Traveling fronts bifurcating from stable layers in the presence of conservation laws

We study traveling waves bifurcating from stable standing layers in systems where a reaction-diffusion equation couples to a scalar conservation law. We prove the existence of weekly decaying traveling fronts that emerge in the presence of a weakly stable direction on a center manifold. Moreover, we show the existence of bifurcating traveling waves of constant mass. The main difficulty is to prove the smoothness of the ansatz in exponentially weighted spaces required to apply the Lyapunov-Schmidt methods. 1


Introduction
In this paper we prove the existence traveling fronts bifurcating from (standing) layers in a class of parabolic systems that couple a scalar conservation law with a scalar reaction-diffusion equation.Our focus here is on systems of the form posed on the real line x ∈ R. Here, the nonlinearities are smooth, a, b, g ∈ C 3 (R), and δ ∈ R is a real parameter.Moreover, a is uniformly elliptic, that is, a(u) ≥ a 0 > 0 for all u ∈ R.
The system (1.1) encompasses a variety of interesting model problems, such as phase-field systems and the Keller-Segel model for chemotaxis with its generalizations [2,5,6,9,14,15].From a theoretical point of view, (1.1) is particularly interesting as a system just slightly more complex than a scalar equation: the steady-state problem can be readily seen to reduce to a scalar equation after integrating the first equation for u as a function of v and substituting the result into the second equation.On the other hand, stability properties of such stationary solutions are slightly more complex than in the scalar case, where only monotone solutions are stable; see [19,20,21,22,23].Interesting dynamics of (1.1) are related to the fact that this system conserves mass ∫ u with suitable decay conditions at x = ±∞.This induces a constraint that, in some circumstances, stabilizes energetically unstable solutions [23], but, on the other hand, complicates the analysis by introducing a "neutral mode".Technically, the linearization at stationary solutions always possesses a neutral eigenfunction related to the constraint, creating in particular neutral essential spectrum for linearized operators.
The simplest example that combines the features mentioned here is the scalar Cahn-Hilliard equation, which conserves mass ∫ u.The steady-state equation reduces to the scalar equation u xx + u − u 3 = µ, with chemical potential µ, and can immediately be analyzed completely.In this most simple version, however, many features of the more complex class of equations (1.1) are not present, in particular the bifurcation to traveling waves that we are interested in here.
In previous work, we have analyzed periodic patterns, spikes (homoclinic), and layer (heteroclinic) stationary solutions of (1.1).While spikes and periodic solutions are always unstable on the real line, layers can be stable in some circumstances.For a layer solution (u * L (x), v * L (x)), we denote by (u ± L , v ± L ) its limits at x = ±∞.Typically, u + L ̸ = u − L , so that layers separate spatial regions with different "mass" u.Varying system parameters, one finds codimension-one situations where u In such a situation, necessarily u L (x) ≡ u ∞ L and b(u ∞ L ) = 0 since ∂ x (v * L ) ̸ = 0 and stability properties of layers change upon perturbing away from this degenerate point.We therefore consider (1.1) with ε-dependent cross-coupling term b(u) → b(u) + ε.
In fact, assuming δ > 0, we showed in [22] that the spectrum of the linearization at a layer solution is contained in Re λ ≤ 0 only if (u On the other hand, (1.1) possesses a Lyapunov function whenever b > 0 [19], so that the boundary of stability can also be seen as the boundary of gradient-like behavior.
Of course, changes of stability are expected to be accompanied by bifurcation of nontrivial solutions.Here, it turns out that the structure of (1.1) prevents a generic saddle-node of layer solutions and layer solutions can typically be continued through such a degenerate point.We emphasize that the change of stability is caused by an eigenvalue crossing from Re λ > 0 into the essential spectrum, Re λ ≤ 0, upon increasing (or decreasing) ε through 0. It is therefore not immediately clear what type of bifurcation to expect.
In different circumstances, crossing of a zero eigenvalue of the linearization at a standing layer induces bifurcation of traveling fronts; see [3,4,7,11,16].In this context, stationary layers are forced by a reflection symmetry in a reaction-diffusion system, and instabilities can occur in a non-variational context.Not surprisingly, given the symmetry, traveling fronts bifurcate in a pitchfork bifurcation with speed s ∼ ε, where ε denotes a typical bifurcation parameter.
In the present context, stationary layers are not enforced by symmetry and there is no a priori reason to expect pitchfork bifurcations.Arguing somewhat intuitively, layers separate regions of different mass concentrations u.Since mass transport is primarily diffusive rather than reactive, it cannot occur at positive, "ballistic" speed.Not surprisingly, traveling front solutions (u(x − st), v(x − st)) therefore have equal asymptotic mass u + = u − .This can be readily seen by integrating the first equation; see Lemma 2.1 for details.As a consequence, traveling fronts may limit on layers with u + = u − in limits where the speed vanishes.
The purpose of this paper is to analyze this somewhat vague and intuitive picture rigorously.Our approach is based on direct Lyapunov-Schmidt methods.We eliminate essential spectrum by the use of exponential weights, which induce negative Fredholm indices.Those can be compensated for by suitable far-field corrections.Complicating the situation compared to previous work [19,22] is the emergence of a weakly stable direction on a center-manifold.We incorporate this weakly stable direction by explicitly correcting in the far-field via a center-manifold solution.In order to preserve differentiability in this ansatz, we use scales of exponential weights related to the proof of smoothness of center manifolds and stable foliations.A similar approach was used in [12,13], albeit exploiting algebraic weights.
The remainder of this introduction will present the main results in a precise formulation.We denote by H k η (R) the Hilbert space of functions u for which u(•) cosh(η•) ∈ H k (R), the usual Hilbert space with square integrable derivatives up to order k; see the end of this introduction for a more formal list of notation used throughout.
Hypothesis 1.Throughout this paper we assume that the functions a, b and g are of class C 3 and that (1.1) has an exponentially convergent layer solution (u * L , v * L )(x) with limits at x = ±∞, u + L = u − L = u ∞ L for some specific δ = δ 0 ̸ = 0.
In the following corollary we compute the expansions of the real valued functions µ * ± and ω * ± and the first order derivatives of the traveling waves profiles u * ± and v * ± .Here and throughout the remainder of this paper, O(•) denotes the Landau symbol, encoding terms of higher order; see the end of this section for a formal definition.

