LOCAL INTEGRAL MANIFOLDS FOR NONAUTONOMOUS AND ILL-POSED EQUATIONS WITH SECTORIALLY DICHOTOMOUS OPERATOR

. We show the existence and C k,γ smoothness of local integral manifolds at an equilibrium point for nonautonomous and ill-posed equations with sectorially dichotomous operator, provided that the nonlinearities are C k,γ smooth with respect to the state variable. C k,γ local unstable integral manifold follows from C k,γ local stable integral manifold by reversing time variable directly. As an application, an elliptic PDE in inﬁnite cylindrical domain is discussed.

1. Introduction. In this paper, we consider local integral manifolds for the nonautonomous differential equation dz(t) dt = Sz(t) + H(t, z(t)), t ∈ R, z(t) ∈ Z, where Z is a Banach space, S : D(S) ⊂ Z → Z is a linear, densely defined and hyperbolic bisectorial operator, and H(t, ·) : O → Z is a smooth map with H(t, 0) = 0 and D z H(t, 0) = 0. Here, O ⊂ Z α is an open set, Z 0 Z, Z α D(S α ) endows with the graph norm which satisfies the relation of continuous embedding D(S) → Z α → Z for α ∈ (0, 1), and S α is the α-fractional power of S. In contrast to Henry's work [9] which focuses on sectorial operator, the linear operator S in (1) has infinite spectrum on both sides of the imaginary axis, the Hille-Yosida theorem follows that it is not the infinitesimal generator of a certain analytic (semi)group on Z and so that the equation (1) does not generate a semi(flow) on Z for a given initial value. So we say that (1) is ill-posed on Z.
Some elliptic PDEs in infinite cylindrical domain can be treated as an ill-posed dynamical system (1) in the unbounded spatial variable. Over the past several decades, invariant manifold theory has been a powerful tool in analyzing the asymptotic behavior of bounded solutions of elliptic equations in the vicinity of an equilibrium, involving autonomous and nonautonomous cases. When S is not hyperbolic, (finite and infinite dimensional) center manifold theory has successfully applied to this kinds of elliptic problem since Kirchgässner [10] pioneered the "spatial dynamics" method-viewing the elliptic problem in infinite cylindrical domain as an ill-posed dynamical system in the unbounded spatial variable-in order to find nontrivial bounded small solutions for elliptic PDEs in infinite cylindrical domain(see Mielke [15][16][17][18][19] and de la Llave [13]). It has been shown that the ill-posed Cauchy problem may not be solvable for arbitrary initial condition, but some solutions can be defined with the initial values in a center manifold if S has the center subspace. When S is hyperbolic, ElBialy in [6] showed the existence of local Lipschitzian stable and unstable manifolds for the following autonomous ill-posed system , dy(t) dt = Ry(t) + g(z(t)), based on the existence of dichotomous mild solutions under the dichotomous initial conditions introduced by Latushkin and Layton [14]. Here X, Y and Z are all Banach spaces, (L, R) is the infinitesimal generator of a hyperbolic and strongly continuous bisemigroup ({e Lt } t≥0 , {e −Rt } t≥0 ), and the nonlinearity (f, g) are locally Lipschitz continuous in z. Especially, ElBialy introduced the so-called dichotomous flow which could recover the symmetry between the past and the future in evolution time such that just reversing time is need rather than providing two separate proofs for stable and unstable manifolds. Using the sufficient condition for a densely defined and hyperbolic bisectorial operator being sectorially dichotomous given by Deng and Xiao [4,Theorem 2.6], and under the dichotomous initial conditions ( z; t 1 , t 2 )|t 1 < t 2 , z = (x 1 , y 2 ) ∈ Z, x(t 1 ) = x 1 ∈ X , y(t 2 ) = y 2 ∈ Y , we can transform the ill-posed equation (1) into the coupled system: dx(t) dt = S + x(t) + F (t, z(t)), dy(t) dt = S − y(t) + G(t, z(t)), x(t 1 ) = x 1 , y(t 2 ) = y 2 , z = x + y ∈ X ⊕ Y = Z, t ∈ [t 1 , t 2 ]. (4) where S + := S| X and −S − := −S| Y are densely defined and sectorial operators on Banach spaces X and Y respectively, F (t, z(t)) = P + H(t, z(t)), G(t, z(t)) = P − H(t, z(t)) and P + (P − ) is the projection of Z onto X (Y) along Y (X ). We are interested in the asymptotic behavior of the dichotomous solution in a small neighborhood of the equilibrium z = 0, and present local stable and unstable integral manifold theorem for the ill-posed system (4). The construction of local stable integral manifold is carried out by the Liapunov-Perron method, and the C k,γ smoothness proof is built on the Lemma 2.1 in Chow and Lu [3] and the Henry's lemma [9,Lemma 6.1.6]. Furthermore, the results about local unstable integral manifold can follow from those about local stable integral manifold by reversing time. Note that the dichotomous solution we study always means dichotomous classical solution. Hence, in contrast to ElBialy's results [6], the solutions we obtained on the manifolds have higher regularity under the same Lipschitz conditions attached to the nonlinear terms. Besides, Deng and Xiao [4] discussed the existence, uniqueness, continuous dependence on the dichotomous initial value, regularity and Z α -estimate of dichotomous solutions for (4), provided that F (t, z) and G(t, z) are locally Hölder continuous in t and locally Lipschitz continuous in z.
