A CLASS OF NONLINEAR SINGULARLY PERTURBED PROBLEMS FOR REACTION DIFFUSION EQUATIONS

2003,23B(3):377-385

..4at~cta?cientia

1t1~JJJ!1m A CLASS OF NONLINEAR SINGULARLY

PERTURBED PROBLEMS FOR REACTION DIFFUSION EQUATIONS 1 Mo Jiaqi ( ,lA;:J.. ) Huzhou Teachers' College, Huzhou 313000, China

Abstract

A class of nonlinear singularly perturbed problems for reaction diffusion equa-

tions are considered. Under suitable conditions, by using the theory of differential inequalities, the asymptotic behavior of solutions for the initial boundary value problems are studied, reduced problems of which possess two intersecting solutions. Key words

Nonlinear; reaction diffusion equation; singular perturbation

2000 MR Subject Classification

35B25; 35K57

1 Introduction Mo studied a class of singularly perturbed problems in [1]-[7]. Now we consider the following nonlinear singularly perturbed reaction diffusion problem: £2Lu - cUt = f(x, U, f),

Bu

=anau + a(x)u = g(x,£),

(t, x) E 17,

a(x) ~ ao > 0, x E

(1)

an,

u = h(x,£), t = 0,

(2)

(3)

where c is a positive small parameter, L is a uniformly elliptic operator

n

L

i,j=l

n

aij~i~j ~ A L~J, V~i

E R, A> 0,

i=l

= {(t, x)lt >

0, x En}, 0. signifies a bounded domain in H" with smooth boundary an, a/an denotes the outward derivative on an. This paper involves a class of nonlinear singularly perturbed reaction diffusion problems, which have two intersecting solutions of the reduced problem. We construct the asymptotic expansion of solution and discuss its asymptotic behavior. It is a very attentive problem in the international academic circles[8]. and V

lReceived July 16, 2001. The Importent Study Profect of the National Natural Science Foundation of China (90211004); The Natural Sciences Foundation of Zhejiang (102009)

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We assume that [HI] aij, i3i' o, t, 9 and h are sufficiently smooth functions with repect to variables in corresponding ranges. [H2 ] The reduced problem (4) i(x,u,O) = 0, x E!l of the problem (1)-(3) has two sufficiently smooth solutions UlO(x) and U20(x).

2

The Formal Composed Stable Outer Solution Let the formal expansion of an outer solution for the pr?blem (1)-(3) for UlO(x) be 00

UI Substituting (5) into (1), developing of c, we obtain

'"

i

LUlic

i.

i=O

(5)

in z, and equating coefficients of the same powers

iu(x, UlO, O)Uli = LUI(i-2) - ii, i = 1,2,···,

(6)

where UlO is a solution of (4) and

ii

= ~ [~i~ z.

oe

]E=O

,i

= 1, 2, ....

It is easy to see that ii, i = 1,2,'"

are determined by functions Ulk , k ::::; i - 1. In the above and below, the values of the terms for negative subscripts are zero. From (6), we may obtain Uli, i = 1,2,'" successively. From (5), we obtain the outer solution U I for the original problem. Let the formal expansion of another outer solution for the problem (1)-(3) for U20(x) be 00

U2 '" L U2ic i.

i=O

(7)

Using the same method for the outer solution U I we can determine U2i , i = 1,2, . ". Then we may decide the outer solution U 2 too. From (5),(7), we now construct the composed stable outer solution for the problem (1)-(3)

L o.«, 00

tj

where U, =

{

»:

i=O

o.; u.. ~ U2i , U2i,

o.. < U2i,

i = 0, 1,2, ....

And let region Sm C !l be a projected hypersurface where :Z=:'o Ulic i = :Z=:'o U2iC i for sufficiently small parameter c. We denote x E !lIm as ~:'O UliC i > :Z=:'O U2ici, and x E !l2m as

""m L..,i=O Uue

i

< ""m L..,i=O Usie i .