Corollary 1.2. Assume Hypothesis 1 and let c
In addition, we have that where W * and Z * solve the equations Here the functions κ j : R → R, j = 1, 2, 3, are defined by Next, we point out that there exists a special class of traveling waves bifurcating from the standing layer with constant mass u ∞ L , under the additional, generic assumption that b ′ (u ∞ L ) ̸ = 0. Indeed, under this additional assumption it is easy to show that some of the traveling waves bifurcating from the standing layer (u ∞ L , v * L ) are particularly simple, having constant mass u ≡ μ(ε).The following theorem characterizes those waves, which should be thought of as a special subfamily of the two-parameter family of waves found in Theorem 1.1, with speed given as a function of the bifurcation parameter ε, rather than allowed as a free parameter.

Theorem 1.3. Assume Hypothesis 1 and suppose that
Then, there exists a locally unique family of traveling fronts with constant mass u, parameterized by ε ∈ (−ε 1 , ε 1 ), bifurcating from the standing layer.The traveling front profile is of the form The functions µ(•) and ψ(•; s, ε) vary smoothly in ε ∈ (−ε 1 , ε 1 ) as elements of R and H 2 η (R), respectively, for some η > 0 sufficiently small.The speed of the traveling waves is given by a function s = s(ε), which has the expansion In addition, we have that , where Y * solves the equation (1.7) Remark 1.4.The proof of Theorem 1.3 is rather standard.For completeness, we briefly explain the main idea.First, we notice that the u-equation of the traveling waves system associated to , this equation can be solved locally using the Implicit Function Theorem.That is, there exists ε 1 > 0 and a Next, we substitute u(x) ≡ µ(ε) into the second equation of (2.1) to obtain the equation To prove the existence result we need to show that there exists a smooth function s : (−ε 1 , ε 1 ) → R and a smoothly varying solution v(•; ε) of (1.8) for s = s(ε).Therefore, the proof reduces to the existence of a traveling wave solution in a standard bistable equation, which is omitted here; see [1] for the relevant arguments.
Together with existence, one is usually interested in the stability of solutions.While this question is of interest for the full two-parameter family found in Theorem 1.1, the analysis in this general setting is quite intricate.We focus here on the special subfamily found in Theorem 1.3.We also restrict to spectral stability of bifurcating traveling front solution.More precisely, we characterize the spectrum of the linearization of (1.1) in the moving frame at a traveling front (u( where (1.10) We point out, that for any ε ∈ (−ε 1 , ε 1 ) the linear operator L(ε) can be considered as a closed linear operator on exponentially weighted spaces L 2 ν (R, C 2 ) for any ν ∈ R. We recall the definition of the essential spectrum: we say that λ belongs to the essential spectrum of L, denoted σ ess (L), if L − λ is not a Fredholm operator with index zero.Theorem 1.5.Assume Hypothesis 1 and suppose b ′ (u ∞ L ) ̸ = 0.Then, the bifurcating traveling fronts obtained by Theorem 1.3 are spectrally stable.More precisely, the following assertions hold true: (ii) The linear operator L(ε) has no eigenvalue with positive real part.
