null Dirichlet boundary condition.Under certain conditions,we prove that the classical solution exists globally and blows up in finite time respectively,and then obtain the uniform blow-up rate in the interior.
基金Project supported by the National Natural Science Foundation of China (Grant No 10274003) and the Department of Science and Technology of China.Acknowledgement We thank Cheng X, Wang C and Wang S for helpful discussion.
文摘The complex Ginzburg-Landau equation (CGLE) has been used to describe the travelling wave behaviour in reaction-diffusion (RD) systems. We argue that this description is valid only when the RD system is close to the Hopf bifurcation, and is not valid when a RD system is away from the onset. To test this, we study spirals and anti-spirals in the chlorite-iodide-malonic acid (CIMA) reaction and the corresponding OGLE. Numerical simulations confirm that the OGLE can only be applied to the CIMA reaction when it is very near the Hopf onset.
文摘By the degree theory on positive cone together with the technique of a priori estimate, the nontrivial equilibrium solutions of a strong nonlinearity and weak coupling reaction diffusion system and the structure of the equilibrium solutions are discussed.
文摘In this paper, the problem of initial boundary value for nonlinear coupled reaction-diffusion systems arising in biochemistry, engineering and combustion_theory is considered.
文摘This paper is concerned with an Initial Boundary Value Problem (IBVP) for a strongly coupled semilinear reaction-diffusion system. By using the upper and lower solutions method and Leray-Schauder fixed point theorem and so on, the authors prove the global existence and uniqueness of a. smooth. solution for this IBVP under some appropriate conditions.
文摘In this paper, we study properties of solutions to doubly nonlinear reaction-diffusion systems with variable density and source. We demonstrate the possibilities of the self-similar approach to studying the qualitative properties of solutions of such reaction-diffusion systems. We also study the finite speed of propagation (FSP) properties of solutions, an asymptotic behavior of the compactly supported solutions and free boundary asymptotic solutions in quick diffusive and critical cases.
文摘Differential method and homotopy analysis method are used for solving the two-dimensional reaction-diffusion model. And the structure of the solutions is analyzed. Finally, the homotopy series solutions are simulated with the mathematical software Matlab, so the Turing patterns will be produced. Overall analysis and experimental simulation of the model show that the different parameters lead to different Turing pattern structures. As time goes on, the structure of Turing patterns changes, and the final solutions tend to stationary state.
文摘Turing demonstrated that spatially heterogeneous patterns can be self-organized, when the two substances interact locally and diffuse randomly. Turing systems have been applied not only to explain patterns observed within the biological and chemical fields, but also to develop image information processing tools. In a twin study, to evaluate the V-shaped bundle of the inner ear outer hair, we developed a method that utilizes a reaction-diffusion system with anisotropic diffusion that exhibited triangular patterns with the introduction of a certain anisotropy strength. In this study, we explored the parameter range over which these periodic triangular patterns were obtained. First, we defined an index for triangular clearness, TC. Triangular patterns can be obtained by introducing a large anisotropy δ, but the range of δ depends on the diffusion coefficient. We found an explanatory variable that can explain the change in TC based on a heuristic argument of the relative distance of the pitchfork bifurcation point between the maximum and minimum anisotropic diffusion function values. Clear periodic triangular patterns were obtained when the distance between the minimum anisotropic function value and pitchfork bifurcation point was over 2.5 times the distance to the anisotropic diffusion function maximum value. By changing the diffusion coefficients or the reaction terms, we further confirmed the accuracy of this condition using computer simulation. Its relevance to diffusion instability has also been discussed.
文摘This paper focuses on the analytical and numerical asymptotical stability of neutral reaction-diffusion equations with piecewise continuous arguments.First,for the analytical solutions of the equations,we derive their expressions and asymptotical stability criteria.Second,for the semi-discrete and one-parameter fully-discrete finite element methods solving the above equations,we work out the sufficient conditions for assuring that the finite element solutions are asymptotically stable.Finally,with a typical example with numerical experiments,we illustrate the applicability of the obtained theoretical results.
基金Project supported by the Postdoctoral Science Foundation of China the Henan Province Natural Science Foundation of China
文摘In this paper, the existence and nonexistence of finite travelling waves (FTWs) for a semilinear degenerate reaction-diffusion system (u<sub>i</sub><sup>αi</sup>t=u<sub>ixx</sub>-multiply from j=1 to N u<sub>j</sub><sup>mij</sup>, x∈R, t】0,i=1,. . . ,N (Ⅰ) is studied. where 0【a<sub>i</sub>【1. mij≥0 and sum from j=1 to N mij】0, i, j=1, . . . ,N .Necessary and sufficient conditions on existence and large time behaviours of FTWs of (Ⅰ) are obtained by using the matrix theory. Schauder’s fixed point theorem, and upper and lower solutious method.
文摘Boundary control for a class of partial integro-differential systems with space and time dependent coefficients is consid- ered. A control law is derived via the partial differential equation (PDE) backstepping. The existence of kernel equations is proved. Exponential stability of the closed-loop system is achieved. Simulation results are presented through figures.
