In this paper the singularly perturbed initial boundary value problems for the nonlocal reaction diffusion system are considered. Unsing the iteration method and the comparison theorem, the existence, uniqueness and i...In this paper the singularly perturbed initial boundary value problems for the nonlocal reaction diffusion system are considered. Unsing the iteration method and the comparison theorem, the existence, uniqueness and its asymptotic behavior of solution for the problem are studied.展开更多
The singularly perturbed initial boundary value problems for the reaction diffusion system are raised. Firstly, under suitable conditions, using a iteration technique, the differential inequalities theorem is construc...The singularly perturbed initial boundary value problems for the reaction diffusion system are raised. Firstly, under suitable conditions, using a iteration technique, the differential inequalities theorem is constructed and introducing two auxiliary functions the existence and uniqueness theorem of solution for the basic reaction diffusion system is proved. Using the singularly perturbed method the formal asymptotic expressions of the solution are constructed with power series theory. By using the comparison theorem the existence and its asymptotic behavior of solution for the original problem are studied. Finally, using method of estimate inequalities, the structure of solutions for the problem is discussed thoroughly in three cases and asymptotic solution of the original problem is given. The asymptotic behavior of solution in the three cases is proved.展开更多
In this paper the singularly perturbed initial boundary value problem for the diffraction reaction diffusion system is considered. Using the comparison principle, the existence, uniqueness and asymptotic behavior of s...In this paper the singularly perturbed initial boundary value problem for the diffraction reaction diffusion system is considered. Using the comparison principle, the existence, uniqueness and asymptotic behavior of solutions for the problem are studied.展开更多
The singularly perturbed initial boudary value problem for the nonlocal reaction diffusion systems was considered. Using iteration method the comparison theorem was obtained. Introducing stretched variable the formal ...The singularly perturbed initial boudary value problem for the nonlocal reaction diffusion systems was considered. Using iteration method the comparison theorem was obtained. Introducing stretched variable the formal asymptotic solution was constructed. And the existence and its asymptotic behavior of solution for the problem were studied by using the method of the upper and lower solution.展开更多
Spatiotemporal structures arising in two identical cells, which are governed by higher autocatalator kinetics and coupled via diffusive interchange of autocatalyst, are discussed. The stability of the unique homogeneo...Spatiotemporal structures arising in two identical cells, which are governed by higher autocatalator kinetics and coupled via diffusive interchange of autocatalyst, are discussed. The stability of the unique homogeneous steady state is obtained by the linearized theory. A necessary condition for bifurcations in spatially non-uniform solutions in uncoupled and coupled systems is given. Further information about Turing pattern solutions near bifurcation points is obtained by weakly nonlinear theory. Finally, the stability of equilibrium points of the amplitude equation is discussed by weakly nonlinear theory, with the bifurcation branches of the weakly coupled system.展开更多
This note studies the global solutions of a semilineear reaction diffusion system which comes from an exothermic themical reaction.This is a complement of paper[1]and gives a positive answer to the question mentioned...This note studies the global solutions of a semilineear reaction diffusion system which comes from an exothermic themical reaction.This is a complement of paper[1]and gives a positive answer to the question mentioned by paper[2].展开更多
In this paper the singularly perturbed initial boundary value problems for the nonlocal reaction diffusion system are considered.Using the iteration method and the comparison theorem, the existence and its asymptotic ...In this paper the singularly perturbed initial boundary value problems for the nonlocal reaction diffusion system are considered.Using the iteration method and the comparison theorem, the existence and its asymptotic behavior of solution for the problem are studied.展开更多
A class of singularly perturbed reaction diffusion systems are considered. Under suitable conditions, using the comparison theorem the asymptotic behavior of solution for the initial boundary value problems is studied.
