Stable manifolds for a class of degenerate evolution equations and exponential decay of kinetic shocks

We construct stable manifolds for a class of degenerate evolution equations including the steady Boltzmann equation, establishing in the process exponential decay of associated kinetic shock and boundary layers to their limiting equilibrium states. Our analysis is from a classical dynamical systems point of view, but with a number of interesting modifications to accomodate ill-posedness with respect to the Cauchy problem of the underlying evolution equation.


Introduction
In this paper we study decay rates at infinity of (possibly) large-amplitude relaxation shocks of kinetic-type relaxation systems on a general Hilbert space H, where A 0 , A are given (constant) bounded linear operator and Q is a bounded bilinear map. More generally, we study existence and properties of stable/unstable manifolds for a class of degenerate evolution equations arising through the study of such profiles. f = f (x, ξ) denoting density at spatial point x of particles with velocity ξ, which, after rescaling by ξ := 1 + |ξ| 2 , can be put in form (1.2), with A is equal to the operator of multiplication by the function ξ 1 / ξ and H an appropriate weighted L 2 space in the variable ξ. In the series of papers [22,23,24,35], Métivier, Texier and Zumbrun obtained existence results for a somewhat larger class of models in the case of shocks with small amplitude ε := u + − u − , in particular yielding exponential decay rates as τ = (x − st) → ±∞; see also the earlier [5,14] in the specific case of Boltzmann's equation. These results were obtained by fixed-point iteration on the whole line, using in an essential way the small-amplitude assumption to construct initial approximations based on the formal fluid-dynamical approximation.
Here, our interest is in treating large-amplitude profiles, without a priori information on the shape of the profile, by dynamical systems techniques that would apply also in the case of boundary layers where the solution is not necessarily defined on the whole line. Our larger goal is to develop dynamical systems tools analogous to those of [8,19,20,21,36,37,38,39,40,41], sufficient to treat 1-and multi-D stability by the techniques of those papers. See in particular the discussion of [38,Remark 4.2.1(4), p. 55], proposing a path toward stability of Boltzmann shock profiles. For this program, the proof of exponential decay rates and the establishment of a stable manifold theorem are essential first steps. For a corresponding center manifold theorem, see [30].
1.1. Assumptions. In [24,Section 4] it is shown that the Boltzman equation with hard-sphere potential can be recast as an equation of form (1.2) where the linear operator A and the nonlinearity Q satisfy the hypotheses (H1)-(H3) below; following [24], we assume these throughout.
Hypothesis (H1) The linear operator A is bounded and self-adjoint on the Hilbert space H. There exists V a proper, closed subspace of H with dim V ⊥ < ∞ and B : H × H → V is a bilinear, symmetric, continuous map such that Q(u) = B(u, u).
To understand the behavior of solutions of (1.3) near the equilibria u ± it is important to study the properties of the linearization of (1.3) along u ± . As pointed out in [24], in many special cases of (1.2), the linear operator Q ′ (u ± ) satisfies the Kawashima condition.
Hypothesis (H2) We say that a bounded linear operator T ∈ B(H) satisfies the Kawashima condition if (i) T is self-adjoint and ker T = V ⊥ ; (ii) There exists δ > 0 such that T |V ≤ −δI V ; (iii) There exists K ∈ B(V) skew-symmetric and γ > 0 such that Re (KA − T ) ≥ γI.
In the hypothesis below we summarize the conditions satisfied by the equilibria u ± .
In many interesting examples the linear operator A is an operator of multiplication by a non-zero function on a certain Hilbert space of functions. One of the goals of this paper is to apply our results to our leitmotif example, the Boltzman equation in the case of nonzero Euler characteristics. Therefore, to prove the main results in this paper we make the additional two hypotheses: Hypothesis (H4) The linear operator A is one-to-one.
Hypothesis (H5) The linear operator P V ⊥ A |V ⊥ is invertible on the finite dimensional space V ⊥ . Here P V ⊥ denotes the orthogonal projection onto V ⊥ .
1.2. Results. Our first task is to study the qualitative properties of the linearization of equation (1.3) about the equilibria u ± . Under Hypothesis (H1) and (H3) we prove that the linear operator L ± = A∂ τ − Q ′ (u ± ) is not invertible when considered as a linear operator on L 2 (R, H), see Remark 2.1 below. In many situations when one is interested in tracing the point spectrum of the linearization along some traveling wave profile, in the case when the linearization is not Fredholm on L 2 (R, H), it is of great interest to study the invertibility properties of the linearization L ± at the equilibria on weighted spaces L 2 η (R, H) with η = 0 small, see e.g. [28,29]. Making a change of variables, one can readily check that L ± is invertible on L 2 η (R, H) if and only if the linear operator L ± η = A∂ τ − Q ′ (u ± ) ∓ ηA is invertible on L 2 (R, H). Our invertibility result reads as follows: Theorem 1.1. Assume Hypotheses (H1) and (H3). Then, there exists η * > 0 such that L ± η = A∂ τ − Q ′ (u ± ) ∓ ηA is invertible with bounded inverse on L 2 (R, H) for all η ∈ (0, η * ).
To prove this theorem we use that the differential operator with constant coefficients L ± η is similar to an operator of multiplication on L 2 (R, H) by an operator-valued function, L ± η (ω) = 2πiωA − Q ′ (u ± ) ∓ ηA. First, we prove that L ± η (ω) is a Fredholm linear operator with index 0 and empty kernel, which allows us to infer that L ± η (ω) is invertible for any ω ∈ R. These steps require two key ingredients: 1) the eigenspaces of the self-adjoint, bounded, linear operator A have a trivial intersection with the orthogonal complement of the image of the bilinear map Q; 2) small perturbations of Q ′ (u ± ) by terms of the form sA with s small enough, are invertible with bounded inverse on the entire space H. Finally, we prove that the function ω → L ± η (ω) −1 is bounded on R, using the Kawashima condition.
