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1 AIMS OES - Author - Decision 頁 1 / 1 212/9/1 Close Sun 1:47:59:916 PM Decision from Editorial Office Subject: Decision of paper 'S-SHAPED BIFURCATION CURVES FOR A COMBUSTION PROBLEM WITH GENERAL ARRHENIUS REACTION-RATE LAWS' Paper ID: CPAA735 Paper Title: S-SHAPED BIFURCATION CURVES FOR A COMBUSTION PROBLEM WITH GENERAL ARRHENIUS REACTION-RATE LAWS Paper Author: Wang, Shin-hwa Dear Professor Wang, Shin-hwa, The review of your article is now complete. I am happy to inform you that your paper is now accepted for publication in CPAA. To expedite the publication of your article and to avoid errors caused in this process, your help is essential in preparing the final tex file of the paper according to the step by step guideline, posted at where you review the instruction, and download necessary files and the copyright form. Here are some specific suggestions: 1. Prepare the tex file according to the instruction and guideline. 2. Use a single tex file including all references. 3. Make sure all figures are of very high resolution, proportional to the page in size, all are placed before the references. 4. Make sure no lines are wider than the width limit. 5. Save the tex file together with all supporting files (without the copyright) as a single zip file. 6. Fill out and sign the copyright form, scan it and save it in PDF. 7. Proofread your final version according to the checklist. 8. At the web page: submit the zip file and the copy right form. On behalf of the editorial board, I would like to thank you for contributing to the journal. I am looking forward to having your continuing support and collaboration. Best regards, The Editorial Office Communications on Pure and Applied Analysis Copyright American Institute of Mathematical Sciences. Close

2 S-SHAPED BIFURCATION CURVES FOR A COMBUSTION PROBLEM WITH GENERAL ARRHENIUS REACTION-RATE LAWS CHIH-YUAN CHEN, KUO-CHIH HUNG AND SHIN-HWA WANG Department of Mathematics, National Tsing Hua University Hsinchu, Taiwan 3, Republic of China A. 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 + ɛu) m e u 1+ɛu =, 1 < x < 1, u( 1) = u(1) =, where the bifurcation parameters λ, ɛ > and < m < 1. We prove that, for ( 4.13 ) m m < 1 for some constant m, the bifurcation curve is S-shaped on the (λ, u )-plane if < ɛ 6 7 ɛsem tr (m), where { ɛ Sem tr (m) = ( 1 1 m 1 4 m ) 2 for < m < 1, m, for m =, is the Semenov transitional value for general Arrhenius kinetics. In addition, for < m < 1, the bifurcation curve is S-like shaped if < ɛ 8 9 ɛsem tr (m). Our results improve and extend those in Wang (Proc. Roy. Soc. London Sect. A, 454 (1998), ) 1. I We study the bifurcation curve and exact multiplicity of positive solutions of the problem with the Dirichlet (Frank-Kamenetskii) boundary conditions d 2 u dx + λ(1 + 2 ɛu)m e u 1+ɛu =, 1 < x < 1, (1.1) u( 1) = u(1) =, where the bifurcation parameters λ, ɛ > and < m < 1. Problem (1.1) is an onedimensional case of an equation arising in combustion theory, which governs the steady-state thermal explosions in a material undergoing an mth-order exothermic reaction. It can be considered as a special case for an n-dimensional Dirichlet problem for the infinite slab. In (1.1), λ, called the Frank-Kamenetskii parameter, is a dimensionless rate of heat production, 2 Mathematics Subject Classification. 34B18, 74G35. Key words and phrases. Positive solution, exact multiplicity, S-shaped bifurcation curve, combustion problem, Frank-Kamenetskii transition value, Semenov transitional value. Corresponding author: Shin-Hwa Wang This work is partially supported by the National Science Council of the Republic of China under grant No M-7-13-MY3. 1

3 2 CHIH-YUAN CHEN, KUO-CHIH HUNG AND SHIN-HWA WANG u is the dimensionless temperature, and ɛ is the reciprocal activation energy parameter or the ambient temperature parameter. The reaction term f(u) (1 + ɛu) m e u 1+ɛu (1.2) is the temperature dependence of the mth-order reaction rate obeying the general Arrhenius reaction-rate law in which heat flow 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 = (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 1.31 m 2.81 in [9]. Criticality (bifurcation) persists as long as the reciprocal activation energy ɛ is smaller than a transitional value ɛ tr. As ɛ approaches ɛ tr, the function f(u) becomes saturated and a transition from criticality to continuity results, see [2]. Boddington et al. [4, Section 4] obtained ɛ F.K. tr (m) < ɛ Sem tr (m) { ( 1 1 m m ) 2 for < m < 1, m, 1 4 for m =, where ɛ F.K. tr (m) is the transitional value for general Arrhenius kinetics under Frank-Kamenetskii boundary conditions (u( 1) = u(1) = ) and ɛ Sem tr (m) is the Semenov transitional value for general Arrhenius kinetics under Semenov boundary conditions (u ( 1) = u (1) = ). Thus, a transition to continuity does occur and there is no Frank-Kamenetskii criticality unless < m < 1 and < ɛ < ɛ Sem tr (m). Accurate transitional values for ɛ F.K. tr (m) have been calculated by numerical quadrature for the infinite 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 define the bifurcation curve of positive solutions of (1.1) S = {(λ, u λ ) : λ and u λ is a solution of (1.1)}. We say that, on the (λ, u )-plane, the bifurcation curve S is S-shaped if S has exactly two turning points at some points (λ, u λ ) and (λ, u λ ) where λ < λ are two positive numbers such that (i) u λ < u λ, (ii) at (λ, u λ ) the bifurcation curve S turns to the left, (iii) at (λ, u λ ) the bifurcation curve S turns to the right. See Figure 1(i) for example. Similarly, we say that, on the (λ, u )-plane, the bifurcation curve S is S-like shaped if S starts from the origin and initially continues to the right, S tends to infinity as λ, and S has at least two turning points.