It is worth pointing out that, the elliptic problem in infinite cylindrical domain formulated by the abstract form (1) could be transformed to a first order system consisting of a pair of semilinear coupled parabolic equations, and it is well known that the investigation of parabolic problem base on the theory of analytic semigroup has a great importance, we refer to the monographs [9,12]. So the assumption of bisectoriality for the linear operator S in the equation (1) make it more realistic. In addition, one often encounter nonlinearities of PDEs depend not only on the unknown state variable but also on its derivatives, so it is reasonable to define the nonlinearities of the equation (1) in Z α which between Z and D(S).
As an application of our results, we study the following elliptic equation in infinite where Ω is an open and bounded subset of R n with smooth boundary, ∇ y is the gradient in the y-variable and ∆ y is the Laplace operator in the y-variable. We shows that the existence and asymptotic behavior of solutions for system (5) under some boundary value conditions. This paper is organized as follows. In Section 2, we first recall some notations and definitions. In Section 3, some hypothesises and lemmas are given. In Section 4, we devote to the C k,γ local stable and unstable integral manifolds. In Section 5, an elliptic equation in infinite cylindrical domain is illustrated to the results.
Let X and Y be Banach spaces and let U be an open subset of X, we denote L(X, Y) by the space of all bounded linear operators from X to Y. Firstly we use the following notations to represent some function spaces: (i) For any integer k ≥ 0, let where D j x denotes the jth differentiation operator with respect to the variable where X j denotes the space X × X × · · · × X j .
For the sectorially dichotomous S and a constant α ∈ (0, 1), we can define αfractional power of −S + , S − and S and obtain the relationship among them, refer to [4, section 3]. We denote (−S + ) α and (S − ) α by the α-fractional powers of −S + and S − , respectively, and denote the balls of radius r around the origin in X α , Y α , Z α respectively, and satisfy that B Zα Then we give the definitions of some dichotomous solutions and invariant manifolds for the ill-posed system (4).
is called dichotomous solution of the ill-posed system (4) for −∞ < t 1 < t 2 < ∞, if it satisfies the following three conditions: (ii) x(t) ∈ D(S + ), y(t) ∈ D(S − ) and z(t) ∈ O for t ∈ (t 1 , t 2 ); (iii) x(t) and y(t) satisfy the first and second equation in (4), respectively, for all t ∈ (t 1 , t 2 ). Moreover, From Definition 2.2, the dichotomous solution z(t) of system (4) satisfies the following dichotomous system of integral equations: If the limits of the integrals in (8) exist, we obtain a dichotomous solution on I = R. In order to distinguishing some asymptotic behaviors of dichotomous solution clearly, two infinitely long dichotomous solutions deserve to be introduced similar to ElBily [6].
Given a neighborhood U ⊂ Z α with size r of 0, we define local stable set W s loc (0, r) ⊂ R × U and the local unstable set W u loc (0, r) ⊂ R × U of the equilibrium 0 as It is trivial to check that W s loc (0, r) and W u loc (0, r) are both invariant to R × U, and the purpose of this paper is to show that they are indeed integral manifolds, before that, we give the definitions of local stable and unstable integral manifolds.
3. Lemmas and hypothesises. First we introduce two lemmas by Chow and Lu [3, Lemma 2.1] and Henry [9, page 151] respectively, which can be used to study C k,γ smoothness of invariant manifolds.
More precisely, Lemma 3.2 says that, a sequence {u n } ⊂ C k,γ (U, Y) and a map u : Then two lemmas for hyperbolic bisectorial operator being sectorially dichotomous are given, see [4]. Consider two linear subspaces Z + , Z − ⊂ Z of Z for hyperbolic bisectorial operator S as follows: Z + = {z ∈ Z : R(λ, S)z has a bounded analytic extension to λ ≥ −h}, Z − = {z ∈ Z : R(λ, S)z has a bounded analytic extension to λ ≤ h}, and two operators A + and A − of the forms Lemma 3.4 (Theorem 2.6, [4]). Let S : D(S) ⊂ Z → Z be a linear, densely defined operator and satisfy (9) and (10). Z ± are defined by (11) and A ± are defined by (12). Then (i) P ± := A ± S are closed and complementary operators on D(P ± ), where (ii) if P + (or P − ) is bounded on D(S), then S is sectorially dichotomous. In this case, P + (or P − ) is the unique bounded extension of A + S(or A − S) from D(S) to Z, and P + (or P − ) is the bounded projection of Z onto Z + (Z − , resp.) along Z − (Z + , resp.).