Next, we show that equations Au ′ = Q ′ (u ± )u and Au ′ = Q ′ (u ± ) ± ηA u are equivalent to an equation of the form u ′ = Su. In each of these cases the linear operator S does not generate a C 0semigroup, but rather a bi-semigroup [3,12]; that is, the linear operator S has the decomposition S = S 1 ⊕ (−S 2 ) on a direct sum decomposition of the entire space H = H 1 ⊕ H 2 , where S j , j = 1, 2, generates a stable C 0 -semigroup on H j , j = 1, 2. We recall that the first order linear differential operators with constant coefficients ∂ τ − S is invertible on function spaces such as L 2 (R, H) if and only if the equation u ′ = Su has an exponential dichotomy on R. We note that for any u 0 ∈ V ⊥ the function u(τ ) = u 0 is a solution of equation Au ′ = Q ′ (u ± )u. Therefore, equation Au ′ = Q ′ (u ± )u does not have an exponential dichotomy on the entire space H; instead it exhibits an exponential dichotomy on a direct complement of the finite dimensional space V ⊥ . To prove this result, we reduce the equation by using the decomposition Indeed, if u is a solution of equation Au ′ = Q ′ (u ± )u, then the pair (h, v) defined by v = P V u and h = P V ⊥ u, satisfies the system We introduce the linear operators S η ± := A −1 Q ′ (u ± ) ± ηA , S r ± =Ã −1 Q ′ 22 (u ± ), whereÃ := (A 22 − A 21 A −1 11 A 12 ). These are densely defined, as we will show later (see Remark 3.2), and indeed generate bi-semigroups. Our dichotomy results are summarized in the following theorem. (i) The bi-semigroup generated by S η ± is exponentially stable on H; (ii) The bi-semigroup generated by S r ± is exponentially stable on V; (iii) Equation Au ′ = Q ′ (u ± ) ± ηA u has an exponential dichotomy on H; (iv) The linear space H can be decomposed into linear stable, center and unstable subspaces, H = V ⊥ ⊕ H s ± ⊕ H u ± such that for any u 0 ∈ V ⊥ the function u(τ ) = u 0 is a solution of (3.2) for any u 0 ∈ H s ± the solution of (3.2) on R + with u(0) = u 0 decays exponentially at + ∞ for any u 0 ∈ H u ± the solution of (3.2) on R − with u(0) = u 0 decays exponentially at − ∞. From this point, we turn our attention towards our main goal, the existence of stable/unstable manifolds of solutions of equation (1.3) near the equilibria u ± . The first step is to show that this equation can be reduced to an equation of the form where D(·, ·) is a bounded, bilinear map, E is negative definite and the linear operator Γ −1 E generates a stable bi-semigroup {T Γ,E s/u (τ )} τ ≥0 . To construct the manifolds, we need to analyze mild solutions of equation (1.5) on R ± , using the results from Theorem 1.1 and Theorem 1.2. Next, we apply, formally, the Fourier transform in (1.5) and then we solve for F u. In this way we obtain that mild solutions of equation (1.5) on say R + satisfy equation Here K Γ,E is the Fourier multiplier defined by the operator-valued function R Γ, where v 0 is a parameter in a dense subspace. Substituting in equation (1.6), we obtain that to construct our center-stable/unstable manifold it is enough to prove existence of solutions of equation ).
An important step of our construction is to find an appropriate subspace of parameters v 0 such that the trajectory T Γ,E s (·)P Γ,E s v 0 belongs to H 1 . To achieve this goal, we use that the linear operators Γ and E are self-adjoint and bounded, hence the bi-semigroup generator Γ −1 E is similar to a multiplication operator by a real valued function bounded from below on some L 2 space. See Section 4 for the details of this construction.
to H ∓ u/s with norm · H , respectively, that are locally invariant under the flow of equation Au ′ = Q(u) and uniquely determined by the property that Finally, we use this result to prove that a large class of profiles converging to equilibria u ± decay exponentially.
1.3. Applications to Boltzmann's equation. As mentioned above, the assumptions (H1)-(H5) of Section 1.1 are abstracted from, and satisfied by, the steady Boltzmann equation with hard sphere collision potential [23], after the change of coordinates f → ξ 1/2 f , Q → ξ 1/2 Q, and ξ := 1 + |ξ| 2 , with A = ξ 1 / ξ . Thus, Theorem 1.3 and Corollary 1.4 apply in particular to this fundamental case. More generally, they apply to Boltzmann's equation with any collision potential, or "cross-section," for which (H1)-(H5) are satisfied, with the crucial aspects being boundedness of the nonlinear collision operator as a bilinear map and spectral gap of the linearized collision operator. This includes in addition the hard cutoff potentials of Grad [5,23].
For the class of admissible cross-sections defined implicitly by (H1)-(H5), Corollary 1.4 implies exponential decay of H 1 loc Boltzmann shock or boundary layer profiles of arbitrary amplitude, so long as such profiles (i) exist, and (ii) are uniformly bounded, and (iii) converge to their endstates in the weak sense that u * − u ± lies in H 1 (R ± , H). This fundamental property, a cornerstone of the dynamical systems approach to stability developed for viscous shock and relaxation waves, had previously been established for kinetic shocks only in the small-amplitude limit [14,23].
However, we do not here establish existence of large-amplitude profiles; indeed, the "structure problem," as discussed by Truesdell, Ruggeri, Boillat, and others [4], of existence and structure of large-amplitude Boltzmann shocks, is one of the fundamental open problems in the theory.

1.4.
Discussion and open problems. In our analysis, the Hilbert structure of H and symmetry of A and Q ′ (u ± ) play an important role. This structure is related to existence of a convex entropy for system (1.2) [6]. In the case of the Boltzmann equation, it is related to increase of thermodynamical entropy and the Boltzmann H-Theorem; see [24, Notes on the proof of Proposition 3.5, point 2].
Further insight may be gained using the invertible coordinate transformation (−E) 1/2 and spectral decomposition of (−E) −1/2 Γ(−E) −1/2 to write the reduced system Γu ′ = Eu + D(u, u) of (1.5) formally as a family of scalar equations indexed by λ, where u λ is the coordinate of u associated with spectrum α λ , real, in the eigendecomposition of (−E) −1/2 Γ(−E) −1/2 , with u 2 H = |u λ | 2 dµ λ , where dµ denotes spectral measure associated with λ and α λ are bounded with an accumulation point at 0. In the first place, we see directly that (Γ∂ τ − E) is boundedly invertible on L 2 (R, H), with resolvent kernel given in u λ coordinates by the scalar resolvent kernel which is readily seen to be integrable with respect to τ , hence bounded coordinate-by-coordinate.
On the other hand, we see at the same time that the operator norm of the full kernel R with respect to L 2 (µ) is with a reverse inequality ≥ C 1 |τ − θ| holding for all (τ − θ) λ = cα λ , in particular for a sequence of values approaching zero, hence is unbounded. If it happens that α λ are continuous and cover all of R, then this holds everywhere. Likewise, a construction as in Example 4.7 shows that (A∂ τ − Q ′ (u ± )) is not boundedly invertible on L ∞ (R, H), motivating our choice of spaces H 1 (R, H), H 1 (R + , H) in the analysis, rather than the usual L ∞ (R, H). This implies, by contradiction, that the operator norm of the resolvent kernel is not only unbounded but non-integrable (cf. [15]). Using the finite-dimensional variation of constants formula scalar mode-by-scalar mode, we may, further, express (1.8) as the fixed point equation (1.11) where Π U and Π S denote projections onto the stable and unstable subspaces determined by sgnα λ . In (1.11) and (1.12) we denote the spectral components of Π S/U g by Π S/U g λ for any g ∈ L 2 (µ), slightly abusing the notation. From this we find after a brief calculation/integration by parts the derivative formula . This is quite different from the usual finite-dimensional ODE or dynamical systems scenario, and explains why we need to take some care in setting up the H 1 (R + , H) contraction formulation.