Remark 1.6.We emphasize that our statement concerns spectral stability, only.Since the essential spectrum touches the imaginary axis, we expect nonlinear stability to be more subtle; see for instance [10] and references therein for nonlinear stability proofs in the case of a layer with a conservation law.
Outline: In Section 2, we prepare the proofs in a sequence of lemmas, in particular setting up a nonlinear equation with far-field corrections, analyzing Fredholm properties of the linearization, and establishing smoothness and thus preparing for Lyapunov-Schmidt reduction.Section 3 exploits those results to prove our main bifurcation result, Theorem 1.1 and expansions in Corollary 1.2.Section 4 contains the proof of Theorem 1.5.

Notations:
For an operator T on a Hilbert space X we use T * , dom(T ), ker T , im T , σ(T ), ρ(T ) and T |Y to denote the adjoint, domain, kernel, range, spectrum, resolvent set and the restriction of T on a subspace Y of X.We divide the spectrum of T into two disjoint sets: σ point (T ), the union of eigenvalues λ for which T − λ is Fredholm with index 0, and σ ess (T ) its complement in σ(T ).The Morse index of a hyperbolic matrix A, denoted i(A), is the dimension of its unstable subspace, which is the generalized eigenspace associated with all eigenvalues λ of A that have Re λ > 0. The usual Lebesgue spaces, the space of bounded uniformly continuous functions and the weighted Lebesgue spaces of vector valued functions are denoted by L p (R, C N ), BU C(R, C N ) and L p (R, C N ; ω(x)dx) respectively.If ω(x) = e 2η|x| for all x ∈ R we denote the L p -weighted space by L p η (R, C N ).Similarly, we define weighted Sobolev spaces W k,p η (R, C N ) and We start by looking for traveling front solutions of the form (u(x − st), v(x − st)) of the system (1.1) under the perturbation b(u We note that the first equation of this system can be integrated once, to obtain the equation where µ ∈ R is a constant.In the next lemma we obtain a necessary condition for any exponentially converging solution of (2.1).
Lemma 2.1.Assume that (u, v) satisfies (2.1) for a fixed s, ε ∈ R and that u, v are exponentially converging, that is, there exist c, η > 0, (2.3) Proof.We define the functions w ± : R → R by From (2.3), we infer that the functions w ± are well-defined, of class C 2 , and Using the second equation of the system (2.1), we obtain that w ′′ ± = −δ 0 u − g(v), which implies that w ′′ ± is bounded.Since, in addition, lim x→±∞ w ± (x) is finite, using Taylor's theorem we infer that lim x→±∞ w ′ ± (x) = 0. Indeed, Solving for u ′ in (2.2), we have that Since lim x→±∞ u(x) is finite, we infer from (2.5) and (2.6) that lim x→±∞ u ′ (x) is finite.From l'Hospital Theorem it follows that the last limit cannot be anything else but 0, that is (2.7) Passing to the limit as x → ±∞ in (2.6), we obtain that u + = u − = µ.Similarly, passing to the limit as x → ±∞ in the second equation of (2.1), we have that δ 0 µ + g(v ± ) = 0.
We are interested in finding traveling waves solutions of (1.1) whose profile at s = ε = 0 is a heteroclinic solution of the system In the next remark we collect a few results that follow immediately from Hypothesis 1; proofs are carried out in [22,Section 2].
Remark 2.2.The following assertions are true: From Lemma 2.1, we note that (2.1) is equivalent to the system (2.9) Next, we note that the equilibria v ± L are robust under small perturbations and the profile of the traveling front satisfies the conditions required by Lemma 2.1.