A class of nonlinear predator prey reaction diffusion systems for singularly pe rturbed problems are considered.Under suitable conditions, by using theory of di fferential inequalities the existence and asymptotic be...A class of nonlinear predator prey reaction diffusion systems for singularly pe rturbed problems are considered.Under suitable conditions, by using theory of di fferential inequalities the existence and asymptotic behavior of solution for in itial boundary value problems are studied.展开更多
The nonlinear predator-prey reaction diffusion systems for singularly perturbed Robin Problems are considered. Under suitable conditions, the theory of differential inequalities can be used to study the asymptotic beh...The nonlinear predator-prey reaction diffusion systems for singularly perturbed Robin Problems are considered. Under suitable conditions, the theory of differential inequalities can be used to study the asymptotic behavior of the solution for initial boundary value problems.展开更多
The problems of the nonlocal boundary conditions for the singularly perturbed reaction diffusion systems are considered. Under suitable conditions, using the comparison theorem the asymptotic behavior of solution for ...The problems of the nonlocal boundary conditions for the singularly perturbed reaction diffusion systems are considered. Under suitable conditions, using the comparison theorem the asymptotic behavior of solution for the initial boundary value problems are studied.展开更多
In this paper we deal with the initial boundary value problem for two classes of reaction diffusion systems with two source terms in bounded domain. Under some assumptions on the exponents and the initial data, applyi...In this paper we deal with the initial boundary value problem for two classes of reaction diffusion systems with two source terms in bounded domain. Under some assumptions on the exponents and the initial data, applying the comparison principle, the maximum prin- ciple and the supersolution-subsolution method, we prove the global existence and blow up of solutions. We also establish some upper blow up rates.展开更多
The nonlinear predator-prey singularly perturbed Robin initial boundary value problems for reaction diffusion systems were considered. Under suitable conditions, using theory of differential inequalities the existence...The nonlinear predator-prey singularly perturbed Robin initial boundary value problems for reaction diffusion systems were considered. Under suitable conditions, using theory of differential inequalities the existence and asymptotic behavior of solution for initial boundary value problems were studied.展开更多
The nonlocal problems for the singularly perturbed reaction diffusion system are considered. Under suitable conditions, using the comparioson theorem the asymptotic behavior of solution for the initial boundary value ...The nonlocal problems for the singularly perturbed reaction diffusion system are considered. Under suitable conditions, using the comparioson theorem the asymptotic behavior of solution for the initial boundary value problems are studied.展开更多
For reaction-diffusion equations in irregular domains with moving boundaries,the numerical stability constraints from the reaction and diffusion terms often require very restricted time step sizes,while complex geomet...For reaction-diffusion equations in irregular domains with moving boundaries,the numerical stability constraints from the reaction and diffusion terms often require very restricted time step sizes,while complex geometries may lead to difficulties in the accuracy when discretizing the high-order derivatives on grid points near the boundary.It is very challenging to design numerical methods that can efficiently and accurately handle both difficulties.Applying an implicit scheme may be able to remove the stability constraints on the time step,however,it usually requires solving a large global system of nonlinear equations for each time step,and the computational cost could be significant.Integration factor(IF)or exponential time differencing(ETD)methods are one of the popular methods for temporal partial differential equations(PDEs)among many other methods.In our paper,we couple ETD methods with an embedded boundary method to solve a system of reaction-diffusion equations with complex geometries.In particular,we rewrite all ETD schemes into a linear combination of specificФ-functions and apply one state-of-the-art algorithm to compute the matrix-vector multiplications,which offers significant computational advantages with adaptive Krylov subspaces.In addition,we extend this method by incorporating the level set method to solve the free boundary problem.The accuracy,stability,and efficiency of the developed method are demonstrated by numerical examples.展开更多
In this paper the singularly perturbed initial boundary value problems for the nonlocal reaction diffusion system are considered. Using the iteration method and the comparison theorem, the existence and its asymptotic...In this paper the singularly perturbed initial boundary value problems for the nonlocal reaction diffusion system are considered. Using the iteration method and the comparison theorem, the existence and its asymptotic behavior of the solution for the problem are studied.展开更多