4 S-SHAPED BIFURCATION CURVES 3 For (1.1) with < m < 1, it has been a long-standing conjecture that, ɛ F.K. tr (m) (m)) is a continuous function such that, on the (m, ɛ)-plane, the bifurcation curve S (< ɛ Sem tr is S-shaped for < ɛ < ɛ F.K. tr when ɛ = ɛ F.K. tr (m), and is monotone increasing for ɛ ɛ F.K. tr (m). In particular, (m), there is a unique turning point. See Figure 1(i) (iii). (This kind of global bifurcation result is useful in understanding the profiles of the solutions to the full exothermic reaction-diffusion system, cf. Mimura & Sakamoto in [1] for details.) Figure 1. Conjectured global bifurcation of bifurcation curve S of (1.1) with < m < and ɛ >. λ 1 = when m = 1. In the case m = (Arrhenius reaction rate), (1.1) becomes to famous perturbed Gelfand problem and ɛ F.K. tr () < 1/4 =.25. Wang [13, Theorem 1.1] used quadrature method (timemap method) to prove that the bifurcation curve is S-shaped for < ɛ < ɛ 1 1/ for some constant ɛ 1, and hence (.222 ) ɛ 1 < ɛ F.K. tr () (<.25). This lower bound for ɛ F.K. tr () was improved to ɛ 2 1/ by Korman & Li [8] by applying a bifurcation theorem of Crandall & Rabinowitz [5]. This lower bound for ɛ F.K. tr () was further improved by Hung & Wang [7] to ɛ 3 1/ for some constant ɛ 3 defined in Hung & Wang [7, Eq. (3.22)]. π 2 ɛ

5 4 CHIH-YUAN CHEN, KUO-CHIH HUNG AND SHIN-HWA WANG Wang [14, Theorems 1.4 and 1.5] proved the following theorem for (1.1) for the shape of the bifurcation curve S and a lower bound of the Frank-Kamenetskii transition value ɛ F.K. tr (m) with < m < 1, m. Theorem 1.1. Consider (1.1) with < m < 1. Then: (i) For < m < 1, the bifurcation curve S is S-shaped on the (λ, u )-plane if < ɛ < max( 1 5, 1 2 ɛsem tr (m)) = { 1, if < m 1(2 1 5).662, 5 2 tr (m), if 1(2 1 5) < m < ɛsem In particular, for m = 1/2, the bifurcation curve S is S-shaped on the (λ, u )-plane if < ɛ < 1 2 ɛsem tr (m = 1/2).3. (ii) For < m <, the bifurcation curve S is S-like shaped on the (λ, u )-plane if < ɛ < 1 2 ɛsem tr (m). In particular, the bifurcation curve S is S-shaped on the (λ, u )- plane for m = 1 and < ɛ < 1 2 ɛsem tr (m) = 1(3 2 2).858, 2 m = 2 and < ɛ < 1 2 ɛsem tr (m) = 1(2 3) Notice that Theorem 1.1(i) implies that max( 1, ɛsem tr (m)) ɛ F.K. tr (m) for < m < 1. (1.3) In this section, we finally note that problem (1.1) is also of mathematical interest for the case m 1. Du [6, Theorems 3.3 and 3.4] proved that, if m > 1, the bifurcation curve S is -shaped on the (λ, u )-plane for ɛ >. In addition, if m = 1, the bifurcation curve S is -shaped for < ɛ < 1 and is a monotone curve for ɛ 1. See Figure 1(iv) (vi). Note that the results of Du [6] cover not only dimension 1 for problem (1.1) but also dimension 2 when the domain is a unit open ball. 2. M 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, for m m < 1, where m 4.13 is a negative constant defined in (6.2), the bifurcation curve S is S-shaped on the (λ, u )-plane if < ɛ 6 7 ɛsem tr (m). Moreover, we prove that (max( 1, ɛsem tr (m)) <) 6 7 ɛsem tr (m) < ɛ F.K. tr (m) for m m < 1. This improves and extends (1.3). In particular, for m = 1/2 (bimolecular reaction rate), we give a better lower bound of the Frank-Kamenetskii transition value ɛ F.K. tr (m = 1/2). In Theorem 2.1(ii), we prove that, for < m < 1, the bifurcation curve S is S-like shaped on the (λ, u )-plane if < ɛ 8 9 ɛsem tr (m); our result improves and extends Theorem 1.1(ii).

6 S-SHAPED BIFURCATION CURVES 5 Figure 2. Mathematical results on the bifurcation curves S of (1.1) with m < 1 and >. m ~ 4:13: Theorem 2.1. (See Figure 2.) Consider (1.1) with (i) For ( 4:13 plane if )m ~ 1 < m < 1. Then: m < 1; the bifurcation curve S is S-shaped on the ( ; kuk1 )< 6 Sem (m): 7 tr In particular, the bifurcation curve S is S-shaped for m = 1=2 and < m = 1 and < m = 2 and < (ii) For < :328; 4 Sem tr (m) = Sem tr (m) = where 6 (3 7 6 (2 7 4 is a constant de ned in (3.45); p 2 2) :147; p 3) :114: 1 < m < 1, the bifurcation curve S is S-like shaped on the ( ; kuk1 )-plane if 8 Sem (m): 9 tr 3. 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: Z p 1 =p [F ( ) F (u)] 1=2 du T ( ) for < < 1, (3.1) 2 Ru where F (u) f (t)dt. So positive solutions u of (1.1) correspond to p kuk1 = and T ( ) =. (3.2)