In particular, we find it necessary to parametrize not by Π S u(0) as is customary in the finitedimensional ODE case, but rather by Π S v 0 := Π S u λ (0) + D λ (u(0), u(0)) where u ′ λ (0) = α −1 λ v 0 . 1.4.1. Relation to previous work. The issue of noninvertibility of A for relaxation systems (1.2) originating from kinetic models and approximations was pointed out in [17,18,38]. This issue has been treated for finite-dimensional systems by Dressler and Yong [7] using singular perturbation techniques; see also [9,10,26]. These analyses concern the case that A has an eigenvalue at zero, and are of completely different character from the analysis carried out here of the case that A has essential spectrum at zero, i.e., an essential singularity; they are thus complementary to ours. In the present, semilinear setting, the case that A has a kernel is particularly simple, giving a constraint restricting solutions (under suitable nondegeneracy conditions) to a certain manifold, on which there holds a reduced relaxation system of standard, nondegenerate, type. For Boltzmann's equation (1.4), Liu and Yu [15] have investigated existence of invariant manifolds in a different (weighted L ∞ (x, ξ)) Banach space setting, using time-regularization and detailed pointwise bounds.
As noted earlier, the treatment of ill-posed equations u ′ − Su = f , and derivation of resolvent bounds via generalized exponential dichotomies, has been carried out in a variety of contexts [1,2,27,31,32,33,34]. The essential difference here is that the corresponding resolvent equation Γu ′ − Eu = f associated with (1.5) rewrites formally in the more singular form for which the singularity Γ −1 enters not only in the generator S but also in the source. Thus, the solution operator is not the one (∂ x − S) −1 deriving from (generalized) exponential dichotomies of the homogeneous flow, but the more singular (Γ∂ x − E) −1 of (1.6), or, formally, the unbounded multiple Γ −1 (∂ x − S) −1 . This explains the new features of unboundedness/nonintegrability of (the operator norm of) the resolvent, alluded to below (1.10).

Open problems.
Our H 1 analysis suggests a number of interesting open questions. The first regards smoothing properties of the profile problem. In the finite-dimensional evolution setting, regularity of solutions is limited only by regularity of coefficients; here, however, that is not true even at the linear level. Certainly, for further (e.g., stability) analysis, we require profiles of at least regularity H 1 , and likely higher. Our arguments can be modified to construct successively smaller stable manifolds in H s (R + ), any s ≥ 1, but for constructing profiles one would like to intersect unstable/stable manifolds that are as large as possible, thus in the weakest possible space. Hence, it is interesting to know, for H 1 profiles of (1.1) defined on the whole line, as opposed to decaying solutions defined on a half line, is further regularity enforced? For small-amplitude profiles, Kawashima-type energy estimates as in [23,24] show that the answer is "yes." A very interesting open question is whether one can find similar energy estimates in the large-amplitude case yielding a similar conclusion. For related analysis in the finite-dimensional case, see [21].
A second question in somewhat opposite direction is "what is the minimal regularity needed to enforce exponential decay?" Specifically, we have shown that solutions of (1.3) that are sufficiently small in H 1 (R + ) must decay pointwise at exponential rate; moreover, they lie on our constructed local H 1 stable manifold. What about solutions that are merely small in L ∞ ? A very interesting observation due to Fedja Nazarov [25] based on the indefinite Lyapunov functional relation u, Au ′ = u, Q ′ (u ± )u − o( u 2 H ) yields the L 2 -exponential decay result e β|·| u(·) ∈ L 2 (R + ) for some β > 0, hence (by interpolation) in any L p , 2 ≤ p < ∞. However, it is not clear what happens in the critical norm p = ∞; it would be very interesting to exhibit a counterexample or prove decay.
usual Sobolev space of X valued functions. The identity operator on a Banach space X is denoted by Id (or by Id X if its dependence on X needs to be stressed). The set of bounded linear operators from a Banach space X to itself is denoted by B(X). The set of bounded, Fredholm linear operators from a Banach space X to itself is denoted by F redh(X). For an operator T on a Hilbert space we use T * , dom(T ), ker T , imT , σ(T ), ρ(T ), R(λ, T ) = (λ − T ) −1 and T |Y to denote the adjoint, domain, kernel, range, spectrum, resolvent set, resolvent operator and the restriction of T to a subspace Y of X. If B : J → B(X) then M B denotes the operator of multiplication by B(·) in L p (J, X) or C b (J, X). If X 1 and X 2 are two subspaces of X, then X 1 ⊕ X 2 denotes their direct (but not necessarily orthogonal) sum. The Fourier transform of a Borel measure µ is defined by Acknowledgement. Thanks to Hari Bercovici, Ciprian Demeter, Narcisse Randrianantoanina and Alberto Torchinsky for helpful discussions regarding Fourier multipliers, and to Fedja Nazarov and Benjamin Jaye for stimulating conversations regarding the L ∞ decay problem.

Invertibility of the linearization at equilibria
In this section we study the invertibility property of the linearization of (1.3) at the equilibria u ± , given by the We can view the differential expression L ± as a densely defined, closed operator on the weighted Throughout this section we assume Hypotheses (H1) and (H3). First, we note that the operator L ± is not invertible on L 2 η (R, H) in the case when η = 0. Remark 2.1. Under assumptions (H1) and (H3), the linear operator L ± is not invertible on L 2 (R, H). Indeed, one can readily check that the linear operator L ± is invertible on L 2 (R, H) with bounded inverse if and only if the operator of multiplication by the continuous, operator valued function L ± (ω) = 2πiωA − Q ′ (u ± ) is invertible on L 2 (R, H) with bounded inverse. From the later we can infer that Q ′ (u ± ) are invertible on H, which contradicts Hypothesis (H3).
Next, we study the invertibility of L ± on the weighted space L 2 η (R, H) for some η small, η = 0. To achieve this goal it is enough to prove that the linear operators H). Again, we consider the differential expressions L ± η as closed linear operators on L 2 (R, H) with their maximal domain equal to the domain of L ± . To prove the desired invertibility properties, we start by proving some preliminary results satisfied by the linear operator A and the bilinear map Q. First, we recall the following result which follows from the Kawashima condition. Proof. Since the linear operator A is self-adjoint we have that σ(A) ⊂ R, which implies that ker(A − λI) = {0}, for any λ ∈ C \ R.
Therefore, to prove the lemma it is enough to prove it for the case when λ ∈ R. From Hypothesis (H2)(iii) it follows that for any h ∈ ker(A − λI) ∩ V ⊥ and λ ∈ R. Moreover, since the linear operator K is skew-symmetric, we obtain that Re Kh, h = 0. From (2.3) we infer that h = 0, proving the lemma.
The next lemma is a crucial preliminary result of this section used to prove the invertibility of the linear operators L ± η .
Lemma 2.3. Assume Hypothesis (H1) and that the linear operator T ∈ B(H) satisfies Hypothesis (H2). Then, there exists η 1 > 0 such that Proof. To begin the proof, we introduce P V and P V ⊥ , the orthogonal projections onto V and V ⊥ , respectively. Since the linear operators A and T satisfy Hypotheses (H1) and (H2), we obtain the following decompositions From Hypothesis (H1) we have that the linear operator A is self-adjoint, therefore, since the decomposition H = V ⊥ ⊕ V is orthogonal, we obtain that the linear operator A 11 is a self-adjoint linear operator on the finite dimensional vector space V ⊥ . It follows that the following orthogonal decomposition holds true: V ⊥ = ker A 11 ⊕ imA 11 . We conclude that there exist y ∈ ker A 11 and z ∈ imA 11 such that (2.7) h = y + z and y ⊥ z.
