Stability analysis and Hopf bifurcation at high Lewis number in a combustion model with free interface
aa r X i v : . [ m a t h . A P ] J a n STABILITY ANALYSIS AND HOPF BIFURCATION AT HIGH LEWISNUMBER IN A COMBUSTION MODEL WITH FREE INTERFACE
CLAUDE-MICHEL BRAUNER, LUCA LORENZI, AND MINGMIN ZHANG
Abstract.
In this paper we analyze the stability of the traveling wave solution for an ignition-temperature, first-order reaction model of thermo-diffusive combustion, in the case of high Lewisnumbers (Le > i is reached. We turn the model to a fully nonlinear problemin a fixed domain. When the Lewis number is large, we define a bifurcation parameter m =Θ i / (1 − Θ i ) and a perturbation parameter ε = 1 / Le. The main result is the existence of acritical value m c ( ε ) close to m c = 6 at which Hopf bifurcation holds for ε small enough. Proofscombine spectral analysis and non-standard application of Hurwitz Theorem with asymptoticsas ε → introduction This paper is devoted to the stability analysis of a unique (up to translation) traveling wavesolution to a thermo-diffusive model of flame propagation with stepwise temperature kineticsand first-order reaction (see [3]) at high Lewis numbers, namely Le >
1. The problem reads inone spatial dimension: ∂ Θ ∂t = ∂ Θ ∂x + W (Θ , Φ) ,∂ Φ ∂t = Le − ∂ Φ ∂x − W (Θ , Φ) . (1.1)Here, Θ and Φ are appropriately normalized temperature and concentration of deficient reactant, x ∈ R denotes the spatial coordinate, t > W (Θ , Φ) is a scaledreaction rate given by (see [3, Section 2, formula (3)]): W (Θ , Φ) = ( A Φ , if Θ ≥ Θ i , , if Θ < Θ i . (1.2)In (1.2), 0 < Θ i < A > i and Le, to be determined hereafter for the purpose of ensuring that the speed of travelingwave is set at unity. Moreover, the following boundary conditions hold at ±∞ :Θ( t, −∞ ) = 1 , Θ( t, ∞ ) = 0 , Φ( t, −∞ ) = 0 , Φ( t, ∞ ) = 1 . (1.3) Mathematics Subject Classification.
Primary: 35R35; Secondary: 35K55, 35B35, 80A25.
Key words and phrases.
Free interface problem; traveling wave solutions; fully nonlinear parabolic systems;stability; Hopf bifurcation; combustion.
In this first-order stepwise kinetics model, Φ does not vanish except as t tends to −∞ . Thus,problem (1.1)-(1.3) belongs to the class of parabolic Partial Differential Equations with dis-continuous nonlinearities. Models in combustion theory and other fields (see, e.g. [2, Section1]) involving discontinuous reaction terms have been used by physicists and engineers for longbecause of their manageability; as a result, elliptic and parabolic PDEs with discontinuousnonlinearities, and related Free Boundary Problems, have received a close attention from themathematical community (see [1, Section 1] and references therein). We quote in particularthe paper [13], by K.-C. Chang, which contains a systematical study of elliptic PDEs withdiscontinuous nonlinearities (DNDE).In this paper, we consider the case of a free ignition interface g ( t ) defined byΘ( t, g ( t )) = Θ i , (1.4)such that Θ( t, x ) > Θ i for x > g ( t ) and Θ( t, x ) < Θ i for x < g ( t ). Formula (1.4) means that theignition temperature Θ i is reached at the ignition interface which defines the flame front. Wepoint out that, in contrast to conventional Arrhenius kinetics where the reaction zone is infinitelythin, the reaction zone for stepwise temperature kinetics is of order unity (thick flame). It isalso interesting to compare the first-order stepwise kinetics with the zero-order kinetics model(see [1, 3, 4]): in the zero-order kinetics, Φ( t, x ) vanishes at a trailing interface and does notappear explicitly in the nonlinear term (see [3, Section 2, formula (4)]).According to (1.4), the system for XXX = (Θ , Φ) reads as follows, for t > x ∈ R , x = g ( t ): ∂ Θ ∂t = ∂ Θ ∂x + A Φ , x < g ( t ) ,∂ Φ ∂t = Le − ∂ Φ ∂x − A Φ , x < g ( t ) , (1.5) ∂ Θ ∂t = ∂ Θ ∂x , x > g ( t ) ,∂ Φ ∂t = Le − ∂ Φ ∂x , x > g ( t ) . (1.6)At the free interface x = g ( t ), the following continuity conditions hold:[Θ] = [Φ] = 0 , (cid:20) ∂ Θ ∂x (cid:21) = (cid:20) ∂ Φ ∂x (cid:21) = 0 , (1.7)where we denote by [ f ] the jump of a function f at a point x , i.e., the difference f ( x +0 ) − f ( x − ).The system above admits a unique (up to translation) traveling wave solution UUU = (Θ , Φ )which propagates with constant positive velocity V . In the moving frame coordinate z = x − V t ,by choosing A = Θ i − Θ i (cid:18) i Le(1 − Θ i ) (cid:19) , (1.8)to have V = 1 and, hence, z = x − t , the traveling wave solution is explicitly given by thefollowing formulae: Θ ( z ) = − (1 − Θ i ) e Θ i − Θ i z , z < , Θ i e − z , z > , TABILITY ANALYSIS AND HOPF BIFURCATION 3 Φ ( z ) = Θ i A (1 − Θ i ) e Θ i − Θ i z , z < , (cid:18) Θ i A (1 − Θ i ) − (cid:19) e − Le z , z > . The goal of this paper is the analysis of the stability of the traveling wave solution
UUU in thecase of high Lewis numbers (Le > pulsating instabilities , i.e., oscillatory behaviorof the flame. This is very unlike cellular instabilities for relatively small Lewis number (Le < uuu of the traveling wave UUU is split as uuu = s dUUUdξ + vvv (“ansatz 1”), in which s is theperturbation of the front g . The largest part of the section is devoted to a thorough study ofthe linearization at 0 of the elliptic part of the parabolic system in a weighted space W whereits realization L is sectorial (see Subsection 2.3 for further details about the use of a weightedspace). Furthermore, we determine the spectrum of L which contains ( −∞ , − ], a parabolaand its interior, the roots of the so-called dispersion relation, and the eigenvalue 0. Thereafter,an important point is getting rid of the eigenvalue 0 which, as it has been already stressed, isgenerated by translation invariance. In Section 3, we use a spectral projection P as well as“ansatz 2” and then derive the fully nonlinear problem (see, e.g. [21]) for : = ( I − P ) + F ( ) . Next, in Sections 4 and 5 we use the bifurcation parameter m defined by m := Θ i − Θ i to investigate the stability of the traveling wave. Simultaneously, as one already noted thatpulsating instability is likely to occur at large Lewis number, it is natural to introduce a smallperturbation parameter ε > ε := Le − , sothat (1.8) reads A = m + εm . The simplest situation arises in the asymptotic case of gaslesscombustion when Le = ∞ , as in [16]. As it is easily seen, as ε →
0, problem (1.5)-(1.6) convergesformally to: ∂ Θ ∂t = ∂ Θ ∂x + A Φ , x < g ( t ) ,∂ Φ ∂t = − A Φ , x < g ( t ) , (1.9) ∂ Θ ∂t = ∂ Θ ∂x , x > g ( t ) , Φ ≡ , x > g ( t ) , (1.10) CLAUDE-MICHEL BRAUNER, LUCA LORENZI, AND MINGMIN ZHANG with conditions [Θ] = [Φ] = 0, (cid:20) ∂ Θ ∂x (cid:21) = 0 at the free interface x = g ( t ). However, the limit freeinterface system (1.9)-(1.10) is only partly parabolic.At the outset, we fix m in Section 4 and let ε tend to 0, which allows to apply the classicalHurwitz Theorem in complex analysis to the dispersion relation D ε ( λ, m ). Our first main result,Theorem 4.2, states that, for 2 < m < m c = 6 and 0 < ε < ε ( m ), the traveling wave UUU isorbitally stable with asymptotic phase and, for m > m c = 6, it is unstable. To give a broadpicture, we take advantage of the regular convergence of the point spectrum as ε → m c = 6. The difficulty is twofold: first, the framework is that of a fully nonlinear problem;second, m is not fixed in the sequence of parameterized analytic functions D ε ( λ, m ) whichprevents us from using Hurwitz Theorem directly. The trick is to find a proper approach tocombining m with ε : to this end we construct a sequence of critical values m c ( ε ) such that m c (0) = m c and apply Hurwitz Theorem to D ε ( λ, m c ( ε )). Proposition 5.1 and Theorem 5.3 arecrucial to prove Hopf bifurcation at m c ( ε ) for ε small enough. Finally, in three appendices, wecollect some formulae and results that we use to prove our main results.2. The linearized operator
In this section, we first derive the governing equations for the perturbations of the travelingwave solution. As usual, it is convenient to transform the free interface problem to a system ona fixed domain. More specifically, we use the general method of [9] that converts free interfaceproblems to fully nonlinear problems with transmission conditions at a fixed interface (see [1]).Then, we are going to focus on the linearized system.2.1.
The system with fixed interface.
To begin with, we rewrite problem (1.5)-(1.7) in anew system of coordinates that fixes the position of the ignition interface at the origin: τ = t, ξ = x − g ( τ ) . Hereafter, we are going to use, whenever it is convenient, the superdot to denote differentiationwith respect to time and the prime to denote partial differentiation with respect to the spacevariable.Then, the system for
XXX = (Θ , Φ) and g reads: ∂ Θ ∂τ − ˙ g ∂ Θ ∂ξ = ∂ Θ ∂ξ + A Φ , ξ < ,∂ Φ ∂τ − ˙ g ∂ Φ ∂ξ =Le − ∂ Φ ∂ξ − A Φ , ξ < , (2.1) ∂ Θ ∂τ − ˙ g ∂ Θ ∂ξ = ∂ Θ ∂ξ , ξ > ,∂ Φ ∂τ − ˙ g ∂ Φ ∂ξ =Le − ∂ Φ ∂ξ , ξ > . (2.2)Moreover, Θ, Φ and their first-order space derivatives are continuous at the fixed interface ξ = 0,thus Θ( · ,
0) = Θ i , [Θ] = [Φ] = 0 , (cid:20) ∂ Θ ∂ξ (cid:21) = (cid:20) ∂ Φ ∂ξ (cid:21) = 0 . (2.3) TABILITY ANALYSIS AND HOPF BIFURCATION 5
In addition, at ξ = ±∞ , Θ and Φ satisfy (1.3).Next, we introduce the small perturbations uuu = ( u , u ) and s , respectively of the travelingwave UUU and of the front g , more precisely, u ( τ, ξ ) = Θ( τ, ξ ) − Θ ( ξ ) ,u ( τ, ξ ) = Φ( τ, ξ ) − Φ ( ξ ) ,s ( τ ) = g ( τ ) − τ. It then follows that the perturbations uuu and s verify the system ∂u ∂τ = ∂ u ∂ξ + ∂u ∂ξ + Au + ˙ s d Θ dξ + ˙ s ∂u ∂ξ , ξ < ,∂u ∂τ = Le − ∂ u ∂ξ + ∂u ∂ξ − Au + ˙ s d Φ dξ + ˙ s ∂u ∂ξ , ξ < , (2.4) ∂u ∂τ = ∂ u ∂ξ + ∂u ∂ξ + ˙ s d Θ dξ + ˙ s ∂u ∂ξ , ξ > ,∂u ∂τ = Le − ∂ u ∂ξ + ∂u ∂ξ + ˙ s d Φ dξ + ˙ s ∂u ∂ξ , ξ > , (2.5)and the corresponding interface conditions obtained from (2.3) are: u ( τ,
0) = 0 , [ u ] = [ u ] = (cid:20) ∂u ∂ξ (cid:21) = (cid:20) ∂u ∂ξ (cid:21) = 0 . (2.6)2.2. Ansatz 1.
