Stable foliations near a traveling front for reaction diffusion systems

. We establish the existence of a stable foliation in the vicinity of a traveling front solution for systems of reaction diﬀusion equations in one space dimension that arise in the study of chemical kinetics and solid fuel combustion. In this way we complement the orbital stability results from earlier papers by A. Ghazaryan, S. Schecter and Y. Latushkin. The essential spectrum of the diﬀerential operator obtained by linearization at the front touches the imaginary axis. In spaces with exponential weights, one can shift the spectrum to the left. We study the nonlinear equation on the intersection of the unweighted and weighted spaces. Small translations of the front form a center unstable manifold. For each small translation we prove the existence of a stable manifold containing the translated front and show that the stable manifolds foliate a small ball centered at the front. ,


Introduction
Traveling fronts are solutions to partial differential equations which move with constant speed without changing their shapes and which are asymptotic to spatially constant steady states. Traveling fronts are important by many reasons and have intensively been studied. We refer to the books and review papers [F, VVV, X] and to more recent sources such as [KP,LW,RM1,RM2,RM3,Sa,TZKS] that contain further bibliography.
In this paper we study the dynamics in the vicinity of traveling fronts for a class of reaction diffusion equations in one space dimension. A typical example arising in combustion theory for solid fuels, cf. [BLR, GLSS, MS], is given by (1.1) where u, v ∈ R, ≥ 0, κ ∈ R, and g(v) = e −1/v for v > 0 and g(v) = 0 for v ≤ 0. These and more general equations covered by our hypotheses often appear in the work on chemical reaction models and in combustion models, see, e.g., [GSM, SMS, SKMS, T, VV]. In such systems the spectrum of the linearization of the equation at the front touches the imaginary axis, cf. [Sa, SS]. To shift the spectrum to the left, one employs exponentially weighted spaces. This idea goes back to [S] and [PW]. However, in weighted spaces one can lose the Lipschitz properties of the nonlinearity. We shall study reaction terms with a certain "product" structure as in (1.1) which allows one to overcome these difficulties. The investigation of this class of nonlinearities was initiated by A. Ghazaryan in [G] and then continued in [GLS1,GLSS,GLS2], see also the review paper [GLS3]. In particular, it was proved in [GLS2] that under appropriate assumptions on the nonlinearity the traveling front is orbitally stable; that is, any solution originating in a small vicinity of the front converges exponentially in the weighted norm to a translation of the front. In this paper we continue the work in [GLS2] now utilizing the theory of invariant manifolds, cf. [BJ, CHT, Lu]. We analyze the dynamics in greater detail by proving in Theorem 4.1 the existence of a stable foliation near the front. Specifically, we observe that the set of all translations of the front serves as a local central unstable manifold consisting of fixed points. Next, using the Lyapunov-Perron method, cf. e.g. [LL,LPS1,LPS2], we establish the existence and the fundamental properties of a locally invariant stable manifold going through each translation of the front. We also show that these manifolds foliate a small neighborhood of the front and therefore each point in the neighborhood belongs to one of them, cf. [BLZ, CHT]. Moreover, the orbit of the point converges to the translation of the front along the stable manifold as proved in [GLS2].
In the construction of the local stable manifolds we have to face the problem that the linearization enjoys good decay properties only in weighted spaces on which the nonlinearity is not locally Lipschitz. To overcome this difficulty, we use both the product structure of the nonlinearity (cf. Hypothesis 2.2) and additional decay properties of the linearization at the limit of the traveling front as ξ → −∞, see Lemmas 3.1 and 3.3.
The paper is organized as follows. In Section 2 we formulate our assumptions and prove several preliminary results. In Section 3 we study the Lyapunov-Perron operator whose fixed points define the stable manifolds. In Section 4 we formulate and prove our main result on the existence of the stable manifolds and discuss two examples.
Notation. Throughout the paper, |·| and ·, · are the Euclidean norm and the scalar product in R n . For a given map f : R m → R k , its differential with respect to y is written as ∂ y f : R m → B(R m , R k ). We let B(E, F) be the set of linear bounded operators between Banach spaces E and F, and abbreviate B(E) = B(E, E). We denote by C a generic constant that may change from one estimate to another, and use T to designate transposition. For a Banach space with norm · , we write B δ ( · ) for the closed ball of radius δ centered at 0.