Moreover, from (2.6) and (2.7) we have that Furthermore, we claim that A 21 y ⊥ v. Since the linear operator A is self-adjoint, from the decomposition (2.5) we have that A * 21 = A 12 . Since, in addition, A * 11 = A 11 and y ∈ ker A 11 , from (2.8) we infer that y, z = 0, proving the claim. Taking scalar product with v in (2.9) we obtain that Since z = −(Ã 11 ) −1 A 12 v from (2.11) it follows that > 0 and we consider s ∈ (−η 1 , η 1 ) \ {0}. From (2.12) and Hypothesis (H1) we infer that (2.13) Remark 2.4. In many interesting applications the linear operator A is a multiplication operator by a scalar, non-zero function on a some function space. Therefore, it makes sense to assume that the linear operator A is one-to-one. This might simplify the proof of Lemma 2.3 a little bit. However, the main idea would be the same, so we choose to formulate the lemma under minimal conditions.
In the next lemma we prove the invertibility of the perturbation by a small multiple of A of Q ′ (u ± ).
Let η * > 0 be the positive constant defined in Lemma 2.5 and fix η ∈ (0, η * ). Taking Fourier Transform, one can readily check that the linear operators L ± η are similar on L 2 (R, H) to the operators of multiplication by the operator valued function L ± η : Since the linear operators A and Q ′ (u ± ) are self-adjoint on H we obtain that In the next couple of lemmas we prove several partial results needed to establish that the linear operator L ± η (ω) is invertible on H for any ω ∈ R and to prove that its inverse is uniformly bounded in ω.
Proof. To simplify the notation, we denote by E η ± = Q ′ (u ± ) ± ηA ∈ B(H). From Hypotheses (H1) and (H3) we have that the linear operators A and Q ′ (u ± ) are self-adjoint on H, which implies that the linear operators E η ± are self-adjoint on H. Moreover, η * > 0 can be chosen small enough such that for all v ∈ V and η ∈ (0, η * ). Here δ = min{δ + , δ − } > 0. The first step towards proving the lemma is to prove that for any ω ∈ R and u ∈ H. Using again the fact that P V is a self-adjoint projector on H, we conclude that Next, we use the estimate (2.22) to prove that im P V L ± η (ω)P V is a closed subspace of H for any ω ∈ R. Indeed, let us fix ω ∈ R, consider {u n } n≥1 a sequence of vectors in H and let y ∈ H be chosen such that y n := P V L ± η (ω)P V u n → y in H as n → ∞. Next, we define the sequence of vectors {v n } n≥1 by v n = P V u n and we note that for all n, m ≥ 1, which proves that the sequence {v n } n≥1 is a fundamental sequence in the closed subspace V. It follows that there exists v * ∈ V such that v n → v * in V as n → ∞. Passing to the limit, we conclude that Next, we prove that ker ) we obtain that P V u = 0, which implies that u ∈ V ⊥ . Thus, we can infer that (2.25) ker P V L ± η (ω)P V = V ⊥ for any ω ∈ R. Moreover, from (2.16) and (2.25) we conclude that (2.26) im for any ω ∈ R. Since V ⊥ is a finite dimensional subspace of H, from (2.24)-(2.26) we conclude that P V L ± η (ω)P V is a bounded linear operator on H with closed range of finite codimension and has a finite dimensional kernel, which implies that P V L ± η (ω)P V is a Fredholm operator on H for all ω ∈ R, proving claim (2.18). In addition, we note that Using again that V ⊥ is a finite dimensional subspace of H we have that the projection P V ⊥ is a finite rank operator, which implies that Finally, from (2.18), (2.27) and (2.28) we infer that L ± η (ω) is a Fredholm operator on H for any ω ∈ R, proving the lemma.
is invertible with bounded inverse on H for any ω ∈ R and η ∈ (0, η * ). Proof. As in the previous lemma, we use the notation E η ± = Q ′ (u ± ) ± ηA. To start the proof of (i), In the case when ω = 0, we denote by v = P V u and h = P V ⊥ u. Moreover, we have that From Hypotheses (H1) and (H3) we have that the linear operators A and E η ± are self-adjoint on H, which implies that Au, u , E η ± u, u ∈ R. Since ω = 0, from (2.29) it follows that Au, u = E η ± u, u = 0, which is equivalent to (2.30) Au, u = Q ′ (u ± )u, u = 0.
From Hypothesis (H3) we have that ker Q ′ (u ± ) = V ⊥ and im Q ′ (u ± ) = V, thus, we conclude that Q ′ (u ± ) = P V Q ′ (u ± )P V . Since the projection P V is self-adjoint, it follows that Hence, v = 0. From Hypothesis (H3) we obtain that Proof of (ii) First, we note that from (2.16) and (i) we have that From Lemma 2.6, (i) and (2.33) we have that the linear operator L ± η (ω) is Fredholm on H, with trivial kernel and its adjoint has trivial kernel, for any ω ∈ R and η ∈ (0, η * ). We infer that L ± η (ω) is invertible with bounded inverse on H for any ω ∈ R and η ∈ (0, η * ), proving the lemma.
Let γ = min{γ + , γ − } > 0, where γ ± are the constants from Hypothesis (H3). From (2.36) and Hypothesis (H3) it follows that for some c > 0, that depends only on η > 0. We infer that which implies that One can readily check that the operator valued function L ± η : R → B(H) is continuous on R. In addition, from Lemma 2.7(ii) we have that L ± η (ω) is invertible for any ω ∈ R. It follows that the operator valued function L ± η (·) −1 is continuous on R which implies that The lemma follows readily from (2.39) and (2.40).
