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EXISTENCE OF TRAVELING-WAVE TYPE SOLUTIONS FOR THE BELOUSOV-ZHABOTINSKII

II

SYSTEM OF EQUATIONS.

UDC 517.9

A. Ya. Kapel'

INTRODUCTION The well-known Belousov-Zhabotinskii

reaction

[i, pp. 87-89]

is described by the sys-

tem o~

~

ot

Ox~

6v Ot

=u(t--u

02U Oxz

--

rv),

buy,

r = const >

O,

(1)

b = const > O.

The quantities u, v correspond to the bromic acid and bromide ion concentrations. This system was derived in [2, pp. 226-229]. It can be regarded as a model for many other more complex biochemical and biological processes. Such processes are characterized, in the planar case, by the presence of circular waves that propagate with some constant speed c [i, pp. 147153]. To these waves there correspond solutions of (i) of the form

u=u(~)>O,

v=v(~)>O,

u(--~)=v(+~)=O,

(2)

~=x+ct,

v(--~)=u(+~)=l,

which are called traveling-wave (TV) type solutions. (i) yields the system of differential equations u"=cu'-u(l--u--r~),

(3)

Substitution of expressions

(2) in Eqs.

(4)

v"=cv'+b~v.

One has t o f i n d t h e s o l u t i o n of s y s t e m (4) under t h e boundary c o n d i t i o n s ( 3 ) .

In [2, pp. 226-

235] results of numerical computations of the TV and its propagation speed are given for a wide range of values of r and b. In [3] it is proved that for any b, 0 0 and c, 0 < c < 2, such that a TV solution exists for Eqs. (I). In [4], the existence of a TV for Eqs. (i) is established in the absence of the diffusion term in the first or second equation. In [5] the existence of a TV is established for an interesting class of locally monotone systems by means of the fixed-point method. In this paper we obtain the following results,

stated as Theorems 1-4.

THEOREM i. For any c > 0, r > 0, b > 0, b ~ i, there exists a solution (2) of system (4) such that u(-~) = v(+~) = 0, u(+~) = i, v0 = v(-~) ~ I/r. The quantity v 0 corresponds

to the bromide ion concentration in front of the wave.

If one makes the substitution v = v0~, then the system retains its original form, with r replaced by rv 0. Hence, Theorem i can be restated as follows: for any b > 0, b z i, c > 0, there exists a value r ~ i for which problem (4), (3) admits a solution. THEOREM 2. Let c e 2 b/~l, with bl = min(b, I), i - b I < r ~ i, where in addition r i there exists a value c for which problem (4), (3) admits a solution. Moreover, here one has that b/2/(r + b)[bz(r + b i 0.5b) g c < 2 / ~ . In [6] one finds proofs of Theorem 2 and 3 for b < i and of the existence of a solution of problem (4), (3) for 0 < b < i, r + b < i, c e 2/i - r. The last case is also considered in [7]. Moscow. Translated from Sibirskii Matematicheskii Zhurnal, Vol. 32, No. 3, pp. 47-59, May-June, 1991. Original article submitted September 8, 1989.

390

0037-4466/91/3203-0390512.50

9

1992 Plenum Publishing Corporation

Let us put w = 1 - u.

Then (4)

takes

w"=cw'+w-w

the

form

2-ru+rvw,

Let us linearize this system near w = 0 (or u = i), v = 0. w" =cw'

(5)

v"=cv'+bu-buw.

+ w-rv,

v" =cv'

We obtain the system

+ bv.

(6)

The solution of (6) which is bounded for ~ § +~ has the form

(7) (8)

w = C t exp k:~ + C2 exp k2~, v =((1--

b ) / r ) C 1 exp ki~,

where k i < 0, k 2 < 0 are the roots of the characteristic

equation of system (6),

( k 2 - c k - b) (k 2 - c k - t) = O, kl = cl2 -- 7c214 + b,

(9) (10)

k2 = cl2 - }"d14 + t.

[k=[

Let 0 ikml. Of interest is only the solution (7), (8) that is positive in a neighborhood of the point ~ § +~. Hence, we must take Ci > 0. Set s = C i e x p k i $ . Then expressions (7), (8) become

w=s

+ ~s ~,

•

k2/k I >

v =((INow l e t

b > 1.

P u t s = C 2 e x p k 2 $ , C2 > 0 , w = s + ls •

I,

2~

C2,C i,

(ii) (12)

b)/r)s.