In the spirit of [9, 18], we introduce the following splitting or ansatz: u ( τ, ξ ) = s ( τ ) d Θ dξ ( ξ ) + v ( τ, ξ ) ,u ( τ, ξ ) = s ( τ ) d Φ dξ ( ξ ) + v ( τ, ξ ) , (2.7)in which v , v are new unknown functions. In a more abstract setting, the ansatz reads uuu ( τ, ξ ) = s ( τ ) dUUUdξ + vvv ( τ, ξ ) , vvv = ( v , v ) . Substituting (2.7) into (2.4)-(2.5), we get the system for uuu and s : ∂v ∂τ = ∂ v ∂ξ + ∂v ∂ξ + Av + ˙ s (cid:18) s d Θ dξ + ∂v ∂ξ (cid:19) , ξ < ,∂v ∂τ = Le − ∂ v ∂ξ + ∂v ∂ξ − Av + ˙ s (cid:18) s d Φ dξ + ∂v ∂ξ (cid:19) , ξ < , (2.8) ∂v ∂τ = ∂ v ∂ξ + ∂v ∂ξ + ˙ s (cid:18) s d Θ dξ + ∂v ∂ξ (cid:19) , ξ > ,∂v ∂τ = Le − ∂ v ∂ξ + ∂v ∂ξ + ˙ s (cid:18) s d Φ dξ + ∂v ∂ξ (cid:19) , ξ > . (2.9) CLAUDE-MICHEL BRAUNER, LUCA LORENZI, AND MINGMIN ZHANG At ξ = 0, it is easy to see that the new interface conditions are:[ v ] = [ v ] = 0 , (cid:20) ∂v ∂ξ (cid:21) = − s (cid:20) d Θ dξ (cid:21) , (cid:20) ∂v ∂ξ (cid:21) = − s (cid:20) d Φ dξ (cid:21) , v ( τ,
0) = − s ∂ Θ ∂ξ (0) . Taking advantage of the conditions d Θ dξ (0) = − Θ i , (cid:20) d Θ dξ (cid:21) = Θ i − Θ i , (cid:20) d Φ dξ (cid:21) = − LeΘ i − Θ i , where we used (1.8) to derive the last condition, it follows that s ( τ ) = v ( τ, i , (cid:20) ∂v ∂ξ (cid:21) = − v ( τ, − Θ i , (cid:20) ∂v ∂ξ (cid:21) = v ( τ, − Θ i . (2.10)Summarizing, the free interface problem (1.5)-(1.6) has been converted to (2 . . v , v and s , with transmission conditions (2.10) at ξ = 0.The next subsections are devoted to the study of the linearized problem (at zero) in an abstractsetting, with simplified notation uuu = ( u, v ) for convenience.2.3. The linearized problem.
Now, we consider the linearization at 0 of the system (2.8)-(2.10), which reads as follows: ∂u∂τ = ∂ u∂ξ + ∂u∂ξ + Av, ξ < ,∂v∂τ = Le − ∂ v∂ξ + ∂v∂ξ − Av, ξ < , (2.11) ∂u∂τ = ∂ u∂ξ + ∂u∂ξ , ξ > ,∂v∂τ = Le − ∂ v∂ξ + ∂v∂ξ , ξ > , (2.12)with the interface conditions[ u ] = [ v ] = 0 , (cid:20) ∂u∂ξ (cid:21) = − u ( τ, − Θ i , (cid:20) ∂v∂ξ (cid:21) = u ( τ, − Θ i . (2.13)Problem (2.11)-(2.12) can be written in the more compact form ∂uuu∂τ = L uuu , where uuu = ( u, v ), L = ∂ ∂ξ + ∂∂ξ Aχ − − ∂ ∂ξ + ∂∂ξ − Aχ − and χ − denotes the characteristic function of the set ( −∞ , W where we analyze the system (2.11)-(2.13). As amatter of fact, the introduction of exponentially weighted spaces for proving stability of travelingwaves has been a standard tool since the pioneering work of Sattinger (see [24]), its role beingto shift the continuous spectrum to the left and, thus, creating a gap with the imaginary axiswhich simplifies the analysis. TABILITY ANALYSIS AND HOPF BIFURCATION 7
Definition 2.1.
The exponentially weighted Banach space W is defined by W = n uuu : e ξ u, e ξ v ∈ C b (( −∞ , C ) , e ξ u, e Le2 ξ v ∈ C b ((0 , ∞ ); C ) , lim ξ → ± u ( ξ ) , lim ξ → ± v ( ξ ) ∈ R o , equipped with the norm: k uuu k W = sup ξ< | e ξ u ( ξ ) | + sup ξ> | e ξ u ( ξ ) | + sup ξ< | e ξ v ( ξ ) | + sup ξ> | e Le2 ξ v ( ξ ) | . In the above definition, C b ( I ; C ) denotes the space of bounded and continuous functions from I to C , I being either the interval ( −∞ ,
0) or (0 , ∞ ). We finally introduce the realization L ofthe operator L in W defined by D ( L ) = (cid:26) uuu ∈ W : ∂uuu∂ξ , ∂ uuu∂ξ ∈ W , [ u ] = [ v ] = 0 , (cid:20) ∂u∂ξ (cid:21) = − u (0)1 − Θ i , (cid:20) ∂v∂ξ (cid:21) = Le u (0)1 − Θ i (cid:27) ,Luuu = L uuu, uuu ∈ W . Remark 2.2.
We observe that, for any Lewis number, the pair dUUUdξ = (cid:18) d Θ dξ , d Φ dξ (cid:19) verifiesSystem (2.11), (2.12), and it belongs to the space W . In other words, dUUUdξ is an eigenfunctionof the operator L associated with the eigenvalue 0.The above remark gives a first justification for the choice of the exponential weights in the def-inition of W . We also stress that, following the same strategy as in the proof of the forthcomingTheorem 2.3 it can be easily checked that the spectrum of the realization of the operator L in thenonweighted space of pairs ( u, v ) such that u , v are bounded and continuous in ( −∞ , ∪ (0 , ∞ ),contains a parabola which is tangent at 0 to the imaginary axis.2.4. Analysis of the operator L . Next theorem is devoted to a deep study of the operator L . For simplicity of notation, for j = 1 , H ,λ = √ λ, H ,λ = q Le + 4Le( A + λ ) , H ,λ = p Le + 4Le λ (2.14)and k j,λ = − − j +1 H ,λ , k j,λ = − Le + ( − j +1 H ,λ , k j,λ = − Le + ( − j +1 H ,λ . (2.15) Theorem 2.3.
The operator L is sectorial and therefore generates an analytic semigroup. More-over, its spectrum has components: (1) ( −∞ , − / ∪ P , where P = { λ ∈ C : a Re λ + b (Im λ ) + c ≤ } with a = (cid:18) − (cid:19) , b = 1Le , c = 2 A + 12 + 8 A − A Le − ;(2) the simple isolated eigenvalue , the kernel of L being spanned by dUUUdξ ; (3) additional eigenvalues given by the solution of the dispersion relation D ( λ ; Θ i , Le) := ( k ,λ − k ,λ )( k ,λ − k ,λ ) (cid:2) − (1 − Θ i ) √ λ (cid:3) + A Le , (2.16) where A is given by (1.8) . CLAUDE-MICHEL BRAUNER, LUCA LORENZI, AND MINGMIN ZHANG
Proof.