Proof of Theorem 1.1. The theorem is a direct consequence of the last three lemmas. Indeed, since the linear operators L ± η are similar on L 2 (R, H) to the operators of multiplication by the operator valued function L ± η , from Lemma 2.7(ii) and Lemma 2.8 we have that the operator of multiplication by the operator valued function L ± η is invertible with bounded inverse. 13

Stable bi-semigroups and exponential dichotomies of the linearization
In this section we continue to study the properties of the linearization of equation (1.3) (with s = 0) at the equilibria u ± . Throughout this section we assume Hypotheses (H1), (H3), (H4) and (H5) and we recall the notation used in several lemmas from the previous section, E η ± = Q ′ (u ± )±ηA, with η ∈ (0, η * ). In particular, we prove that equation is equivalent to an equation of the form u ′ = Su, where the linear operator S generates a stable bi-semigroup. Here, η ∈ (0, η * ) and η * > 0 is defined in Section 2. We recall that a linear operator generates a bi-semigroup on a Banach or Hilbert space X, if there exist two closed subspaces X j , j = 1, 2, of X, invariant under S, such that X = X 1 ⊕X 2 and S |X 1 and −S |X 2 generate C 0 -semigroups on X j , j = 1, 2. We say that the bi-semigroup is stable if the two semigroups are stable. It is well-known, see e.g. [11,12], that the invertibility of L ± on L 2 (R, H) is equivalent to the exponential dichotomy on H of equations From Remark 2.1 we have that the linear operators L ± defined in (2.1) are not invertible on L 2 (R, H). Moreover, we note that for any u 0 ∈ V ⊥ the constant function u(τ ) = u 0 is a solution of equation (3.2). In this section we also prove that equations (3.2) exhibit an exponential dichotomy on a direct complement of the finite dimensional space V ⊥ . Using the decomposition We note that Hypothesis (H5) holds if and only if the linear operator A 11 is invertible on V ⊥ . Integrating the first equation, we obtain that solutions u = (h, v) that decay to 0 at ±∞, satisfy the conditions To prove that equations (3.2) have an exponential dichotomy on a complement of V ⊥ it is enough to show that equation where the linear operator S generates a stable bi-semigroup on V. In the following lemma we show that the reduced equation (3.6) can be treated similarly to equation (3.1) since their operator valued, constant coefficients share many properties. This lemma will allow us to treat these two equations in a unified way. (i) The linear operatorÃ : Proof. (i) Since the linear operator A is self-adjoint, from decomposition (3.3), we obtain that A * 11 = A 11 , A * 22 = A 22 and A * 12 = A 21 , which implies thatÃ is self-adjoint. To show thatÃ is oneto-one, we consider v ∈ kerÃ and denote by h = −A −1 11 A 12 v ∈ V ⊥ . Using again the decomposition (3.3), one can readily check that Assertion (ii) follows immediately from Hypothesis (H3) since Q ′ 22 (u ± ) ≤ −δ ± I V . To prove (iii) and (iv), we note that sinceÃ and Q ′ 22 (u ± ) are self-adjoint we have that . We obtain that It follows thatL (ω) is one-to-one and imL (ω) is closed in V for any ω ∈ R. Moreover, from (3.7) we infer that kerL (ω) * = {0} for any ω ∈ R, proving (iii). Assertion (iv) is a consequence of (3.8).
In what follows we focus our attention on equations of the form Hypothesis (S) We assume that bounded linear operators Γ, E ∈ B(X) satisfy the following conditions: (i) Γ is self-adjoint and one-to-one; (ii) The linear operator E is self-adjoint and invertible with bounded inverse on X; (iii) The linear operator 2πiωΓ − E is invertible on X for any ω ∈ R; (iv) sup ω∈R (2πiωΓ − E) −1 < ∞. is closed and densely defined. Indeed, since the linear operators Γ and E are bounded, one can readily check that the graph of S Γ,E is a closed subspace of X × X. Next, we show that the domain of S Γ,E is dense. We note that from Hypothesis (S)(i) we have that the linear operator Γ is selfadjoint and ker Γ = {0}, which proves that imΓ is a dense subspace of X. Fix u ∈ H. Since imΓ is dense in X it follows that there exists {w n } n≥1 a sequence of elements of X such that Γw n → Eu as n → ∞. We define the sequence {u n } n≥1 by u n = E −1 Γw n . We note that u n ∈ dom(S Γ,E ) for all n ≥ 1. Moreover, from Hypothesis (S) (ii) we have that the linear operator E is invertible with bounded inverse on X, therefore, we infer that u n → u as n → ∞, proving that the domain of S Γ,E is dense in X. Proof. Since Γ ∈ B(X) is self-adjoint and one-to-one, we have that the linear operator Γ is similar to a multiplication operator on an L 2 space. Therefore, there exists {X n } n≥1 a sequence of closed subspaces of X, invariant under Γ, such that X n , Γ |Xn is invertible with bounded inverse on X n for any n ≥ 1. Since E is invertible with bounded inverse, we conclude that for any n ≥ 1 the subspace G n = E −1 X n is closed in X. Moreover, from (3.11) we have that X = ∞ n=1 G n . Since EG n = X n and Γ −1 |Xn is bounded for any n ≥ 1 we have that n=1 G n we infer that S Γ,E is generating a bi-semigroup on X, proving the lemma.
To prove that the bi-semigroup generated by S Γ,E is stable, we use the methods from [12]. In the next lemma we prove that S Γ,E is hyperbolic and the basic estimates satisfied by the norm of the resolvent operators. To formulate the lemma, we introduce the operator valued function Lemma 3.4. Assume Hypothesis (S). Then, the following assertions hold true: Proof. Assertion (i) follows from Hypothesis (S)(iii) and the definition of S Γ,E in (3.10). Indeed, since ΓS Γ,E u = Eu for any u ∈ dom(S Γ,E ) one readily checks that for any u ∈ dom(S Γ,E ). Moreover, since the linear operators Γ and E are bounded, we have that for any u ∈ X. It follows that L Γ,E (ω) −1 Γu ∈ dom(S Γ,E ) and for any u ∈ X, proving (i).
Proof of (ii). Using the same argument used to prove the resolvent equation, one can show that for any ω 1 , ω 2 ∈ R. Setting ω 2 = 0 and multiplying the equation by E from the right we obtain that for any ω ∈ R. Assertion (ii) follows readily from Hypothesis(S)(iv) and (3.15).
Next, we define S Γ,E : dom(S Γ,E ) ⊂ L 2 (R, X) → L 2 (R, X) by (S Γ,E u)(τ ) = u ′ (τ ) − S Γ,E u(τ ) with its natural, maximal domain given by To prove that the linear operator S Γ,E generates a stable bi-semigroup, we need to prove that the linear operator S Γ,E defined in (3.16) is invertible on L 2 (R, X). To be able to study the properties of the operator S Γ,E , we start by describing its domain using a frequency domain formulation. To prove the next lemma, we recall a classical result on the commutative properties of the Fourier transform with closed linear operators.
Lemma 3.6. Assume Hypothesis (S). Then, the domain of S Γ,E is equal to the set of all u ∈ L 2 (R, X) for which there exists f ∈ L 2 (R, X) such that In the case the above equation holds then S Γ,E u = f .