We denote by E 0 with norm |·| 0 either the Sobolev space H 1 or the space BU C of bounded uniformly continuous functions on R with vector values, and by E α with norm |·| α the respective space of (exponentially) weighted functions, see (2.12). Let |·| β be the norm on the intersection space E β := E 0 ∩E α ; i.e., |y| β := max{|y| 0 , |y| α }.

The setting
We consider the system of reaction diffusion equations x) ∈ R n , and R : R n → R n is a C 3 function satisfying additional properties listed below. Passing in (2.1) to the moving coordinate frame ξ = x − ct and redenoting ξ again by x, we arrive at the nonlinear equation We discuss the wellposedness of this system in Remark 2.3.
Hypothesis 2.1. We assume that for some velocity c ∈ R the system (2.2) admits a stationary solution Y 0 ∈ C 3 (R); i.e, (2.1) possesses the traveling front solution . It is also required that Y 0 (x) converges to the end states Y ± as x → ±∞ exponentially; i.e., we can and will assume that Y − = 0 (and we then drop the tilde).
We further assume that the nonlinear term R in (2.1) and (2.2) has the following product structure.
Hypothesis 2.2. The nonlinear term R belongs to C 3 (R n , R n ). In appropriate variables Y = (U, V ) T with U ∈ R n1 , V ∈ R n2 and n 1 + n 2 = n, we have for a constant n 1 × n 1 matrix A 1 .
In other words, we suppose that where the maps R j belong to C 2 (R n , R nj ) and satisfy R j (U, 0) = 0 for j ∈ {1, 2} and U ∈ R n1 . Note that condition (2.4) yields R(0, 0) = R(Y − ) = 0. We also split Let q ∈ R. We write Y q (x) = Y 0 (x − q) for the shifted wave. Since (2.2) is translationally invariant, Y q is again a steady state solution of (2.2) and thus yields a traveling wave solution for (2.1). Linearizing (2.2) at Y q (that is, substituting Y q + Y instead of Y in (2.2)), we arrive at the equation Here, ∂ Y is the differential with respect to Y ∈ R n and the nonlinear term F q : R n → R n is written as The linearization of (2.2) at Y − = (0, 0) T is given by Below we impose conditions on L 0 at q = 0; i.e., on the linearization at the original traveling wave Y 0 . We further consider L q for |q| ≤ q 0 with some q 0 > 0, which will be fixed sufficiently small in the final theorem. The shifted wave Y q decays as in Hypothesis 2.2 with the same exponents ω ± and constants C only depending on q 0 . Assumption (2.4) also yields the formulas with the differential expressions (2.10) Remark 2.3. We consider the equations (2.2) and (2.5) on the space E 0 which is either the Sobolev space H 1 (R) n or the space of bounded uniformly continuous functions BU C(R) n . It is straightforward to check that the nonlinearites R and F q are Lipschitz on bounded subsets of E 0 . For the differential expressions L q and L − defined in (2.5) and (2.7), respectively, we denote by L q and L − the differential operators on E 0 on their natural domain D defined as follows. For E 0 = H 1 (R) n , the domain D of L q and of L − consists of the vector functions Y = (Y j ) n j=1 whose components Y j belong to H 3 (R) if d j > 0 and to H 2 (R) if d j = 0. For E 0 = BU C(R) n , we choose the domain analogously with H 3 (R) replaced by BU C 2 (R) and H 2 (R) replaced by BU C 1 (R), the spaces of differentiable functions which are bounded and have bounded, uniformly continuous derivatives. The operators L q and L − generate strongly continuous semigroups {T q (t)} t≥0 and {S(t)} t≥0 on E 0 , respectively, cf. e.g. [GLS1,§2.2].