Remark 2.3. There exists
The conclusions of the remark follow by applying the Implicit Functions Theorem to the function Next, we note that (2.9) can be rewritten as the first order system (2.10) The Jacobian of the left-hand side of (2.10) at any of the equilibria described above is given by and ε ∈ R small enough the matrix J ± (µ, ε) has three algebraically simple eigenvalues given by λ = 0 and λ = ± T , which at s = 0 is simply given by the curve of equilibria δ 0 u + g(v) = 0, w = 0.The dynamics on the center manifold W ± (µ, s, ε) are hence determined by the dynamics of the u-component, ] . (2.11) We denote by u ± c (•; µ, ω, s, ε) the solution of (2.11) defined for ±x ≥ 1 with initial condition u ± c (±1; µ, ω, s, ε) = ω.This solution possesses the expansion 12) The other components of the solution of (2.10) on the center manifold W ± (µ, s, ε) satisfy the following expansions: Next, we collect some of the properties of the center manifold solutions (u ± c , v ± c , w ± c ) needed in the sequel.We are especially interested in the boundedness and growth properties of these solutions for µ, ω ∼ u ∞ L and s, ε ∼ 0. Remark 2.4.Differentiating in (2.11)-(2.13)we have that the following assertions hold true: (i) There exists ε 0 > 0 and s 0 > 0 such that the functions T is stable within the center manifold W ± (µ, s, ε), solutions that converge to the equilibrium converge with uniform exponential rate for x → ±∞.Therefore, when s ≥ 0, we use the ansatz (2.16) while for the case s ≤ 0, we use the ansatz (2.17)Here we used the definition Next, we substitute the ansatz (2.16)-(2.17)into (2.10),we obtain two equations where We note that the functions F ± are not of class C 1 .To overcome this issue, we formally expand the functions F ± as follows: (2.20) ) are defined by (2.21) while ) is the remainder, satisfying the condition (2.24) Next, we focus our attention on the properties of the functions from the decomposition (2.20).
Remark 2.5.Since the layer (u ∞ L , v * L ) converges exponentially at ±∞, we infer that there exists Lemma 2.6.Then, the functions Proof.Since the functions (u ± c , v ± c , w ± c ) are the center manifold solutions of (2.10) used in the ansatz (2.16)-(2.17),we conclude that F ± (0, 0, 0, µ, ω, s, ε) is a smooth function with compact support.Thus, we have that there exist The functions f ± j , j = 1, 2, 3, can be expressed in terms of the functions a, b, g, the center manifold solutions (u ± c , v ± c , w ± c ), the cut-off functions χ ± and the variables µ, ω, s, ε.From Remark 2.4(i) we have that In addition, from Remark 2.4(i) we conclude that for any q ∈ {µ, ω, s, ε} the partial derivatives ∂ q f ± j grow polynomially for x → ±∞.Since the center manifold solutions (u ± c , v ± c , w ± c ) are solutions of (2.10), it follows that for any q ∈ {µ, ω, s, ε} the partial derivatives ∂ x ∂ q f ± j grow polynomially for x → ±∞.We infer that for any θ > 0 there exits M θ > 0 such that 23), (2.27), (2.28), Remark 2.5 and Lemma A.1 we obtain that the functions Proof.Since the functions N ± are defined as the second order remainder in the decomposition (2.20), we have that ] .
(2.29)Here Z 2 is defined by be expressed in terms of the functions a, b, g, the center manifold solutions (u ± c , v ± c , w ± c ), the cut-off functions χ ± and the variables µ, ω, s, ε.Therefore, from Remark 2.4 we infer that In addition, we have that for any θ > 0 there exits M θ > 0 such that The lemma follows shortly from (2.29) and (2.32).
Next, we study the Fredholm properties of the linear operators L ± (µ, ω, s, ε).First, we note that where , where In the next lemma we show that the operator T is Fredholm and we compute its index.
In the next lemma we describe the kernels of T and of T * , the L 2 -adjoint of T .Here, we consider the operator T * as a closed, densely defined linear operator on . The following assertions are true: Proof.(i) To find ker T we solve the system (2.37) Solving the first equation of (2.37) we obtain that From the second and third equations of (2.37) we obtain the equation Since equation (2.38) is the variational equation of