7 6 CHIH-YUAN CHEN, KUO-CHIH HUNG AND SHIN-HWA WANG Thus, studying of the exact number of positive solutions of (1.1) is equivalent to studying the shape of the time map T (α) on (, ). Also, proving that the bifurcation curve S is S-shaped (resp. S-like shaped) on the (λ, u )-plane is equivalent to proving that T (α) has exactly two (resp. at least two) critical points, a local maximum at some α and a local minimum at some α > α, on (, ). See Figure 1(i). The following lemma contains some basic properties of the time map T (α), which follows from Wang [14, Proposition 1.2]. Lemma 3.1. Consider (3.1) with < m < 1. If < ɛ < ɛ Sem tr (m), then lim T (α) = and lim α + T (α) =. (3.3) α 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]. Lemma 3.2. Consider (1.1) with < m < 1 and m. If < ɛ < ɛ Sem tr (m), then: (i) If < m < 1, then f (u) = f(u) (1 + ɛu) 4 [m(m 1)ɛ4 u 2 + 2ɛ 2 ( 1 + m ɛm + ɛm 2 )u +(1 2ɛ + 2ɛm + ɛ 2 m ɛ 2 m 2 )] (3.4) > on (, C), = when u = C, < on (C, ) where C = m + m ɛm + ɛm 2. (3.5) ɛ 2 (1 m)m (ii) If < m <, then > on (, C) (D, ), f (u) = when u = C, D, < on (C, D) where C = m + m ɛm + ɛm 2 ɛ 2 (1 m)m < D = 1 1 m + m ɛm + ɛm 2. (3.6) ɛ 2 (1 m)m For < m < 1 and < ɛ < ɛ Sem tr (m), thus by Lemma 3.2(i), it is easy to check that f(u) = (1 + ɛu) m e u 1+ɛu satisfies the following properties (P1) (P3): (P1) f() > (positone) and f(u) > on (, ).

8 S-SHAPED BIFURCATION CURVES 7 (P2) f is convex-concave on (, ); that is, f has exactly one positive inflection point at C = 1+ 1 m+m ɛm+ɛm 2 such that ɛ 2 (1 m)m > on [, C), f (u) = when u = C, < on (C, ). (P3) f is asymptotic sublinear at ; that is, lim u (f(u)/u) =. Next we define the following function H(u) = 3 u tf(t)dt u 2 f(u) for u. (3.7) For < m < 1, since f satisfies (P1) (P3), thus our main result in Theorem 2.1 in the case < m < 1 is based upon the following Lemma 3.3 proved by Hung & Wang [7, Theorem 2.1]. Lemma 3.3. Consider (1.1) with < m < 1. If < ɛ < ɛ Sem tr (m), then: (i) If H(C), then the bifurcation curve S is S-shaped on the (λ, u )-plane. (ii) If H(u ) for some u >, then the bifurcation curve S is S-like shaped on the (λ, u )-plane. For T (α) in (3.1), we compute that where T (α) = 1 2 2α α θ(u) 2F (u) uf(u) = 2 θ(α) θ(u) du, (3.8) 3/2 [F (α) F (u)] u f(t)dt uf(u). (3.9) Thus by (3.9), we compute that, there exist positive numbers A < C < B such that where θ (u) = f(u) uf (u) = f(u) (1 + ɛu) 2 [ (1 m)ɛ 2 u 2 + ( 1 + 2ɛ mɛ)u + 1 ] A = 1 2ɛ + ɛm 1 4ɛ + 2ɛm + ɛ 2 m 2 2ɛ 2 (1 m) and we also compute that > on [, A) (B, ), = when u = A, B, < on (A, B) By (3.12) and Lemma 3.2(i) (ii), we notice that: (3.1) < B = 1 2ɛ + ɛm + 1 4ɛ + 2ɛm + ɛ 2 m 2, 2ɛ 2 (1 m) (3.11) θ (u) = uf (u). (3.12)

9 8 CHIH-YUAN CHEN, KUO-CHIH HUNG AND SHIN-HWA WANG (i) If < m < 1 and < ɛ < ɛ Sem tr (m), then θ (u) = uf (u) < on (, C), = when u = C, > on (C, ). (ii) If < m < and < ɛ < ɛ Sem tr (m), then < on (, C) (D, ), θ (u) = uf (u) = when u = C, D, > on (C, D). (3.13) (3.14) For H(u) in (3.7), we compute that Moreover, by (3.15) and (3.1), H (u) = uf(u) u 2 f (u) = uθ (u) for u, (3.15) H (u) = θ (u) + uθ (u) for u. (3.16) > on (, A) (B, ), H (u) = uθ (u) = when u = A, B, < on (A, B). (3.17) The following lemma contains some properties of θ(u), which basically follow from Wang [14, Lemmas 3.3, 3.6, 4.2 and 4.5] after some slight modification; we omit the proof. Lemma 3.4 is useful in studying the auxiliary function H(u). Lemma 3.4. Consider (1.1) with < m < 1 and m. If < ɛ < ɛ Sem tr (m), then: (i) For fixed m < 1 and m, θ (C) is a strictly increasing function of ɛ (, ɛ Sem tr (m)); that is, θ (C) >. (3.18) ɛ (ii) If < m < 1, then θ (C) >, (3.19) and θ (u) changes sign exactly once on (, C) and changes sign exactly once on [C, ). (iii) If < m <, then θ (C) >, θ (D) <, (3.2) and θ (u) changes sign exactly once on (, C) and changes sign exactly once on [C, D). Lemma 3.5. Consider (1.1) with fixed m < 1 and m. If < ɛ 6 7 ɛsem tr (m), then H (C) > C 2.