Since the proof is rather lengthy, we split it into four steps. In the first two steps, weprove properties (1) and (3). Step 3 is devoted to the proof of property (2). Finally, in Step 4,we prove that the operator L is sectorial in W .For notational convenience, throughout the proof, we set I := Z ∞ f ( s ) e − k s ds, I := Z −∞ f ( s ) e − k s ds, I := Z −∞ f ( s ) e − k s ds, I := Z −∞ f ( s ) e − k s ds, I := Z ∞ f ( s ) e − k s ds, for any fixed fff = ( f , f ) ∈ W , where, here and Step 1 to 3, we simply write k j instead of k j,λ to enlighten the notation. Step 1 . To begin with, we prove that the interval ( −∞ , − /
4] belongs to the point spectrumof L . We first assume that λ ≤ − Le / > k ) = Re( k ) = − /
2, Re( k ) = Re( k ) = − Le / uuu defined by u ( ξ ) = (cid:26) c e k ξ + c e k ξ , ξ < ,c e k ξ + c e k ξ , ξ ≥ , v ( ξ ) = (cid:26) , ξ < ,c e k ξ + c e k ξ , ξ ≥ , (2.17)belongs to W and solves the equation λuuu − L uuu = for any choice of the complex parameters c , c , c , c , c and c . Since there are only four boundary conditions to impose to guarantee that uuu ∈ D ( L ), the resolvent equation λuuu − L uuu = is not uniquely solvable in W . Thus, λ belongsto the point spectrum of L .Next, we consider the case when λ ∈ ( − Le / , − / k ) = Re( k ) = − /
2, however, Re( k ) + Le / >
0, Re( k ) + Le / <
0. Thanks to the fact that e Le2 ξ v ( ξ ) shouldbe bounded in (0 , ∞ ), the constant c in (2.17) is zero, whereas the constants c , c , c , c c are arbitrary. As above, the resolvent equation λuuu − Luuu = cannot be solved uniquely.Consequently, we conclude that ( −∞ , − /
4] belongs to the point spectrum of the operator L .From now on, we consider the case when λ / ∈ ( −∞ , − / k ) + 1 / >
0, Re( k ) +1 / <
0, Re( k ) + Le / > k ) + Le / <
0. Similarly to the previous procedure, usingthe formulae (A.4), (A.5) and (A.2) as well as the fact that the functions ξ e ξ u ( ξ ) and ξ e Le2 ξ v ( ξ ) should be bounded in R and in (0 , ∞ ) respectively, the constants c , c , c can bedetermined explicitly and they are given by c = 1 H ,λ Z −∞ ( Av ( s ) + f ( s )) e − k s ds, c = 1 H ,λ I , c = Le H ,λ I . We now consider formula (A.3). Since Le >
1, it follows that Re( k ) + 1 / <
0. Moreover,we observe that the inequality Re( k ) + 1 / ≤ λ ∈ P . Indeed, fix any λ ∈ ◦ P , the interior of P , so that Re( k ) + 1 / <
0, and take f ( ξ ) = (cid:26) e − ξ , ξ < , , ξ ≥ , f ≡ R . In such a case, the more general solution, uuu ∈ W , to the equation λuuu − L uuu = fff is given by u ( ξ ) = c e k ξ and v ( ξ ) = c e k ξ for ξ ≥
0, whereas v ≡ −∞ ,
0) and u ( ξ ) = c e k ξ +2 H − ,λ (2 e − ξ − e k ξ ) for ξ <
0. Note that k = k for λ ∈ ◦ P . Imposing the boundary conditions,we deduce that c = c = 0, c = − H − ,λ and k c = 2 H − ,λ k , which is clearly a contradiction. TABILITY ANALYSIS AND HOPF BIFURCATION 9
We conclude that the domain ◦ P and, consequently, its closure belong to the continuous spectrumof L . Summarizing, property (1) in the statement of the theorem is established. Step 2 . Here, we consider the equation λuuu − L uuu = fff for fff ∈ W and values of λ which are notin ( −∞ , − / ∪ P . For such λ ’s and j = 1 , k j − ) + 12 > , Re( k j ) + 12 < , Re( k ) + Le2 > , Re( k ) + Le2 < . (2.18)We first assume that k = k . Imposing that the function uuu defined by (A.4)-(A.3) belongs to W , we can uniquely determine the constants c , c , c and c and we get u ( ξ ) = c e k ξ + e k ξ H ,λ Z ξ f ( s ) e − k s ds + e k ξ H ,λ Z ξ −∞ f ( s ) e − k s ds + AH ,λ (cid:26)(cid:18) e k ξ k − k − e k ξ − e k ξ k − k (cid:19) c + Le H ,λ (cid:20)(cid:18) e k ξ − e k ξ k − k − e k ξ k − k (cid:19) Z ξ f ( s ) e − k s ds + e k ξ k − k Z ξ f ( s ) e − k s ds + (cid:18) e k ξ − e k ξ k − k + e k ξ k − k (cid:19) Z ξ −∞ f ( s ) e − k s ds + e k ξ k − k Z ξ f ( s )( e − k s − e − k s ) ds + ( k − k ) e k ξ ( k − k )( k − k ) Z ξ −∞ f ( s ) e − k s ds (cid:21)(cid:27) , (2.19) v ( ξ ) = (cid:18) c + Le H ,λ Z ξ f ( s ) e − k s ds (cid:19) e k ξ + Le e k ξ H ,λ Z ξ −∞ f ( s ) e − k s ds, (2.20)for ξ <
0. Note that k − k = 0 (see Appendix A). For ξ >
0, we get u ( ξ ) = e k ξ H ,λ Z ∞ ξ f ( s ) e − k s ds + (cid:18) c + 1 H ,λ Z ξ f ( s ) e − k s ds (cid:19) e k ξ , (2.21) v ( ξ ) = Le e k ξ H ,λ Z ∞ ξ f ( s ) e − k s ds + (cid:18) c + Le H ,λ Z ξ f ( s ) e − k s ds (cid:19) e k ξ . (2.22)Imposing the boundary conditions, we obtain the following linear system for the unknowns c , c , c and c : A ( k − k ) H ,λ − − k Ak ( k − k ) H ,λ i − − k k − Θ i − k c c c c = F F F F , (2.23)where F = − A Le( k − k ) H ,λ H ,λ I − H ,λ I + 1 H ,λ I − A Le( k − k )( k − k )( k − k ) H ,λ H ,λ I ; F = Le H ,λ I − Le H ,λ I ; F = − A Le k ( k − k ) H ,λ H ,λ I − k H ,λ I + 1 H ,λ (cid:18) k + 11 − Θ i (cid:19) I + A Le k ( k − k )( k − k ) H ,λ I ; F = Le k H ,λ I − Le k H ,λ I − Le(1 − Θ i ) H ,λ I . This system is uniquely solvable if and only if D ( λ ; Θ i , Le) = [Le( k − k )] − D ( λ ; Θ i , Le), thedeterminant of the matrix in left-hand side of (2.23), does not vanish, where D ( λ ; Θ i , Le) isdefined in (2.16). Hence, the solutions to the equation D ( λ ; Θ i , Le) = 0 are elements of thepoint spectrum of L . Property (3) is proved. On the other hand, as it is easily seen, if λ / ∈ ( −∞ , − / ∪ P is not a root of the dispersion relation, then it is easy to check that the function uuu given by (2.19)-(2.23) belongs to D ( L ), so that λ is an element of the resolvent set of operator L . Finally, we consider the case when k = k , which gives λ = λ ± := − A LeLe − ± i √ A Le(Le − − (seeAppendices A and B). It is easy to check that this pair of conjugate complex numbers does notbelong to P . It thus follows that u for ξ ≥ v for ξ ∈ R are still given by (2.20), (2.21) and(2.22). On the other hand, for ξ < u is given by u ( ξ ) = c e k ξ − Ac H ,λ ξe k ξ + e k ξ H ,λ Z ξ f ( s ) e − k s ds + e k ξ H ,λ Z ξ −∞ f ( s ) e − k s ds + A Le e k ξ H ,λ H ,λ Z ξ ( s − ξ ) f ( s ) ds − A Le e k ξ H ,λ H ,λ Z −∞ f ( s ) e − k s ds + A Le e k ξ H ,λ H ,λ Z ξ f ( s ) e − k s ds + A Le e k ξ H ,λ H ,λ Z ξ −∞ f ( s ) e − k s ds + AH ,λ (cid:26) e k ξ k − k c + Le H ,λ (cid:20) e k ξ k − k Z ξ −∞ f ( s ) e − k s ds − e k ξ k − k Z ξ f ( s ) e − k s ds + ( k − k ) e k ξ ( k − k )( k − k ) Z ξ −∞ f ( s ) e − k s ds (cid:21)(cid:27) . Notice that sup ξ< e ξ | u ( ξ ) | < ∞ ; therefore, uuu belongs to W . Imposing the boundary conditions,we get a linear system for the unknowns ( c , c , c , c ), whose matrix is the same as in (2.23).Since the determinant is not zero when λ = λ ± (see Appendix B) and the first- and second-orderderivatives of uuu belong to WWW , we conclude that λ ± are in the resolvent set of operator L . Step 3 . Now, we proceed to show that 0 is an isolated simple eigenvalue of the operator L . Inview of the previous steps, in a neighborhood of λ = 0 the solution uuu = R ( λ, L ) fff of the equation λuuu − Luuu = fff is given by (2.19)-(2.22) for any fff ∈ W , where c = Le( k − k ) D ( λ ; Θ i , Le) (cid:26)(cid:20) ( k − k )(1 − Θ i )Le − A ( k − k ) H ,λ (cid:21) I + k − k Le H ,λ I − A ( k − k )( k − k )( k − k ) H ,λ I + AH ,λ H ,λ (cid:18) k − k k − k − k − k k − k (cid:19) I − A ( k − k ) H ,λ I (cid:27) ,c = Le( k − k ) D ( λ ; Θ i , Le) (cid:26) I + I − A Le( k − k )( k − k ) I + 1 H ,λ (cid:20) ( k − k ) (cid:2) − H ,λ (1 − Θ i ) (cid:3) + A Le k − k (cid:21) I + (cid:2) − H ,λ (1 − Θ i ) (cid:3) I (cid:27) , TABILITY ANALYSIS AND HOPF BIFURCATION 11 c = Le( k − k ) D ( λ ; Θ i , Le) (cid:26) H ,λ (cid:18) Ak − k + k − k Le (cid:19) I + ( k − k )(1 − Θ i )Le I − A ( k − k )(1 − Θ i )( k − k )( k − k ) I + A (1 − Θ i ) H ,λ (cid:18) k − k k − k − k − k k − k (cid:19) I − A (1 − Θ i ) k − k I (cid:27) ,c = Le( k − k ) D ( λ ; Θ i , Le) (cid:26) I + I − A Le( k − k )( k − k ) I + (cid:20) − H ,λ (1 − Θ i )+ A Le( k − k )( k − k ) (cid:21) I + (cid:20) A Le( k − k ) H ,λ + [1 − H ,λ (1 − Θ i )] (cid:18) k − k H ,λ (cid:19)(cid:21) I (cid:27) . As it is immediately seen, the function D ( · ; Θ i , Le) is analytic in a neighborhood of λ = 0,which is simple zero of such a function, and the other functions appearing in (2.19)-(2.22) areholomorphic in a neighborhood of λ = 0. Hence, we conclude that zero is a simple pole ofthe resolvent operator R ( λ, L ). Since dUUUdξ belongs to the kernel of L (see Remark 2.2) and thematrix in (2.23) has rank three at λ = 0, this function generates the kernel, so that the geometricmultiplicity of the eigenvalue λ = 0 is one. This is enough to conclude that λ = 0 is a simpleeigenvalue of L . Property (2) is established and the spectrum of L is completely characterized. Step 4 . In order to prove that L is sectorial, it is sufficient to show that there exist twopositive constants C and M such that k R ( λ, L ) k L ( W ) ≤ C | λ | − , Re λ ≥ M. (2.24)Without loss of generality, we can assume that k ,λ = k ,λ and the conditions in (2.18) are allsatisfied if Re λ ≥ M . Throughout this step, C j denotes a positive constant, independent of λ and fff ∈ W .We begin by estimating the terms H j,λ ( j = 1 , , | H ,λ | ≥ Re( H ,λ ) = s | Le + 4Le( A + λ ) | + Le + 4Le( A + Re λ )2 ≥ p | λ | (2.25)for any λ ∈ C with positive real part. Since H ,λ and H ,λ can be obtained from H ,λ , by taking,(Le , A ) = (1 ,
0) and (Le , A ) = (Le ,
0) respectively, we also deduce that | H ,λ | ≥ Re( H ,λ ) ≥ p | λ | , | H ,λ | ≥ Re( H ,λ ) ≥ p | λ | (2.26)for the same values of λ . Thanks to (2.25) and (2.26), we can easily estimate the terms I j ( j = 1 , . . . , k ) + 1 / >
0, we obtain | I | = (cid:12)(cid:12)(cid:12)(cid:12) Z ∞ f ( s ) e − k s ds (cid:12)(cid:12)(cid:12)(cid:12) ≤ sup ξ> e ξ | f ( ξ ) | Z ∞ e − Re( H ,λ ) s ds ≤ C | λ | − k fff k W . The other terms I j can be treated likewise and we get P j =2 | I j | ≤ C | λ | − k fff k W for every fff ∈ W and λ ∈ C with positive real part.Next, we turn to the function D ( · ; Θ i , Le). We observe that | D ( λ ; Θ i , Le) | ≥ [(1 − Θ i ) p | λ | − | k ,λ − k ,λ || k ,λ − k ,λ | − A Lefor any λ ∈ C . Taking (2.25) and (2.26) into account, we can show that C p | λ | ≤ | k ,λ − k ,λ | + | k ,λ − k ,λ | ≤ C p | λ | (2.27) for λ ∈ C with sufficiently large positive real part. Hence, for such values of λ ’s we can continuethe previous inequality and get | D ( λ ; Θ i , Le) | ≥ C | λ | . (2.28)Similarly, | k ,λ − k ,λ | ≤ C p | λ | for any λ with positive real part and | k ,λ − k ,λ | ≥ | H ,λ | − | H ,λ | − Le − ≥ r Le | λ | − r | λ | − Le − ≥ C p | λ | , (2.29)if Re λ is sufficiently large. From (2.25)-(2.29) we infer that | c | + | c | + | c | + | c | ≤ C | λ | − forany λ ∈ C with Re( λ ) ≥ M and a suitable positive constant M . Further, observing that | k ,λ − k ,λ | + | k ,λ − k ,λ | ≥ C p | λ | , | k ,λ − k ,λ | ≤ C p | λ | , we are now able to estimate the functions u and v in (2.19)-(2.22) and show that (2.24) holdstrue. The proof is complete. (cid:3) Remark 2.4.