Standard results then show the local wellposedness of (2.5) in E 0 for initial values y 0 in the domain of L q , where the (classical) solutions belong to C 1 ([0, t 0 ), E 0 ) and take values in D. They are given by Duhamel's formula See e.g. Theorems 6.1.4 and 6.1.6 in [P]. A function Y ∈ C([0, t 0 ), E 0 ) satisfying (2.11) is called a mild solution of (2.5). This concept is strictly weaker than that of classical solvability. We mostly work with mild solutions. Similar remarks apply to (2.2) and the differential expression D∂ xx + c∂ x equipped the same domain D. Approximating a given initial value y 0 ∈ E 0 in E 0 by functions in D, we see that all mild solutions of (2.2) are given by Y q + Y (t) where Y (t) solves (2.11). 3 Let α = (α − , α + ) ∈ R 2 . We say that γ α : R → R is a weight function of class α if γ α is C 2 , γ α (x) > 0 for all x ∈ R, and γ α (x) = e α−x for x ≤ −x 0 and γ α (x) = e α+x for x ≥ x 0 for some x 0 > 0. We shall always assume that 0 < α − < −ω − and 0 ≤ α + < ω + , (2.12) where ω ± are the exponents mentioned in (2.3). Given such a pair α = (α − , α + ), we introduce the weighted space The differential expressions L q , L − etc. equipped with their natural domains define operators in E α which are denoted by L q,α , L − α etc. (cf. Remark 2.3). On the spectrum of L 0,α , we impose the following assumptions.
Here the essential spectrum Sp ess (A) of a closed densely defined operator contains all points in the spectrum Sp(A) which are not isolated eigenvalues of finite algebraic multiplicity. We discuss various consequences of the above hypothesis which are important for our proofs.
Remark 2.5. We claim that assertions (a) and (b) in Hypothesis 2.4 are satisfied for E 0 = H 1 (R) n or E 0 = BU C(R) n if and only if they hold when E 0 is replaced by the space L 2 (R) n and E α by the space L 2 α (R) n of functions u with γ α u ∈ L 2 (R) which is endowed with the norm |u| α = |γ α u| L 2 . Indeed, the "if" part of the claim above is proved in Lemma 3.8 of [GLS2]. So we assume Hypothesis 2.4 for E 0 = H 1 (R) n or E 0 = BU C(R) n . Then assertion (a) of this hypothesis for E 0 = L 2 (R) n is true since the right-hand boundary of the essential spectra of L 0,α is the same for all three spaces by [GLS2,Lemma 3.5]. To show assertion (b) for E 0 = L 2 (R) n , we assume that L 0,α on L 2 α (R) n has an isolated eigenvalue λ of finite algebraic multiplicity with Re λ ≥ 0. By means of the isomorphism u(·) → γ(·)u(·) between L 2 α (R) n and L 2 (R) n we obtain a differential operatorL in L 2 (R) n which is similar to L 0,α in L 2 α (R) n , cf. [GLS2,Eqn. (3.2)], and hence possesses the unstable isolated eigenvalue λ, too. Palmer's Dichotomy Theorem in [Pa] says that the first order system corresponding to the second order eigenvalue problem forL admits exponential dichotomies on R − and R + . Arguing as in the proof of Lemma 3.8 of [GLS2], we see that the respective eigenfunction Z decays exponentially as x → ±∞. It thus belongs to BU C(R) n , and also to H 1 (R) n since Z x can be bounded by Z itself due to the eigenvalue equation, see (3.3) in [GLS2]. As a result,L in H 1 (R) n or BU C(R) n has the unstable eigenvalue λ and therefore also L 0,α in E α . Hypothesis 2.4 now shows that λ = 0, completing the proof of the claim. 3 Lemma 2.6. Assume that Hypothesis 2.4 holds. Then assertions (a) and (b) in Hypothesis 2.4 are satisfied by the operator L q,α instead of L 0,α and by the function Proof. The operators L q,α and L 0,α are similar via the transformation Y → Y (·−q) which also maps Y into Y q . The assertions then easily follow.
