Singular perturbation for the weakly nonlinear reaction diffusion equation with boundary perturbation

In this paper, a class of nonlinear singularly perturbed initial boundary value problems for reaction diffusion equations with boundary perturbation are considered under suitable conditions. Firstly, by dint of the regular perturbation method, the outer solution of the original problem is obtained. Secondly, by using the stretched variable and the expansion theory of power series the initial layer of the solution is constructed. And then, by using the theory of differential inequalities, the asymptotic behavior of the solution for the initial boundary value problems is studied. Finally, using some relational inequalities the existence and uniqueness of solution for the original problem and the uniformly valid asymptotic estimation are discussed.     

The nonlinear nonlocal singularly perturbed problems for reaction diffusion equations

A class of nonlinear nonlocal for singularly perturbed Robin initial boundary value problems for reaction diffusion equations is considered. Under suitable conditions, firstly, the outer solution of the original problem is obtained, secondly, using the stretched variable, the composing expansion method and the expanding theory of power series the initial layer is constructed, finally, using the theory of differential inequalities the asymptotic behavior of solution for the initial boundary value problems are studied and educing some relational inequalities the existence and uniqueness of solution for the original problem and the uniformly valid asymptotic estimation is discussed. Foundation items: the National Natural Science Foundation of China (10071048); the “Hunfred Talents Project” by Chinese Academy of Sciences Biography: Mo Jia-qi (1937−)     

Singularly perturbed solution to semilinear reaction diffusion equations with two parameters

A class of singularly perturbed initial boundary value problems for semilinear reaction diffusion equations with two parameters is considered, Under suitable conditions and using the theory of differential inequalities, the existence and the asymptotic behavior of the solution to the initial boundary value problem are studied.     

Singularly perturbed reaction diffusion equations with time delay

Abstract A class of initial boundary value problems of differential-difference equations for reaction diffusion with a small time delay is considered. Under suitable conditions and by using the stretched variable method, a formal asymptotic solution is constructed. Then, by use of the theory of differential inequalities, the uniform validity of the solution is proved.     

Singularly perturbed solution for weakly nonlinear equations with two parameters

A class of singularly perturbed boundary value problems of weakly non- linear equation for fourth order on the interval[a,b]with two parameters is considered. Under suitable conditions,firstly,the reduced solution and formal outer solution are con- structed using the expansion method of power series.Secondly,using the transformation of stretched variable,the first boundary layer corrective term near x=a is constructed which possesses exponential attenuation behavior.Then,using the stronger transfor- mation of stretched variable,the second boundary layer corrective term near x=a is constructed,which also possesses exponential attenuation behavior.The thickness of second boundary layer is smaller than the first one and forms a cover layer near x=a. Finally,using the theory of differential inequalities,the existence,uniform validity in the whole interval[a,b]and asymptotic behavior of solution for the original boundary value problem are proved.Satisfying results are obtained.     

A class of singular perturbation solutions to semilinear equations of fourth order

A class of singularly perturbed boundary value problems for semilinear equations of fourth order with two parameters are considered. Under suitable conditions, using the method of lower and upper solutions, the existence and the asymptotic behavior of the solution to the boundary value problem are studied, In the present paper, the solution to the original singularly perturbed problem with two parameters has only one boundary layer.     

Singular perturbation of boundary value problem of systems for quasilinear ordinary differential equations

In this paper,we study the singular perturbation of boundary value problem of systemsfor quasilinear ordinary differential equations:x′=f(t,x,y,ε),εy″=g(t,x,y,ε)y′ h(t,x,y,ε),x(0,ε)=A(ε),y(0,ε)=Bε,y(1,ε)=C(ε)where xf.y,h,A,B and C belong to R″and a is a diagonal matrix.Under the appropriateassumptions,using the technique of diagonalization and the theory of differentialinequalities we obtain the existence of solution and its componentwise uniformly validasymptotic estimation.     

On two-dimensional large-scale primitive equations in oceanic dynamics （Ⅱ）

The initial boundary value problem for the two-dimensional primitive equations of largescale oceanic motion in geophysics is considered sequetially. Here the depth of the ocean is positive but not always a constant. By Faedo-Galerkin method and anisotropic inequalities, the existence and uniqueness of the global weakly strong solution and global strong solution for the problem are obtained. Moreover, by studying the asymptotic behavior of solutions for the above problem, the energy is exponential decay with time is proved.     

On two-dimensional large-scale primitive equations in oceanic dynamics(Ⅱ)

The initial boundary value problem for the two-dimensional primitive equations of largescale oceanic motion in geophysics is considered sequetially.Here the depth of the ocean is positive but not always a constant.By Faedo-Galerkin method and anisotropic inequalities,the existence and uniqueness of the global weakly strong solution and global strong solution for the problem are obtained.Moreover,by studying the asymptotic behavior of solutions for the above problem,the energy is exponential decay with time is proved.     