.40)
Here A T denotes the transpose of the matrix A. This system is equivalent to (2.41) From the second and the third equation we obtain that that is, ϕ is an exponentially localized solution of (2.38), the variational equation of (2.39).It We conclude that φ satisfies the first order differential equation (2.43) We infer that there exists c ∈ R such that , proving the lemma.
Next, we introduce the functions )) defined by (2.44) In the next lemma we enumerate the properties of the linear operator Proof.First, we prove that the linear operators L ± (u ∞ L , u ∞ L , 0, 0) are onto.From (2.44) one readily checks that Thus, to prove that the linear operators L ± (u ∞ L , u ∞ L , 0, 0) are onto, it is enough to show that From Lemma 2.9(ii)-(iii) we have that where From (2.47) we conclude that (2.46) holds true provided that the matrix In order to evaluate the scalar products above, we use (2.22) and (2.48).We distinguish between the two cases: In this case the two vectors that span ker

.50)
We conclude that This case is similar to Case 1. From (2.48) we have that ker T * is spanned by (2.52) Again, we conclude that From (2.51) and (2.53) we obtain that (2.49) holds true, which implies that the linear operators L ± (u ∞ L , u ∞ L , 0, 0) are onto.To finish the proof of lemma we show that ker L ± (u ∞ L , u ∞ L , 0, 0) is one dimensional.Indeed, from (2.44), (2.46) and Lemma 2.9(i) one readily checks that Next, we are going to analyze the linear operators L ± (µ, ω, s, ε) in further detail.First, we note that )) are defined by

.56)
In the next lemma we prove the invertibility of the linear operators L ± (µ, ω, s, ε) with µ, ω ∼ u ∞ L and s, ε ∼ 0. To formulate this result we introduce the Hilbert spaces where the symbol V ⊖ W = Z refers to an arbitrary choice of a complement Z of W in V .
(2.73) From (2.66) and (2.72) it follows that the functions are differentiable and Analyzing the argument from (2.73) in detail, we infer that the operator ∂ y L † ± (y) in (2.75) is understood as a derivative (limit) in the B(H )) we have that this operator can be understood as the derivative (limit) in the B(H )) topology.However, from (2.66) we know that the derivative exists also in the B(H )) topology.Therefore we have that η is an extension of the operator ∂ y L † ± (y) .

Existence of weakly decaying fronts -proof of Theorem 1.1 and Corollary 1.2
We prove the main result of this paper on bifurcation of traveling waves from standing layers.
To start the proof of Corollary 1.2, we differentiate Γ ± with respect to q.From (2.24), ( .63) we have that Since ∂ (u,p) Γ ± (0, 0, 0) = diag(Id η , I 2 ) we conclude that Introducing the notation we obtain from (3.8) that Taking the L 2 -scalar product with U j , j = 1, 2, defined in (2.48), we conclude from (2.47) that Using the definition of the invertible matrices Q ± given in (2.49), it follows that Next, we evaluate the L 2 scalar products from the right hand side of (3.11).From (2.22) and (2.48) we obtain that (3.12) Multiplying the first equation by e −c∞v * L and integrating it, we obtain that Solving for Υ ± s in the second equation of (3.14) and substituting the result into the third equation of (3.14), we obtain from (3.16) that ) . (3.17) Next, we note that From the definition of the function κ 1 in (1.5), one readily checks that which proves that equation (3.18) has a unique solution denoted W * .It follows that To finish the proof of the corollary we compute (Φ ± ε , Ψ ± ε , Υ ± ε ) T .Using again the definition of T from (2.33)-(2.34)from (2.22) and (3.9) we obtain that ) . (3.21) We note that the system (3.21) is almost identical to (3.14), the only difference being the term where κ 2 and κ 3 are defined in (1.5).Solving for Υ ± ε in the second equation of (3.21) and substituting the result into the third equation of (3.21), we obtain from (3.22) that ) . (3.23) Similar to (3.18), we note that Ψ Moreover, from the definition of the function κ 2 and κ 3 in (1.5) we infer that which proves that equation (3.24) has a unique solution denoted Z * .Thus, we have that