Remark 2.7. Assume Hypothesis 2.4. Lemma 2.6 says that λ = 0 is an isolated simple eigenvalue for L q,α . We let P c q denote the spectral projection for L q,α in E α onto ker L q,α = span{Y q }. Basic operator theory (see, e.g., [DL,Lemma 2.13]) yields that ran(I Eα − P c q ) = ker P c q = ran(L q,α ). Moreover, the one-dimensional projection P c q is given by for an element ζ q in ker L * q,α which is also one dimensional, cf. [K,Theorem IV.5.13]. As in the proof of Lemma 2.6, the operators L * q,α and L * 0,α are similar and therefore the norms of ζ q ∈ E * α are bounded uniformly for |q| ≤ q 0 . Also, in view of Lemma 3.3 in [GLS2], the first three derivatives of the shifted wave Y q are bounded by Ce −ω−ξ for ξ ≤ 0 and by Ce −ω+ξ for ξ ≥ 0 with ω ± from Hypothesis 2.2 and constants C only depending on q 0 . We conclude that As a consequence, P c q induces maps . We use the same notation P c q and P s q on all these spaces and their norms are uniformly bounded for |q| ≤ q 0 . The projections further satisfy for |p|, |q| ≤ q 0 and a constant independent of p and q. In fact, (2.5) yields and similarly for E 0 = H 1 . These estimates can easily be transferred to the resolvents on a sufficiently small circle around 0 which implies the claim (2.14). 3 Remark 2.8. To provide extra information, we now determine ζ q from (2.13) as a solution of a differential equation. Remark 2.5 yields that Hypothesis 2.4 is also true if we replace E 0 by L 2 (R). We first determine ζ q for the operator L * q,α acting on the dual L 2 α (R) * of the space L 2 α (R) of functions with the exponential weight γ α . We recall that the operator γ α : , is an isometric isomorphism. Moreover, L 2 α (R) * van be identified with L 2 space with the weight 1/γ α , where the duality map between L 2 α (R) and L 2 α (R) * is given by the usual (real) L 2 scalar product. Hence, the adjoint operator γ * α : L 2 (R) → L 2 α (R) * coincides with the multiplication operator by γ α . The We note that the dimension of the kernels is preserved by similarity and duality. The functional ζ q ∈ ker L * q,α from (2.13) is then represented by ζ q = γ α Z q where Z q ∈ L 2 (R) belongs to ker γ −1 α L * q,α γ α . In other words, Z q ∈ L 2 (R) is the unique (up to a normalization) solution on R of the differential equation γ −1 α L * q,α γ α Z q = 0. Reasoning as in the proof of Lemma 3.8 in [GLS2] or in Remark 2.5 we conclude that the solution Z q decays exponentially to zero as x → ±∞. Moreover, Z q is the translation Z 0 (· − q) of Z 0 , and the decay of the function Z q is thus uniform in q for |q| ≤ q 0 . Formula (2.13) now yields for all Y ∈ L 2 α (R), where Z q is the exponentially decaying function normalized such that π q (Y q ) = 1.
Finally, returning to the cases E 0 = H 1 (R) n or E 0 = BU C(R) n , we notice that π q (·) is a bounded functional on E α in both cases. Using also the decay properties of Y q recalled in Remark 2.7, we confirm from (2.17) once again that P c q is a bounded operator from both E β and E α into E β , with uniform constants for q ∈ [−q 0 , q 0 ]. 3 Remark 2.9. Let B q be multiplication operator induced by the matrix valued function B q (·) from (2.8). Lemma 8.2 of [GLS2] says that B q belongs to B(E α , E 0 ). As in assertion (3) of this lemma, one also sees that Inspecting the proofs, we see that the constants do not depend on p and q, but on q 0 . 3 The operators L q and L q,α generate strongly continuous semigroups on E 0 and E α , respectively, which are both denoted by {T q (t)} t≥0 , see e.g. [GLS1,§2.2]. By Lemma 2.6, there are numbers Lemma 3.13 of [GLS2] then yields the exponential decay see also [GLS1]. The constant C can be chosen unform in q because of the transformation used in the proof of Lemma 2.6. Also the operators L − and L − α generate strongly continuous semigroups on E 0 and E α , designated by {S(t)} t≥0 . Since the multiplication operator B q is bounded on these spaces, formula (2.8) implies the variation of constant formula The upper triangular structure of the operator L − indicated in (2.9) implies an analogous representation of the semigroup Here {S 1 (t)} t≥0 and {S 2 (t)} t≥0 are the semigroups generated by the operators L (1) and L (2) from (2.10), respectively. On these semigroups we impose the following assumptions.
