A new type of localized oscillatory pattern is presented in a two-layer coupled reaction-diffusion system under conditions in which no Hopf instability can be discerned in either layer.The transitions from stationary ...A new type of localized oscillatory pattern is presented in a two-layer coupled reaction-diffusion system under conditions in which no Hopf instability can be discerned in either layer.The transitions from stationary patterns to asynchronous and synchronous oscillatory patterns are obtained.A novel method based on decomposing coupled systems into two associated subsystems has been proposed to elucidate the mechanism of formation of oscillating patterns.Linear stability analysis of the associated subsystems reveals that the Turing pattern in one layer induces the other layer locally,undergoes a supercritical Hopf bifurcation and gives rise to localized oscillations.It is found that the sizes and positions of oscillations are determined by the spatial distribution of the Turing patterns.When the size is large,localized traveling waves such as spirals and targets emerge.These results may be useful for deeper understanding of pattern formation in complex systems,particularly multilayered systems.展开更多
Using the sign-invariant theory, we study the nonlinear reaction-diffusion systems. We also obtain some new explicit solutions to the nonlinear resulting systems.
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 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
We investigate the Turing-like wave instability of the uniform oscillator in oscillatory mediums using theoretical and flumerical methods. A propagating wave pattern originated at the corner of the system emerges when...We investigate the Turing-like wave instability of the uniform oscillator in oscillatory mediums using theoretical and flumerical methods. A propagating wave pattern originated at the corner of the system emerges when the uniform oscillator becomes unstable via Thring-like wave instability. Bifurcations from periodically propagated wave patterns to quasi-periodically propagated wave patterns, then to spatiotemporal chaos occur, as the system size increases from the instability threshold of the uniform oscillator.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
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
In this paper, we consider the existence of pullback random exponential attractor for non-autonomous random reaction-diffusion equation driven by nonlinear colored noise defined onR^(N) . The key steps of the proof ar...In this paper, we consider the existence of pullback random exponential attractor for non-autonomous random reaction-diffusion equation driven by nonlinear colored noise defined onR^(N) . The key steps of the proof are the tails estimate and to demonstrate the Lipschitz continuity and random squeezing property of the solution for the equation defined on R^(N) .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
In the one-dimensional space, traveling wave solutions of parabolic differential equations have been widely studied and well characterized. Recently, the mathematical study on higher-dimensional traveling fronts has a...In the one-dimensional space, traveling wave solutions of parabolic differential equations have been widely studied and well characterized. Recently, the mathematical study on higher-dimensional traveling fronts has attracted a lot of attention and many new types of nonplanar traveling waves have been observed for scalar reaction-diffusion equations with various nonlinearities. In this paper, by using the comparison argument and constructing appropriate super- and subsolutions, we study the existence, uniqueness and stability of three- dimensional traveling fronts of pyramidal shape for monotone bistable systems of reaction-diffusion equations in R3. The pyramidal traveling fronts are characterized as either a combination of planar traveling fronts on the lateral surfaces or a combination of two-dimensional V-form waves on the edges of the pyramid. In particular, our results are applicable to some important models in biology, such as Lotk,u-Volterra competition-diffusion systems with or without spatio-temporal delays, and reaction-diffusion systems of multiple obligate mutualists.展开更多
This paper is devoted to the investigation of stability for a class of coupled impulsive Markovian jump reaction-diffusion systems on networks(CIMJRDSNs). By using graph theory, a systematic method is provided to cons...This paper is devoted to the investigation of stability for a class of coupled impulsive Markovian jump reaction-diffusion systems on networks(CIMJRDSNs). By using graph theory, a systematic method is provided to construct global Lyapunov functions for the CIMJRDSNs. Based on Lyapunov functions and stochastic analysis method, some novel stability principles associated with the topology property of the networks are established.展开更多
This paper deals with a reaction-diffusion system with nonlinear absorption terms and boundary flux. As results of interactions among the six nonlinear terms in the system, some sufficient conditions on global existen...This paper deals with a reaction-diffusion system with nonlinear absorption terms and boundary flux. As results of interactions among the six nonlinear terms in the system, some sufficient conditions on global existence and finite time blow-up of the solutions are described via all the six nonlinear exponents appearing in the six nonlinear terms. In addition, we also show the influence of the coefficients of the absorption terms as well as the geometry of the domain to the global existence and finite time blow-up of the solutions for some cases. At last, some numerical results are given.展开更多
This paper is concerned with the asymptotic stability of planar waves in reaction-diffusion system on Rn, where n 2. Under initial perturbation that decays at space infinity, the perturbed solution converges to planar...This paper is concerned with the asymptotic stability of planar waves in reaction-diffusion system on Rn, where n 2. Under initial perturbation that decays at space infinity, the perturbed solution converges to planar waves as t →∞. The convergence is uniform in Rn. Moreover, the stability of planar waves in reaction-diffusion equations with nonlocal delays is also established by transforming the delayed equations into a non-delayed reaction-diffusion system.展开更多
基金Project supported by the National Natural Science Foundation of China(Grant Nos.12275065,12275064,12475203)the Natural Science Foundation of Hebei Province(Grant Nos.A2021201010 and A2024201020)+3 种基金Interdisciplinary Research Program of Natural Science of Hebei University(Grant No.DXK202108)Hebei Provincial Central Government Guiding Local Science and Technology Development Funds(Grant No.236Z1501G)Scientific Research and Innovation Team Foundation of Hebei University(Grant No.IT2023B03)the Excellent Youth Research Innovation Team of Hebei University(Grant No.QNTD202402)。
文摘A new type of localized oscillatory pattern is presented in a two-layer coupled reaction-diffusion system under conditions in which no Hopf instability can be discerned in either layer.The transitions from stationary patterns to asynchronous and synchronous oscillatory patterns are obtained.A novel method based on decomposing coupled systems into two associated subsystems has been proposed to elucidate the mechanism of formation of oscillating patterns.Linear stability analysis of the associated subsystems reveals that the Turing pattern in one layer induces the other layer locally,undergoes a supercritical Hopf bifurcation and gives rise to localized oscillations.It is found that the sizes and positions of oscillations are determined by the spatial distribution of the Turing patterns.When the size is large,localized traveling waves such as spirals and targets emerge.These results may be useful for deeper understanding of pattern formation in complex systems,particularly multilayered systems.