So far, the Lyapunov direct method is still the most effective technique in the study of stability for ordinary differential equations and stochastic differential equations. Due to the shortage of the corresponding It...So far, the Lyapunov direct method is still the most effective technique in the study of stability for ordinary differential equations and stochastic differential equations. Due to the shortage of the corresponding It6 formula, this useful method has not been popularized in stochastic partial differential equations. The aim of this work is to try to extend the Lyapunov direct method to the It6 stochastic reaction diffusion systems and to establish the corresponding Lyapunov stability theory, including stability in probablity, asymptotic stability in probablity, and exponential stability in mean square. As the application of the obtained theorems, this paper addresses the stability of the Hopfield neural network and points out that the main results ob- tained by Holden Helge and Liao Xiaoxin et al. can be all regarded as the corollaries of the theorems presented in this paper.展开更多
In this paper the initial-boundary value problem for nonlocal singularly perturbed reaction diffusion system are considered.Using the iterative method and the compa- rison theorem,the existence and asymptotic behavior...In this paper the initial-boundary value problem for nonlocal singularly perturbed reaction diffusion system are considered.Using the iterative method and the compa- rison theorem,the existence and asymptotic behavior of the solution for the problem are studied.展开更多
In this paper the singularly perturbed initial boundary value problems for a nonlocal reaction diffusion system are considered. Using the iteration method and the comparison theorem, the existence and asymptotic behav...In this paper the singularly perturbed initial boundary value problems for a nonlocal reaction diffusion system are considered. Using the iteration method and the comparison theorem, the existence and asymptotic behavior of solutions for the problem are studied.展开更多
This article deals with an inverse problem of reconstructing two time independent coefficients in the reaction diffusion system from the final time space discretized measurement using the optimization method with the ...This article deals with an inverse problem of reconstructing two time independent coefficients in the reaction diffusion system from the final time space discretized measurement using the optimization method with the help of the smooth interpolation technique.The main objective of the article is to analyse the asymptotic behavior of the solution of the inverse problem for the linearly coupled reaction diffusion system with respect to the homogeneous Dirichlet boundary condition.展开更多
基金The project is supported by The National Natural Science Foundation of China(10071048)"Hundred People Project" of Chinese Academy of Sciences
文摘In this paper the singularly perturbed initial boundary value problems for the nonlocal reaction diffusion system are considered. Unsing the iteration method and the comparison theorem, the existence, uniqueness and its asymptotic behavior of solution for the problem are studied.
基金Supported by the National Natural Science Foundation of China (40676016 and 10471039)the National Program for Basic Science Researches of China (2003CB415101-03 and 2004CB418304)+1 种基金the Key Project of the Chinese Academy of Sciences (KZCX3-SW-221)E-Insitutes of Shanghai Municipal Education Commission (N.E03004)
文摘The singularly perturbed initial boundary value problems for the reaction diffusion system are raised. Firstly, under suitable conditions, using a iteration technique, the differential inequalities theorem is constructed and introducing two auxiliary functions the existence and uniqueness theorem of solution for the basic reaction diffusion system is proved. Using the singularly perturbed method the formal asymptotic expressions of the solution are constructed with power series theory. By using the comparison theorem the existence and its asymptotic behavior of solution for the original problem are studied. Finally, using method of estimate inequalities, the structure of solutions for the problem is discussed thoroughly in three cases and asymptotic solution of the original problem is given. The asymptotic behavior of solution in the three cases is proved.
基金The NNSF(40676016 10471039)of China,the National Key Project for Basics Research(2003CB415101-03 and 2004CB418304) the Key Project of the Chinese Academy of Sciences(KZCX3-SW-221) in part by E-Insitutes of Shanghai Municipal Education Commission(N.E03004).
文摘In this paper the singularly perturbed initial boundary value problem for the diffraction reaction diffusion system is considered. Using the comparison principle, the existence, uniqueness and asymptotic behavior of solutions for the problem are studied.