Hypothesis 2.10. The strongly continuous semigroup {S 1 (t)} t≥0 is bounded and the semigroup {S 2 (t)} t≥0 is uniformly exponentially stable on E 0 ; that is, for some ρ > 0 and all t ≥ 0.

The Lyapunov-Perron operator
In this section we introduce the Lyapunov-Perron operator associated with the nonlinear equation (2.5) and show that it is a contraction of a small ball in a certain space of functions u : R → E 0 ∩ E α . First, we establish the main technical estimates for the nonlinearity F q : R n → R n defined in (2.6).
We finally compute Again we infer that This completes the proof of the lemma.
Remark 3.2. It follows from the observations after Remark 2.9 that the realization of L q in E β = E 0 ∩ E α generates a strongly continuous semigroup. The Lipschitz properties proved in the above lemma thus imply that the semilinear equation (2.5) is locally wellposed also in E β , cf. Remark 2.3. 3 We next establish basic properties of the Lyapunov-Perron operator Φ q (y, z 0 ) defined by where |q| ≤ q 0 and z 0 ∈ E 0 ∩ E α = E β satisfies for some δ 0 > 0. Here we use that P c q maps into the kernel of the generator of {T q (t)} t≥0 , see Remark 2.7, so that the semigroup is just the identity on the range of P c q and we can omit it in the second integral in (3.9). For a continuous map y = (u, v) : R → E β = E 0 ∩ E α we define the norms y ω,α = sup where ω > 0 is specified below and α = (α − , α + ) is given by (2.12). Let δ > 0. Then (B δ , · ) is the set of continuous functions y : R → E 0 ∩ E α such that y := max ( y ω,α , y 0,0 , v ω,0 ) ≤ δ. (3.11) We recall from Hypothesis 2.10 and (2.18) the exponential estimates for t ≥ 0. For technical reasons (see the next proof), if necessary we have to modify these exponents such that 0 < ω < ρ < ν.
We thus have shown that Φ q (·, z 0 ) leaves the ball B δ ( · ) invariant if first δ > 0 and then δ 0 > 0 are chosen small enough.

Stable manifolds
For a small q 0 > 0 and each q ∈ [−q 0 , q 0 ], we now construct a function φ q : ran(P s q ) → P c q whose graph contains Y q and it is a stable manifold M s q for the system (2.2). We further prove that the sets M s q satisfy the standard properties of stable manifolds and that they foliate a small neighborbood of Y 0 .
Let δ, δ 0 > 0 be sufficiently small and q 0 > 0. Take |q| ≤ q 0 and z 0 ∈ ran(P s q ) ∩ B δ0 (| · | β ). Lemma 3.3 then yields a unique function y q z0 : R + → E β which belongs to B δ ( · ) and is a fixed point of the Lyapunov-Perron operator Φ q (·, z 0 ); that is, for t ≥ 0. At t = 0 we obtain the identity (4.2) In this notation, we have y q z0 (0) = z 0 + φ q (z 0 ) so that y q z0 (0) belongs to the graph graph δ0 φ q of φ q over the small neighborhood ran(P s q ) ∩ B δ0 (| · | β ) of 0. Adding and substracting the term t 0 P c q F q (y q z0 (τ )) dτ , we deduce from (4.1) that the fixed point y = y q z0 of the Lyapunov-Perron operator satisfies the equation Consequently, y = y q z0 is the mild solution of the nonlinear equation (2.5) in B δ ( · ), and the function Y q +y solves (2.2) in the mild sense, cf. Remark 2.3. By uniqueness, y q 0 is the 0 function. Let alsoz 0 belong to ran(P s q ) ∩ B δ0 (| · | β ). Taking a sufficiently small δ > 0 in (3.14), we deduce the estimates For a number η > 0 to be fixed below, the stable manifold M s q is then defined by where |q| ≤ q 0 and Y 0 + B η (| · | β ) is the closed ball in E β = E α ∩ E 0 with radius η and centered at the original traveling wave Y 0 .
(i) Each M s q is a Lipschitz manifold in E β . If Y (0) ∈ M s q and the mild solution Y (t; Y (0)) of (2.2) belongs to Y 0 + B η (| · | β ) for some t ≥ 0, then Y (t; Y (0)) is contained in M s q .