10 S-SHAPED BIFURCATION CURVES 9 Proof of Lemma 3.5. Let < m < 1 and m. By (3.1) and (3.5), we compute and obtain that = θ (C) ɛ= 6 (1 m) 7 ɛsem tr (m) [ m(m 1+ 1 m) ( 1+ 1 m) 2 (1 m) ] m [ m+( 2+ 1 m)m ] 6 m 7 1 m (m 1+ 1 m) 2 e m[12( 1+ is an increasing function of m < 1, see Figure 3. 1 m)+( m)m+m 2 ] 6( 1+ 1 m) 2 (m 1+ 1 m) We then compute that Thus Figure 3. Graph of θ (C) when ɛ = 6 7 ɛsem tr (m) for < m < 1. lim m 1 θ (C) ɛ= 6 e7/6 7 ɛsem tr (m) = θ (C) ɛ= 6 7 ɛsem tr (m) < (.535). and hence by (3.18) and (3.21), for < ɛ 6 7 ɛsem tr (m), we have that for < m < 1, (3.21) θ (C) < 1 2. (3.22) Then by (3.15) and (3.22), we obtain that H (C) = Cθ (C) > C. The proof of Lemma is complete. Lemma 3.6. Consider (1.1) with < m < 1 and m. If < ɛ 6 7 ɛsem tr (m), then C > 13A. 5 Proof of Lemma 3.6. For < m < 1 and m, we compute and obtain that C 13 m( m 3m + 16mɛ 3m 2 ɛ) + 13m 2 1 4ɛ + 2mɛ + ɛ A = 2 m 2 > 5 1(1 m)m 2 ɛ 2

11 1 CHIH-YUAN CHEN, KUO-CHIH HUNG AND SHIN-HWA WANG if K (13m 2 1 4ɛ + 2mɛ + ɛ 2 m 2 ) 2 [ m( m 3m + 16mɛ 3m 2 ɛ) ] 2 >. (3.23) Notice that, if < m < 1 and m 8, then we obtain that 5 where K = 4m 4 ( m + 4m 2 )(ɛ ˆɛ)(ɛ ˇɛ) ˆɛ = m(8 8 1 m 16m + 15m 1 m + 8m 2 ) + 5mF 1 2 2m 2 ( m + 4m 2 ) ˇɛ = m(8 8 1 m 16m + 15m 1 m + 8m 2 ) 5mF 1 2 2m 2 ( m + 4m 2 ) >, <, (see Figure 4) and F = (1 m) 3648m (1 m)m 48(1 m)m m (1 m)m (1 m)m 2 628m 3 >. Observe that Figure 4. Graphs of 6 7 ɛsem tr (m), ˆɛ and ˇɛ for m < m + 4m 2 { < on ( 8 5, 1), > on (, 8 5 ). (3.24) Thus by (3.23) and (3.24), we obtain the following results (i) (iii), see Figure 4. (i) For m ( 8, 1) and ˆɛ < < ɛ ɛsem tr (m) < ˇɛ, we have that K = 4m 4 ( m + 4m 2 )(ɛ ˆɛ)(ɛ ˇɛ) >. This implies C > 13 A by (3.23). 5

12 S-SHAPED BIFURCATION CURVES 11 (ii) For m = 8 and < ɛ ( ɛsem tr (m) = 6 45 ) (.125), we compute that K = > (note that ( 34.36) >. + ( ) ɛ 625 ( ) ( ) < ) 625 This implies C > 13 A by (3.23). 5 (iii) For m (, 8) and < ɛ ɛsem tr (m) < ˇɛ < ˆɛ, we have that K = 4m 4 ( m + 4m 2 )(ɛ ˆɛ)(ɛ ˇɛ) >. This implies C > 13 A by (3.23). The proof of Lemma is complete. Lemma 3.7. Consider (1.1) with < m < 1 and m. If < ɛ < ɛ Sem tr (m), then: (i) If < m < 1, then H (C) > and there exists a number J (, C) such that < on (, J), H (u) = when u = J, > on (J, ). (3.25) (ii) If < m <, then H (C) > and H (D) <, and there exist two numbers J 1 (, C) and J 2 (C, D) such that < on (, J 1 ) (J 2, D], H (u) = when u = J 1, J 2, > on (J 1, J 2 ). Proof of Lemma 3.7. First by (3.16) and (3.4), we compute that (3.26) H (u) = 2θ (u) + uθ (u) (3.27) = ue u 1+ɛu (1 + ɛu) m 6 {m ( 1 m 2) ɛ 6 u 4 + 3m (1 m) (2ɛ + mɛ + 1) ɛ 4 u 3 +3 (1 m) ( m 2 ɛ 2 + 4mɛ 2 + 2mɛ + 2ɛ + 1 ) ɛ 2 u 2 +[ ( 1 9m m 2) mɛ 3 + ( 12 9m 3m 2) ɛ 2 3mɛ 1]u +3m (1 m) ɛ (1 m) ɛ 3} ue u 1+ɛu (1 + ɛu) m 6 Q(u) (3.28)

13 12 CHIH-YUAN CHEN, KUO-CHIH HUNG AND SHIN-HWA WANG where Q(u) = m ( 1 m 2) ɛ 6 u 4 + 3m (1 m) (2ɛ + mɛ + 1) ɛ 4 u 3 Moreover, we computed that where +3 (1 m) ( m 2 ɛ 2 + 4mɛ 2 + 2mɛ + 2ɛ + 1 ) ɛ 2 u 2 +[ ( 1 9m m 2) mɛ 3 + ( 12 9m 3m 2) ɛ 2 3mɛ 1]u +3m (1 m) ɛ (1 m) ɛ 3. Q() = 3m (1 m) ɛ (1 m) ɛ 3 = 3m (1 m) (ɛ µ)(ɛ ν) (3.29) µ = m m m(1 m) and ν = m 1 1 m. m(1 m) (i) Let < m < 1 and < ɛ < ɛ Sem tr (m). By (3.27), (3.13) and (3.19), we have that H (C) = 2θ (C) + Cθ (C) = Cθ (C) >, and hence by (3.28), Q(C) >. (3.3) It is easy to check that ν < < ɛ Sem tr (m) < µ. Thus by (3.29), we have that We compute that Q() <. (3.31) Q (u) = m ( 1 m 2) ɛ 6 u 4 + 3m (1 m) (2ɛ + mɛ + 1) ɛ 4 u 3 +3 (1 m) ( m 2 ɛ 2 + 4mɛ 2 + 2mɛ + 2ɛ + 1 ) ɛ 2 u 2 > (3.32) since it is easy to see that all coeffi cients of terms u 4, u 3 and u 2 are all positive. Hence Q(u) is always concave up on (, ). So by (3.3) (3.32), there exists a number J (, C) such that Q(u) < on (, J), = when u = J, > on (J, ). Thus by (3.28), (3.25) holds. (ii) Let < m < and < ɛ < ɛ Sem tr (m). By (3.27), (3.14) and (3.2), we have that Thus by (3.28), we obtain that H (C) = 2θ (C) + Cθ (C) = Cθ (C) >, H (D) = 2θ (D) + Dθ (D) = Dθ (D) <. Q(D) < < Q(C). (3.33) It is easy to check that ɛ Sem tr (m) < µ < v. Thus by (3.29), we have that Q() <. (3.34)

