Nonlinear
Analysis.
Theory. Methods
Vol. 22, No. 12, pp. 1475-1485, 1994
Copyright
0 1994 Elsevier Science Lfd
Printed in Great Britain. All rights reserved
0362-546X/94
$7.00+
.OO
&Applications,
Pergamon
ON S-SHAPED BIFURCATION
SHIN-HWA
CURVES
WANG
Department of Mathematics, National Tsing Hua University, Hsinchu, Taiwan 300, Republic of China
(Received 26 August 1992; received in revised form 24 February
1993; received for publication
Key words and phrases:
Nonnegative solution, time map, bifurcation
two-point boundary-value problem.
curve, bifurcation
16 June 1993)
diagram,
1. INTRODUCTION
IN THIS paper we study mainly the bifurcation
two-point boundary-value problem
of nonnegative
CYU
-u”(x) = Aj(u(x)) = A exp cY+z4
i
1’
solutions of the nonlinear
-1 <x<
1,
(l.lh
u(-1) = u(1) = 0,
where U(X) is the temperature, 1 1 0 is the Frank-Kamenetskii parameter, and (Y> 0 is the
activation energy. This is the one-dimensional case of a problem of some importance in the
theory of combustion. This problem in general form is written as
-Au=Aexp
u=o
CYU
I o+U 1
in Sz,
(1.2),
on XJ,
where Sz c R” (n 2 1) is bounded with smooth boundary aQ. Readers interested in the
derivation of the problem are referred to the work by Bebernen [l], particularly Section 2.
Relevant work on the analysis of this problem has been done by Brown et al. [2], Shivaji [3-61,
Hastings and McLeod [7], Sattinger [8], Parks [9] and Tam [lo].
Clearly u = 0 is a solution of (1 .l), when 2 = 0. It follows from the contraction mapping
theorem that (l.l), always has a solution for A sufficiently small. Moreover, let
S = {(A,u): 1 2 0 and u is a nonnegative solution of (1 . l)J.
It can be shown by a quadrature technique introduced by Laetsch [l l] that S defined above is
a curve in II?+x C([--1, 11) joining (0,O) to 00 (see [2]).
For problem (1. l), , Brown et al. [2] use the quadrature technique to prove that the function
U
H(u) =
exp(cu.s/(cr + s)) ds - *u exp(au/(cr + u))
.r0
=
F(u)
-
(1.3)
+uf(u)
(i) is monotonically increasing in u if CY5 4, and, thus, there exists only a unique nonnegative solution for each I > 0,
1475
1476
S.-H.
WANG
(ii) takes negative values for all CYgreater than or equal to some CY,,= 4.25, which shows that
the bifurcation curve for (1. l), has at least two turning points for all (YL CY~.Moreover, they
use numerical evaluation of the quadrature to show that the bifurcation curve is S-shaped for
01> 4.07 and is a monotone curve for cx < 4.07 in the (A, Ilull,)-plane. However, they are not
able to give a proof (see [2, 51 for details). We say the bifurcation curve for (1. 1)x is a monotone
curve, it means that the bifurcation curve for (1. l), has no turning points.
Shivaji [3] proves that, for (1.2), and (I.l)x, for every Q > 0 there exists only a unique
nonnegative solution for small and large values of A.
Hastings and McLeod [7, theorem 41 study a problem from catalysis which is equivalent to
(1 .l),. Their result implies that, for CYsufficiently large, the bifurcation curve for (l.l)x is
S-shaped; that is, there exist I_,(a), A*(a) such that (1. 1)x has a unique nonnegative solution for
0 < d < &(cY), A > A*(Q), exactly three nonnegative solutions for A,(a) < A < A*(cY),and
exactly two nonnegative solutions for ,I = A*(a) and A*(a); see Fig. l(c). All these results and
other numerical evaluation suggest that there exists & > 0 such that:
(i) for 0 < Q < 6, the bifurcation curve for (l.l), is a monotone curve. Moreover, the
bifurcation curve has a unique inflection point at (1, /iill,) such that the slope of the tangent
line of the bifurcation curve at (I, Ilfill_) is positive (see Fig. l(a));
(ii) for Q!= G, the bifurcation curve for (l.l), is a monotone curve. Moreover, the
bifurcation curve has a unique inflection point at (1, Iliill_) with a vertical tangent line there
(see Fig. l(b));
(iii) for CY> &, the bifurcation curve for (l.l)x is S-shaped. Moreover, the bifurcation curve
s has a unique inflection point at (X, I(fill,) such that the slope of the tangent line of the
bifurcation curve S at (x, Il~ll_) is negative (see Fig. l(c)).