(ii) For each Y (0) ∈ M s q there exists a solution Y (t; Y (0)) of (2.2) which exists for all t ≥ 0 and satisfies |Y (t; The following lemma will be used in the proof of Theorem 4.1. Recall the definition of the ball B δ ( · ) in (3.11).
(b) y can be extended to a global mild solution of (2.5) in B δ ( · ), and it is the fixed point y q z0 of the Lyapunov-Perron operator Φ q (·, z 0 ) from (3.9). (c) y can be extended to a global mild solution of (2.5) in B δ ( · ).
exists. Since y solves (4.3) and T q (t − τ ) is the identity on ran(P c q ), we can write using again Lemma 3.3 and (c). The definition of Φ q (y, z 0 ) in (3.9) then yields Due to (c) and (3.11), the functions y and Φ q (y, z 0 ) tend to 0 in E α as t → ∞, and hence z c = 0. Equation (4.6) thus implies y = Φ q (y, z 0 ) so that (a) is a consequence of the observations after (4.2).
Proof of Theorem 4.1. Recall from Remark 2.3 that all mild solutions of (2.2) are given by y + Y q for a mild solution y of (2.5).
Take t 0 > 0 such that It is easy to see that y(t 0 + ·) still belongs to B δ ( · ) and that it is the mild solution of (2.5) with the initial value y(t 0 ). Moreover, Remark 2.7 (in particular, that P s q ∈ B(E β )) and (2.15) yield if we choose η > 0 and q 0 small enough. (Note that the constants are uniform for q in compact intervals and independent of η.) Therefore, Using again (2.15), we can estimate Possibly decreasing η, q 0 > 0, we deduce from conditions (a)-(c) the inequality (3.11) for y and from Remark 2.7 the estimate |P s q (Y (0) − Y q )| β ≤ δ 0 . Lemma 4.2 now yields that y(0) ∈ graph δ0 φ q , proving (iii).
(iv). By Theorem 3.14 in [GLS2], we can fix a sufficiently small radius η > 0 such that for each point Y (0) in the ball Y 0 + B η (| · | β ; Y 0 ) there exists a shift q = q(Y (0)) such that the solution Y (·; Y (0)) of (2.2) satisfies properties (a)-(c) of item (ii). We remark that in Theorem 3.14 we can choose the same number δ > 0 as in the current proof and exponents 1 ν, ρ > ω which are different from our exponents ν and ρ in (3.13). Item (iii) then implies that Y (0) is contained in M s q . If Y (0) is also an element of M s q for someq ∈ [−q 0 , q 0 ], then the corresponding solution y would converge both to Y q and Yq as t → ∞, and so q =q. Hence, (iv) holds.
(v). Let |q|, |q| ≤ q 0 and z 0 ∈ B δ0 (| · | β ). The maps q → P c q ∈ B(E κ , E β ), q → P s q ∈ B(E κ ) and q → B q ∈ B(E α , E 0 ) are Lipschitz for κ ∈ {β, α} due to (2.14) and Remark 2.9. Lemma 3.7 of [GLS2] implies that γ α Y 0 and γ −1 α Y 0 are bounded. Using (2.6) and (2.15), we then deduce the estimate for all Y ∈ E κ and κ ∈ {0, α}. In view of (4.2), for (v) it remains to check that the map q → y q z0 =: y q is Lipschitz for · . Since y q is the fixed point, we infer from (3.9) the identity y q − yq = Φ q (y q , z 0 ) − Φq(y q , z 0 ) + Φq(y q , z 0 ) − Φq(yq, z 0 ). By (3.20), the second difference on the right hand side is bounded by Cδ y q − yq and can thus be absorbed by the left hand side possibly after decreasing δ > 0 once more. To control the other difference, we note that the bounded perturbation theorem and (2.16) imply that q → T q (t) ∈ B(E κ ) is Lipschitz for κ ∈ {0, α} and uniformly for t ≥ 0 in compact sets, see Corollary 3.1.3 of [P]. To extend this property to R + , let t ∈ (n, n + 1]. We write (1))Tq(k)P s q + Tq(t − n)(P s q − P s q )Tq(n)P s q . In the exponential decay estimate (2.18) for T q (t)P s q we can replace ν by a slightly larger number, see Lemma 3.13 of [GLS2]. This and the above mentioned facts lead to the inequality T q (t)P s q − Tq(t)P s q B(Eα) ≤ Ce −νt |q −q|, t ≥ 0.