We next set up a local coordinate system (r, s) near Sm for sufficiently small c. Define the coordinate of every point Q in the neighborhood near Sm in the following way: The coordinate

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[r], (Irl :S S) is the distance from the point Q to the Sm, where r = 0 describes the hypersurface Sm, r < 0 characterzes points in n 1m , r > 0 represents points in n 2m , and S is small enough such that normal at every point of Sm does not intersect each other in this S-neighborhood n J near Sm' S = (Sl' S2,' .. ,Sn-d is a nonsingular coordinate system on the (n - I)-dimensional manifold Sm' The coordinate S of the point Q is equal to the coordinate S of the point P at which the normal through the point Q intersects the hypersurface Sm' It is obvious that, if S is sufficiently small, then (r, s) represents a local coordinate system in S-neighborhood nJ. We also assume that [H3 ] Sm is a simple smooth closed Jordan connected region in nand m

fu(x, L

UlQc i, 0) > 0, fu(x, L

i=O

i=O

m

m

fu(x'LUlQCi,O)

U20c i, 0)

< 0, x

< 0, fu(x, L U20 c i , 0) > 0, x

i=O

E

n 1mUan,

E n 2mUan.

i=O

We define the function W m by

It follows from [H1 J, [H2 ] that

fu(x, Wm,c) {

> 0,

= 0,

(8)

Furthermore, we assume that [H4 ] fu(x, Wm,c) 2: K;/rlm+l, x E nJ, where K; is a positive constant and (r,s) is the local coordinate system in n J. Notice that the composed stable outer solution U may not satisfy the boundary and initial conditions (2),(3), so that we need to construct boundary and initial layer functions v and w.

3

The Boundary Layer Term

We first construct the boundary layer v. Set up a local coordinate system (p, ¢) near an too. Define the coordinate in the po-neighborhood p o near an with the same way as the local coordinate system (r, s) near Sm, where Po is small enough such that the po-neighborhood p o near an and the S-neighborhood n J of Sm do not intersect each other. In the po-neighborhood npo near an : 0 :S p:::; Po,

n

n

(9)

where the exprenssions of the coefficients of L in (9) are omitted.

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We lead into the variables of multiple scales[9,1O] on 0 ~ P ~ Po:

h(p, ¢) _

= ---, P = P, ¢ = ¢, e

U

where h(p, ¢) is a function to be determined. For convenience, we still substitute p for From (9), we have

L

1

1

= c2Ko + -K 1 +Ko, c

where

p below. (10)

02 o K = annh~ ou 2 ' 02

K, = 2annh pouop

n-l

+L

02 anihp OUO¢i

.

_

0

+ (annh pp + bnhp) OU '

t=l

From (9) and (10), let h( p, 'f' "') and the solution u be

u

-l -

P

a

J ann1(PI, "') dPI, 'f'

= U(x,c) +v(u,p,¢,c).

Let

(11)

00

v"" LVi(U,p,¢)ci.

(12)

i=O

Substituting (10),(11) and (12) into (1),(2), expanding nonlinear terms in e, and equating the coefficients of same powers of e, we obtain (13)

ova OU = g(x, 0), x For i = 1,2"", we have

02Vi ou 2

Eon.

(14)

= fu(x, Uo + va, 0) + Fi ,

OVi OU = G, - BUi -

x

l ,

(15)

Eon,

(16)

where F i and G, are successively determined functions. From the hypertheses [H 1]-[H3 ] and (8), it is easy to see that there are solution Vi i = 0,1"" of the problem (13)-(14) and (15)-(16), and satisfy estimation near the boundary on

Vi = C, exp( -I\:i exp u) = C i( where C, and I\:i > 0 are some constants. Thus we have

-1\:/\ e

00

ii "" LVici, i=O

0

< e ~ 1,

(17)

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381

where (18) while 'l/Jl (p) is a sufficiently smooth function and

={

'l/Jl(P)

4

I,

0::; p::; (1/3)po,

_

0, ((2/3)po::; P::; Po) U(O\O ::; P::; Po).

The Initial Layer Term We now introduce a stretched variable[lO] t r--

(19)

- e'

and let the solution of the problem (1),(3) be

u = U +v+w(r,x,e), where

W

(20)

is an initial layer term possessing the form 00

W rv LWi(r,x)ei.