C z _< 0.

Then expressions

(7),

(8)

E = CliC ~ ~ 0,

x = lq/k 2 >1,

take

the

form (13)

v = ((I - b)/r)ks".

(14)

I n S e c . 1 we p a s s t o t h e v a r i a b l e s in system (5). N e a r s = 0 we c o n s t r u c t the solution in the form of a series i n p o w e r s o f s a n d i s K, t h e f i r s t terms of which have the form (ii)-(14). In Sec. 2 we work in the phase space u, v, p = u ~, q = v', taking u as the independent variable. Using the maximum principle we show that the quantities v, p/(l - u) are monotonic in u and we obtain a priori estimates. In Sec. 3 we show that the solution is monotonic in the parameter I. In Sec. 4 we prove the existence of the solution of the boundary-value problem in phase space by making use of the results of Secs. 1-3 and the considerations in Sec. 2 of [8]. The proof amounts to selecting values of I and c for which the required boundary conditions are satisfied.

For b = 1 the roots of Eq. (9) have multiplicity tained by letting b § i, b < i. !.

larger than I.

The solution is ob-

CONSTRUCTION OF THE SOLUTION IN THE NEIGHBORHOOD OF ~ = +~

For the sake of convenience let us write system (4) in the new variables z = (r/(l b))v, b ~ i, q = w - z, s = C e x p k g , where C = Ci, k = ki if b < i, and C = C2, k = k 2 for b > I. Then system (4) takes on the form k2s2z " -F blsz' - bz = - b z ( z + ~]) ,

bi = rain(b, I ) ,

(i.1)

k2S2~l " -}- blS~l ' - ~1 = - ~1 ( z -[- rl) .

We seek a solution of (i.i) in the form Z=

?]

r"~

i• zi(x),

" "" 11 -- ~.~'~7L~s~,:q~(@

i=0

The parameters

(1.2)

i=0

I and ~ are defined in (i0) for b i.

Let us formally substitute expressions (1.2) in Eqs. (i.I) and equate the left- and right-hand coefficients of lis i

1

~ P(h)

'

p > 0 and p ( u )

tl 2

I zq chq = "27 (,0 "

( 2.30 )

'o

from (2.30) and (2.27) we obtain

(u) > ~22/2 [:)(0) b - r~' (0) + cb + c;.v (0) !. Finally,

is in-

(2.31), (2.20), and (2.29)

(2.31)

yield the needed estimate

/)O~)>Yzz2(t--zz), Suppose p(0) = 0. Then also q(0) = 0. (2.27) takes on the form

7 = e , ' 2 [ b + c ~ ( b + !)1.

(2.32)

Let us estimate v(0) from below.

Equality

1

(2.33)

eb + cry (0) = b I ~' (1 - ,:) du. 0

Now from (2.33) and (2.17) we deduce that cb + cry(0) > b/21kl, whence,

after elementary

transformations, bl--mi~(b,

v(O)>(b/rb:)(t/4-b:+(I/2)yl/4+b:/c~), 3.

i).

(2.34)

MONOTONICITY OF THE SOLUTION IN THE PARAMETER k

Let us consider the family of solutions (2.7) for X ~ 0. Let Xl < X2 ~ 0, Pi(U) = p(u, I i, c), qi(u) = q(u, li, c), vi(u) = v(u, li, c), i = i, 2, Ap = P2 -- P1~ Aq = q2 - ql, Av = v 2 -- v z. From (2.1) we obtain for the functions Pi the relation

dAp du

u Ap(I--u--rv,)-P pip2

ru AV. ~ ' p~

(3,1)

395

Also,

(2.22) and (2.2) imply the following relations among Pi, qi, vi:

d_Au du

P2

Av + g (u),

(3.2)

1

,P.--7 p~ (,~) P~ (,q) ap (,,~) e,,~ + ---~p., b - ~--~. /.

(3.3)

u,

From (1.8)-(1.13)

it follows that in a sufficiently small neighborhood of u = I, u < i,

Ap > O, Let us show t h a t t h e s e i n e q u a l i t i e s (3.1) for ap and (3.2) for av.

i U, (]-- Ul)

Au < O.

(3.4)

a r e p r e s e r v e d as long as Pl > O, P2 > O.