基金This work was supported by the National Natural Science Foundation of China(Nos.41801029,11701041)the Natural Science Basic Research Plan in Shaanxi Province of China(No.2020JM-223).
文摘This paper is concerned with the spreading speeds of time dependent partially degenerate reaction-diffusion systems with monostable nonlinearity.By using the principal Lyapunov exponent theory,the author first proves the existence,uniqueness and stability of spatially homogeneous entire positive solution for time dependent partially degenerate reaction-diffusion system.Then the author shows that such system has a finite spreading speed interval in any direction and there is a spreading speed for the partially degenerate system under certain conditions.The author also applies these results to a time dependent partially degenerate epidemic model.
基金supported by the National Natural Science Foundation of China (Nos. 40676016, 40876010)the Knowledge Innovation Program of Chinese Academy of Sciences (No. KZCX2-YW-Q03-08)+1 种基金the LASG State Key Laboratory Special Fundthe E-Institute of Shanghai Municipal Education Commission (No. E03004)
文摘In this paper, the nonlocal nonlinear reaction-diffusion singularly perturbed problems with two parameters are studied. Using a singular perturbation method, the structure of the solutions to the problem is discussed in relation to two small parameters. The asymptotic solutions of the problem are given.
基金The first and the third authors are partially supported by HKBU FRG grants and the Hong Kong Research Grant CouncilThe second author is partially supported by the Hong Kong RGC grant(No.201710).
文摘This work is concerned with the numerical simulations for two reactiondiffusion systems,i.e.,the Brusselator model and the Gray-Scott model.The numerical algorithm is based upon a moving finite element method which helps to resolve large solution gradients.High quality meshes are obtained for both the spot replication and the moving wave along boundaries by using proper monitor functions.Unlike[33],this work finds out the importance of the boundary grid redistribution which is particularly important for a class of problems for the Brusselator model.Several ways for verifying the quality of the numerical solutions are also proposed,which may be of important use for comparisons.
基金Supported by the National Natural Science Foundation of China(61563026)
文摘In this article, a two-species predator-prey reaction-diffusion system with Holling type-IV functional response and subject to the homogeneous Neumann boundary condition is regarded. In the absence of the spatial diffusion, the local asymptotic stability, the instability and the existence of Hopf bifurcation of the positive equilibria of the corresponding local system are analyzed in detail by means of the basic theory for dynamical systems. As well, the effect of the spatial diffusion on the stability of the positive equilibria is considered by using the linearized method and analyzing in detail the distribution of roots in the complex plane of the associated eigenvalue problem. In order to verify the obtained theoretical predictions, some examples and numerical simulations are also included by applying the numerical methods to solve the ordinary and partial differential equations.
基金supported by the Educational Department Foundation of Fujian Province of China(Nos. JA08140 and A0610025)the Scientific Research Foundation of Zhejiang University of Scienceand Technology (No. 2008050)the National Natural Science Foundation of China (No. 50679074)
文摘Singular perturbation reaction-diffusion problem with Dirichlet boundary condition is considered. It is a multi-scale problem. Presence of small parameter leads to boundary layer phenomena in both sides of the region. A non-equidistant finite difference method is presented according to the property of boundary layer. The region is divided into an inner boundary layer region and an outer boundary layer region according to transition point of Shishkin. The steps sizes are equidistant in the outer boundary layer region. The step sizes are gradually increased in the inner boundary layer region such that half of the step sizes are different from each other. Truncation error is estimated. The proposed method is stable and uniformly convergent with the order higher than 2. Numerical results are given, which are in agreement with the theoretical result.
基金supported by Hundred-Talent Project of Chinese Academy of Sciencesby the National Natural Science Foundation of China for Young Scientist
文摘The stochastic cracking and healing behaviors of reaction-diffusion growth of thin filmswere studied by means of Markov processes analysis. We chose the thermal growth ofoxide scales on metals as an example of reaction-diffusion growth. The thermal growthof oxide films follows power law when no cracking occurs. Our results showed that thegrowth kinetics under stochastic cracking and healing conditions was different fromthat without cracking. It might be altered to either pseudo-linear or pseudo-power lawsdependent upon the intensity and frequency of the cracking of the films. When thehoping items dominated, the growth followed pseudo-linear law; when the diffusionalitems dominated, it followed pseudo-power law with the exponentials lower than theintrinsical values. The numerical results were in good agreement with the meassuredkinetics of isothermal and cyclic oxidation of NiAl-0.1 Y (at. %) alloys in air at 1273K.
基金supported by NSF of China(11401478)Gansu Provincial Natural Science Foundation(145RJZA220)
文摘This paper is concerned with the stability of traveling wavefronts for a population dynamics model with time delay. Combining the weighted energy method and the comparison principle, the global exponential stability of noncritical traveling wavefronts (waves with speeds c 〉 c*, where c=c* is the minimal speed) is established, when the initial perturbations around the wavefront decays to zero exponentially in space as x → -∞, but it can be allowed arbitrary large in other locations, which improves the results in[9, 18, 21].