(21)

i=O

Substituting (19)-(21) into (1)-(3), we have

oWo ar

_

_

+ f(Uo + Vo + Wo, 0) - f(Uo + Vo, 0) = 0,

(22)

wolr=t=o = h(x,O), OWi

or

_

(23)

_ -

+ f(Uo + Vo + Wo, 0) = L[Ui- 2 + Vi-2 + Wi-2] + Pi,

i = 1,2,···,

(24) (25)

where Pi and Hi are successively determined functions. From (22),(23) and (24),(25), we can yield Wi, i = 0, 1, ... , which possess initial layer behavior as follows:

(26) where C i and "K;i > 0 are some constants. Thus we have initial layer term (21). From (20), we can construct the following formal asymptotic expansion of the solution u for the original problem (1)-(3): 00

U rv

L[U i +Vi +wi]e i, (t,x) E V +oV,O < e« 1,

(27)

i=O

where

Uli(x)

+ 'l/J2 (r/e) (U2i (x) - Uli(x)), x

E

0 0,

Uli(x),

x E (Olm + OOlm)\Oo,

U2i (x),

X

E (02m

+ a0 2 m)\00,

(28)

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while (T, s) is the local coordinate system in 0 0 and 'l/J2(C;) is a sufficiently smooth function such that

'l/J2(") =

{

a,

,,~ -1,

E (0,1),

-1 <.«

1,

" 2: 1.

< 1,

It is obvious that from (28) 2::::0 Uisi is a sufficiently smooth function.

2::::0U iSi

in the form

m

m

LUisi

= LUisi + W(x,s).

i=O

i=O

Then, by taking into account 2::::0 U2iS i - 2:::: 0 UliS i that for 0 0 = 0c' (i.e. 6 = e), W(x,s) satisfies

_

If we represent

)

W(x,s =

= O(iTlm+I)

in 0 0 , it is easy to show

{O(Sm+I), x E Oc, 0, x E (0

+ aO)\Oc'

Moreover, we have

UiSi] + O(sm+!), <'L(U)~{ S2S2 L[2::::0 L[2::::0 UiSi] + O(sm+2),

5

x E 0c'

XE (0 + aO)\Oc'

(29)

The Main Result

Now we prove that (27) is uniformal valid asymptotic expansion. Theorem Under the hypotheses [HI]-[I1.iJ, there exists a solution u of the singularly perturbed problem (1)-(3) for the nonlinear reaction diffusion equation, possessing a uniformaly

valid asymptotic expansion (27) in (t, x) E V + av for a « e < 1. Proof We first construct two auxiliary functions a and /3:

(30)

/3

= Ym + I'Sm+!,

(31)

where I' is a positive constant, large enough, which will be decided below, and m

Ym == L[Ui

+ Vi + Wi]Si.

i=O

Now we prove that a and tively.

/3

are lower and upper solutions for the problem (1)-(3) respec-

Obviously, a and /3 are continuously differentiable with respect to t, x in VIm + aVIm and twice continuously differentiable with respect to t, x in VIm USm and in V2m + av2m. And we have

/3, (t, x) E (V + aV). /3 are smooth functions

a ~

From (28), we know that a and

(32) near Sm' Then we have

aa aa a/3 a/3 aT 1+0 - aT 1-0 2: 0, aT 1+0 - aT 1-0 ~ 0, x E Sm,

(33)

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where a/or denotes differentiation with respect to the inner normal of Sm. From (26),(27), we also have that, there are positive constants M 1 and M 2 for x E all, such that Bo.

= BYm

m

m

-

Bbc m + 1 ]

i=l m

= g(x, 0) + L

GiC

i

m

= B[L Ulic i] + B[L Vici] + B[L Wici] i=l

+ .1\11cm +1 -

a(xhc m +1

i=O

ao/,c m +1

i=l

::; g(x,c)

+ (M 1 + M 2 ) C m +l

Thus selecting /' 2: (M 1

-

ao/,c m + 1 .

+ M 2 ) / ao we have Bo.::; g(x,c), x

Analogously, for /' 2: (M 1

+ kh)/ao,

Eon.

(34)

we can prove that

B(3 2: g(x,c), x E all.

(35)

And we have that, there is a positive constant M 3 such that

Thus selecting /' 2: M 3 , we obtain o.(O,x,c)::; h(x,c).

(36)

Analogously, for /' 2: .1\13 we can prove that (3(0, x, c) 2: h(x, c).

(37)

Now we prove that

We only check that (3 satisfies the inequality (39). The inequality (38) for a. may be checked by the same way. We divide the discussion of (39) into three cases as follows.

(i) x E lib == nc. From the hypotheses [Hd-[H 4 J, (28),(29) and Vi = 0, i < e ::; C1, there are positive constants M 4 and M 5 such that

°

0,1,···, for small enough c,

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m

~ f(x, UO , 0) + 2)LUi- 2 - fu(x, UO , O)Ui - j;]e i i=O

OWO fh - f(x, Uo + wo, 0) - f(x, Uo, 0)]

+ [-

+~ ~ [+(M4 Selecting I' 2: (M4

OWi 07 - fu(x, Ui,

+ M5

+ M 5 )/ !\; , we may

-

+ wo, O)Wi + L[Ui- 2 + Wi-2] + -] F, e i

= (M4 + M 5 -

!\;I')em+l

!\;1')em+ 1.

prove the inequality (39).