Let us solve

We obtain

-- ~rL,1

i

ru~

i u2(t--u,~)[l--,~l(U2)]

du2dul '

(3.5)

and

Au=Av(uo) exp

--du

1+

p~(~)

"UO

-p~ - (u) dU2 dup

g(u 0exp

~/0

(3.6)

ul

Suppose Av(5 I) = 0, Av < 0 for El 0 for E 2 0 for E l ~ u < i. In conjunction with formula (3.3) and the inequality Vm'(U) < 0, this implies that g(u) > 0 for E I ~ u < i. This last inequality and (3.6) imply Av(UI) < 0. The contradiction we have reached proves inequalities (3.4). Suppose Pl > O, P2 > 0 for 0 O, then from (3.5) we

Ap(O)> O.

(3.7)

If v1(0) > l/r, then (3.7) again holds. This is readily verified using (2.23), (2.24) for Pm and (3.5) for u0 = 0. Suppose p2(0) > 0. In (3.6) put u = 0, u0 = i. We obtain

Av(O)< 0. Now suppose p2(0) = 0. we have

(3.8)

Then also p0(0) = 0, ql(0) = q2(0) = 0.

By (2.27), for Pi, qi, vi

1

Av(0)=~-b I u ( l _ _ u ) A J _ d u < 0 " rc

p 0

i.e.,

i n e q u a l i t y ( 3 . 8 ) a l s o h o l d s in t h e case where p2(O) = O. 4.

PROOF OF THE EXISTENCE OF A SOLUTION TO THE BOUNDARYVALUE PROBLEM

Let us consider the equation

dZ du

where f'(u) is piecewise continuous

=

c --

1~"~,

p

c =

const

>

O,

(4.1)

for 0 ~ u ~ i.

LEMMA i. Let f(u) > 0 for 0 0, f'(1) < 0, f'(u) f'(0) for 0 0 for 0 0 if c
0 and p2(u) clp du

with

f~ and f= c o n t i n u o u s

> 0 be solutions

h (~)

- - ,P

ci

0 c=, p~ (0) ~> p~ (0),

for 0 ~ u ~ uo,

> p=(u)

Proof.

From

(4.2)

f~ ~ 0 for 0 ~ u ~ ~, 0 < ~ ~ u0,

h(u)>h(~) Then pz(u)

of the e q u a t i o n s

and

fo= 0 < ~ 0 for u 0 0, p2(u) > 0 satisfy Eqs. (4.2) for u 0 p1(u) for u 0 0, b > 0, b x 1 there exists c) > 0 for u 0 < u < i, p(u0, ~0, c) = 0, u 0 ~ 0.

a value

X = ~0 such that p(u,

Xo,

The proof follows from (1.9), (i.ii) , and (1.13). One has to put i 0 = -so i - ~ , K = ks/9 k I if b < i, and 10 = -(I/K)s0 1 - K K = kl/k 2 if b > i, where so is s u f f i c i e n t l y small, s o > 0. The q u a n t i t y u 0 c o r r e s p o n d s to the value s = s o . T H E O R E M 4.1. For any c > 0, r > 0, b > 0, b ~ i, there exists a value ~ = 10(c) such that p(u, 10(c), c) > 0 for 0 0 in some interval 0 i.

(4~5)

From (4.5), (1.5), (1.6), and (1.13) it follows that p(u, 0, c) satisfies the e q u a t i o n dp/ du = c - (blu(l - u))/p. By Lemma i, the function p = p(u, 0, c) is defined e v e r y w h e r e in the interval 0 0 for 0 < u < i, p(0, 0, c) ~ 0. Hence, the set of all values p r o v i d e d by Lemma 4 is bounded above. Let 10(c) be the s u p r e m u m of this set. From the very d e f i n i t i o n of 10(c) it follows that for I > %0(c) the solution (2.7) is defined for 0 yu2(l - u), where 7 does not depend on ~. By this estimate and the continuous dependence of the solution on i and u, the curve p = p(u, X0(c), c) intersects the u axis for u < 1 only at the origin, as needed. LEM]iA 5.