Traveling fronts with constant mass profile
In this section we prove that the bifurcating traveling fronts with constant mass obtained by Theorem 1.3 are stable.Throughout this section we assume in addition that b ′ (u ∞ L ) ̸ = 0. First, we focus our attention on computing the essential spectrum.Using the results from Theorem 1.3 one can readily check that : where the matrix-valued functions D(•, •), M (•, •) and N (•, •) are continuous and bounded.D(x, s) is a diagonal matrix and thus, invertible, and the matrix-valued function D where Fix ε ∈ (−ε 1 , ε 1 ).Since D(•, ε) and D −1 (•, ε) are continuous and bounded, we infer that . Fredholm properties of the latter can be inferred from [8, Chapter 5, Thm A2]: ) From (4.3) and (4.4) we conclude that the essential spectrum of L(ε) consists of the union of three graphs where Since g ′ (v ± L ) < 0 and the functions g ′ , v ± and µ are continuous, it follows that we can choose . We note that λ ∈ σ ess (L(ε)) ∩ iR if and only λ = λ 0 (τ ; ε) and τ = 0, which implies that λ = 0, proving (i).
To start the proof of (ii), we note that the operator L(ε) has a lower-triangular block structure, which implies that the eigenvalue problem L(ε)(u, v) T = λ(u, v) T decouples as follows: ) is in divergence form, we have that L 11 (ε) has no eigenvalue with positive real part.Arguing for a contradiction, assume L(ε) 11 has an eigenvalue with positive real part.Them, there exists a solution u of the equation u t = L 11 (ε)u exponentially growing in time and exponentially localized in space, which implies that ∥u(t)∥ L 1 would be growing exponentially as t → ∞.Using the fact that ∫ u is conserved Since f ∈ C 2 (R × I) we conclude that for any x ∈ R there exists y j n (x), ỹj n (x) ∈ I such that for all x ∈ R, n ≥ 1 and j = 1, . . ., m. From (ii) we obtain that for all x ∈ R, n ≥ 1 and j = 1, . . ., m. From (A.2) and (A.3) and Lebesgue's Dominated Convergence Theorem we obtain that 1 tn (F (y +t n e j )−F (y)) → ∂ y j f (•, y) as n → ∞ in H 1 −γ (R), proving that the partial derivatives of F exist and ∂ y j F (y) = ∂ y j f (•, y) for all y ∈ I.
To finish the proof of lemma we have to prove that the partial derivatives of F are continuous.Let y ∈ I and {y n } n≥1 such that y n → y as n → ∞.We note that From (ii) for θ = γ 2 we have that In what follows we denote by c > 0 a generic positive constant.To prove the next lemma we recall the following result.(R) ≪ 1.From (A.9) we infer that the M α ′ •ϕ 0 , the operator of multiplication by Integrating with respect to x we infer that ) ) . (A.14) From (A.12) and (A.To finish the proof of lemma we have to prove that DF α is continuous on A p .We fix again ψ 0 ∈ A p and let ψ ∈ A p with ∥ψ − ψ 0 ∥ H 1 − γ 2 (R) ≪ 1.Since A p ⊂ {ψ ∈ L ∞ (R) : ∥ψ∥ ∞ ≤ M + p}, from (A.13) and Remark A.2 we estimate (DF α )(ψ) − (DF α )(ψ 0 ) (A.17) the first equation.Multiplying again the first equation by e −c∞v * L , integrating it and arguing as in (3.15)-(3.16),we conclude that

Acknowledgment: AS acknowledges support under grants NSF DMS-1311740 and DMS- 1612441. AP acknowledges support by through a Summer Research Grant by College of Arts and Science, Miami University. 2 Setting up the bifurcation problem -weakly decaying trav- eling fronts
are of class C 1 for any γ > 0. Assertion (2.25) follows from Remark 2.4(iii) and the definitions of the functions F ± and R ± in (2.19) and (2.23), respectively.