Summing up, (v) is true.
To conclude, we briefly mention two motivating examples borrowed from [GLS3] that fit our setting. More details can be found in the papers [GLSS] and [GLS2], respectively. We stress, however, that for this type of examples the Hypotheses 2.1, 2.2 and 2.4 (a) can rigorously be verified not in all cases while the absence of the unstable eigenvalues required in Hypothesis 2.4 (b) is usually checked only numerically for certain ranges of the parameter values.
Example 4.3. Gasless combustion. A simple combustion model in one space dimension has been mentioned in the Introduction and is given by the system ∂ t u = ∂ xx u + vg(u), ∂ t v = −βvg(u), where g(u) = e − 1 u if u > 0 and g(u) = 0 if u ≤ 0. In this system, u is the temperature, v is the concentration of unburned fuel, g is the unit reaction rate, and β > 0 is a constant parameter. This system was a primary guiding example in [G,GLS1,GLSS,GLS2,GLS3]. One motivation for looking at this well-studied problem, in which the reactant does not diffuse, was heat-enhanced methods of oil recovery in which the reactant is coke contained in the rock formation, see [AY]. The value u = 0 represents the ignition temperature and is also taken to be the background temperature, at which the reaction does not take place.
Clearly, Hypothesis 2.2 is satisfied here. One looks for traveling waves Y 0 = (u 0 , v 0 ) such that Y − = (u − , 0) with u − > 0, Y + = (0, 1), and (u 0 (x), v 0 (x)) approaches these end states exponentially as x → ±∞. For each β > 0 there is a unique c > 0 for which such a wave exists, cf. [GLS3,§3.2]. This wave represents a combustion front that leaves behind of it high temperature u − = 1/β and no fuel, while in front of it temperature is 0 and there is fuel, with concentration normalized to 1. As discussed in Paragraph 3.2 of [GLS3], Hypothesis 2.10 is true and Hypothesis 2.4 can be verified (partly numerically) for small β > 0.
We note the lack of diffusion in the second equation which inspired the linear Lemma 3.13 in [GLS2] used to derive the exponential decay (2.18) from the spectral assumptions in Hypothesis (2.4), and the form of the nonlinear term in this and related problems which inspired the triangular and product structure of the nonlinearity in the current paper that follows from Hypothesis 2.2.
Example 4.4. Exothermic-endothermic chemical reactions. A model in which two chemical reactions occur at rates determined by temperature was studied in [SMS, SKMS], see also [GLS2]. One reaction is exothermic (produces heat), the other is endothermic (absorbs heat). The system reads ∂ t y 1 = ∂ xx y 1 + y 2 f 2 (y 1 ) − σy 3 f 3 (y 1 ), (4.8) ∂ t y 2 = d 2 ∂ xx y 2 − y 2 f 2 (y 1 ), (4.9) ∂ t y 3 = d 3 ∂ xx y 3 − τ y 3 f 3 (y 1 ). (4.10) Here y 1 is the temperature, y 2 is the quantity of an exothermic reactant, and y 3 is the quantity of an endothermic reactant. The parameters σ and τ are positive, and there are positive constants a i and b i such that f i (u) = a i e − b i u for u > 0 and f i (u) = 0 for u ≤ 0. In [SMS, SKMS] it is shown numerically that in certain parameter regimes there exist traveling wave solutions Y 0 of (4.8)-(4.10) with speed c > 0 and the end states Y − = (1 − σ τ , 0, 0) and Y + = (0, 1, 1). Moreover, both end states are approached at an exponential rate, the zero eigenvalue of the linearization is simple, and there are no other eigenvalues in the right half plane. A rigorous motivation for the existence of such traveling wave is also given in [GLS2,Section 9.2]. Assuming the existence of the traveling wave with these properties, the remaining hypotheses of the current paper are easy to verify.