We are not able to show the above conjecture. However, we are able to find a number
(Y*= 4.4967 defined later in (2.21) such that for cy > a!*, the bifurcation curve for (l.l)x is
S-shaped, which gives a partial answer to the above conjecture. Hence, under the assumption
of the above conjecture, CY*provides a nice upper bound for &; note that 4 is a lower bound for
c?. Note that in actual problems CYtypically may range from 20 to 100 [9, lo]. Thus, our result
works well for actual problems physically.
THEOREM 1. For (II > CY*= 4.4967,
defined in (2.21).
(a) Oca<&
the bifurcation
curve for (l.l)x
(b) a= 8
Fig. 1. Conjectured bifurcation diagrams.
is S-shaped,
(c) aA
where (Y* is
S-Shaped bifurcation curves
1411
The restriction to one dimension arises because we intend to use the autonomous nature of
the problem in that case to reduce the problem to a quadrature. More precisely, our method is
based upon more detailed analysis of the quadrature technique used by Brown et al. [2] to show
that the bifurcation curve for (l.l)x is roughly S-shaped for (Y large enough. That is, we
investigate the time map T defined in (1.6).
Multiplying (l.l), by u’(x) and integrating, we obtain
b’(x)12
___
2
+ IzF(u(x)) = constant,
(1.4)
where F(u) = jif(s) d.s.
Since we are dealing with positive solutions, U(X)has to be symmetric with respect to x = 0
and u’(x) < 0 for 0 < x < 1. Thus, llulloJ= u(0). If we let p = u(O) and substitute x = 0 in
(1.4), then we have
U’(X) = -J2A[F(p)
- F(U)],
O<x<l.
(1.5)
Integrating (1.5) in [0, 11, we obtain
fl
=
2-l/2
du
p
.i
0 JF(P)
:=
(1.6)
T(p).
- F(u)
Now solutions u of (l.l)x correspond to ll~ll~ = p and T(p) = x/A. Thus, to show that the
bifurcation curve for (l.l), is S-shaped in the (A, [lull,)-plane is equivalent to showing that the
time map T(p) defined by (1.6) has exactly two critical points in (0,~).
Recalling one formula and one estimate in [12, p. 2731, we write (1.7) and (1.11) listed below.
T’(p)
= 2-3’2
p
s
B(P) - W)
o
(AF)3’2
$J
p ’
(1.7)
where AF = F(p) - F(u) and
0(x) (= 2H(x)) = 2F(x) - xf(x)
(1.8)
O’(x) = f(x) - Xf’(X),
(1.9)
(see (1.3)). Hence, (1.8) gives
P(x) = -xf”(x).
T”(p) + 2 T’(p)
P
;‘(;;;)$)
> 2-‘/2
P
(1.10)
du,
(1.11)
where
P(X)
=
X~J(X)
-
e(x).
(1.12)
By (1.10) we have
p’(x) = -xZf”(x).
(1.13)
Formula (1.7) and estimate (1.11) are useful in our analysis of the time map T.
To show that the bifurcation curve for (1 .l)x is S-shaped in the (A, Ilul[,)-plane is equivalent
to showing that the time map T(p) has exactly two critical points in the ([lull,, L)-plane. We
show the following theorem.
1478
S.-H. WANG
THEOREM2. For (Y> 01* = 4.4967 defined in (2.21), the time map T(p) satisfies
(i)
T(0) = 0,
(1.14)
(ii)
lim T(p) = co,
P-m
(1.15)
(iii)
T(p) has exactly two critical points in (0, co).