14 S-SHAPED BIFURCATION CURVES 13 In addition, we compute that Q( 1 ɛ ) = 1 ɛ >. (3.35) We consider the following three cases as follows: Case (i). 1 < m <. In this case, Q(u) is a quartic polynomial with negative leading coeffi cient m (1 m 2 ) ɛ 6. Thus by (3.34) and (3.35), Q(u) has (exactly) two distinct negative zeros. In addition, by (3.33) and (3.34), there exist two numbers J 1 (, C) and J 2 (C, D) such that < on (, J 1 ) (J 2, D], Q(u) = when u = J 1, J 2, > on (J 1, J 2 ). (3.36) Case (ii). m = 1. In this case, Q(u) is a cubic polynomial with negative leading coeffi cient 6(ɛ + 1)ɛ 4. Thus by (3.33) and (3.34), Q(u) has (exactly) one negative zeros. In addition, by (3.33) and (3.34), there exist two numbers J 1 (, C) and J 2 (C, D) such that the result in (3.36) holds. Case (iii). < m < 1. In this case, Q(u) is a quartic polynomial with positive leading coeffi cient m (1 m 2 ) ɛ 6. Thus by (3.33) and (3.34), Q(u) has (exactly) one negative zeros. In addition, by (3.33) and (3.34), there exist two numbers J 1 (, C) and J 2 (C, D) such that the result in (3.36) holds. Hence by (3.28), for Cases (i) (iii), (3.26) always holds. The proof of Lemma 3.7 is complete. Lemma 3.8. Consider (1.1) with < m < 1 and m. If < ɛ 6 7 ɛsem tr (m), then H(C). Proof of Lemma 3.8. Let < m < 1 and m. By Lemma 3.7 and (3.17), for < ɛ 6 7 ɛsem tr (m), there exists a number M (, C) such that the function H satisfies: (H ) (u) < for u (, M), (H ) (u) > for u (M, C), H () =, (H ) () >, H (C) <, (H ) (C) <. (3.37) Let U = (A, ), P = (C, ), Q = (C, H (C)). Then the tangent line of y = H (u) at U = (A, ) intersects the line through the points P and Q at some point V = (C, C ). There are four cases to be considered as follows:

15 14 CHIH-YUAN CHEN, KUO-CHIH HUNG AND SHIN-HWA WANG Figure 5. Four possible graphs of H (u) on [; C]. (i) H (A) < and H (C) (ii) H (A) < and H (C) > C. (iii) H (A) =. (iv) H (A) >. C. Case (i). H (A) < and H (C) C. By Lemma 3.6, C > 13 A > 2A: In addition, by 5 (3.37) the convexity of H (u) on (; C); it is easy to see that Z C Z 2A Z A H (t)dt = H(A) H(C): H (t)dt < H (t)dt < < H(A) = A A So H(C) < : Case (ii). H (A) < and H (C) > C. By (3.15), (3.1), (3.13) and (3.14), we obtain that H (u) = u (u) A (u) A () = Af () = A for u A: (3.38) By Figure 5(ii) and Lemma 3.5, we have that area( U P Q) = (C Thus by (3.38), (3.39) and Figure 5(ii), Z A Z 2 H (t)dt < A and A) [ H (C)] C 2 AC > : 2 4 C H (t)dt > area( U P Q) > (3.39) C2 4 A 5 Moreover, by Lemma 3.6, we obtain that A < 13 C and Z A Z C AC C 2 4A2 + AC H(C) = H (t)dt + H (t)dt < A2 + = 4 4 A AC C2 < : 1 2 C < : 169

