THE POINTED LAYER SOLUTION OF SINGULAR PERTURBATION FOR NONLINEAR EVOLUTIONEQUATIONS WITH TWO PARAMETERS

OUYANG Cheng,CHEN Xian-feng,MO Jia-qi

(1.Faculty of Science,Huzhou University,Huzhou 313000,China)

(2.Department of Mathematics,Shanghai Jiaotong University,Shanghai 200240,China)

(3.Department of Mathematic,Anhui Normal University,Wuhu 241003,China)

THE POINTED LAYER SOLUTION OF SINGULAR PERTURBATION FOR NONLINEAR EVOLUTION
EQUATIONS WITH TWO PARAMETERS

OUYANG Cheng1,CHEN Xian-feng2,MO Jia-qi3

(1.Faculty of Science,Huzhou University,Huzhou 313000,China)

(2.Department of Mathematics,Shanghai Jiaotong University,Shanghai 200240,China)

(3.Department of Mathematic,Anhui Normal University,Wuhu 241003,China)

In this paper,the nonlinear singular perturbation problem for the evolution equations is studied.The outer solution and corrective terms of the pointed,boundary and initiallayers for the solution are constructed.By using the fi xed point theorem,the uniformly validity of solution to the problem ia proved and the results of the study for the singular perturbation with two parameters is extended.

pointed layer;singular perturbation;evolution equation

1 Introduction

The nonlinear singular perturbation evolution equations are an important target in the mathematical,engineering mathematics and physical etc.circles.Many approximate methods were improved.Recently,many scholars did a great of work,such as de Jager et al.[1],Barbu et al.[2],Hovhannisyan et al.[3],Graef et al.[4],Barbu et al.[5], Bonfoh et al.[6],Faye et al.[7],Samusenko[8],Liu[9]and so on.Using the singular perturbation and other’s theorys and methods the authoes also studied a class of nonlinear singular perturbation problems[10–24].In this paper,using the specialand simple method, we consider a class of the evolution equation.

Now we studied the following singular perturbation evolution equations initial-boundary value problem with two parameters

where

εand µ are small positive parameters,x=(x1,x1,···,xn) ∈ Ω,Ω is a bounded region inℜn, ∂ Ω denotes boundary of Ω for class C1+α,where α ∈ (0,1)is H¨older exponent,T0is a positive constant large enough,f(t,x,u)is a disturbed term,L signifies a uniformly elliptic operator.

Hypotheses that

[H1] σ = ε/µ as µ → 0;

[H2] αij,βiwith regard to x are H¨older continuous,g and hiare suffi ciently smooth functions in correspondence ranges;

[H3]f is a suffi ciently smooth functions in correspondence ranges except x0∈ Ω;

[H4]f(t,x,u) ≤ −c ＜ 0,(x/=x0),where C ＞ 0 is a constant and for f(t,x,u)=0, there exists a solution

2 Construct Outer Solution

Now we construct the outer solution of problems(1.1)–(1.3).

The reduced problem for the original problem is

From hypotheses,there is a solution U00(t,x)(x/=x0)to equation(2.1).And there is a U00(t,x)which satisfies f(t,x0,U00(t.x0))=0.

Let the outer solution U00(t,x)to problems(1.1)–(1.3),and

Substituting eq.(2.2)into eq.(1.1),developing the nonlinear term f in ε,and µ,and equating coeffi cients of the same powers of εiµj(i,j=0,1,···,i+j/=0),respectively.We can obtain Uij(t,x),i,j=0,1,···,i+j/=0.Substituting U00(t,x)and Uij(t,x),i,j= 0,1,···,i+j/=0 into eq.(2.2),we obtain the outer solution U(t,x)to the original problem. But it does not continue at(t,x0)and it may not satisfy the boundary and initialconditions (1.2)–(1.3),so that we need to construct the pointed layer,boundary layer and initial layer corrective functions.

3 Construct Pointed Layer Corrective Term

Set up a local coordinate system(ρ,φ)near x0∈ Ω.Define the coordinate of every point Q in the neighborhood of x0with the following way:the coordinate ρ(≤ ρ0)is the distance from the point Q to x0,where ρ0is smallenough.The φ =(φ1,φ2,···,φn−1)is a nonsingular coordinate.

In the neighborhood of x0:(0 ≤ ρ ≤ ρ0) ∈ Ω,

where

We lead into the variables of multiple scales[1]on(0 ≤ ρ ≤ ρ0) ⊂ Ω:

where h(ρ,φ)is a function to be determined.For convenience,we still substitute ρ,φ for ~ρ,~φ below respectively.From eq.(3.1),we have

while

and K1,K2are determined operators and their constructions are omitted. and the solution u of originalproblems(1.1)–(1.3)be

where V1is a pointed layer corrective term.And

Substituting eqs.(3.1)–(3.4)into eq.(1.1),expanding nonlinear terms in σ and µ,and equating the coeffi cients of like powers of σiµj,respectively,for i,j=0,1,···,we obtain

where Gij(i,j=0,1,···,i+j/=0)are determined functions.From problems(3.5)–(3.6),we can have v100,From v100and eqs.(3.7)–(3.8),we can obtain solutions v1ij(i,j= 0,1,···,i+j/=0),successively.