(1.16)
2. THE PROOF OF THEOREM 2
In this section, we show theorem 2. The results (1.14) and (1.15) are quite well known. Thus,
to complete the proof of theorem 2, it suffices to show (1.16). We investigate the time map 7’(p)
defined in (1.6). First, by easy computations, we have the following lemma.
LEMMA1. For cy > 4, f has a unique inflection point at .x = C = $CY(CX
- 2) in (0,oo). Moreover,
f”(x)
> 0
for x E (0, C),
f”(x) = 0
for x = C,
f”(x) < 0
for x E (C, a).
(2.1)
By (1.8)-(1.10) we have the following easy corollary of lemma 1.
LEMMA2. For CY> 4, e(O) = 0, e’(0) = 1, lim 0(x) = 00, and 0 has a unique inflection point at
X-m
x = C = &(a - 2) in (0, a). Moreover,
V(x) < 0
for x E (0, C),
V’(x) = 0
for x = C,
U(x) > 0
for x E (C, 03).
(2.2)
LEMMA3. For CY> CX*,0(C) < 0.
Lemma 3 is the key lemma to the proof of theorem 2. We first use lemma 3 to show theorem
2 then turn to the proof of lemma 3.
Lemmas 2 and 3 say that the graph of 6’is typically depicted in Fig. 2. Let A be the first positive critical point of 8, B the first positive zero of 8, D the second positive critical point of 8,
and E the point in (D, 00) such that B(E) = O(A).
Suppose 01 > CY*.Then by (1.7), it is easy to see that
T’(p) > 0
for P E (0, Al,
T’(P) < 0
for p E [B, 0,
T’(P) > 0
for p E [E, 00).
(2.3)
So the time map T has at least one critical point in (A, B), and at least one critical point in
(D, E). Thus, to complete the proof of (1.16) and, hence, of theorem 2, it suffices to show that
(i) T has exactly one critical point, a relative maximum, in (A, B),
(ii) T has exactly one critical point, a relative minimum, in (D, E).
SShaped
bifurcation
Fig. 2. The graph
curves
1479
of 0 for 01 > 01*.
We first consider the time map Tin (A, B). We knowf(x) > 0 in (0, C). In addition, by (2. l),
f”(x) = -(l/x)&‘(x) > 0 in (0, C); recall that C is the unique inflection point off in (0,oo).
Thus, a result of Laetsch [ll, theorem 3.21 implies that T has at most one critical point in
(0, C), and, hence, T has exactly one critical point, a relative maximum in (A, B).
We then consider the time map T in (D, E). In (l.ll), by (1.12), (1.13) and (2.1), we have
P(O) = 0,
$9(D) = D&(D) - 0(D) = --0(D) > 0.
$0’(x) = -xZf”(x) < 0
for x E (0, C),
V’(C) = 0,
p’(x) = -x2f”(x) > 0
for x E (C, ~0).
Thus,
P(P) - V(U) > 0
for D < p < E, 0 -c u < p,
which implies that
T”(p) + ;T’(p)
> 0
for D < p < E.
(2.4)
This implies that T has exactly one critical point, a relative minimum, in (D, E).
The proof of theorem 2 is now complete.
We now show lemma 3 as follows.
Proof of lemma 3. We compute that
e’(x) = f(x)
F(x)
- xf ‘(x) = [exp(olx/(ar + x))]
= -xf”(x)
= [exp(oLx/(a! + x))]
x2 - (r2x + 2cYx+ Q2
’
(a + x)2
a2x(2x - (Y2+ 2a)
(01 + x)4
’
(2.5)
(2.6)
S.-H. WANG
1480
and
P(x)
= [exp(a/(a
[-4x3 + (r(5~ - 6)x2 + (~~(4 - a)x + a3(2 - cr)]
+ x))] &
2
=
z?(x).