文摘The singularly perturbed initial boudary value problem for the nonlocal reaction diffusion systems was considered. Using iteration method the comparison theorem was obtained. Introducing stretched variable the formal asymptotic solution was constructed. And the existence and its asymptotic behavior of solution for the problem were studied by using the method of the upper and lower solution.
基金the National Natural Science Foundation of China(No.60574075)
文摘Spatiotemporal structures arising in two identical cells, which are governed by higher autocatalator kinetics and coupled via diffusive interchange of autocatalyst, are discussed. The stability of the unique homogeneous steady state is obtained by the linearized theory. A necessary condition for bifurcations in spatially non-uniform solutions in uncoupled and coupled systems is given. Further information about Turing pattern solutions near bifurcation points is obtained by weakly nonlinear theory. Finally, the stability of equilibrium points of the amplitude equation is discussed by weakly nonlinear theory, with the bifurcation branches of the weakly coupled system.
文摘This note studies the global solutions of a semilineear reaction diffusion system which comes from an exothermic themical reaction.This is a complement of paper[1]and gives a positive answer to the question mentioned by paper[2].
基金Supported by Important Study Project of the National Natural Science Foundation of China( 90 2 1 1 0 0 4 ) and by the"Hundred Talents Project" of Chinese Academy of Sciences
文摘In this paper the singularly perturbed initial boundary value problems for the nonlocal reaction diffusion system are considered.Using the iteration method and the comparison theorem, the existence and its asymptotic behavior of solution for the problem are studied.
文摘A class of singularly perturbed reaction diffusion systems are considered. Under suitable conditions, using the comparison theorem the asymptotic behavior of solution for the initial boundary value problems is studied.
基金Supported by important study project of the National Natural Science Foundation of China(9 0 2 1 1 0 0 4 ) and by the"Hundred Talents'Project"of Chinese Academy of Sciences
文摘A class of nonlinear predator prey reaction diffusion systems for singularly pe rturbed problems are considered.Under suitable conditions, by using theory of di fferential inequalities the existence and asymptotic behavior of solution for in itial boundary value problems are studied.
文摘The nonlinear predator-prey reaction diffusion systems for singularly perturbed Robin Problems are considered. Under suitable conditions, the theory of differential inequalities can be used to study the asymptotic behavior of the solution for initial boundary value problems.
文摘The problems of the nonlocal boundary conditions for the singularly perturbed reaction diffusion systems are considered. Under suitable conditions, using the comparison theorem the asymptotic behavior of solution for the initial boundary value problems are studied.
基金supported by the National Natural Science Foundation of China(11471087)the China Postdoctoral Science Foundation(2013M540270)+1 种基金the Heilongjiang Postdoctoral Foundation(LBH-Z13056,LBHZ15036)the Fundamental Research Funds for the Central Universities
文摘In this paper we deal with the initial boundary value problem for two classes of reaction diffusion systems with two source terms in bounded domain. Under some assumptions on the exponents and the initial data, applying the comparison principle, the maximum prin- ciple and the supersolution-subsolution method, we prove the global existence and blow up of solutions. We also establish some upper blow up rates.
基金Important Study Project of the NationalNatural Science F oundation of China( No.90 2 110 0 4),and"Hun-dred Talents Project"of Chinese Academy of Sciences
文摘The nonlinear predator-prey singularly perturbed Robin initial boundary value problems for reaction diffusion systems were considered. Under suitable conditions, using theory of differential inequalities the existence and asymptotic behavior of solution for initial boundary value problems were studied.
文摘The nonlocal problems for the singularly perturbed reaction diffusion system are considered. Under suitable conditions, using the comparioson theorem the asymptotic behavior of solution for the initial boundary value problems are studied.