For any c > 0, r > 0, b > 0, b = 1 one has the

inequality

v(0, lo(c), c ) ~ t/r. sects

(4.6)

Proof. L e t t < Xo(c ) and l e t u 0 be a s i n T h e o r e m 4 . 1 . The c u r v e p = p ( u , t h e u a x i s a t t h e p o i n t uo a t an a n g l e ~ ~ ~ / 2 w i t h t h e p o s i t i v e l y - o r i e n t e d

From this c i r c u m s t a n c e and Eq. (2.1) it follows u(l - u)(l - rv). The last i n e q u a l i t y implies

that

f(u0)

~ 0, where

f(u)

1, c ) i n t e r u axis.

= u(! - u - rv) =

~(uo, %, c ) ~ t/r. Let I + %0(c), X < ~0(c). v(u0, ~, c) § v(0, lo(c), value

(4.7)

Then by the c o n t i n u i t y of the s o l u t i o n in u and I we have u 0 § 0, c) = v(0, 10(c), c). In the limit (4.7) yields (4.6).

LEMMA 6. For any r > 0, b > 0, b x i, c, cl, Ii = I(ci) such that

0 < c I ~ c < 2 b/~1, there

exists

a unique

397

(o, ~ (o0, o) = (l - o~/4) ~.

(4.8)

Proof. We have v(0, 0, c) = (i - bl)/r < (i - c12/4)/r. By Lemma 5, v(0, 10(c), c) i/r > (I - c12/4)/r, w h i c h in view of the fact that v is continuous and strictly decreasing in I implies the assertion of the lemma. LEMMA 7. Let i - b I < r < I, b I = min(b, I), b ~ i, and there exists a value i = If(c) such that p(u, If(c), c) > 0 for 0 < u < i, v(0, ii(c), c) = i. Then for 0 < u < i one has the inequalities

pl(u) 0 for u < i, i = i, 2, f~(u) = (i - r)u(l -

Proof. The function v = v(u, 11(c), c)/(l - u) is d e c r e a s i n g in u, v = i for u = 0, = i for u = 0, v = (I - bl)/r for u = i. Therefore, (i - bl)/r < v < i for 0 < u < i, whence f2(u) < f(u) < f2(u) for 0 < u < i, where f(u) = u(l - u)(l - rv). By Lemma 3, these inequalities imply inequalities (4.9). T H E O R E M 4.2.

Let i - b I < r 0,

Then there exists a unique value

398

~ (~) ' du

0

i - ba < r ~ I/(b + I),

(4.12)

o ~ 2 ~ I - r.

(4.13)

I = It(c)

such that

p(~,7.,(c),c)>0np~0 K U The constants

for 0 < ZZ< ~Zl, ~ > 0,

U~ > 0.

(4.19)

K and u I depend only on b and ci.

Proof. Put fl(u) = u(l - u - rv(u, 1o, c)) = u(l - u)(l - rv(u, 1 o, c)), f2(u) = u(l u)(l - rv(u, 11(cl) , c)), u(l - u)(l - rv(u, ll(cl) , c)), where 11(c l) is as in Lemma 6. From (4.8), (4.18), and the fact that v is d e c r e a s i n g in I it follows that l 0 < ll(Cl). This implies v(u, 11(ci) , c) < v(u, Io, c), and c o n s e q u e n t l y fl(u) < f2(u). Similarly to (4.15) we have f2(u) ~ f3(u) = u(l - u - rv0 + rv0bu/(l - rv0)) for 0 O, d p 3 / d u = c~ - f 3 / P 3 f o r 0 < u ~ u l , p 3 ( O ) = O. cause, by (4.8), c I = 2/f3'(0). We consider that P3 is fixed.

(4.20)

The s o l u t i o n P3 e x i s t s We also have

pS(Z~)~ K~g for 0 ~ [g ~ ~Zl. Inequalities

(4.20),

(4.21)

be-

(4.21)

imply (4.19).

T H E O R E M 4.4. For any r > i, b > 0 there exists a value c > 0 for which the boundaryvalue p r o b l e m (2.1)-(2.5) has a solution. Proof. Let b ~ i, and let i = 1o(c) small c > 0 we have

be as in T h e o r e m 4.1.

v(O, lo(c), c ) > t > f/r.

By (2.34)~

for s u f f i c i e n t l y

(4.22)

Also, p(0, 10(c) , c) = 0. On the other hand, v(0, 0, c) = (i - bl)/r < l. Moreover, p(0, I, c) is strictly increasing in I > 10(c ) thanks to (4.22) (see Sec. 2), and p(0, 0, c) > 0 for c < 2 / ~ . Consequently, for a r b i t r a r y s u f f i c i e n t l y small c > 0 there is a unique value I = 12(c) such that

399

v(O, X2(c), c ) = l ,

p(O, ~2(c), c ) > O .