[exp(ax/(ol + x))] ~
(CY“, x)6
(2.7)
To show 19(c) < 0 for Q > (Y*, we study P(x) in (2.7) and, hence, we study the cubic
polynomial
R(x) = -4x3 + cy(5cr - 6)x2 + a3(4 - a)x + a3(2 - a)
(2.8)
defined above in (2.7) in the interval (0, C). We compute that
R(O) = cY3(2- CY)< 0,
R(C) = R&(cX - 2)) = &?(a
(2.9)
- 2) > 0
(2.10)
for (Y> 01* > 4. In addition, by observing (2.8), (2.9) and (2. lo), the cubic polynomial R(x) has
exactly one zero in (0, C); that is, R(x) changes sign exactly once in (0, C). Thus, in (2.7)
(2.11)
P’(x) changes sign exactly once in (0, C).
Furthermore,
in (2.5), for x = C = &(a
- 2),
x2 - a2x + 2cY.x+ (Y2= &x3(4 - (Y)< 0
for LY> (Y*> 4. Thus, in (2.5)
e’(c) < 0
for a > cP > 4.
(2.12)
By (2.2), (2.5)-(2.12), we obtain the following lemma.
LEMMA4. For (Y> (Y*, e’(0) = 1, e’(C) < 0, 8’ is strictly decreasing in (0, C), Y”(0) < 0,
P”(C) > 0, and 8” changes sign exactly once in (0, C).
Note that lemma 4 implies that 8’ changes sign exactly once in (0, C).
By (2.5), it can be easily solved that A = &[a
computation shows that, for (I! > cr*,
- 2 - (a2 - ~cx)“~]. Then
an easy
R(A) = R(&[(Y - 2 - (cY2- 4#2])
> 0 if (Y*< (Y< 4.5,
= +[--405 + 12~~ + 05(a2 - ICY)-“’ - (r3((r2 - ~cY)~“] = 0 if o = 4.5,
< 0 if 4.5 < 01< co.
Thus, by (2.7) we obtain the next lemma.
LEMMA 5. For (Y> (Y*,
> 0 if a* < (Y< 4.5,
en’(A) = e~(++
- 2 - (a2 - 4~)“~])
= 0 if 01 = 4.5,
< 0 if 4.5 < 01< 00.
(2.13)
1481
S-Shaped bifurcation curves
LEMMA6. For CY> (Y*, C > 2A.
Proof of lemma 6. We compute that
c - 2A = +cXy(,- 2) - a[cY - 2 - (a2 - 4cY)“2]
= &Y[2(J - 4cY)“2 - (a! - 2)].
Since
[2(a2 - 4a)“2]2 - ((u - 2)2 = 4((r2 - 4a) - (f.Y2- 4a + 4)
= 3cY2- 12a!-4>0
for (II >
6 + 4J3
3
and (Y*> (6 + 4~/3)/3 = 4.3094, lemma 6 follows.
In the following, for the convenience of discussion, we divide the interval OI*< a < co into
two subintervals (I) 4.5 I CY< w and (II) a* < CY< 4.5.
(I) For 4.5 I (Y< m, by lemmas 4 and 5, there exists M with C > A4 L A such that
(P)“(X) < 0
for x E (0, M),
@3’)“(X)> 0
for x E (M, C),
e’(0) = 1,
e’(A) = 0,
(e’)‘(x) < 0
S’(C) < 0,
(2.14)
for x E (0, C).
Let U = (A, 0), P = (C, 0), Q = (C, e’(C)); see Fig. 3. Then the tangent line of P(x) at
U = (A, 0) will intersect the line through the points P and Q at some point V = (C, B’(N)).
There are two subcases to be considered.
Subcase (A). e’(C) I e’(N). In this subcase, by lemma 6, C > 2A > 0 and by (2.14) the
convexity of 8’ in (0, C), it is easy to see that
C
2A
A
e'(t) dt
o<
i0
e’(t) dt < -
< sA
eyt) dt;
sA
e
1
(a) O’(C)
(b)W&43'(N)
Q’(N)
Fig. 3. The graph of 0’ in (0, C) for 4.5 5 (Y< ~0; U = (A, 0), P = (C, 0), Q = (C, 0’(C)),
v = (C, B’(N)).
1482
S.-H.