It is worth pointing out that, as Le → ∞ , the set P degenerates into a verticalline Re λ = − Θ i (1 − Θ i ) − − /
2. In the limit case, the system is partly parabolic and thesemigroup is not analytic, see, e.g., [17, Section 1, p. 2435].3.
The fully nonlinear problem
Our goal in this section is to get rid of the eigenvalue 0 and then derive a new fully nonlinearproblem. We recall that the eigenvalue 0 is related to the translation invariance of the travelingwave. In a first step, we use here a method similar to that of [12] or [21, p. 358].3.1.
Ansatz revisited: elimination of the eigenvalue . It is convenient to write System(2.4)-(2.5) with notation uuu = ( u , u ), UUU = (Θ , Φ ), see Section 2.1, in an abstract form:˙ uuu = Luuu + ˙ sUUU ′ + ˙ suuu ′ . (3.1)Note that, in view of (2.6), uuu ( τ, · ) belongs to D ( L ) for each τ . Since 0 is an isolated simpleeigenvalue of L , we can introduce the spectral projection P onto the kernel of L , defined by P fff = h fff , eee ∗ i UUU ′ for every fff ∈ W and a unique eee ∗ ∈ W ∗ , the dual space of W , such that h UUU ′ , eee ∗ i = 1. For further use, we recall that P commutes with L on D ( L ). We are going toapply the projections P and Q = I − P to System (3 .
1) to remove the eigenvalue 0.
Ansatz 2.
We split uuu into uuu ( τ, · ) = P uuu ( τ, · ) + Quuu ( τ, · ) = p ( τ ) UUU ′ + ( τ, · ), i.e., u ( τ, ξ ) = p ( τ ) d Θ dξ ( ξ ) + w ( τ, ξ ) , (3.2) u ( τ, ξ ) = p ( τ ) d Φ dξ ( ξ ) + w ( τ, ξ ) , where p ( τ ) = h uuu ( τ ) , eee ∗ i and = ( w , w ). Clearly, ( τ, · ) ∈ Q ( D ( L )) for each τ . It follows from(3 .
1) that ˙ p = ˙ s + ˙ s h uuu ′ , eee ∗ i , ˙ = + ˙ sQuuu ′ , (3.3)a Lyapunov-Schmidt-like reduction of the problem. We point out that the above proceduregenerates a new ansatz slightly different from ansatz 1 (see (2.7)) that helps us determine thefunctional framework. TABILITY ANALYSIS AND HOPF BIFURCATION 13
Thanks to new ansatz 2, we are going to derive an equation for in the space W . Now, thespectrum of the part of L in Q ( W ) does not contain the eigenvalue 0.3.2. Derivation of the fully nonlinear equation.
To get a self-contained equation for , weneed to eliminate ˙ s from the right-hand side of the second equation in (3.3). For this purpose,we begin by evaluating the first component of (3.3) at ξ = 0 + to get ∂w ∂τ ( · , + ) =( ) ( · , + ) + ˙ s ( Quuu ′ ) ( · , + )=( ) ( · , + ) + ˙ s ∂u ∂ξ ( · , + ) + ˙ s h uuu ′ , eee ∗ i Θ i . (3.4)Next, we observe that the function w is continuous (but not differentiable) at ξ = 0, sinceboth uuu and UUU ′ are continuous at ξ = 0. Therefore, evaluating (3.2) at ξ = 0 and recalling that u ( τ,
0) = 0 (see (2.6)), we infer that w ( τ,
0) = Θ i p ( τ ). Differentiating this formula yields ∂w ∂τ ( · ,
0) = ˙ p Θ i = ˙ s Θ i + ˙ s h uuu ′ , eee ∗ i Θ i , (3.5)From (3 .
4) and (3 . s Θ i = ( ) ( · , + ) + ˙ s ∂u ∂ξ ( · , + ) . (3.6)To get rid of the spatial derivatives of u from the right-hand side of (3.6), we use (3.2) to write ∂u ∂ξ ( · , + ) = p d Θ dξ (0 + ) + w ′ ( · , + ) = w ( · ,
0) + w ′ ( · , + ) . (3.7)Plugging (3 .
7) into (3 . s = ( ) ( · , + )Θ i − w ( · , − w ′ ( · , + ) , (3.8)which can be regarded as a underlying second-order Stefan condition , see [10]. Hence, replacingit in (3 . = + ( ) ( · , + )Θ i − w ( · , − w ′ ( · , + ) Quuu ′ = + ( ) ( · , + )Θ i − w ( · , − w ′ ( · , + ) Q (cid:18) w ( · , i U ′′ U ′′ U ′′ + w ′ w ′ w ′ (cid:19) , which is a fully nonlinear parabolic equation in the space W written in a more abstract form: = + F ( ) ∈ Q ( D ( L )) . (3.9)and is going to be the subject of our attention. Note that Equation (3.9) is fully nonlinear sincethe function F depends on also through the limit at 0 + of . Moreover, the operator L issectorial in Q ( W ). Hence, we can take advantage of the theory of analytic semigroups to solveEquation (3.9). We refer the reader to [21, Chapter 4] for further details. Stability of the traveling wave solution
This section is devoted to the analysis of the stability of the traveling wave solution
UUU . Here,stability refers to orbital stability with asymptotic phase s ∞ . From now on, we focus on theasymptotic situation where the Lewis number, Le, is large and, in this respect, we use thenotation ε = 1 / Le to stand for a small perturbation parameter. Simultaneously, we assumethat Θ i is close to the burning temperature normalized at unity, which is physically relevant(see [3, Section 3.2, Fig. 5]). More specifically, we restrict Θ i to the domain < Θ i < m := Θ i / (1 − Θ i ) as the bifurcation parameter which runs inthe interval (2 , ∞ ), due to the choice of Θ i . With the above notation, A = m + εm and the dispersion relation D ( λ ; Θ i , Le) (see (2.16)) in Section 2 reads: D ε ( λ ; m ) = − (cid:0)p ε ( m + εm + λ ) + √ ελ (cid:1) × (cid:18) ε [ p ε ( m + εm + λ ) − √ λ (cid:19)(cid:18) − √ λ m (cid:19) + m + εm . (4.1)This section is split into two parts. First, we study the stability of the null solution of thefully nonlinear equation (3.9). Second, we turn our attention to the stability of the travelingwave.4.1. Stability of the null solution of (3.9).
To begin with, we recall that the spectrum ofthe part of L in W Q := Q ( W ) is the set (cid:0) −∞ , − (cid:3) ∪ P ∪ { λ ∈ C \ { } : D ε ( λ ; m ) = 0 } . As we will show, the roots of the dispersion relation D ε ( · ; m ) are finitely many. As a consequence,there is a gap between the spectrum of this operator and the imaginary axis (at least for ε smallenough). In view of the principle of linearized stability, the main step in the analysis of thestability of the null solution of Equation (3.9) is a deep insight in the solutions of the dispersionrelation. More precisely, we need to determine when they are all contained in the left halfplaneand when some of them lie in the right halfplane.The limit critical value m c = 6 will play an important role in the analysis hereafter. Theorem 4.1.