Proof. To start the proof, we denote by D Γ,E the set of all u ∈ L 2 (R, X) for which there exists f ∈ L 2 (R, X) such that (3.17) holds. First, we show that dom( 16) we have that u ∈ H 1 (R, X), which implies that I R u ∈ L 2 (R, X). Here, we recall the notation I R as the identity function on R. In addition, we have that Γf = Γu ′ − Eu. It follows that From Hypothesis(S)(iii) we have that we can solve (3.18) for u. Moreover, from Lemma 3.4(i) we obtain that proving that dom(S Γ,E ) ⊆ D Γ,E . To prove the converse inclusion, we consider u ∈ dom(S Γ,E ) and f ∈ L 2 (R, X) such that (3.17) holds true. From Lemma 3.4(ii) we have that for almost all ω ∈ R. Since f, f ∈ L 2 (R, X) we conclude that I R u ∈ L 2 (R, X), which implies that u ∈ H 1 (R, X). Given that u and f satisfy equation (3.17), we obtain that u(ω) ∈ dom(S Γ,E ) for almost all ω ∈ R and Since u, u ′ − f ∈ L 2 (R, X) from Remark 3.5(ii) and (3.21) we infer that u(τ ) ∈ dom(S Γ,E ) for almost all τ ∈ R and u ′ − S Γ,E u = f ∈ L 2 (R, X). It follows that u ∈ dom(S Γ,E ) and S Γ,E u = f , proving that D Γ,E ⊆ dom(S Γ,E ) and the lemma. Proof. (i) First, we note that the R(2πiω, S Γ,E ) is a bounded linear operator on X for any ω ∈ R. Moreover, from Lemma 3.4(ii) we have that the operator valued function R(2πi·, S Γ,E ) is bounded on R. From (3.17) we infer that S Γ,E is a closed linear operator. To prove that the domain of the linear operator S Γ,E is dense in L 2 (R, X), we fix h ∈ dom(S Γ,E ) ⊥ and f ∈ L 2 (R, X).
Next, we define u f = F −1 M R(2πi·,S Γ,E ) f . One can readily check that u f ∈ dom(S Γ,E ). From (3.17) we obtain that Given that (3.22) holds true for any f ∈ L 2 (R, X) we conclude that M R(2πi·,S Γ,E ) * F h = 0. Since the R(2πiω, S Γ,E ) * is an one-to-one linear operator on X and the Fourier Transform is invertible on L 2 (R, X), we infer that h = 0, proving (i). Proof of (ii). From equation (3.17) we have that F u = M R(2πi·,S Γ,E ) F S Γ,E u for any u ∈ dom(S Γ,E ). Since the Fourier Transform is invertible on L 2 (R, X), and the operator valued function R(2πi·, S Γ,E ) is bounded on R, one can readily check that S Γ,E is invertible with bounded inverse and S −1 Γ,E = F −1 M R(2πi·,S Γ,E ) F . Assertion (iii) is a direct consequence of (ii) and the main result in [12].
To conclude this section, we use Theorem 3.7 to prove that equations (3.1) and (3.6) exhibit exponential dichotomy on H and V, respectively. We recall the definition of the linear operators . Proof of Theorem 1.2. From Hypothesis (H1) and the results proved in Section 2 we infer that the linear operators A and Q ′ (u ± ) ± ηA satisfy Hypothesis (S). Indeed, the conditions listed in Hypothesis (S)(i)-(iv) follow, respectively, from Hypothesis (H1), Lemma 2.5, Lemma 2.7(ii) and Lemma 2.8. Similarly, from Lemma 3.1 we have that the linear operatorsÃ = A 22 − A 21 A −1 11 A 12 and Q ′ 22 (u ± ) satisfy Hypothesis (S). Assertions (i) and (ii) follow directly from Theorem 3.7(iii). Assertion (iii) is a direct consequence of (i). Since equation (3.2) is equivalent to the system (3.5), we infer that assertion (iv) follows readily from (ii). Moreover, if we denote the stable/unstable spaces of equation (3.6) by V s/u ± , then the stable/unstable subspaces of equation (3.2) are given by the formula One can readily check that H = V ⊥ ⊕ H s ± ⊕ H u ± , proving the corollary.

Solutions of general steady relaxation systems
In this section we analyze the qualitative properties of solutions of the steady equation in H satisfying lim τ →±∞ u(τ ) = u ± and its linearization along u ± . In particular, we are interested in describing the smoothness properties of these solutions. Also, it is interesting to consider all of these equations on R ± , respectively. Making the change of variable w ± (τ ) = u(τ ) − u ± in (4.1) we obtain the equations (4.2) Aw ± τ (τ ) = 2B(u ± , w ± (τ )) + Q(w ± (τ )). Here, we recall that Q(u) = B(u, u) is bilinear, symmetric, continuous on H. Moreover, since the range of the bilinear map B is contained in V, denoting by h ± = P V ⊥ w ± and v ± = P V w ± , we obtain that equation (4.2) is equivalent to the system ). Integrating the first equation and using that lim τ →±∞ w ± (τ ) = 0, we obtain that solutions w ± = (h ± , v ± ) of (4.3) satisfy the condition h ± = −A −1 11 A 12 v ± . Plugging in the second of (4.3) we obtain that to prove the existence of a center-stable manifold around the equilibria u ± , it is enough to prove the existence of a center-stable manifold around equilibria P V u ± of equation . We note that it is especially important to study the solutions of equations (4.3) and (4.4) close to ±∞, therefore we focus our attention on their solutions on R ± , rather than the entire line. To study these equations we use the properties of stable bi-semigroups. We recall that if a linear operator S generates a stable bi-semigroup, then the linear operator −S generates a stable bi-semigroup as well. Making the change of variables τ → −τ in (4.4), we obtain an equation that can be handled in the same way as the original equation, as shown in [12,Section 4]. Therefore, to understand the limiting properties of solutions of equations (4.4) at ±∞, we need to understand the limiting properties in equations of the form Here the pair of bounded linear operators (Γ, E) on a Hilbert space X satisfies Hypothesis (S) and D : X × X → X is a bounded, bi-linear map. In addition, the linear operator E is negative definite.
In what follows we denote by R Γ,E : R → B(X) the operator valued function R Γ,E (ω) = (2πiωΓ− E) −1 . The bi-semigroup generated by S Γ,E = Γ −1 E on X is denoted by {T Γ,E s/u (τ )} τ ≥0 on X Γ,E s/u , the stable/unstable subspaces of X invariant under Γ −1 E. In addition, we denote by P Γ,E s/u the projections onto X Γ,E s/u parallel to X Γ,E u/s , associated to the decomposition X = X Γ,E s ⊕ X Γ,E u . In the sequel we use that the function R Γ,E satisfies A first step towards understanding equation (4.5), is to study the perturbed equation for some function f ∈ L 1 loc (R + , X) or f ∈ L 2 loc (R + , X). For a function g defined on a proper subset of R we keep the same notation g to denote its extension to R by 0. (iii) The function u is a smooth/mild solution of (4.7) on R + of (4.7) if u is a smooth/mild solution of (4.7) on [0, τ 1 ] for any τ 1 > 0.
Our definition of mild solutions follows [12, Section 2], where it is shown that the frequency domain reformulation given in (4.8) is much easier to handle than the classical approach where one defines the mild solution by simply integrating equation (4.7). We note that by taking Fourier transform in (4.7) and integrating by parts, it is easy to verify that smooth solutions of equation are also mild solutions.
Next, we define the linear operator K Γ,E : L 2 (R, X) → L 2 (R, X) by K Γ,E f = F −1 M R Γ,E F f . Here we recall that M R Γ,E denotes the multiplication operator on L 2 (R, X) by the operator valued function R Γ,E . From Hypothesis (S)(iv) we have that sup ω∈R R Γ,E (ω) < ∞, which proves that K Γ,E is well defined and bounded on L 2 (R, X).