16 S-SHAPED BIFURCATION CURVES 15 Case (iii). H (A) =. By (3.37), we obtain that H (C) > C. Applying the same arguments in Case (ii), we are able to obtain that H(C) <. Case (iv). H (A) >. By (3.37), we obtain that H (C) > C. Applying the same arguments in Case (ii), we are able to obtain that H(C) <. In all Cases (i) (iv), we obtain that H(C) <. The proof of Lemma 3.8 is complete. The following lemma follows immediately by slight modification of the proof of Hung & Wang [7, Theorem 2.1]; we omit the proof. Lemma 3.9. Consider (1.1) with < m <. If < ɛ 6 7 ɛsem tr (m), the bifurcation curve S is S-shaped if θ(d) > θ(a) (> ). (3.4) Lemma 3.1. Consider (1.1) with ( 4.13 ) m m <. If < ɛ 6 7 ɛsem tr (m), then θ(d) > θ(a) (> ). The proof of Lemma 3.1 is put in Appendix. Lemma Consider (1.1) with fixed m < 1 and m. If < ɛ 8 9 ɛsem tr (m), then H (C) > 9 C. 25 The proof of Lemma 3.11 follow by the similar arguments in the proof of Lemma 3.5; we omit the proof. Lemma Consider (1.1) with < m < 1 and m. If < ɛ 8 9 ɛsem tr (m), then C > 12 A and B > 4A. 5 The proof of Lemma 3.12 follow by the same arguments in the proof of Lemma 3.6; we omit the proof. Lemma Consider (1.1) with < m < 1 and m. If < ɛ 8 9 ɛsem tr (m), then: (i) H (B) >, (ii) H(B). Proof of Lemma (i) By (3.27), (3.14) and (3.4), we compute that and H (B) = 2θ (B) + θ (B) > θ (B) (3.41) θ (u) = e u 1+ɛu (1 + ɛu) m 6 [( 1 + 2ɛ 2ɛm + ɛ 2 m ɛ 2 m 2 ) (1 4ɛ + 3ɛm 3ɛ 2 m 2ɛ 3 m + 3ɛ 2 m 2 + ɛ 3 m 2 + ɛ 3 m 3 )u +ɛ 2 (5 6ɛ 3m + 12ɛm 6ɛm 2 + 3ɛ 2 m 2 3ɛ 2 m 3 )u 2 ɛ 4 (4 7m + 2ɛm + 3m 2 5ɛm 2 + 5ɛm 3 )u 3 ɛ 6 m(1 m) 2 u 4 ] e u 1+ɛu (1 + ɛu) m 6 R(u) (3.42)

17 16 CHIH-YUAN CHEN, KUO-CHIH HUNG AND SHIN-HWA WANG where R(u) = ( 1 + 2ɛ 2ɛm + ɛ 2 m ɛ 2 m 2 ) By (3.11), we compute that where R(A + B) = (1 4ɛ + 3ɛm 3ɛ 2 m 2ɛ 3 m + 3ɛ 2 m 2 + ɛ 3 m 2 + ɛ 3 m 3 )u +ɛ 2 (5 6ɛ 3m + 12ɛm 6ɛm 2 + 3ɛ 2 m 2 3ɛ 2 m 3 )u 2 ɛ 4 (4 7m + 2ɛm + 3m 2 5ɛm 2 + 5ɛm 3 )u 3 ɛ 6 m(1 m) 2 u 4. = 1 [ 4 13ɛ + 1ɛ 2 + (3 2ɛ 3ɛ 2 )ɛm + ɛ 3 m 2] (m 1) 2 ɛ ɛ 2 (m 1) 2 (m m 1)(m m 2 ) (3.43) m 1 = 3 + 2ɛ + 3ɛ2 + (ɛ 1) 9 1ɛ + 9ɛ 2 2ɛ 2, m 2 = 3 + 2ɛ + 3ɛ2 (ɛ 1) 9 1ɛ + 9ɛ 2 2ɛ 2. Notice that 8 9 ɛsem tr (m) is a strictly increasing function of m < 1, and we can solve ɛ = 8 9 ɛsem tr (m) as m = m = 4( 2+3 2ɛ). We thus find that m 9ɛ > m 1 > m 2 for < ɛ 8, see 9 Figure 6. Thus for < ɛ 8 9 ɛsem tr (m), we have that R(A + B) > by (3.43). Thus by (3.42), θ (A + B) >. (3.44) Figure 6. Graphs of m 1 and m 2 for < ɛ 1, and of ɛ = 8 9 ɛsem tr (m) for < ɛ 8/9. Case (I). If < m < 1, we obtain θ (B) > immediately by Lemma 3.4(ii) and (3.44). Thus by (3.41), H (B) >. Case (II). If < m <, it can be checked that C < B < A + B < D. Thus we obtain θ (B) > immediately by Lemma 3.4(iii) and (3.44). So by (3.41), H (B) >.

18 S-SHAPED BIFURCATION CURVES 17 (ii) The proof of H(B) for < ɛ 8 9 ɛsem tr (m) follows by Lemma 3.13(i) and the similar arguments in the proof of H(C) for < ɛ 6 7 ɛsem tr (m) in Lemma 3.8; we omit the proof. The proof of Lemma 3.13 is complete. The following lemma follows by the same arguments of Hung & Wang [7, Lemma 3.5]; we omit the proof. Lemma Consider (1.1) with m = 1/2. Then there exists a positive constant ɛ 4 < (m) = satisfying ɛ Sem tr H(C(ɛ 4 )) = and H(C(ɛ)) < for all < ɛ < ɛ 4. (3.45) (Numerical simulation shows that ɛ 4.328). 4. P T 2.1 (i) For < m < 1, if < ɛ 6 7 ɛsem tr (m), then we obtain that H(C) by Lemma 3.8. Thus by Lemma 3.3(i), the bifurcation curve S is S-shaped on the (λ, u )-plane. For ( 4.13 ) m m <, if < ɛ 6 7 ɛsem tr (m), we obtain that θ(d) > θ(a) by Lemma 3.1. Thus by Lemma 3.9, the bifurcation curve S is S-shaped on the (λ, u )-plane. In particular, for m = 1/2, the result in Theorem 2.1(i) follows by Lemma Hence Theorem 2.1(i) holds. (ii) For < m < 1, if < ɛ 8 9 ɛsem tr (m), then we obtain that H(B) by Lemma 3.13(ii). Thus by Lemma 3.3(ii), the bifurcation curve S is S-like shaped on the (λ, u )- plane immediately. For < m <, if < ɛ 8 9 ɛsem tr (m), we obtain that H(B) by Lemma 3.13(ii). Recall T (α) defined in (3.8). Applying the same arguments used in Hung & Wang [7, Theorem 2.2], it can be proved that T (B) <. So by (3.3) obtained by Lemma 3.1, the time map T (α) has at least two critical points on (, ). So the bifurcation curve S is S-like shaped on the (λ, u )-plane. Hence Theorem 2.1(ii) holds. The proof of Theorem 2.1 is complete. 5. F R We finish this paper by giving next two remarks. Remark 5.1. For each fixed m (, 1), by applying similar arguments as we did in Lemma 3.14; that is, Hung & Wang [7, Lemma 3.5], it can be shown that the following assertion (i) and (ii) hold: (i) There exists a constant ɛ m ( 8 9 ɛsem tr (m), ɛ Sem tr (m)) satisfying H(C(ɛ m)) = and H(C(ɛ)) < for all < ɛ < ɛ m. (ii) There exists a constant ɛ m (ɛ m, ɛ Sem tr (m)) satisfying H(B(ɛ m)) = and H(B(ɛ)) < for all < ɛ < ɛ m. Thus by Lemma 3.3, for < m < 1, on the (λ, u )-plane, the bifurcation curve S is S-shaped if < ɛ ɛ m, and is S-like shaped if ɛ m < ɛ ɛ m.