The following properties are satisfied. (i)
Let m ∈ (2 , m c ) be fixed. Then, there exists ε = ε ( m ) > such that, for ε ∈ (0 , ε ) ,the null solution of the fully nonlinear problem (3.9) is stable with respect to perturbationsbelonging to Q ( D ( L )) . (ii) Let m > m c be fixed. Then, there exists ε = ε ( m ) small enough such that, for ε ∈ (0 , ε ) ,the null solution of (3.9) is unstable with respect to perturbations belonging to Q ( D ( L )) .Proof. To begin with, we observe that the functions D ε ( · , m ) are holomorphic in C \ ( −∞ , − / limit dispersion relation D ( · , m ) defined by D ( λ ; m ) = −
12 [2( m + λ ) + 1 + √ λ ] (cid:18) − √ λ m (cid:19) + m = √ λ − m ) [4 λ − ( m − √ λ + m + 2] , TABILITY ANALYSIS AND HOPF BIFURCATION 15 as ε → + . The solutions of the equation D ( λ ; m ) = 0 are λ = 0, for all m , and the roots of thesecond-order polynomial 4 λ + (6 m − m ) λ + 2 m , whose real part is not less than − ( m + 2) / λ , = a ( m ) ± ib ( m ), where a ( m ) = ( m − m ) and b ( m ) = ( m − p | m − m | , if m ∈ (2 ,
8) and real solutions λ , = a ( m ) ± b ( m ) otherwise. Thecoefficient a ( m ) is negative whenever 2 < m < m >
6. It can be easily checkedthat Re( λ , ) ≥ − ( m + 2) / m ∈ (2 , ∞ ), so that λ , solve the equation D ( λ ; m ) = 0.In particular, there are two conjugate purely imaginary roots λ , = ±√ i at m = 6.We can now prove properties (i) and (ii).(i) Fix ρ > λ , and radius ρ is contained in { Re z < }\ ( −∞ , − ]. Hurwitz Theorem (see, e.g., [14, Chapter 7, Section 2]) and the aboveresults show that there exists ε > ε ∈ (0 , ε ), D ε ( λ ; m ) admits exactly twoconjugate complex roots λ , ( ε ) in the disk | λ − λ i | < ρ and λ i ( ε ) converges to λ i , as ε → i = 1 ,
2. Therefore, all the elements of the spectrum of the part of operator L in W Q have negative real parts, which implies that the operator norm of the restriction to W Q of theanalytic semigroup e τL generated by L , decays to zero with exponential rate as t → ∞ . Now,the nonlinear stability follows from applying a standard machinery: the solution of Equation(3.9), with initial datum in a small (enough) ball of Q ( D ( L )) centered at zero, is given bythe variation-of-constants-formula ( τ, · ) = e τL + Z τ e ( τ − s ) L F ( ( s, · )) ds, τ > . Applying the Banach fixed point theorem in the space X αω = (cid:26) ∈ C ([0 , ∞ ); WWW Q ) : sup σ ∈ (0 , σ α k k C α ([ σ, D ( L )) < ∞ : τ e ωτ ( τ, · ) ∈ C α ([1 , ∞ ); D ( L )) (cid:27) , endowed with the natural norm, where α is fixed in (0 ,
1) and ω is any positive number less thanthe real part of λ ( ε ), allows us to prove the existence and uniqueness of a solution of (3.9),defined in (0 , ∞ ) such that k ( τ, · ) k WWW + k ( τ, · ) k WWW ≤ Ce − ωτ k k D ( L ) for τ ∈ (0 , ∞ ) and somepositive constant C , which yields the claim. For further details see [21, Chapter 9].(ii) For m > m c , we use again Hurwitz Theorem to show that there exists ε = ε ( m ) > D ε ( λ, m ) = 0 admits a solution with positive real part if ε ∈ (0 , ε ).More precisely, it admits a couple of conjugate complex roots with positive real parts, if m < m = 8, and two real solutions if m >
8. For these values of ε , the restrictionof the semigroup e τL to W Q exhibits an exponential dichotomy, i.e., there exists a spectralprojection P + which allows to split W Q = P + ( W Q ) ⊕ ( I − P + )( W Q ). The semigroup e τL decays to zero with exponential rate when restricted to ( I − P )( W Q ), whereas the restrictionof e τL to P + ( W Q ) extends to a group which decays to zero with exponential rate as τ → −∞ .Again with a fixed point technique, we can prove the existence of a nontrivial backward solution zzz of the nonlinear equation (3.9), defined in ( −∞ ,
0) such that k zzz ( τ, · ) k WWW + k Lzzz ( τ, · ) k WWW ≤ C ω e ωτ for τ ∈ ( −∞ ,
0) and any ω positive and smaller than the minimum of the positive real parts ofthe roots of the dispersion relation. The sequence ( zzz n ) defined by zzz n = zzz ( − n, · ) vanishes in D ( L )as n → + ∞ and the solution n to (3.9) subject to the initial condition n (0 , · ) = zzz n exists atleast in the time domain [0 , n ], where it coincides with the function zzz ( · − n, · ). Thus, the normof k n k C ([0 ,n ]; WWW Q ) is positive and far way from zero, uniformly with respect to n ∈ N , whence the instability of the trivial solution of (3.9) follows. Again, we refer the reader to [21, Chapter9] for further results. (cid:3) Stability of the traveling wave.
We can now rewrite the results in Theorem 4.1 in termsof problem (2.1)-(2.3).
Theorem 4.2.
The following properties are satisfied. (i)
For m ∈ (2 , m c ) fixed, there exists ε = ε ( m ) > such that, for ε ∈ (0 , ε ) , the travelingwave solution UUU is orbitally stable with asymptotic phase s ∞ ( see (4.2)) , with respect toperturbations belonging to the weighted space D ( L ) . (ii) For m > m c fixed, there exists ε = ε ( m ) small enough such that, for ε ∈ (0 , ε ) , thetraveling wave UUU is unstable. with respect to perturbations belonging to the weighted space D ( L ) .Proof. (i) Let us fix ∈ Q ( D ( L )) with k k D ( L ) small enough, so that Theorem 4.1(i) can beapplied. Denote by the classical solution to Equation (3.9) which satisfies the initial condition (0 , · ) = = ( w , , w , ). Observe that, since p = Θ − i w ( · ,
0) (see Subsection 3.1) it followsthat the problem (3.1), subject to the initial condition uuu (0 , · ) = Θ − i w , UUU ′ + , admits a uniqueclassical solution ( uuu, s ), where uuu decreases to zero as τ → ∞ , with exponential rate. Moreover,using (3.8) it is immediate to check that s ( τ ) converges to s ∞ = Z ∞ ( ) ( τ, + )Θ i − w ( τ, − w ′ ( τ, + ) dτ, (4.2)as τ → ∞ (assuming for simplicity that g vanishes at τ = 0). We point out that s ∞ depends onthe initial condition.Coming back to problem (2.1)-(2.3) with initial condition XXX (0) = uuu + UUU and g (0) = 0, weeasily see that the solution XXX = (Θ , Φ) is defined by
XXX = pUUU ′ + + UUU = Θ − i w ( · , UUU ′ + + UUU ,g ( τ ) = τ + Z τ ( ) ( σ, + )Θ i − w ( σ, − w ′ ( σ, + ) dσ, τ ≥ . From this formula and the above result, the claim follows at once.(ii) The proof is similar to that of property (i) and, hence, it is left to the reader. (cid:3) Hopf bifurcation
This section is devoted to investigating the dynamics of the perturbation of the traveling wavein a neighborhood, say (6 − δ, δ ), of the limit critical value m c = 6 (see Section 4). As regardsparameter m , the situation is more complicated than in Section 4 when it was fixed. Now, thedispersion relation D ε ( λ ; m ) can be seen as a sequence of analytic functions parameterized by m .The main difficulty here is that Hurwitz Theorem does not a priori apply, particularly becauseof the lack of uniformity of D ε ( λ ; m ) with respect to ε and m . We especially find a properapproach to combining m with ε : we construct in Proposition 5.1 a sequence of critical values m c ( ε ) such that m c (0) = m c and apply Hurwitz Theorem to the sequence D ε ( λ, m c ( ε )). Thisproposition will be crucial for proving the existence of a Hopf bifurcation (see Theorem 5.3). TABILITY ANALYSIS AND HOPF BIFURCATION 17
Local analysis of the dispersion relation.
We look for the roots of the dispersionrelation , see (4.1), in a neighborhood of m c = 6 and of λ = ± i √
3, for ε > D ε ( λ ; m ) = 0 into a much more useful form. Replac-ing p ε ( m + εm + λ ) + √ ελ by 4 ε ( m + εm )( p ε ( m + εm + λ ) − √ ελ ) − with some straightforward algebra we obtain the equivalent equation √ ελ −
11 + m p ε ( m + εm + λ ) √ λ + 1 + εm m √ λ = ε λ m + 1 − ε. (5.1)If we denote by ζ the right-hand side of (5.1) and setΣ =1 + 4 ελ + 2 + 6 εm + 5 ε m + 4 ελ (1 + m ) (1 + 4 λ ) , Σ = 1 + 4 λ (1 + m ) (cid:20) (2 + 6 εm + 5 ε m + 4 ελ )(1 + 4 ελ ) + [1 + 4 ε ( m + εm + λ )](1 + εm ) (1 + m ) (1 + 4 λ ) (cid:21) , Σ = [1 + 4 ε ( m + εm + λ )](1 + εm ) (1 + m ) (1 + 4 ελ )(1 + 4 λ ) . Squaring both sides of (5.1) and rearranging terms we get the equation ζ − Σ = 2 √ λ m (cid:26) √ ελ [1 + εm − p ε ( m + εm + λ )] − εm m √ λ p ε ( m + εm + λ ) (cid:27) . (5.2)Squaring both sides of (5.2) and rearranging terms gives( ζ − Σ ) − = 8 √ ελ (1 + 4 λ )(1 + m ) (cid:20) [1 + 4 ε ( m + εm + λ )](1 + εm )1 + m √ λ − (1 + εm ) m p ε ( m + εm + λ ) √ λ − (1 + εm ) √ ελ p ε ( m + εm + λ ) (cid:21) . (5.3)Finally, squaring both sides of (5.3) and using (5.2), we conclude that [( ζ − Σ ) − ] − ζ = 0 or, equivalently, P ( λ ; m, ε ) = 0, where P ( · ; m, ε ) is a seventh-order polynomial (seeAppendix C for the expression of the coefficients of the polynomial).Finding the eigenvalues of P ( · ; m, ε ) is quite challenging. The Routh-Hurwitz criterion (see,e.g., [15, Chapter XV]) gives relevant information on the eigenvalues without computing themexplicitly, in particular whether the eigenvalues lie in the left halfplane Re λ <
0, by computingthe Hurwitz determinants ∆ j ( j = 1 , . . . ,
6) associated with P ( λ ; m, ε ). Unfortunately, ourdouble-squaring method produces spurious eigenvalues which render Routh-Hurwitz criterioninefficient. However, Orlando’s formula (see [15, Chapter XV, 7]), a generalization of the well-known property for the sum of the roots of a quadratic equation, establishes a relation betweenthe leading Hurwitz determinant ∆ and the sums of all different pairs of roots of P ( λ ; m, ε ). In particular, ∆ = 0 in the case when either 0 is a double eigenvalue (i.e., 0 is an eigenvaluewith algebraic multiplicity two) or two eigenvalues are purely imaginary and conjugate.The following one is the main result of this subsection. Proposition 5.1.
There exist ε > and δ > , and a unique function m c : (0 , ε ) → (6 − δ, δ ) with m c (0) = 6 , such that the polynomial e P ( λ ; ε ) := P ( λ ; m c ( ε ) , ε ) has exactly one pair of purelyimaginary roots ± iω ( ε ) , with ω ( ε ) > . Moreover, ω ( ε ) converges to √ as ε tends to . We first need a preliminary technical lemma:
Lemma 5.2.