(4.23)

Consider the set of all values of c for which (4.23) holds. From relations (4.23), Lemma 8, and the monotonicity of v and p in the variable % (see Sec. 2) it follows that c < 2/b I. Thus, the upper bound of the set in question is as follows: co ~ 2 / ~ . For c = c o (4.23) falls since the solution satisfying (4.23) is continuous in c. Therefore, (4.23) holds along some sequence of values c = c n + co, c n < co. By Lemma 9,

p(U, ~.2(Cn), cn)>gv~ for

O %3 = const, with %3 independent of n. Thus, ~3 < %l(Cn) < 0. Consequently, there exists a subsequence c n § co (we use for it the previous notation) such that ~2(Cn) + 44. Then, by the continuity of the solution, Pn(U) ~ p(u, %l(cn), c n) + p(u, ~4, Co), q ~ q(u, ~l(Cn), Cn) § q(u, ~4, c0), Vn(U) ~ v(u, ~1(Cn), c n) § v(u, ~4, Co) uniformly in u on the segment [0, i]. The solutions Pn, qn, Vn obey the a priori estimates (2.32), (4.24), which are independent of n. Therefore, the limit solution is defined everywhere in the interval 0 0, then it is not difficult to show that v(0, ~4, co) = i. To this end one has to use Eqs. (2.1)-(2.3) and the a priori estimates (2.32), (4.24). Since the last case is impossible, p(0, %4, co) = 0. Let us show that then v(0, ~4, co) = i. From the equality p(0, X4, co) = 0 it follows that q(0, X~, c o ) = 0 (see Sec. 2). By (2.27), Pn, qn, Vn and the limit solution satisfy the relations 1

p~ (0) b -- rq~ (0) = b ~ u (t -- zd. du, .

p ~ (u)

(4.25)

0 1

c~

" P 0

( u , - - - du. ~4' co)

(4.26)

In (4.25) let us pass to the limit n + ~. Using (4.26), (4.24), (2.32), we obtain v(0, ~4, c o ) = i. In the case b = i the theorem is proved exactly as Theorem 4.2, with the aid of Lemma 9, the a priori estimates (2.32), (4.24), and p < P2 (with P2 as in Lemma 7) and the uniform boundedness of Pu' in b and u. Theorem 4.1, Lemma 5, and Theorems 4.2-4.4 imply the Theorems 1-4 formulated in the introduction (cf. [9, Sec. 2]). The lower bound for c given in Theorem 4 follows from (2.34) and v(0) = I. One can show that X4 = X0(c0) and c o < 2/b z .

i. 2. 3. 4. 5.

LITERATURE CITED A . M . Zhabotinskii, Concentration Autooscillations [in Russian], Nauka, Moscow (1974). J . D . Murray, Nonlinear Differential Equations in Biology. Lectures on Models [Russian translation], Mir, Moscow (1983). W . C . Troy, "The existence of traveling wave front solutions of a model of the BelousovZhabotinskii chemical reaction," J. Differential Equations, 36, No. l, 89-98 (1980). R . G . Gibbs, "Traveling waves in the Belousov-Zhabotinskii reaction," SIAM J. Appl. Math., 38, 422-444 (1980). A . I . Vol'pert and V. A. Vol'pert, "Application of the theory of rotation of vector fields to the study of wave solutions of parabolic equations," Trudy Mosk. Mat. Obshch.,

5_.22, 58-109 (1989). 6. 7.

8.

400

Ya. I. Kanel', "Existence of traveling-wave type solutions for the Belousov-Zhabotinskii system of equations," Differents. Uravn., 26, No. 4, 625-660 (1990). Qi Xiao Ye and Ming Xin Wang, "Traveling fron wave solution of Noyes-Field system for Belousov-Zhabotinskii reaction," Nonlinear Anal. Theory Meth. Appl., ii, 1289-1302 (1987). A . N . Kolmogorov, I. G. Petrovskii, and N. S. Piskunov, "Study of a diffusion equation that is related to the growth of a quantity of matter, and its application to a biological problem," Byul. Mosk. Gos. Univ., Ser. A, Mat. Mekh., i, No. 6, 1-26 (1937).