WANG
see Fig. 3(a). Thus, in this subcase (A)
(2.15)
Subcase (B). S’(C) > e’(N). In this subcase we obtain
A
t!?‘(t)
dt < Area(AOUW),
(2.16)
P(t)
dt > Area(AUPQ),
(2.17)
.i0
c
_
sA
where 0 is the origin and W is the intersection point of the tangent line of the function 0’(x)
at the point U = (A, 0) with the positive (Y-axis; see Fig. 3(b). We compute that
cy2
Area(AUPQ)
= i(C - A)(-e’(C))
- 2a - cY((r2- 4cY)l/2
(II (a2
4oc)“2
= [exp(ol - 2)]i(a - 4)(or2 - 401)“‘.
Let
G,(a) = Area(AOUW)
X
CY5[CY2-
- Area(AUPQ)
2a -
cY(cY -
=
- 2a - c&Y2 - 4a) l/2
exp CY’
a - (a2 - 4ay
[
(
>I
2 - (cr2 - 4#2)][Q!
[a2 -
a(c?
-
- 2 - (cY2-
4cp]3
4cy2]4
- [exp(a! - 2)]$((~ - 4)((_y2- 4~)“‘.
It can be easily shown that G,(a) < 0 for (YL 4.5. Hence, in this subcase (B), by (2.16) and
(2.17), we obtain
e(C) = j;&(t)dt
= j)‘T(t)dt
< Area(AOUW)
+ j)i”(t)dt
- Area(AUPQ)
= Go
< 0
for Q!I 4.5.
(2.18)
(II) For IX*< CY< 4.5, by lemmas 4 and 5, there exists M with 0 < M < A such that
(eyyx) < 0
for x fz (0, M),
(eyyx) <
for x E (M, C),
e’(o) = 1,
0
ey_4) = 0,
(eyyx) < 0
see Fig. 4.
e’(c) < 0,
for x E (0, C);
(2.19)
S-Shaped bifurcation curves
1483
Fig. 4. The graph of 8’ in (0, C) for 01*< 01< 4.5; ZJ = (A, O), P = (C, O), Q
= CC,e'(c)).
Thus, it is easy to see that
A
8'(t)
dt < A = &X[CX
- 2 - (a2 - 401)“~],
W(t)
dt > Area(AUPQ)
i0
C
-
= *(C - A)(-F(C))
= [exp(a - 2)]$((r - 4)((r2 - 401)~‘~.
iA
Let
G2(01)= $CY[CX
- 2 - (01~- ~cY)“~] - [exp(a - ~)]+(cY- ~)(cY~- 4~)“‘.
(2.20)
It can be shown that there is a number CX*= 4.4967, the unique zero of G2(o) in (4, cx)), such
that
'32(a) >
for 4 < (Y< a*,
0
G2(cr*) = 0,
G,(a)
So
0(C) = i:s.(t)dt
= {:F(t)dt
<
0
(2.21)
for CX*< (Y< 4.5.
+ i;B’(t)dt
< G2(a) < 0
for CY*< u < 4.5.
(2.22)
Combining (2.15), (2.18) and (2.22), we obtain e(C) < 0 for (II> (Y*.The proof of lemma 3 is
now complete.
3. A GENERALIZED
THEOREM
We remark that our method used to show theorem 2 in Section 2 can be generalized for
general nonlinear two-point boundary-value problems of the form
-u”(X) = A.&x)),
-1 < x < 1,
U(-1) = U(1) = 0,
(3-l),
where I 5: 0 and f-is defined on [0, r) for some r, 0 < r I 00 and satisfies
(fl) YE C’([O, r)), f(U) > 0 for 0 5 u < r;
(f2) there exists C > 0 such that f;‘(x) > 0 in (0, C), p(C)
= 0, and F(x) < 0 in (C, r);
1484
(f3a)
(f3b)
S.-H.
WANG
r = 00 and there exists C,, such that Y(U) I C, for all u 2 0; or
r < CDand f”~ C’([O, r]) withf(r) = 0.
Modifying the proof of theorem 2, we are able to show the next theorem.
THEOREM
(f4)
3 (cf. [6]). Suppose that, in addition to (fl),
(f2), and (f3a) or (f3b), f-satisfies
e(C) = 2&C) - Cf(C) < 0 (where P(U) = {i_?(.s)ds).