There exist υ > and ε ∗ > such that, for all m in the interval [3 ,
7] ( to fixideas ) , ε ∈ (0 , ε ∗ ) and any purely imaginary root iυ of P ( · ; m, ε ) , with υ > , it holds that < υ < υ .Proof. We observe that, if iυ is a root of P ( · ; m, ε ), then, in particular, the imaginary part of P ( iυ ; m, ε ), i.e., the term − a υ + a υ − a υ + a υ vanishes.A straightforward computation (see Appendix C) reveals thatIm P ( iζ ; m, ε ) = − ε − ε ζ − ε ( m + 3 m + 2) ζ + O ( ε ) ζ − m − m − m − ζ + O ( ε ) ζ + a ζ, for every ζ >
0, where we denote by O ( ε k ) terms depending only on ε such that the ratio O ( ε k ) /ε k stays bounded and far away from zero for ε in a neighborhood of zero. Since m + 3 m + 2 and2 m − m − m − m ∈ [3 , ∞ ), we can estimate | Im P ( iζ ; m, ε ) | ≥ [8( m + 3 m + 2) − O ( ε )] εζ +[128(2 m − m − m − − O ( ε )] ζ − K | ζ | , where K := max {| a ( m, ε ) | : m ∈ [3 , , ε ∈ (0 , } . Hence, we can determine ε ∗ > | Im P ( iζ ; m, ε ) | ≥ m − m − m − ζ − K | ζ | , m ∈ [3 , , ε ∈ (0 , ε ∗ ) . (5.4)The right-hand side of (5.4) diverges to ∞ as ζ → + ∞ . From this it follows that there exists υ > | Im P ( iζ ; m, ε ) | > ζ > υ and this clearly implies that υ ≤ υ . (cid:3) Proof of Proposition 5.1.
We split the proof into two steps.
Step 1 . First, we prove the existence of a function m c with the properties listed in thestatement of the proposition. For this purpose, we consider the sixth-order Hurwitz deter-minant ∆ ( m, ε ) associated with the polynomial P ( λ ; m, ε ). It turns out that ∆ ( m, ε ) = ε m C e ∆ ( m, ε ) for some positive constant C . As ε → e ∆ ( · , ε ) converges to the function ∆ ,which is defined by∆ ( m ) = − m + 8 m + 97 m + 42 m − m − m − m − m + 19913 m + 31292 m − m − m − m − m − m + 4666 m + 2628 m + 500 m + 24 . Noticing that ∆ (6) = 0 and ddm ∆ (6) >
0, it then follows from the Implicit Function Theoremthat there exist ε ∈ (0 , ε ∗ ), with ε ∗ given by Lemma 5.2, δ > m c :(0 , ε ) → (6 − δ, δ ) with m c (0) = 6, such that e ∆ ( m c ( ε ) , ε ) = 0 and ∂∂m e ∆ ( m c ( ε ) , ε ) > ε ∈ (0 , ε ). Then, upon an application of Orlando formula, it follows that either 0 is a doubleroot of e P ( λ ; ε ) or there exists at least one pair ± ω ( ε ) i (with ω ( ε ) >
0) of purely imaginary
TABILITY ANALYSIS AND HOPF BIFURCATION 19 roots of e P ( λ ; ε ) for every ε ∈ (0 , ε ). The first case is ruled out, since 0 is not a root of e P ( λ ; ε ).Indeed, a ( m, ε ) converges to a positive limit as ε tends to 0. Step 2 . Next, we prove that ± ω ( ε ) i is the unique pair of purely imaginary roots of thepolynomial e P ( λ ; ε ) for every ε ∈ (0 , ε ). For this purpose, we begin by observing that e P ( · ; ε )converges, locally uniformly in C as ε →
0, to the fourth-order polynomial e P , defined by e P ( λ ) = − λ + 1)( λ − λ + 3) for every λ ∈ C . By Hurwitz Theorem, four roots of e P ( λ ; ε ),say λ ( ε ), λ ( ε ), λ ( ε ) and λ ( ε ) converge respectively to λ (0) = − , λ (0) = 12 , λ (0) = √ i and λ (0) = −√ i . More precisely, for r > λ i ( ε ) ( i = 1 , . . . ,
4) is simple in theball B ( λ i (0) , r ) for ε ∈ (0 , ε ) (up to replacing ε with a smaller value if needed). Assume bycontradiction that there exists a positive infinitesimal sequence { ε n } such that, for any n ∈ N ,( λ ( ε n ) , λ ( ε n )) is another pair of purely imaginary and conjugate roots of e P ( λ ; ε n ), differentfrom ± ω ( ε n ) i . By Lemma 5.2, ν ( ε n ) = | λ ( ε n ) | ≤ υ for every n ∈ N . Take a subsequence { ε n k } such that ν ( ε n k ) converges as k → ∞ . The local uniform convergence in C of e P ( · ; ε n )to e P implies that ν ( ε n k ) tends to √ k → ∞ . Since the limit is independent of the choiceof subsequence { ε n k } , we conclude that ν ( ε n ) converges to √ n → ∞ . Next, thanks toHurwitz Theorem and the fact that λ ( ε ), λ ( ε ) converge to √ i, −√ i respectively, the pair( λ ( ε n k ) , λ ( ε n k )) coincides with ( λ ( ε n k ) , λ ( ε n k )) in B ( √ i, r ) × B ( −√ i, r ). This contradictsthe fact that λ ( ε n k ) , λ ( ε n k ) are both simple. Up to replacing ε with a smaller value if needed,we have proved that ( ω ( ε ) i, − ω ( ε ) i ) is the unique pair of conjugate eigenvalues of e P ( · ; ε ) and λ ( ε ) = ω ( ε ) i for every ε ∈ (0 , ε ). The proof is now complete. (cid:3) Hopf bifurcation theorem.
For fixed 0 < ε < ε , ε and δ given by Proposition 5.1, letus consider the fully nonlinear problem (3.9), where now we find it convenient to write F ( ; m )instead of F ( ) to make much more explicit the dependence of the nonlinear term F on thebifurcation parameter m . According to Proposition 5.1, the bifurcation parameter m has acritical value m c ( ε ) ∈ (6 − δ, δ ). We intend to prove that a Hopf bifurcation occurs at m = m c ( ε ) if ε is small enough. For m close to m c ( ε ), we are going to locally parameterize m and by a parameter σ ∈ ( − σ , σ ). To emphasize this dependence, we will write e m ( σ ) and e ( · , · ; σ ). Theorem 5.3.
For any fixed α ∈ (0 , , there exists ˜ ε ∈ (0 , ε ) , such that whenever ε ∈ (0 , ˜ ε ) is fixed, the following properties are satisfied. (i) There exist σ > and smooth functions e m , ρ : ( − σ , σ ) → R , e : ( − σ , σ ) → C α ( R ; WWW ) ∩ C α ( R ; Q ( D ( L ))) , satisfying the conditions e m (0) = m c , ρ (0) = 1 and e ( · , · ; 0)= 0 . In addition, e ( · , · ; σ ) is not a constant if σ = 0 , and e ( · , · ; σ ) is a T ( σ ) -periodic so-lution of the equation e τ ( · , · ; σ ) = QL e ( · , · ; σ ) + F ( e ( · , · ; σ ); e m ( σ )) , τ ∈ R , where T ( σ ) = 2 πρ ( σ ) ω − and ω = ω ( ε ) is defined in Proposition . . (ii) There exists η such that if m ∈ (6 − δ , δ ) , ¯ ρ ∈ R and ∈ C α ( R ; WWW ) ∩ C α ( R ; Q ( D ( L ))) is a π ¯ ρω − -periodic solution of the equation τ = + F ( ; m ) such that k k C α ( R ; WWW ) + k k C α ( R ; Q ( D ( L ))) + | ¯ m | + | − ¯ ρ | ≤ η , then there exist σ ∈ ( − σ , σ ) and τ ∈ R such that m = e m ( σ ) , ¯ ρ = ρ ( σ ) and = e ( · + τ , · ; σ ) . Proof.
We split the proof into two steps.
Step 1.
Here, we prove that there exists ε > ± ω ( ε ) i are simple eigenvaluesof L (and, hence, of the part of L in WWW Q = Q ( WWW )) for every ε ∈ (0 , ε ] and there are noother eigenvalues on the imaginary axis, i.e., we prove that this operator satisfies the so-calledresonance condition.To begin with, let us prove that ± ω ( ε ) i are eigenvalues of L . In view of Theorem 2.3, we needto show that they are roots of the dispersion relation (4.1). For this purpose, we observe that thefunction e D ε := D ε ( · ; m c ( ε )) converges to e D locally uniformly in the strip { λ ∈ C : | Re λ | ≤ ℓ } (for ℓ small enough), where e D ( λ ) = − λ − √ λ λ ) √ λ + 1 + 4 λ ] , λ ∈ C . The function e D has just one pair of purely imaginary conjugate roots ±√ i . Hurwitz theoremshows that there exists r > B ( √ i, r ) contains exactly one root λ ( ε ) of e D ε for each ε small enough. By the proof of Proposition 5.1, we know that there exists r > ω ( ε ) i is the unique root of e P in the ball B ( √ i, r ). Clearly, λ ( ε ) is a root of the polynomial e P and, Hurwitz theorem also shows that λ ( ε ) converges to √ i as ε → + . Therefore, for ε small enough, both λ ( ε ) and ω ( ε ) i belong to B ( √ i, r ) and, hence, they do coincide. The sameargument shows that − ω ( ε ) i is also a root of e D ε . We have proved that there exists ε ≤ ε suchthat ω ( ε ) i and − ω ( ε ) i are both eigenvalues of L of every ε ∈ (0 , ε ]. In particular, ± ω ( ε ) i aresimple roots of the function e D ε and there are no other eigenvalues of L on the imaginary axis.To conclude that ± ω ( ε ) i are simple eigenvalues of L for each ε ∈ (0 , ε ], we just need to checkthat their geometric multiplicity is one. For this purpose, we observe that the proof of Theorem2.3 shows that the eigenfunctions associated with the eigenvalues ± ω ( ε ) i are given by u ( ξ ) = c e k ξ + AH ,λ (cid:18) e k ξ k − k − e k ξ − e k ξ k − k (cid:19) c , v ( ξ ) = c e k ξ , ξ < ,u ( ξ ) = c e k ξ , v ( ξ ) = c e k ξ , ξ ≥ k j = k j, ± ω ( ε ) i and the constants c , c , c and c are determined through the equation(2.23) (with λ = ± ω ( ε ) i ) where F = . . . = F = 0. Since the rank of the matrix in (2.23) isthree at λ = ± ω ( ε ) i , it follows at once that the geometric multiplicity of ± ω ( ε ) i is one. Step 2:
Now, we check the nontransversality condition. We begin by observing that, for every ε ∈ (0 , ε ], the function D ε is analytic with respect to λ and continuously differentiable withrespect to m in B ( √ i, r ) × (6 − δ, δ ), where r is such that the ball B ( √ i, r ) does not intersectthe half line ( −∞ , − / ω ( ε ) i, m c ( ε ))for ε small enough. In this respect, we need to show that the λ -partial derivative of D ε does notvanish at ( λ ( ε ) , m c ( ε )). To this aim, we observe thatlim ε → + ∂D ε ∂λ ( ω ( ε ) i, m c ( ε )) = ∂D ∂λ ( √ i,
6) = 5 √ i − . Therefore, there exists ε ≤ ε such that, if ε ∈ (0 , ε ], the λ -partial derivative of D ε at( ω ( ε ) i, m c ( ε )) does not vanish. Then, it follows from the Implicit Function Theorem that for each ε ∈ (0 , ε ], there exist δ ε > r ε < r and a C -mapping λ ε : ( m c ( ε ) − δ ε , m c ( ε )+ δ ε ) → B ( √ i, r ε ),such that D ε ( λ ε ( m ) , m ) = 0 for all m ∈ ( m c ( ε ) − δ ε , m c ( ε ) + δ ε ) and λ ε (6) = ω ( ε ) i . TABILITY ANALYSIS AND HOPF BIFURCATION 21
As a consequence, there are two branches of conjugate isolated and simple eigenvalues, λ ε ( m )and λ ε ( m ), which cross the imaginary axis respectively at ± ω ( ε ) i for m = m c ( ε ).It remains to determine the sign of the real part of the derivative of λ ε at m = m c ( ε ). Sincelim ε → + ∂λ ε ∂m ( m c ( ε )) = − (cid:18) ∂D ∂m ( √ i, (cid:19)(cid:18) ∂D ∂λ ( √ i, (cid:19) − = 34 + √ i there exists ε ≤ ε such that the real part of the derivative of λ ε is positive at m c ( ε ) for any ε ∈ (0 , ε ]. which completes the proof of Step 2.Applying [21, Theorem 9.3.3], the claims follow with ˜ ε = ε . (cid:3) Bifurcation from the traveling wave.