From the hypotheses,it is easy to see that v1ij(i,j=0,1,···)possesses boundary layer behavior

where δij＞ 0(i,j=0,1,···)are constants.

For convenience,we still substitute v1ijfor v1ijbelow.Then from eq.(3.4),we have the pointed layer corrective term V1near(0 ≤ ρ ≤ ρ0) ⊂ Ω.

4 Construct Boundary Layer Corrective Term

Now we set up a localcoordinate systemin the neighborhood near ∂ Ω :0 ≤ ρ ≤ ρ0as Ref.[9],where.In the neighborhood of ∂ Ω :0 ≤ ρ ≤ ρ0,

where

We lead into the variables of multiple scales[1]on

while

where V2is a boundary layer corrective term.And

Substituting eq.(4.4)into eqs.(1.1)and(1.2),expanding nonlinear terms in ε and σ, and equating the coeffi cients of like powers of εiσj(i,j=0,1,···).And we obtain

From problems(4.5)–(4.6),we can have v200.And from eqs.(4.7),(4.8),we can obtain solutions v2ij(i=0,1,···,i+j/=0)successively.Substituting into eq.(4.4),we obtain the boundary layer corrective function V2for the original boundary value problems(1.1)–(1.3).

From the hypotheses,it is easy to see that v2ij(i,j=0,1,···)possesses boundary layer behavior

For convenience,we still substitute v2ijforbelow.Then from eq.(4.4)we have the boundary layer corrective term V2near

5 Construct Initial Layer Corrective Term

The solution u of originalproblems(1.1)–(1.3)be

where W is an initial layer corrective term.Substituting eq.(5.1)into eqs.(1.1)–(1.3),we

have

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

Substituting eqs.(2.2),(3.4),(4.4)and(5.6)into eqs.(5.2)–(5.5),expanding nonlinear terms in εand µ,and equating the coeffi cients oflike powers of εiµj,respectively,for i,j=0,1,···, we obtain

where Gij(i,j=0,1,···,i+j/=0)are determined functions.From problems(5.7)–(5.10),we can have w00,From w00and eqs.(5.11)–(5.14),we can obtain solutions wij(i,j= 0,1,···,i+j/=0)successively.

From the hypotheses,it is easy to see that wij(i,j=0,1,···)possesses initial layer behavior

where~δij＞ 0(i,j=0,1,···)are constants.

Then from eq.(5.15)we have the initialcorrective term W.

From eq.(5.1),thus we obtain the formal asymptotic expansion of solution u for the nonlinear singular perturbation evolution equations initial-boundary value problems(1.1)–

(1.3)with two parameters

6 The Main Result

Now we prove that this expansion(5.16)is a uniformly valid in Ω and we have the following theorem

TheoremUnder hypotheses[H1]− [H4],then there exists a solution u(t,x)of the nonlinear singular perturbation evolution equation initial-boundary value problems(1.1)–(1.3)with two parameters and holds the uniformly valid asymptotic expansion(5.16)for ε and µ in(t,x) ∈ [0,T0]× Ω.

ProofWe now get the remainder term R(t,x)of the initial-boundary value problems (1.1)–(1.3).Let

where

Using eqs.(2.2),(3.9),(4.9),(5.15),(6.1),we obtain

The linearized differential operator L reads

and therefore

For fixed ε,µ,the normed linear space N is chosen as

with norm

and the Banach space B as

with norm

From the hypotheses we may show that the condition

of the fixed point theorem[1,2]is fulfilled where l−1is independent of εand µ,i.e.,L−1is continuous.The Lipschitz condition of the fixed point theorem become

where C1,C2and C3are constants independent of ε and µ,this inequality is valid for all

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

From eq.(6.1),we have

The proof of the theorem is completed.

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两参数非线性发展方程的奇摄动尖层解

欧阳成1, 陈贤峰2, 莫嘉琪3
(1.湖州师范学院理学院, 浙江 湖州 313000)
(2.上海交通大学数学系, 上海 200240)
(3.安徽师范大学数学系, 安徽 芜湖 241003)

本文研究了一类具有非线性发展方程奇摄动问题.引入伸长变量和多重尺度,构造了初始边值问题外部解和尖层、边界层和初始层校正项,得到了问题形式解. 利用不动点定理,证明了问题的解的一致有效性.推广了对两参数的奇摄动问题的研究结果.

尖层;奇摄动;发展方程

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A < class="emphasis_bold">Article ID:0255-7797(2017)02-0247-10

0255-7797(2017)02-0247-10

∗Received date:2015-06-18 Accepted date:2015-11-30

Foundation item:Supported by the National Natural Science Foundation of China(11371248) and the Natural Science Foundation of Zhejiang Province,China(LY13A010005).

Biography:Ouyang Cheng(1962–),female,born at Deqing,Zhejiang,master,professor,major in Applied Mathematics.