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. 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
In this paper,we investigate the existence of periodic solutions of a semi-ratiodependent predator-prey diffusion system with functional responses and time delays in a two-patch environment on time scales by using a c...In this paper,we investigate the existence of periodic solutions of a semi-ratiodependent predator-prey diffusion system with functional responses and time delays in a two-patch environment on time scales by using a continuation theorem based on coincidence degree theory.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 paper considers the Cauchy problem of the following convection diffusion system [GRAPHICS] with initial data [GRAPHICS] A global existence result is established by employing the techniques of F. B. Weissler and t...This paper considers the Cauchy problem of the following convection diffusion system [GRAPHICS] with initial data [GRAPHICS] A global existence result is established by employing the techniques of F. B. Weissler and the energy method. Here a,b,epsilon > 0 are constants.展开更多
The asymptotic behaviour of solutions for general partly dissipative reaction-diffusion systems in Rn is studied. The asymptotic compactness of the solutions and then the existence of the global attractor are proved i...The asymptotic behaviour of solutions for general partly dissipative reaction-diffusion systems in Rn is studied. The asymptotic compactness of the solutions and then the existence of the global attractor are proved in L2(Rn )× L2(Rn ) .展开更多
This paper deals with an initial boundary value problem for the strongly coupledreaction-diffusion systems with a full matrix of diffusion coefficients. The global existence ofsolutions is proved by using the techniqu...This paper deals with an initial boundary value problem for the strongly coupledreaction-diffusion systems with a full matrix of diffusion coefficients. The global existence ofsolutions is proved by using the techniques based on invariant regions, Lyapunov functionalmethods, and local Lp prior estimates independent of time.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
In this paper, by the method of upper and lower solutions, we establish the existence of the non-trivial nonnegative periodic solutions for a class of degenerate diffusion system arising from dynamics of biological gr...In this paper, by the method of upper and lower solutions, we establish the existence of the non-trivial nonnegative periodic solutions for a class of degenerate diffusion system arising from dynamics of biological groups.展开更多
In this paper, we prove some properties of solutions to a class of degener-ate diffusion systems which arise from modeling interacting evolution of two biological groups.
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
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.
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
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.
基金Supported by the Foundation for Scientific Research Projects of Education Department of Heilongjiang Province(11553058)
文摘In this paper,we investigate the existence of periodic solutions of a semi-ratiodependent predator-prey diffusion system with functional responses and time delays in a two-patch environment on time scales by using a continuation theorem based on coincidence degree theory.
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 paper considers the Cauchy problem of the following convection diffusion system [GRAPHICS] with initial data [GRAPHICS] A global existence result is established by employing the techniques of F. B. Weissler and the energy method. Here a,b,epsilon > 0 are constants.
文摘The asymptotic behaviour of solutions for general partly dissipative reaction-diffusion systems in Rn is studied. The asymptotic compactness of the solutions and then the existence of the global attractor are proved in L2(Rn )× L2(Rn ) .
基金Supported by the Henan Innovation Project for University Prominent Research Talents (2003KJCX008)
文摘This paper deals with an initial boundary value problem for the strongly coupledreaction-diffusion systems with a full matrix of diffusion coefficients. The global existence ofsolutions is proved by using the techniques based on invariant regions, Lyapunov functionalmethods, and local Lp prior estimates independent of time.
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In this paper, by the method of upper and lower solutions, we establish the existence of the non-trivial nonnegative periodic solutions for a class of degenerate diffusion system arising from dynamics of biological groups.
文摘In this paper, we prove some properties of solutions to a class of degener-ate diffusion systems which arise from modeling interacting evolution of two biological groups.
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.
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.