As in Subsection 4.2, we rewrite the resultsin Theorem 5.3 in terms of problem (2.1)-(2.3). As above, ε is fixed in (0 , ˜ ε ); therefore, thetraveling wave UUU depends only on m , which itself is parameterized by σ ∈ ( − σ , σ ). Accordingly,the traveling wave reads e UUU ( . ; σ ).The following theorem expresses that there exists a bifurcated branch bifurcating from thetraveling wave at the bifurcation point m c ( ε ). The proof can be obtained arguing as in the proofof Theorem 4.2. Hence, the details are skipped. Theorem 5.4.
For each σ ∈ ( − σ , σ ) , the problem (2.1) - (2.3) admit a non trivial solution ( e XXX ( · , · ; σ ) , e g ( · ; σ )) defined by: e XXX ( · , · ; σ ) = Θ − i e w ( · , σ ) e UUU ′ ( · ; σ ) + e ( · , · ; σ ) + e UUU ( · ; σ ) , e g ( τ ; σ ) = τ + τT ( σ ) Z T ( σ )0 ( L e ( r, · ; σ )) ( σ, + )Θ i − e w ( r, σ ) − e w ′ ( r, + ; σ ) dr + e h ( τ ; σ ) , τ ∈ R . where e XXX ( · , · ; 0) = e UUU ( . ; 0) , e is defined by Theorem . . The function e h ( · ; σ ) belongs to C α ( R ) .Moreover, e XXX ( · , · ; σ ) and e h ( · ; σ ) are periodic with period T ( σ ) = 2 πρ ( σ ) ω − . At the bifurcationpoint, the “virtual period” is T (0) = 2 πω − . We refer to, e.g., [20, 23] for solutions which are periodic modulo a linear growth.
Acknowledgments
L.L. greatly acknowledges the School of Mathematical Sciences of the University of Scienceand Technology of China for the warm hospitality during his visit. M.M.Z. would like to thankthe Department of Mathematical, Physical and Computer Sciences of the University of Parmafor the warm hospitality during her visit. The authors wish to thank Peter Gordon, CongwenLiu and Gregory I. Sivashinsky for fruitful discussions.
References [1] D. Addona, C.-M. Brauner, L. Lorenzi, W. Zhang,
Instabilities in a combustion model with two free interfaces (submitted). Available on ArXiv: arXiv:1807.02462.[2] R.K. Alexander, B.A. Fleishman,
Perturbation and bifurcation in a free boundary problem , J. DifferentialEquations (1982), 34-52.[3] I. Brailovsky, P.V. Gordon, L. Kagan, G.I. Sivashinsky, Diffusive-thermal instabilities in premixed flames:Stepwise ignition-temperature kinetics , Combust. Flame (2015), 101-124.[4] C.-M Brauner, P.V. Gordon, W. Zhang,
An ignition-temperature model with two free interfaces in premixedflames , Combust. Theory Model. (2016), 976-994. [5] C.-M. Brauner, L. Hu, L. Lorenzi, Asymptotic analysis in a gas-solid combustion model with pattern forma-tion , Chin. Ann. Math. Ser. B (2013), 65-88.[6] C.-M. Brauner, J. Hulshof, L. Lorenzi, Stability of the Travelling Wave in a D weakly nonlinear Stefanproblem , Kinet. Relat. Models (2009), 109-134.[7] C.-M. Brauner, J. Hulshof, L. Lorenzi, Rigorous derivation of the Kuramoto-Sivashinsky equation in a Dweakly nonlinear Stefan problem , Interfaces Free Bound. (2011), 73-103.[8] C.-M. Brauner, J. Hulshof, L. Lorenzi, G. Sivashinsky, A fully nonlinear equation for the flame front in aquasi-steady combustion model , Discrete Contin. Dyn. Syst. (2010), 1415-1446.[9] C.-M. Brauner, J. Hulshof, A. Lunardi, A general approach to stability in free boundary problems , J. Differ-ential Equations (2000), 16-48.[10] C.-M. Brauner, L. Lorenzi,
Local existence in free interface problems with underlying second-order Stefancondition , Rev. Roumaine Math. Pures Appl. (2018), 339-359.[11] C.-M. Brauner, L. Lorenzi, G.I. Sivashinsky, C.-J. Xu, On a strongly damped wave equation for the flamefront , Chin. Ann. Math. Ser. B (2010), 819-840.[12] C.-M. Brauner, A. Lunardi, C. Schmidt-Lain´e, Stability of travelling waves with interface conditions , Non-linear Anal. (1992), 465-484.[13] K.-C. Chang, The obstacle problem and partial differential equations with discontinuous nonlinearities , Comm.Pure Appl. Math. (1980), 117-146.[14] J.B. Conway, Functions of one complex variable, Springer-Verlag, 1978.[15] F.R. Gantmakher, The theory of matrices, Reprint of the 1959 translation. AMS Chelsea Publishing, Provi-dence, RI, 1998.[16] A. Ghazaryan, C.K.R.T. Jones, On the stability of high Lewis number combustion fronts , Discrete Contin.Dyn. Syst. (2009), 809-826.[17] A. Ghazaryan, Y. Latushkin, S. Schecter, Stability of traveling waves for a class of reaction-diffusion systemsthat arise in chemical reaction models , SIAM J. Math. Anal. (2010), 2434-2472.[18] L. Lorenzi, A free boundary problem stemmed from combustion theory. I. Existence, uniqueness and regularityresults , J. Math. Anal. Appl. (2002), 505-535.[19] L. Lorenzi,
A free boundary problem stemmed from combustion theory. II. Stability, instability and bifurcationresluts , J. Math. Anal. Appl. (2002), 131-160.[20] L. Lorenzi,
Bifurcation of codimension two in a combustion model , Adv. Math. Sci. Appl. (2004), 483-512.[21] A. Lunardi, Analytic Semigroups and Optimal Regularity in Parabolic Problems, Birkh¨auser, Basel, 1996.[22] B.J. Matkowsky, G.I. Sivashinsky, An asymptotic derivation of two models in flame theory associated withthe constant density approximation , SIAM J. Appl. Math. (1979), 686-699.[23] G. Namah, J.-M. Roquejoffre, Convergence to periodic fronts in a class of semilinear parabolic equations ,NoDEA Nonlinear Differential Equations Appl. (1997), 521-536.[24] D.H. Sattinger, On the stability of waves of nonlinear parabolic systems , Adv. Math. (1976), 312-355.[25] G.I. Sivashinsky, On flame propagation under condition of stoichiometry , SIAM J. Appl. Math. (1980),67-82. Appendix A. General solution to the equation λuuu − L uuu = fff Here, we collect the expression of the more general classical solution to the equation λuuu −L uuu = fff when fff = ( f , f ) is a continuous function and λ ∈ C . We preliminarily note that, sinceLe >
1, the equation k ,λ = k ,λ has no complex solutions λ . The equation k ,λ = k ,λ admitstwo complex conjugate solutions λ ∗ j = − A Le + ( − j i p A Le(Le − − , j = 1 , , (A.1)whose real part is negative. Moreover, the equation k ,λ = k ,λ admits no complex solutions.Also the equation k ,λ = k ,λ admits no solutions. Indeed, squaring twice the equation H ,λ + H ,λ = Le − λ ∗ and λ ∗ as solutions, which would imply that k ,λ = k ,λ . Obviously,this can not be the case. TABILITY ANALYSIS AND HOPF BIFURCATION 23
Setting uuu = ( u, v ), it turns out that, for any fff = ( f , f ) ∈ W and λ = { λ ∗ , λ ∗ } , the generalclassical solution to the equation λuuu − L uuu = fff is given by u ( ξ ) = (cid:18) c − H ,λ Z ξ ( Av ( s )+ f ( s )) e − k ,λ s ds (cid:19) e k ,λ ξ + (cid:18) c + 1 H ,λ Z ξ ( Av ( s )+ f ( s )) e − k ,λ s ds (cid:19) e k ,λ ξ = (cid:26) c − AH ,λ (cid:20) e ( k ,λ − k ,λ ) ξ − k ,λ − k ,λ c + e ( k ,λ − k ,λ ) ξ − k ,λ − k ,λ c (cid:21) + A Le H ,λ H ,λ (cid:20) e ( k ,λ − k ,λ ) ξ k ,λ − k ,λ Z ξ f ( s ) e − k ,λ s ds − e ( k ,λ − k ,λ ) ξ k ,λ − k ,λ Z ξ f ( s ) e − k ,λ s ds + k ,λ − k ,λ ( k ,λ − k ,λ )( k ,λ − k ,λ ) Z ξ f ( s ) e − k ,λ s ds (cid:21) − H ,λ Z ξ f ( s ) e − k ,λ s ds (cid:27) e k ,λ ξ + (cid:26) c + AH ,λ (cid:20) e ( k ,λ − k ,λ ) ξ − k ,λ − k ,λ c + e ( k ,λ − k ,λ ) ξ − k ,λ − k ,λ c (cid:21) + A Le H ,λ H ,λ (cid:20) e ( k ,λ − k ,λ ) ξ k ,λ − k ,λ Z ξ f ( s ) e − k ,λ s ds − e ( k ,λ − k ,λ ) ξ k ,λ − k ,λ Z ξ f ( s ) e − k ,λ s ds − k ,λ − k ,λ ( k ,λ − k ,λ )( k ,λ − k ,λ ) Z ξ f ( s ) e − k ,λ s ds (cid:21) + 1 H ,λ Z ξ f ( s ) e − k ,λ s ds (cid:27) e k ,λ ξ , (A.2) v ( ξ ) = (cid:18) c − Le H ,λ Z ξ f ( s ) e − k ,λ s ds (cid:19) e k ,λ ξ + (cid:18) c + Le H ,λ Z ξ f ( s ) e − k ,λ s ds (cid:19) e k ,λ ξ (A.3)for ξ < u ( ξ ) = (cid:18) c − H ,λ Z ξ f ( s ) e − k ,λ s ds (cid:19) e k ,λ ξ + (cid:18) c + 1 H ,λ Z ξ f ( s ) e − k ,λ s ds (cid:19) e k ,λ ξ , (A.4) v ( ξ ) = (cid:18) c − Le H ,λ Z ξ f ( s ) e − k ,λ s ds (cid:19) e k ,λ ξ + (cid:18) c + Le H ,λ Z ξ f ( s ) e − k ,λ s ds (cid:19) e k ,λ ξ , (A.5)for ξ ≥
