S-SHAPED BIFURCATION CURVES FOR A COMBUSTION PROBLEM WITH GENERAL ARRHENIUS REACTION-RATE LAWS

We study the bifurcation curve and exact multiplicity of positive solutions of the combustion problem with general Arrhenius reaction-rate laws { u′′(x) + λ(1 + u)e u 1+ u = 0, − 1 < x < 1, u(−1) = u(1) = 0, where the bifurcation parameters λ, > 0 and −∞ < m < 1. We prove that, for (−4.103 ≈) m̃ ≤ m < 1 for some constant m̃, the bifurcation curve is S-shaped on the (λ, ||u||∞)-plane if 0 < ≤ 67 Sem tr (m), where Sem tr (m) = { ( 1− √ 1−m m ) 2 for −∞ < m < 1, m 6= 0, 1 4 for m = 0, is the Semenov transitional value for general Arrhenius kinetics. In addition, for −∞ < m < 1, the bifurcation curve is S-like shaped if 0 < ≤ 89 Sem tr (m). Our results improve and extend those in Wang (Proc. Roy. Soc. London Sect. A, 454 (1998), 1031—1048.)


Introduction
We study the bifurcation curve and exact multiplicity of positive solutions of the problem with the Dirichlet (Frank-Kamenetskii) boundary conditions 8 < : u is the dimensionless temperature, and is the reciprocal activation energy parameter or the ambient temperature parameter. The reaction term is the temperature dependence of the mth-order reaction rate obeying the general Arrhenius reaction-rate law in which heat ‡ow is purely conductive, see e.g. [1,2,3,12]. Problem (1.1) was studied mainly for physically important range of numerical exponent m < 1, and particularly, for m = 2 (sensitized reaction rate), m = 0 (Arrhenius reaction rate) and m = 1=2 (bimolecular reaction rate) (see e.g. [1,2,3,4]). Reaction reports have been given with 2 m 2:67 in [11]. Reaction reports have also been given with 10:31 m 2:81 in [9].
Criticality (bifurcation) persists as long as the reciprocal activation energy is smaller than a transitional value tr . As approaches tr , the function f (u) becomes "saturated" and a transition from criticality to continuity results, see [2]. Boddington  where F.K. tr (m) is the transitional value for general Arrhenius kinetics under Frank-Kamenetskii boundary conditions (u( 1) = u(1) = 0) and Sem tr (m) is the Semenov transitional value for general Arrhenius kinetics under Semenov boundary conditions (u 0 ( 1) = u 0 (1) = 0). Thus, a transition to continuity does occur and there is no Frank-Kamenetskii criticality unless 1 < m < 1 and 0 < < Sem tr (m). Accurate transitional values for F.K. tr (m) have been calculated by numerical quadrature for the in…nite slab; i.e. for (1.1). We note that in the previous numerical work, especially that by Boddington et al. in [1,2,3,4]; they found an S-shaped bifurcation diagram and three solutions for some parameter values.
We de…ne the bifurcation curve of positive solutions of (1.1) S = f( ; ku k 1 ) : 0 and u is a solution of (1.1)g .
We say that, on the ( ; kuk 1 )-plane, the bifurcation curve S is S-shaped if S has exactly two turning points at some points ( ; ku k 1 ) and ( ; ku k 1 ) where < are two positive numbers such that (i) ku k 1 < ku k 1 , (ii) at ( ; ku k 1 ) the bifurcation curve S turns to the left, (iii) at ( ; ku k 1 ) the bifurcation curve S turns to the right.
See Figure 1(i) for example. Similarly, we say that, on the ( ; kuk 1 )-plane, the bifurcation curve S is S-like shaped if S starts from the origin and initially continues to the right, S tends to in…nity as ! 1, and S has at least two turning points.
For (1.1) with 1 < m < 1, it has been a long-standing conjecture that, F.K. tr (m) (< Sem tr (m)) is a continuous function such that, on the (m; )-plane, the bifurcation curve S is S-shaped for 0 < < F.K. tr (m), and is monotone increasing for F.K. tr (m). In particular, when = F.K. tr (m), there is a unique turning point. See Figure 1(i)-(iii). (This kind of global bifurcation result is useful in understanding the pro…les of the solutions to the full exothermic reaction-di¤usion system, cf. Mimura & Sakamoto in [10] for details.) Figure 1. Conjectured global bifurcation of bifurcation curve S of (1.1) with 1 < m < 1 and > 0. 1 = 2 p when m = 1.

Main result
The main result in this paper is next Theorem 2.1 which improves and extends Theorem 1.1. In Theorem 2.1(i), we prove that, form m < 1, wherem 4:103 is a negative constant de…ned in (6.2), the bifurcation curve S is S-shaped on the ( ; kuk 1 )-plane if 0 < tr (m = 1=2). In Theorem 2.1(ii), we prove that, for 1 < m < 1, the bifurcation curve S is S-like shaped on the ( ; kuk 1 )-plane if 0 < Sem tr (m):

Lemmas
To prove our main result, we modify the time-map techniques developed recently in Hung & Wang [7]. The time map formula which we apply to study (1.1) takes the form as follows: Thus, studying of the exact number of positive solutions of (1.1) is equivalent to studying the shape of the time map T ( ) on (0; 1). Also, proving that the bifurcation curve S is S-shaped (resp. S-like shaped) on the ( ; kuk 1 )-plane is equivalent to proving that T ( ) has exactly two (resp. at least two) critical points, a local maximum at some and a local minimum at some > , on (0; 1). See Figure 1(i). The following lemma contains some basic properties of the time map T ( ), which follows from Wang [14,Proposition 1.2].
In the following Lemma 3.2, we study the concavity of f (u); which is used to study the shape of the bifurcation curve S: Lemma 3.2 follows from Wang [14, Lemmas 2.1 and 2.2].
The following lemma follows immediately by slight modi…cation of the proof of Hung & Wang [7, Theorem 2.1]; we omit the proof.   Sem tr (m); then C > 12 5 A and B > 4A: The proof of Lemma 3.12 follow by the same arguments in the proof of Lemma 3.6; we omit the proof. : We thus …nd that m 0 > m 1 > m 2 for 0 <  The proof of Theorem 2.1 is complete.

Final Remarks
We …nish this paper by giving next two remarks.  Remark 5.2. For Lemma 3.9, we observe that: (i) For 20 < m < 0 and 0 < 6 7 Sem tr (m); numerical simulation shows that assertion (3.40) holds, and hence our arguments are able to show that the bifurcation curve S is S-shaped on the ( ; kuk 1 )-plane.