Then the bifurcation
curve of positive solutions for (3.1), is S-shaped.
In addition to problem (1. l), , theorem 3 can apply for several examples of one dimensional
problems giving rise to S-shaped bifurcation curves; see [2, pp. 478-479; 13; 151. For an easy
example, consider
--u” = A(1 + u + u2 - &US),
-1 < x < 1,
24-l)
= u(1) = 0,
(3.2),
where L 2 0, E > 0, which is studied by Crandall and Rabinowitz [14]. It is easy to see that the
functionf=
1 + u + 24’ - eu3 satisfies (fl), (f2), and (f3b) with r = 00 and with a unique
inflection point at C = l/3&. Note that numerical evaluation of (3.2),, which is comparatively
easy in the case as F(u) = u + 3~ + f~’ - $su4 is known explicitly, shows that the bifurcation
curve is S-shaped for E < 0.175 [2]. Applying theorem 3, we obtain the next theorem.
THEOREM
4. For 0 < E < (54)-i’* = 0.1361, the bifurcation
curve for (3.2), is S-shaped.
The proof of theorem 4 is similar to and easier than that of theorem 2. It can be computed
that
-1 + 54E2
<o
162~~
for 0 < e < (54)-i’*.
Thus, (f4) is also satisfied for 0 < E < (54)- 1’2 and, hence, theorem 4 follows.
We finally note that much of the computation in this paper has been checked by using the
symbolic manipulator Mathematics.
Acknow~edgemenfs-The
author thanks Professor
of Brown et al. [2] by which this paper is motivated,
problem.
Klaus Schmitt
and Professor
for bringing to the attention of the author the work
Song-Sun Lin for many valuable discussions on this
REFERENCES
BEBERNEN J., Solid fuel combustion-some
mathematical
problems, Rocky Mount. J. M&h. 16, 417-433 (1986).
BROWN K. J., IBRAHIM M. M. A. & SHIVAJI R., S-Shaped bifurcation
curves, NonlinearAnalysis 5,475-486 (1981).
SHIVAJI R., Uniqueness results for a class of positone problems, Nonlinear Analysis 7, 223-230 (1983).
SHIVAJI R., Remarks on an S-shaped bifurcation
curve, J. math. Annlysis Applic. 111, 374-387 (1985).
SHIVAJI R., A note on the persistence of an S-shaped bifurcation
curve, Applicable Anulysis 24, 175-179 (1987).
SHIVAJI R., A comparison
of two methods on non-uniqueness
for a class of two-point boundary
value problems,
Applicable Analysis 27, 173-180 (1988).
7. HASTINGS S. P. & MCLEOD J. B., The number of solutions to an equation from catalysis, Proc. R. Sot. Edinb.
Sect. A 101, 15-30 (1985).
8. SATTINGER D. H., A nonlinear parabolic systems in the theory of combustion,
Q. uppl. Math. 33, 47-61 (1975).
9. PARKS J. R., Criticality
criteria for various configurations
of a self-heating
chemical as functions of activation
energy and temperature
of assembly, J. them. Phys. 34, 46-55 (1961).
1.
2.
3.
4.
5.
6.
S-Shaped
bifurcation
curves
1485
10. TAM K. K., Construction
of upper and lower solutions for a problem in combustion
theory. .Z. math. Analysis
Applic. 69, 131-145 (1979).
11. LAETSCH T., The number of solutions of a nonlinear two point boundary value problem, Indiana Univ. mnth. J.
20, l-13 (1970).
12. SMOLLERJ. & WASSERMANA., Global bifurcation
of steady-state
solutions, J. diff. Eqns 39, 269-290 (1981).
13. WANG S.-H. & LEE F.-P., Bifurcation
of an equation from catalysis theory, Nonlinear Analysis (to appear).
14. CRANDALL M. & RABINOWITZ H., Bifurcation,
perturbation
of simple eigenvalues, and linearized stability, Archs
ration. Mech. Analysis 52, 161-180 (1973).
15. Lu C., Multiple steady states in a biochemical
system, SIAM J. mafh. Analysis 18, 1771-1783 (1987).