The Singularly Perturbed Problems for Nonlinear Nonlocal Disturbed Evolution Equations with Two Parameters?

FENG Yi-hu,WU Qin-kuan,XU Yong-hong,MO Jia-qi

(1-Department of Electronics and Information Engineering,Bozhou College,Bozhou,Anhui 236800;2-Department of Mathematics&Physics,Nanjing Institute of Technology,Nanjing,Jiangsu 211167;3-Department of Mathematics&Physics,Bengbu College,Bengbu,Anhui 233030;4-Department of Mathematics,Anhui Normal University,Wuhu,Anhui 241003)

1 Introduction

The nonlinear singularly perturbed disturbed evolution equations are a very attractive target in the physical and engineering mathematics etc.Many approximate methods have been developed,including the boundary layer methods,to solve the equations.Recently,many scholars have done a great deal of work on this topic,such as de Jager and Jiang[1],Barbu and Morosanu[2],Hovhannisyan and Vulanovic[3],Graef and Kong[4],Barbu and Cosma[5],Bonfohet al[6],Fayeet al[7],Samusenko[8],Liu[9]and so on.Using the singular perturbation and other methods Moet al[10-19]also considered a class of singularly perturbed nonlinear problems.In this paper,using the special and simple singularly perturbed theory,we study a class of the nonlinear initial boundary value problems.

Now we consider the nonlocal singularly perturbed disturbed evolution equations initial boundary value problem with two parameters as follows

where

x=(x1,x2,···,xn)∈?,? denotes a bounded region inRn,?? signifies a boundary of? for classC1+α(α∈(0,1))is H¨older exponent,T0is a large enough positive constant,μf(t,x,Tu)is a disturbed term,Lis a uniformly elliptic operator,Tis a integral operator,andKis a continuous function.

We need the following hypotheses:

[H1]:σ=ε/μ→0 asε→0;

[H2]: The second order partial derivations ofαij,βiwith regard toxare H¨older continuous,andgandhiwith regard tot,xare H¨older continuous,with regard toε,μand they are sufficiently smooth functions in correspondence ranges;

[H3]:f(t,x,v)is a sufficiently smooth bounded function with regard to variables in correspondence ranges andf(t,x,v)≤?c<0,wherec>0.

2 Construct outer solution

From hypothesis[H3]and the theory of Fredholm integral equation,the reduced problem for the original problem has a solution.

LetU(t,x)be the outer solution to the problem(1)–(4),and let

Substituting(5)and(6)into(1),developingfinεandμ,equating coefficients of like powers ofεiμj,respectively,fori,j=0,1,···,i+j?=0,we obtain

where

In the above and below equations,the values of terms for the negative subscript are zeroes.From theU00and(6),(7),we obtain the outer solution to the original problem as

But it may not satisfy the boundary and initial conditions(2)–(4),so that we need to construct the boundary layer and initial layer functions.

3 Construct boundary layer term

Set up a local coordinate system(ρ,?)as[10],where?=(?1,?2,···,?n?1).In the neighborhood of?? :0≤ρ≤ρ0,

where

and the constructions of coefficientsani,aij,an,ajare omitted.

We lead into the variables of multiple scales[1,2]in 0≤ρ≤ρ0:whereh(ρ,?)is a function to be determined.For convenience,we still substituteρforbelow.From(9),we have

where

and the constructions ofK11andK12are omitted,too.

Lethρ= √and the solutionuof the original problem(1)–(4)be

whereVis a boundary layer term.Substituting(11)into(1),(2),we have

Substituting(8),(14)and(10)into(12),(13),expanding nonlinear terms inσandμ,and equating the coefficients of like powers ofσiμj,respectively,fori,j=0,1,···,we obtain

whereGij,i=0,1,···,i+j?=0 are determined functions.From the problems(15)and(16),we havev00.Fromv00and(17)and(18),we can obtain solutionsvij(i,j=0,1,···,i+j?=0).

From the hypotheses,it is easy to see thatvij(i=0,1,···)possesses boundary layer behavior

whereδij>0,i,j=0,1,···are constants.

Letis a sufficiently smooth function in 0≤ρ≤ρ0,which satisfies:ψ(ρ)=1,as 0≤ρ≤(1/3)ρ0andψ(ρ)=0,asρ≥(2/3)ρ0.

For convenience,we still substitutevijforas below.Then from(14)we have the boundary corrective termVnear??.

4 Construct initial layer term

The solutionuof the original problem(1)–(4)is

whereWis an initial layer term.Substituting(20)into(1)–(4),we have

We lead into a stretched variable[1,2]:τ=t/εand let

Substituting(8),(14),(10)and(25)into(21)–(24),expanding nonlinear terms inεandμ,and equating the coefficients of like powers ofεiμj,respectively,fori,j=0,1,···,we obtain

whereare determined functions.From the problems(26)–(29),we havew00,Fromw00and the(30)–(33),we can obtain solutionswij,i,j=0,1,···,i+j?=0 successively.

From the hypotheses,it is easy to see thatwij,i,j=0,1,···,possesses boundary layer behavior

whereare constants.

Letfor convenience,we still substitutebelow.Then from(25)we have the initial corrective termW.

From(8),(14),(25),we obtain the formal asymptotic expansion of solution for the nonlinear nonlocal singularly perturbed disturbed evolution equations initial boundary value problem with two parameters(1)–(4):

5 The main result

Now,we prove that the expansion(35)is uniformly valid in ? and we have the following theorem:

TheoremUnder the hypotheses[H1]–[H3],there exists a solutionu(t,x)of the nonlinear nonlocal singularly perturbed disturbed evolution equations initial boundary value problem with two parameters(1)–(4)and holds the uniformly valid asymptotic expansion(35)for

ProofWe can prove that(35)is a uniformly valid asymptotic expansion[1,2].

We get the remainder termR(t,x)the initial boundary value problem(1)–(4).Let

where

Using(36)we obtain

As to the proof of the validity of the approximation(35),it is possible to use the fi xed point theorem(see[1,2]).

The linearized di ff erential operatorreads

and therefore

where 0<θ<1.For fixedε,the normed linear spaceNis chosen as

with norm

and the Banach spaceBis

with norm

From the hypotheses we may show that the condition

of the fixed point theorem is fulfilled,wherel?1is independent ofεandμ,and thusL?1is continuous.The Lipschitz condition of the fixed point theorem becomes

whereC1,C2andCare constants independent ofεandμ,this inequality is valid for all

p1,p2in a ballKN(r)with∥r∥≤1.Finally,we obtain the result that the remainder term exists and moreover

From(36),we obtain

The proof of the Theorem is completed.

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