To prove our results we need to understand the properties of the Fourier multiplier defined by K Γ,E . Our first goal in this section is to show that the definition we use for mild solutions of equation (4.7) can be seen as an extension of the classical variation of constants formula. To prove such a result we need to understand some of the smoothing properties of K Γ,E . Lemma 4.3. Assume Hypothesis (S). Then, we have that Γ(K Γ,E f )(·) ∈ C 0 (R, X) for any f ∈ L 2 (R, X).
Proof. Let f ∈ L 2 (R, X) and g = K Γ,E f . To prove the lemma we note that it is enough to show that Γg ∈ L 1 (R, X). Using the definition of K Γ,E we have that From Lemma 3.4 and the definition of E and its associated bi-semigroup, we have that (4.11) From (4.10) and (4.11) we conclude that Γg ∈ L 1 (R, X), proving the lemma. Now, we are ready to prove that (4.8) is a generalization of the variation of constants formula.  Proof. First, we prove that any mild solution u ∈ L 2 (R, X) of (4.7) satisfies equation (4.12), provided Γu ∈ C 0 (R + , X) . Since χ [0,τ 1 ] → χ [0,∞) simple as τ 1 → ∞, from the Lebesgue dominated Convergence theorem we obtain that uχ [0,τ 1 ] → u and f χ [0,τ 1 ] → f in L 2 (R + , X) ֒→ L 2 (R, X) as τ 1 → ∞. Since the linear operators F and K Γ,E are continuous on L 2 (R, X) we conclude that Moreover, since u is a solution of (4.7) on [0, τ 1 ] for all τ 1 > 0 we have that Since Γu ∈ C 0 (R + , X) from (4.14) it follows that From (4.13) and (4.15) we infer that Taking inverse Fourier transform, from Remark 4.2(ii) we obtain that Next, we prove that equality holds true in (4.16) for any τ ≥ 0. Indeed, multiplying the equation by Γ from the left, we obtain that Γu = ΓT Γ,E s (·)P Γ,E s u(0) + Γ(K Γ,E f )(·) almost everywhere on R + . Since Γu is continuous on R + , {T Γ,E s (τ )} τ ≥0 is a strongly continuous semigroup, and from Lemma 4.3 we have that Γ(K Γ,E )(·) is continuous, we infer that the equality Γu = ΓT Γ,E s (·)P Γ,E s u(0) + Γ(K Γ,E f )(·) holds everywhere on R + . Since Γ is one-to-one on X, by Hypothesis (S)(i), it follows that equation (4.12) holds true. 20 To finish the proof of lemma, we prove that under the assumption that f ∈ L 2 (R + , X), any function u satisfying equation (4.12) belongs to L 2 (R, X), Γu ∈ C 0 (R + , X) and is a mild solution of (4.7) on [0, τ 1 ] for any τ 1 > 0. Indeed, since {T Γ,E s (τ )} τ ≥0 is a stable C 0 -semigroup on X and K Γ,E is well-defined and bounded on L 2 (R, X), one can readily check that u belongs to L 2 (R + , X). Moreover, from Lemma 4.3 and (4.12) we conclude that Γu ∈ C 0 (R + , X).
To better understand the solutions of equation (4.5) we need to further study the Fourier multiplier K Γ,E : in particular we are interested in finding suitable subspaces of L 2 (R, X) that are invariant under K Γ,E . To achieve this, we first use that the linear operators Γ and E are selfadjoint to describe the structure of the bi-semigroup generator S Γ,E .

Lemma 4.5. Assume Hypothesis (S) and assume that the linear operator E is negative-definite. Then, the bi-semigroup generator S Γ,E is similar to an operator of multiplication by some realvalued, bounded from below, measurable function H
Proof. Since the linear operator E is bounded, self-adjoint, invertible and negative-definite, we have that E = (−E) 1 2 is a bounded, self-adjoint, invertible linear operator on X. One can readily check that Since the linear operator Γ and E are self-adjoint, we obtain that the linear operator ES Γ,E E −1 is self-adjoint. It follows that the linear operator ES Γ,E E −1 is unitarily equivalent to an operator of multiplication on some L 2 space. Therefore, there exists a measure space (Λ, µ), a real-valued function H Γ,E : Λ → R and a unitary, bounded, linear operator V Γ,E : µ)). To finish the proof of lemma, we need to prove that the function H Γ,E is bounded from below. From Theorem 3.7 we have that the linear operator S Γ,E generates a stable bi-semigroup. From (4.22) we conclude that there exists ν = ν(Γ, E) > 0 such that Since the spectrum of any multiplication operator on L 2 spaces is given by its essential range, we conclude that The representation (4.22) holds true when we modify the function H Γ,E on a set of µ-measure 0, therefore we can assume from now on that the inequality (4.24) is true for any λ ∈ Λ.
We note that the main idea used to obtain the representation (4.22) is based on the unitary equivalence of self-adjoint operators to multiplication operators, which is spectral in nature. Thus, it is natural to refer to functions in L 2 (Λ, µ) as spectral components of the generator S Γ,E . Lemma 4.6. Assume Hypothesis (S) and assume that the linear operator E is negative definite. Then, the following assertions hold true: (i) The linear operators U Γ,E and E satisfy the identity The operator-valued function R Γ,E has the following representation for any ω ∈ R, where R Γ,E : R → B(L 2 (Λ, µ)) is given by (iii) The bi-semigroup generated by the linear operator S Γ,E has the representation (4.29) , for any τ ≥ 0.
To better understand the properties of the Fourier multiplier K Γ,E , we analyze the properties of the Fourier multiplier K Γ,E : L 2 (R, L 2 (Λ, µ)) → L 2 (R, L 2 (Λ, µ)) defined by K Γ,E f = F −1 M R Γ,E F f . From Hypothesis (S)(iv) and (4.26) we conclude that which implies that the linear operator K Γ,E is well-defined and bounded on L 2 (R, L 2 (Λ, µ)). Moreover, we note that the operator-valued functions R Γ,E and R Γ,E are differentiable, and from Hypothesis (S)(iv) and (4.35), respectively, we have that From the Mikhlin-Hormander multiplier theorem we conclude that the Fourier multipliers K Γ,E and K Γ,E are well-defined and bounded on L p (R, X) and L p (R, L 2 (Λ, µ)), respectively, for any p ∈ (1, ∞). In the case of differential equations on finite dimensional spaces one proves the existence of the center-stable manifold by using a fixed point argument on L ∞ (R + , X) or C b (R + , X). In the example below, we prove that the Fourier multiplier K Γ,E (and thus, K Γ,E ) is not a bounded, linear operator on L ∞ (R + , X). Therefore, to prove the existence result of a center-stable manifold of solutions of equation (4.1), we need to find a proper subspace of L ∞ (R + , X) invariant under K Γ,E .