19 18 CHIH-YUAN CHEN, KUO-CHIH HUNG AND SHIN-HWA WANG Remark 5.2. For Lemma 3.9, we observe that: (i) For 2 < m < and < ɛ 6 7 ɛsem tr (m), numerical simulation shows that assertion (3.4) holds, and hence our arguments are able to show that the bifurcation curve S is S-shaped on the (λ, u )-plane. (ii) However, for m < negatively large enough, assertion (3.4) do not hold. For example, let m = 25 and < ɛ <.1, then numerical simulation shows that θ(d(ɛ)) <. Thus our arguments does not apply to show that the bifurcation curve S is S-shaped on the (λ, u )-plane. Further investigation is needed. R [1] T. Boddington, C.-G. Feng and P. Gray, Disappearance of criticality in thermal explosion under Frank- Kamenetskii boundary conditions, Combust. Flame, 48 (1982), [2] T. Boddington, C.-G. Feng and P. Gray, Thermal explosion, criticality and the disappearance of criticality in systems with distributed temperatures. I. Arbitrary Biot number and general reaction-rate laws, Proc. R. Soc. Lond. A, 39 (1983a), [3] T. Boddington, C.-G. Feng and P. Gray, Thermal explosion and the theory of its initiation by steady intense light, Proc. R. Soc. Lond. A, 39 (1983b), [4] T. Boddington, P. Gray and C. Robinson, Thermal explosion and the disappearance of criticality at small activation energies: exact results for the slab, Proc. R. Soc. Lond. A, 368 (1979), [5] M. G. Crandall and P. H. Rabinowitz, Bifurcation, perturbation of simple eigenvalues and linearized stability, Arch. Rational Mech. Anal., 52 (1973), [6] Y. Du, Exact multiplicity and S-shaped bifurcation curve for some semilinear elliptic problems from combustion theory, SIAM J. Math. Anal., 32 (2), [7] K.-C. Hung and S.-H. Wang, A theorem on S-shaped bifurcation curve for a positone problem with convexconcave nonlinearity and its applications to the perturbed Gelfand problem, J. Differential Equations, 251 (211), [8] P. Korman and Y. Li, On the exactness of an S-shaped bifurcation curve, Proc. Amer. Math. Soc., 127 (1999), [9] G. P. Miller, The structure of a stoichiometric CCI4 CH4 air flat flame, Combust. Flame, 11 (1995), [1] M. Mimura and K. Sakamoto, Multi-dimensional transition layers for an exothermic reaction diff usion system in long cylindrical domains, J. Math. Sci. Univ. Tokyo, 3 (1996), [11] A. L. Sánchez, A. Liñán and F. A. Williams, Chain-branching explosions in mixing layers, SIAM J. Appl. Math., 59 (1999), [12] K. Taira, Semilinear elliptic boundary-value problems in combustion theory, Proc. Roy. Soc. Edinburgh Sect. A, 132 (22), [13] S.-H. Wang, On S-shaped bifurcation curves, Nonlinear Anal., 22 (1994), [14] S.-H. Wang, Rigorous analysis and estimates of S-shaped bifurcation curves in a combustion problem with general Arrhenius reaction-rate laws, Proc. R. Soc. Lond. A, 454 (1998), A Proof of Lemma 3.1. We divide the proof into the following four steps 1 4:

20 S-SHAPED BIFURCATION CURVES 19 Step 1. f(d/2) f(d). Step 2. D > 4. Step 3. θ(d) > 2. Step 4. 2 > θ(a) >. Thus we obtain θ(d) > 2 > θ(a) > and the proof of Lemma 3.1 is complete by Steps 3 and 4. Proof of Step 1. By (1.2) and (3.6), we compute that = f(d/2) ɛ f(d) [ m 3 (1 m) 2 1 m+ 1 m 1+ 1 m+m 2 ɛ m(1+ɛ) > for m < and < ɛ 6 7 ɛsem tr ] m e m(m 1)( 1 1 m+m mɛ+m 2 ɛ) (1 m+ 1 m)[1+ 1 m+m 2 ɛ m(1+ɛ)] 2 m [ m + m 2 ɛ m(1 + ɛ) ] 2 [ ɛ m 1 + ] 1 m m(1 m) (m) < m m. (6.1) m(1 m) Thus f(d/2) f(d) ɛ = into f(d/2) f(d) is a strictly increasing function of ɛ (, 6 7 ɛsem tr (m)] for fixed m <. Setting, we obtain that f(d/2) f(d) = 2 m e m(1 m)(1 m+ ɛ= 1 m) (1 m+ 1 m) 2 > 1 on ( m, ), = 1 when m = m, < 1 on (, m), (6.2) where constant m Thus by (6.1) and (6.2), for m m < and < ɛ 6 7 ɛsem tr (m), we have that f(d/2) f(d). Proof of Step 2. Let m m < and < ɛ 6 7 ɛsem tr (m). By (3.6), we compute that D 4 = 4m 1 mɛ 2 + m 1 mɛ + 1 m + 1 m = 4 (ɛ ξ)(ɛ η) (6.3) 1 mɛ 2 ɛ2 where ξ = m 1 m + 16m 16m 1 m + 17m 2 m 3 8m m + 1 η = m 1 m 16m 16m 1 m + 17m 2 m 3 8m m + 1 >, <.