文摘For reaction-diffusion equations in irregular domains with moving boundaries,the numerical stability constraints from the reaction and diffusion terms often require very restricted time step sizes,while complex geometries may lead to difficulties in the accuracy when discretizing the high-order derivatives on grid points near the boundary.It is very challenging to design numerical methods that can efficiently and accurately handle both difficulties.Applying an implicit scheme may be able to remove the stability constraints on the time step,however,it usually requires solving a large global system of nonlinear equations for each time step,and the computational cost could be significant.Integration factor(IF)or exponential time differencing(ETD)methods are one of the popular methods for temporal partial differential equations(PDEs)among many other methods.In our paper,we couple ETD methods with an embedded boundary method to solve a system of reaction-diffusion equations with complex geometries.In particular,we rewrite all ETD schemes into a linear combination of specificФ-functions and apply one state-of-the-art algorithm to compute the matrix-vector multiplications,which offers significant computational advantages with adaptive Krylov subspaces.In addition,we extend this method by incorporating the level set method to solve the free boundary problem.The accuracy,stability,and efficiency of the developed method are demonstrated by numerical examples.
基金Supported by the National Natural Science Foundation of China(No.90111011,No.10471039)the National Key Project for Basics Research(2003CB415101-03 and 2004CB418304)+1 种基金the Key Project of the Chinese Academy of Sciences(KZCX3-SW-221)in part by E-Insitutes of Shanghai Municipal Education Commission(N.E03004).
文摘In this paper the singularly perturbed initial boundary value problems for the nonlocal reaction diffusion system are considered. Using the iteration method and the comparison theorem, the existence and its asymptotic behavior of the solution for the problem are studied.
基金Supported by the National Natural Science Foundation of China(Grant No.60574042)
文摘So far, the Lyapunov direct method is still the most effective technique in the study of stability for ordinary differential equations and stochastic differential equations. Due to the shortage of the corresponding It6 formula, this useful method has not been popularized in stochastic partial differential equations. The aim of this work is to try to extend the Lyapunov direct method to the It6 stochastic reaction diffusion systems and to establish the corresponding Lyapunov stability theory, including stability in probablity, asymptotic stability in probablity, and exponential stability in mean square. As the application of the obtained theorems, this paper addresses the stability of the Hopfield neural network and points out that the main results ob- tained by Holden Helge and Liao Xiaoxin et al. can be all regarded as the corollaries of the theorems presented in this paper.
基金Supported by the National Natural Science Foundation of China (40676016 and 10471039)the National Key Project for Basics Research (2003CB415101-03 and 2004CB418304)+1 种基金the Key Project of the Chinese Academy of Sciences (KZCX3-SW-221)in part by E-Insitutes of Shanghai Municipal Education Commission (N.E03004).
文摘In this paper the initial-boundary value problem for nonlocal singularly perturbed reaction diffusion system are considered.Using the iterative method and the compa- rison theorem,the existence and asymptotic behavior of the solution for the problem are studied.
基金Supported by the National Natural Science Foundation of China(No.40676016)the National Key Project for Basics Research(2003CB415101-03 and 2004CB418304)+2 种基金the Key Project of the Chinese Academy of Sciences(KZCX3-SW-221)in part by E-Insitutes of Shanghai Municipal Education Commission(N.E03004)the National Science Foundation from the Education Bureau of Anhui Province(No.KJ2007A013).
文摘In this paper the singularly perturbed initial boundary value problems for a nonlocal reaction diffusion system are considered. Using the iteration method and the comparison theorem, the existence and asymptotic behavior of solutions for the problem are studied.
基金supported by the Council of Scientific and Industrial Research(CSIR),India(No.09/472(0143)/2010-EMR-I)
文摘This article deals with an inverse problem of reconstructing two time independent coefficients in the reaction diffusion system from the final time space discretized measurement using the optimization method with the help of the smooth interpolation technique.The main objective of the article is to analyse the asymptotic behavior of the solution of the inverse problem for the linearly coupled reaction diffusion system with respect to the homogeneous Dirichlet boundary condition.