0. Here, H i,λ ( i = 1 , ,
3) and k j,λ ( j = 1 , . . . ,
6) are defined by (2.14)-(2.15).If λ ∈ { λ ∗ , λ ∗ } , then k ,λ = k ,λ . Hence, in the definition of u for ξ <
0, the term − A ( e k ,λ − k ,λ − H ,λ ( k ,λ − k ,λ ) c + A Le H ,λ H ,λ (cid:20) e ( k ,λ − k ,λ ) ξ k ,λ − k ,λ Z ξ f ( s ) e − k ,λ s ds + k ,λ − k ,λ ( k ,λ − k ,λ )( k ,λ − k ,λ ) Z ξ f ( s ) e − k ,λ s ds (cid:21) should be replaced by − AH ,λ c ξ − A Le H ,λ H ,λ Z ξ ( s − ξ ) f ( s ) e − k ,λ s ds − A Le e ( k ,λ − k ,λ ) ξ H ,λ H ,λ ( k ,λ − k ,λ ) Z ξ f ( s ) e − k ,λ s ds. Appendix B. On the equality k ,λ = k ,λ Here, we show that the solutions of the equation k ,λ = k ,λ , i.e., the complex numbersgiven by (A.1), are not solutions of the dispersion relation. Since (Le + 4Le( A + λ ∗ j )) / =Le − λ ∗ j ) / , it is easy to see that D ( λ ∗ j , Θ i , Le) = 0 if and only if √ Le + 4 λ Le (cid:20) ± i √ A Le √ Le − i − (cid:18) − A LeLe − ± i √ A Le √ Le − (cid:19)(cid:21) =2 A Le − (cid:18) Le ± i √ A Le √ Le − (cid:19)(cid:20) ± i √ A Le √ Le − i − (cid:18) − A LeLe − ± i √ A Le √ Le − (cid:19)(cid:21) . (B.1)Squaring both sides of (B.1) and identifying real and imaginary parts of the so obtainedequation, after some long but straightforward computation we get the following system for Leand Θ i : Θ i + A Le+16(Θ i − A Le (Le − − A LeLe − i (Θ i − i − − Θ i Le+4(Θ i − A Le Le − , A Le(Θ i − i −
3) + (Le − i − i − Le + 2Θ i Le) = 0 . (B.2)First, we consider the second equation in (B.2). Replacing A with its value given by (1.8)and solving the so obtained equation with respect to Le, we obtain that there are no positivesolutions if Θ i = 1 / i ∈ (0 , \ { / } , then the equation has two real solutionsLe ± = 20Θ i − i − ± (400Θ i − i + 169Θ i + 14Θ i + 1) i − . A straightforward computation reveals that Le − > i ≤ /
2, whereas Le + > i ∈ (cid:0) Θ i , (cid:1) , where the value Θ i = (4 + √ / ≈ .
724 will play a significantrole hereafter.Now, we go back to the first equation in (B.2). Replacing A by its value, given by (1.8), andtaking Le = Le ± , we get the following equation p (Θ i ) = (signum(1 − i ))(1 − Θ i ) q (Θ i ) q i − i + 169Θ i + 14Θ i + 1 (B.3)for Θ i ∈ (0 , / ∪ (Θ i , p (Θ i ) = − i + 296896Θ i − i + 1041468Θ i − i + 218492Θ i − i − i − i − ,q (Θ i ) =1920Θ i − i + 19164Θ i − i + 2174Θ i + 251Θ i + 8 . It follows from the next lemma that (B.3) admits no solutions in the set (0 , / ∪ (Θ i ,
1) and,consequently, the solutions of k ,λ = k ,λ are not zeros of the dispersion relation. Lemma B.1.
Function q is positive in (0 , / and negative in (Θ i , . On the contrary, p isnegative in (0 , / and positive in (Θ i , .Proof. Since the proof is easy but rather technical, we sketch it. In what follows, we denoteby c positive constants which may vary from line to line. Similarly, by p k and q k we denotepolynomials of degree k , which may vary from estimate to estimate. TABILITY ANALYSIS AND HOPF BIFURCATION 25
We begin by considering the function q . For Θ i ∈ (0 , / q by 13364Θ i , so that q (Θ i ) > Θ i (13364Θ i − i +2174Θ i + 251) + 8 and the right-hand side of the previous inequality is not less than − i + 8,so that q is positive in (0 , / i ∈ (cid:0) Θ i , (cid:1) things are a bit trickier. Obviously, it suffices to prove that q is negativein (7 / , q (7 / <
0, we can estimate q < q − q (7 /
10) =: q in such an interval and q (Θ i ) 2) = c (1 − i ) p (Θ i ) for every Θ i ∈ (0 , / 2) and p is negative in (0 , / p (Θ i ) is negative for each Θ i ∈ (0 , / i ∈ (cid:0) Θ i , (cid:1) . Since Θ i > / 25 =: e Θ i , we can limit ourselves toproving that p is negative in ( e Θ i , p (Θ i )
75] then we estimate Θ ki ≤ · − k for k = 4 , 6, Θ ki ≥ · − k for k = 1 , , , 5, and conclude that p and, hence, p is negative in [0 . , . i ∈ (0 . , p (Θ i ) < p (Θ i ) − p (3 / 4) = c (4Θ i − p (Θ i ). Iterating this procedure, we concludethat p (Θ i ) < (4Θ i − p (Θ i ) and the polynomial p is negative in (0 . , i ∈ (0 . , . i = 0 . p (Θ i ) < ( p (Θ i ) − p (Θ i )) < c (Θ i − Θ i ) p (Θ i ) < c (Θ i − Θ i )( p (Θ i ) − p (Θ i ))= c (Θ i − Θ i ) p (Θ i ) ≤ c (Θ i − Θ i ) ( p (Θ i ) − p ( e Θ i )) = c (Θ i − e Θ i )(Θ i − Θ i ) p (Θ i )and observe that p is negative in [0 . , . p is negative in this interval as well.Summing up, we have proved that p is negative in ( e Θ i , 1) as claimed. This concludes theproof. (cid:3) Appendix C. The coefficients of the polynomial P ( · ; m, ε )We collect here the expression of the coefficients a i = a i ( m, ε ) ( i = 0 , , . . . , 7) of the poly-nomial P ( λ ; m, ε ) = a λ + a λ + a λ + a λ + a λ + a λ + a λ + a , which appears in Subsection 5.1. They are given by a = 2 ( ε − ε ; a = − ( ε − ε ) [(5 ε − ε + 1) m + 2( ε + 1) m + 4]; a = ( ε − ε ) (cid:2) ε (59 ε − ε + 17 ε − m + 4 ε (15 ε + 15 ε + 17 ε + 1) m + 4( ε + 2)( ε + 9 ε + 5 ε + 1) m − ε − ε − ε − m − ε − ε + 2) (cid:3) ; a = 2 (cid:2) − ε (5 ε − ε + ε + 7 ε − m − ε (59 ε − ε + 74 ε + 8 ε − m − ε (4 ε +27 ε +24 ε +37 ε +6 ε − m +4 ε (4 ε +20 ε − ε − ε − ε − m +4(9 ε +27 ε − ε − ε − ε − m +4( ε +17 ε − ε − ε − m − ε − (2 ε +1) (cid:3) ; a = 2 (cid:2) ε (9 ε − ( ε − m + 8 ε ( ε − ε + 5 ε − ε + 3) m +8 ε (21 ε − ε − ε − ε +27 ε − m − ε (34 ε +42 ε +113 ε +81 ε − ε − m − ε +96 ε +75 ε +176 ε +42 ε − ε − m +16 ε (6 ε − ε − ε − ε − m +16(29 ε − ε − ε − ε − m + 32( ε − ε + 14 ε + 3) m − ε − (cid:3) ; a = 2 m (cid:2) ε ( ε − (9 ε + ε − m + ( ε − ε )(38 ε + 46 ε − ε + 3) m + (36 ε + 33 ε − ε − ε + 54 ε − m − (8 ε − ε + 169 ε + 233 ε + 25 ε − ε − m − (60 ε + 110 ε + 320 ε + 129 ε − ε − m − ε +37 ε +41 ε +7 ε − m +2(4 ε − ε − ε − m +4(5 ε − ε − (cid:3) ; a = 2 m (cid:2) ε ( ε − ( ε + 1)(2 ε − m + ( ε − ε )(16 ε + 58 ε − ε − ε + 3) m + (16 ε + 72 ε − ε − ε + 42 ε + 17 ε − m + 2(20 ε − ε − ε − ε + 41 ε + 5) m − (2 ε + 1)(18 ε + 81 ε + 80 ε − m − ε + 31 ε + 20 ε − m − ε − ε + 1) m − − ε ) (cid:3) ; a = 2 ( m + m ) (cid:2) ε ( ε − ε − m + ε (2 ε − ε − ε − m + (2 ε − ε + 4 ε + 1) m − ε − ε + 1) m + 2(1 − ε ) (cid:3) . School of Mathematical Sciences, University of Science and Technology of China, Hefei 230026(China), and Institut de Math´ematiques de Bordeaux, Universit´e de Bordeaux, 33405 TalenceCedex (France).Dipartimento di Scienze Matematiche, Fisiche e Informatiche, Plesso di Matematica e Informati-ca, Universit`a di Parma, Parco Area delle Scienze 53/A, I-43124 Parma (Italy)School of Mathematical Sciences, University of Science and Technology of China, Hefei 230026(China). E-mail address : [email protected] E-mail address : [email protected] E-mail address ::