基金National Natural Science Foundation of China under Grant Nos.10447007 and 10671156the Natural Science Foundation of Shaanxi Province of China under Grant No.2005A13
文摘Using the sign-invariant theory, we study the nonlinear reaction-diffusion systems. We also obtain some new explicit solutions to the nonlinear resulting systems.
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We investigate the Turing-like wave instability of the uniform oscillator in oscillatory mediums using theoretical and flumerical methods. A propagating wave pattern originated at the corner of the system emerges when the uniform oscillator becomes unstable via Thring-like wave instability. Bifurcations from periodically propagated wave patterns to quasi-periodically propagated wave patterns, then to spatiotemporal chaos occur, as the system size increases from the instability threshold of the uniform oscillator.
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supported by the NSFC(12271141)supported by the Fundamental Research Funds for the Central Universities(B240205026)the Postgraduate Research and Practice Innovation Program of Jiangsu Province(KYCX24_0821).
文摘In this paper, we consider the existence of pullback random exponential attractor for non-autonomous random reaction-diffusion equation driven by nonlinear colored noise defined onR^(N) . The key steps of the proof are the tails estimate and to demonstrate the Lipschitz continuity and random squeezing property of the solution for the equation defined on R^(N) .
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.
基金supported by National Natural Science Foundation of China (Grant Nos. 11371179 and 11271172)National Science Foundation of USA (Grant No. DMS-1412454)
文摘In the one-dimensional space, traveling wave solutions of parabolic differential equations have been widely studied and well characterized. Recently, the mathematical study on higher-dimensional traveling fronts has attracted a lot of attention and many new types of nonplanar traveling waves have been observed for scalar reaction-diffusion equations with various nonlinearities. In this paper, by using the comparison argument and constructing appropriate super- and subsolutions, we study the existence, uniqueness and stability of three- dimensional traveling fronts of pyramidal shape for monotone bistable systems of reaction-diffusion equations in R3. The pyramidal traveling fronts are characterized as either a combination of planar traveling fronts on the lateral surfaces or a combination of two-dimensional V-form waves on the edges of the pyramid. In particular, our results are applicable to some important models in biology, such as Lotk,u-Volterra competition-diffusion systems with or without spatio-temporal delays, and reaction-diffusion systems of multiple obligate mutualists.
基金supported by the National Natural Science Foundation of China under Grant Nos.61473097,11301090the State Key Program of Natural Science Foundation of China under Grant No.U1533202+2 种基金Shandong Independent Innovation and Achievements Transformation Fund under Grant No.2014CGZH1101Civil Aviation Administration of China under Grant No.MHRD20150104Guangxi Natural Science Foundation under Grant No.2016JJA110005
文摘This paper is devoted to the investigation of stability for a class of coupled impulsive Markovian jump reaction-diffusion systems on networks(CIMJRDSNs). By using graph theory, a systematic method is provided to construct global Lyapunov functions for the CIMJRDSNs. Based on Lyapunov functions and stochastic analysis method, some novel stability principles associated with the topology property of the networks are established.
基金the National Natural Science Foundation of China(No.10471022,10771032)the Natural Science Foundation of Jiangsu province BK2006088.
文摘This paper deals with a reaction-diffusion system with nonlinear absorption terms and boundary flux. As results of interactions among the six nonlinear terms in the system, some sufficient conditions on global existence and finite time blow-up of the solutions are described via all the six nonlinear exponents appearing in the six nonlinear terms. In addition, we also show the influence of the coefficients of the absorption terms as well as the geometry of the domain to the global existence and finite time blow-up of the solutions for some cases. At last, some numerical results are given.
文摘This paper is concerned with the asymptotic stability of planar waves in reaction-diffusion system on Rn, where n 2. Under initial perturbation that decays at space infinity, the perturbed solution converges to planar waves as t →∞. The convergence is uniform in Rn. Moreover, the stability of planar waves in reaction-diffusion equations with nonlocal delays is also established by transforming the delayed equations into a non-delayed reaction-diffusion system.