(ii) x E Dp o ' As 0 ~ P ~ (1/3)po, from the hypotheses [H1]-[H3 ], (8),(18),(28),(29) and Vi = Vi, Z = 0,1"", for small enough e, 0 < e ~ el, there are positive constants M 6 and Cl > 0 such that e 2 L{3 - e{3t - f(x, (3, c) m 2 = e L ~[Ui [

+

[f(X,

+ Vi + Wi]e i

]

m

m

- e ~(Wi)tei - f(x, ~[Ui + Vi

~[Ut + Vi + Wi]ei,e) -

f(x,

+ Wi]e i, c)

~[Ui + Vi + Wi]e i + l'e m+ ,c)] 1

m

~ f(x, U«, 0) + L[LU1(i-2) - fu(x, Ue, O)Uli

-

fi]e i

i=O

+ [02 ouVO 2 -

+~ + [-

+ Va, 0) + f(x, UlO , 0)]

f(x, UlO

[~:2i -

fu(x, UlO

owo 07 - f(x, Ui,

+~ ~ [-

+ Va, O)Vi -

-

Cl1'E

m

i

+ Va + Wo, 0) + f(x, Uo + Va, 0)]

OWi 07 - fu(x, Uo + Vo

+M6 E m + 1

Fi] e

+1 = (M 6

+ WO, O)Wi + L[Ui-2 + Vi-2 + Wi-2] + -] Fi -

C11') E

m

E

i

+l .

Selecting I' 2: M 6 / Cl , we may prove the inequality (39). As (1/3)po ~ P ~ Po, from Vi, uu; i = 0,1"" being exponentially decay and asymptotically tending to zero for (17) and (26) or Vi = 0, i = 0,1" . " by the same argument as above, we can show that (39) holds too. (iii) x E D\(D p o UDo == Dc))' From the hypotheses [H1]-[H3],(8),(18), and Vi = 0, i = 0,1"", for small enough e, o < E ~ El, and by using the same method of (i), there are positive constants M 7 and C2 > 0 such that

Selecting I' 2:

Md C2,

we may prove the inequality (39).

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From (32)-(39), we know that a and f3 are lower and upper solutions for the problem (1)-(3) respectively. Thus from [8J, we obtain

a(x,c) ::; u(x,c) ::; f3(x,c), (t,x,c) E (V

+ BV)

X

[O,Cl].

From (30) and (31), we obtain m

u(x, t,c) =

2:)U i + Vi + Wi]c i + O(cm+l), i=O

t 2: 0, x E 0,

°< c«

l.

The proof of the theorem is completed. References

2 3 4 5 6

Mo Jiaqi. The singularly perturbed problem for combustion reaction diffusion. Acta Math Appl Sinica, 2001,17: 255-259 Mo Jiaqi, Ouyang Cheng. A class of nonlocal boundary value problems of nonlinear elliptic systems in unbounded domains. Acta Math Sci, 2001,21B: 93-97 Mo Jiaqi. A class of singularly perturbed reaction diffusion integral differential system. Acta Math Appl Sinica, 1999,15: 19-23 Mo Jiaqi. A class of singularly perturbed problems with nonlinear reaction diffusion equation. Advances in Math, 1998, 27: 53-58 Mo Jiaqi. The nonlocal boundary value problems of nonlinear elliptic systems in unbounded domains. Appl Math Comput, 1997,86: 115-121 Mo Jiaqi. A singualrly perturbed nonlinear boundary value problem. J Math Anal Appl, 1993,178: 289-293

7 Mo Jiaqi, Singular perturbation for a class of nonlinear reaction diffusion systems. Science in China, 1989, 32: 1306-1315

8 Butuzov V F, Nefedov N N, Schneider K R. Singularly perturbed elliptic problems in the case of exchange of stabilities. J Differential equs, 2001,169: 373-395 9 Holmes M H. Introduction to Perturbation Methods. Texts in Applied Mathematics 20. New York: Springer-Verlag, 1991 10 De Jager E M, Jiang Furu. The Theory of Singular Perturbations. Amsterdam: North-Holland, 1996