Asymptotic theory of initial value problems for nonlinear perturbed Klein-Gordon equations

Introduction InthispaperasymptotictheoryofthefollowinginitialvalueproblemforanonlinearKlein Gordonequationisconsidered.tt-Δ =εF(t,x,,ε),t>0,x∈R2,(0,x,ε)=0(x,ε),t(0,x,ε)=1(x,ε),x∈R2,(1)where(t,x)isarealvaluedunknownfunction,Δ=2i     

The Asymptotic Behavior of Solution for the Singularly Perturbed Initial Boundary Value Problems of the Reaction Diffusion Equations in a Part of Domain

Ｔｈｅａｕｔｈｏｒｓｓｔｕｄｉｅｄａｃｌａｓｓｏｆｓｉｎｇｕｌａｒｌｙｐｅｒｔｕｒｂｅｄｐｒｏｂｌｅｍｓｉｎ [1 ] -[7] .Ｎｏｗｗｅｒａｉｓｅａｃｌａｓｓｏｆｓｉｎｇｕｌａｒｌｙｐｅｒｔｕｒｂｅｄｐｒｏｂｌｅｍｓｏｎａｐａｒｔｏｆｄｏｍａｉｎ .Ｃｏｎｓｉｄｅｒｔｈｅｆｏｌｌｏｗｉｎｇｉｎｉｔｉａｌｂｏｕｎｄａｒｙｖａｌｕｅｐｒｏｂｌｅｍｆｏｒｔｈｅｒｅａｃｔｉｏｎｄｉｆｆｕｓｉｏｎｅｑｕａｔｉｏｎｓ　　ｕｔ-λε(ｘ) ( μ(ｕ)ｕｘ) ｘ Ｋｘ(ｕ) ｆ(ｘ ,ｔ,ｕ) =0 ,(ｔ,ｘ) ∈ ( 0 ,Ｔ)× ( ( 0 ,α) ∪ (…     

The corner solution for quasilinear differential equation with two parameters

I.IntroductionWeconsiderinthispaperboundaryvalueproblemofquasilineardifferentialequationby# pi(x,y)y' g(x,y)=0,a相似文献   

Interior layer behavior of boundary value problems for second order vector equation of elliptic type

In this paper, making use of the theory of partial differential inequalities, we will investigate the boundary value problems for a class of singularly perturbed second order vector elliptic equations, and obtain the existence and asymptotic estimation of solutions, involving the interior layer behavior, of the problems described above.     

SINGULARLY PERTURBED NONLINEAR BOUNDARY VALUE PROBLEM FOR A KIND OF VOLTERRA TYPE FUNCTIONAL DIFFERENTIAL EQUATION

IntroductionThereweresomeresultsofstudyingonboundaryvalueproblemsforfunctionaldifferentialequation[1～6 ]byemployingthetoplolgicaldegreetheoryandsomefixedpointprinciplesinrecentyears.Buttheworktostudyboundaryvalueproblemsfordelaydifferentialequationwithsmallparameterbymeansofthetheoryofsingularperturbationrarelyappeared[7～11].Thereasonforitisthattheworktoconstructtheuppersolutionandlowersolutionforthecaseofdifferentialequationwithdelayisdifficult.Theauthorhasstudiedakindofboundaryvalueproblem…     

SINGULAR PERTURBATION OF INITIAL BOUNDARY VALUE FROBLEMS OF REACTION DIFFUSlON EQUATION WITH DELAY

ＳＩＮＧＵＬＡＲＰＥＲＴＵＲＢＡＴＩＯＮＯＦＩＮＩＴＩＡＬＢＯＵＮＤＡＲＹＶＡＬＵＥＦＲＯＢＬＥＭＳＯＦＲＥＡＣＴＩＯＮＤＩＦＦＵＳｌＯＮＥＱＵＡＴＩＯＮＷＩＴＨＤＥＬＡＹＺｈａｎｇＸｉａｎｇ（张祥）（ＡｎｈｕｉＮｏｒｍａｌＵｎｉｖｅｒｓｉｔｙ）Ｗｕ...     

A class of singularly perturbed reaction diffusion systems

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. Project supported by the National Natural Science Foundation of China     

Exact analysis of Burgers equation on semiline with flux condition at the origin

The initial boundary value problem for the Burgers equation in the domain x 0, t > 0 with flux boundary condition at x = 0 has been solved exactly. The behaviour of the solution as t tends to infinity is studied and the “asymptotic profile at infinity” is obtained. In addition, the uniqueness of the solution of the initial boundary value problem is proved and its inviscid limit as → 0 is obtained.     

Boundary value problems for second order singularly perturbed differential equations

In this paper the existence and uniqueness of solutions of boundary value problems (which contains the Robin's problem) is discussed by using the upper and lower solution. In addition, the asymptotic estimation of the solution is given as well.     

A class of boundary value problems for third-order differential equation with a turning point

A class of boundary value problems for a third-order differential equation with a turning point is considered. Using the method of multiple scales and others, the uniformly valid asymptotic expansion of solution for the boundary value problem is constructed.     

Singular perturbation of initial boundary value problems of reaction diffusion equation with delay

In this paper we consider the initial-boundary value problems for a class of general singularly perturbed delay reaction diffusion equations which are often met in applications, such as biomathematics and biochemistry. Applying the method of composite expansion we construct the formally asymptotic solution of the problem described. With the help of theory of upper and lower solutions we prove the uniformly validity of the formal solution and the existence of solution of the original problem. Project Supported by the Anhui Normal University Youth Natural Science Foundation of China