21 2 CHIH-YUAN CHEN, KUO-CHIH HUNG AND SHIN-HWA WANG Figure 7. Graphs of 6 7 ɛsem tr (m), ξ and η for ( 4.13 ) m m <. Since < ɛ 6 7 ɛsem tr (m), we obtain that η < < ɛ 6 7 ɛsem tr (m) < ξ for m m <, see Figure 7. Thus by (6.3), we have that D > 4. Proof of Step 3. Let m m < and < ɛ 6 7 ɛsem tr (m). First we compute that f() = 1 and f (u) = f(u) ( mɛ 2 (1 + ɛu) 2 u + mɛ + 1 ) > on [, ρ), = when u = ρ = (1+mɛ), (6.4) mɛ 2 < on (ρ, ). It is easy to show that D/2 < ρ < D; we omit the proof. Thus by (6.4), we have that and f(u) > 1 on (, D/2) (6.5) f (D/2) >, f (D) <. (6.6) By Step 1, (6.5) and (6.6), we obtain the following Figure 8 on two possible graphs of f(u) with f(d/2) f(d). Figure 8. Two possible graphs of f(u) with f(d/2) f(d). (i) f(d) 1. (ii) f(d) > 1.

22 S-SHAPED BIFURCATION CURVES 21 Case (i). Suppose f(d) 1 (see Figure 8(i)). Then by (3.9), (6.5) and Step 2, we obtain that D θ(d) = 2 = = > D D D f(t)dt Df(D) f(t)dt + f(t)dt + f(t)dt > D D D/2 [f(t) f(d)] dt + [f(t) f(d)] dt f(t)dt > D/2 > 2. D f(d)dt Df(D) Case (ii). Suppose f(d) > 1 (see Figure 8(ii)). Then by (6.4) and (6.6), there exists a positive number D D/2 such that f( D) = f(d). Thus by (3.9), (6.5) and Step 2, we obtain that D θ(d) = 2 f(t)dt Df(D) ( D D/2 = 2 f(t)dt + ( D/2 f(d)dt + ( D = 2 f(t)dt + D D D/2 D f(d)dt + D/2 f(d)dt f(d)dt + D ) D/2 D D f(d)dt + D D [f(t) f(d)]dt [f(t) f(d)]dt ) ) > D f(t)dt + D/2 D f(d)dt > D 1dt + D/2 D 1dt = D/2 > 2. Proof of Step 4. Let m m < and < ɛ 6 7 ɛsem tr (m). First we show that θ(a(ɛ)) >. ɛ By (1.2), (3.9) and (3.1), we have that θ(a(ɛ)) = θ(ɛ, A(ɛ)) and θ(ɛ, A(ɛ)) =. In u

23 22 CHIH-YUAN CHEN, KUO-CHIH HUNG AND SHIN-HWA WANG addition, we compute that ɛ θ(a(ɛ)) = θ(ɛ, A(ɛ)) + θ(ɛ, A(ɛ)) A(ɛ) ɛ u ɛ = θ(ɛ, A(ɛ)) [ ɛ ( u )] = 2 f(t)dt uf(u) ɛ u=a(ɛ) [ u ] t(m t + ɛmt) u(m u + ɛmu) = 2 f(t) dt uf(u) (1 + ɛt) 2 (1 + ɛu) 2 u=a(ɛ) [ u ] = 2 f(t)dt u f(u) u=a(ɛ) = θ(a(ɛ)) (6.7) where u θ(u) 2 f(t)dt u f(u) and We compute that θ() = and f(u) u(m u + ɛmu) f(u). (1 + ɛu) 2 θ (u) = f(u) u f (u) = u2 f(u) (1 + ɛu) 4 ɛ2 ( 1 + m + mɛ m 2 ɛ) (u u 1 ) (u u 2 ) > for < u < u 2 (and since m <, 1 + m + mɛ m 2 ɛ < ) where u mɛ2 2m 2 ɛ ɛ 2 4mɛ 2 2ɛ 2 (1 m mɛ + m 2 ɛ) u mɛ2 2m 2 ɛ ɛ 2 4mɛ 2 2ɛ 2 (1 m mɛ + m 2 ɛ) for m m < and < ɛ 6 7 ɛsem tr (m); we omit the proof. So >, < θ(u) > for < u < u2. (6.8) It is easy to check that < A(ɛ) = 1 2ɛ + mɛ 1 4ɛ + 2mɛ + m 2 ɛ 2 2ɛ 2 (1 m) < u 2 (6.9) for m m < and < ɛ 6 7 ɛsem tr (m); we omit the proof. Thus by (6.7) (6.9), ɛ θ(a(ɛ)) = θ(a(ɛ)) >, (6.1) and hence θ(a(ɛ)) is a strictly increasing function of ɛ (, 6 7 ɛsem tr (m)] for fixed m <.

24 S-SHAPED BIFURCATION CURVES 23 Figure 9. Graph of θ(a) when ɛ = 6 7 ɛsem tr (m) for ( 4.13 ) m m <. We find that 2 > θ(a) ɛ= 6 7 ɛsem tr (m) for m m <, (6.11) see Figure 9. (Notice that 2 > θ(a) ɛ= 6 7 ɛsem tr (m) for m = m.) Thus by (6.1) and (6.11), we obtain that 2 > θ(a). Moreover, we compute that θ() =. Thus by (3.1), we obtain that θ(a) >. The proof of Lemma 3.1 is complete. address: s @m99.nthu.edu.tw, kchung@mx.nthu.edu.tw, shwang@math.nthu.edu.tw