Next, we look for a pointwise convolution-like formula for the Fourier multiplier K Γ,E . In what follows we use the following notation: for any function f ∈ L ∞ (R + , L 2 (Λ, µ)) we use the simplified notation f (τ, y) for f (τ ) (y) for any τ ≥ 0 and λ ∈ Λ.
Next, we study if the Sobolev space H 1 (R + , X) is invariant under K Γ,E . First, we note that g ∈ H 1 (R + , X) if and only if g ∈ L 2 (R + , X) and the function ω → 2πiω g(ω) − g(0) belongs to L 2 (R, X). Here, we recall that if a function g is defined on a proper subset of R, we use the same 24 notation to denote its extension by 0 to the hole line. Using Hypothesis (S)(iv) we can show that the space H 1 (R, X) is invariant under K Γ,E . However, by using the same argument, we can check that for any x ∈ X \ dom(|S Γ,E | 1/2 ). Our goal is to prove the existence of an H 1 center-stable manifold by using a fixed point argument on equation (4.12) for f = D(u, u). Since H 1 (R + , X) is not invariant under K Γ,E , we need to rearrange the equation first by adding a correction term to K Γ,E . We parameterize equation (4.12) as follows: we look for solutions u satisfying u(0) = v 0 − E −1 f (0) for some v 0 to be chosen later. Therefore, equation (4.12) is equivalent to .
In what follows we prove that K mod Γ,E f ∈ H 1 (R + , X) for any f ∈ H 1 (R + , X) and compute its derivative.
Lemma 4.8. Assume Hypothesis (S). Then, K mod Γ,E f ∈ H 1 (R + , X) for any f ∈ H 1 (R + , X) and Proof. To prove our general result, we prove it for functions in a dense subset of H 1 (R + , X). We introduce the subspace H 1 Γ = {f : R + → X : there exists g ∈ H 1 (R + , X) such that f (τ ) = Γg(τ ) for any τ ≥ 0}. One can readily check that H 1 Γ is a dense subspace of H 1 (R + , X). To prove the lemma we need to compute K mod Γ,E f . Let g ∈ H 1 (R + , X) and f = Γg. From the definition of the Fourier multiplier K Γ,E , from Remark 4.2(ii) we obtain that for any ω ∈ R, which implies that K Γ,E f = G Γ,E * g. It follows that T Γ,E u (s)P Γ,E u g(s)ds for any τ < 0. (4.48) . Therefore, we obtain that (4.49) for any ω ∈ R. (4.50) From (4.48), (4.49) and (4.50) we conclude that for any ω ∈ R. In addition, from (4.48) we have that Since g ∈ H 1 (R + , X) we have that 2πiω g(ω) = g(0) + g ′ (ω) for any ω ∈ R. From (4.51) and (4.52) it follows that (4.53) , from (4.54) we obtain that From Using Remark 4.2(ii), (4.53) and (4.55) we infer that for any ω ∈ R. Arguing similarly as in (4.47), we have that K Γ,E f ′ = G Γ,E * g ′ , which implies that for any τ < 0. Since G Γ,E (τ )x = 0 for any τ ≥ 0 and any x ∈ X Γ,E u , from (4.56) and (4.57) we conclude that (4.58) 2πiω Since f ′ ∈ L 2 (R + , X) ֒→ L 2 (R, X) and K Γ,E is bounded on L 2 (R, X) from (4.58) we infer that K mod Γ,E f ∈ H 1 (R + , X) and (K mod Γ,E f ) ′ = (K Γ,E f ′ ) |R + for any f ∈ H 1 Γ . Next, we fix f ∈ H 1 (R + , X) and let {f n } n≥1 be a sequence of functions in H 1 Γ such that f n → f as n → ∞ in H 1 (R + , X). From Remark 4.2(i) we obtain that for any n ≥ 1. Since the Fourier multiplier K Γ,E is bounded on L 2 (R, X), from (4.59) we conclude To finish the proof of lemma, we need to prove (4.46). Indeed, since K Γ,E is bounded on L 2 (R, X) we have that for any f ∈ H 1 (R + , X), proving the lemma.
Next, we analyze the invariance properties of weighted spaces under K mod Γ,E . In particular, we are interested in checking whether the weighted Sobolev space To prove this result we need the following lemma: Lemma 4.10. Assume Hypothesis (S) and let ψ ∈ H 2 (R) be a smooth scalar function. Then, the following identity holds: Proof. To start the proof of lemma, we first justify that the left hand side of equation (4.64) is well defined. Indeed, since ψ ∈ H 2 (R) we have that ψ, ψ ′ ∈ L ∞ (R). Moreover, from Remark 4.2(i) it follows that G Γ,E and thus G * Γ,E are exponentially decaying, operator valued functions, which implies that G * Γ,E * f ∈ L 2 (R, X) for any f ∈ L 2 (R, X). We conclude that ψf + ψ ′ (G * Γ,E * f ) ∈ L 2 (R, X) for any f ∈ L 2 (R, X).
Remark 5.8. When proving results on existence of nonlinear center-stable/unstable manifolds in the case of differential equation on finite dimensional spaces, the manifolds can be expressed as graphs of C 1 functions from H s/u to H u/s ⊕ H c , where H s , H u and H c are the linear stable, unstable and center subspaces of the linearization along the equilibria at +∞. In our case we can prove a similar result assuming Hypothesis (S) and that the linear operator E is negative-definite. Indeed, from the definitions of the function J Γ,E or some other dense subspace of X, but we do not pursue this here.
Using Theorem 5.7 we can now prove the main result of this paper, the existence of centerstable and center-unstable manifolds of equation Au τ = Q(u) near the equilibria u ± at ±∞. We recall that the linear operator S r ± = A 22 − A 21 A −1 11 A 12 −1 Q ′ 22 (u ± ) generates a stable bisemigroup on V (Theorem 1.2(iv)) and that equation Au ′ = Q ′ (u ± )u has an exponential trichotomy on H, with stable/unstable subspaces denoted H s/u ± and center subspace V ⊥ (Theorem 1.2(vi)). Moreover, we recall that the pair (Γ, E) = (A 22 − A 21 A −1 11 A 12 , Q ′ 22 (u ± )) satisfies Hypothesis (S) by Theorem 1.2(ii) and Q ′ 22 (u ± ) is negative-definite by Hypothesis (H3). In this case we have that X Γ,E Proof of Theorem 1.3. Making the change of variables w ± = u−u ± in equation Au τ = Q(u) and denoting by h ± = P V ⊥ w ± and v = P V w ± , we obtain the systemÃv ± τ = Q ′ 22 (u ± )v ± + D(v ± , v ± ), h ± = −A −1 11 A 12 v ± , as shown in Section 4. HereÃ = A 22 − A 21 A −1 11 A 12 and the bilinear map , is bilinear and bounded on V. Since the linear operator S r ± generates a stable bi-semigroup on V, the theorem follows from Theorem 1.2, Theorem 5.7 and Remark 5.8.
Next, we show how we can use Theorem 5.7 to prove that solutions u * of equation Au τ = Q(u) that converge at ±∞ to equilibria u ± decay exponentially at ±∞.