Based on the Lemma 3.4, there exist a pair of bounded, closed and complementary projections P + and P − on Z and two closed subspaces X , Y of Z such that Z = X ⊕ Y holds, where P + : Z → X and P − : Z → Y. Actually, X and Y have the forms of Z + and Z − respectively. Moreover, S decomposes with respect to Z = X ⊕ Y. Namely, S admits the block matrix representation S = , and X and Y are both S-invariant subspaces, where S + and S − are defined respectively by In particular, Furthermore, some results of S + and −S − and the semigroups generated by them as follows.
for t ≥ 0, and for t > 0.
In particular, by the spectral mapping theorem [7, Corollary 4.3.12], T + (t) and and To study the existence and C k,γ smoothness of local integral manifolds, we give further hypothesis for the nonlinear map H of the system (1) provided that Lemma 3.4 holds. Hypothesis 3.6. Let F (t, z(t)) =: P + H(t, z(t)), G(t, z(t)) =: P − H(t, z(t)), F and G are uniformly Hölder continuous in t, 1], and L 0,1 (r), L 0,1 (r), L k,γ (r), L k,γ (r) are all positive constants depending on r.
(i) For k = 0 and γ = 1, 4. Local stable and unstable integral manifolds. In this section, we study local stable and unstable integral manifolds of equation (1) with dichotomous initial condition (3) provided that Lemma 3.4 holds, i.e., S in (1) is a sectorially dichotomous operator, which is equivalent to considering system (4). The local stable and unstable integral manifolds theorem for system (4) are as follows.
Theorem 4.1. Assume that Hypothesis 3.6 is satisfied for the ill-posed system (4), and L k,γ (r), L k,γ (r) are sufficiently small. Then is a unique C k,γ local stable integral manifold of the ill-posed system (4). Moreover, the infinitely long forward dichotomous solutions on W s loc (0, r) take the form is a unique C k,γ local unstable integral manifold of the ill-posed system (4). Moreover, the infinitely long backward dichotomous solutions on W u loc (0, r) take the form To prove the Theorem 4.1, we first give some lemmas in the following.
Lemma 4.2. Assume that the nonlinear term F (t, z(t)) and G(t, z(t)) of system (4) on arbitrary finite time interval [t 1 , t 2 ] satisfy the Hypothesis 3.6(i), and We define a mapping Ψ : E → E as follows: that is, Now we check the well-definedness of mapping Ψ firstly, so we consider the formulas (21) and (22). Obviously, and θ − 0,1 < 1/2. Thus, Ψu Z < r, and then (Ψu)(t) ∈ B Z r . From [1, Corollary 3.7.21], and since T + (t) and T − (−t) are contraction semigroups, we know that . On the other hand, we claim that there exists constants and which proves that where Indeed, the continuity of u(t) and the Hypothesis 3.6 (i) imply that F (t, S −α u(t)) and G(t, S −α u(t)) are bounded on [t 1 , t 2 ], there exist two positive constants N 1 and N 2 such that for t ∈ [t 1 , t 2 ]. Besides, assume that there exists β 1 , β 2 and h 0 such that . We discuss that whether (23) and (24) hold in two cases of α.
Hence, Ψ maps E into itself. In the following, we show that Ψ is a contraction map on E, which prove that Ψ has a unique fixed point on E.
Indeed, for u, v ∈ E, By induction, we have . R(n) < 1 for n being large enough, one can apply the extension of the contraction mapping theorem to Ψ on E to obtain that there exists a unique fixed point u ∈ E of the mapping Ψ, that is, Ψu = u. From (25) and the fact that the composition of Hölder continuous functions is a Hölder continuous function, we know that F (S −α u(t)) and G(S −α u(t)) are uniformly Hölder continuous in t on (t 1 , t 2 ). Let we consider the following linear non-homogeneous system for t ∈ (t 1 , t 2 ): where f (t) = F (t, S −α u(t)) and g(t) = G(t, S −α u(t)) are continuous in t. By [20,Corollary 4.3.3], the linear non-homogeneous system (33) has a unique solution ; Y) is given respectively by and It remains to show that which proves that the function z(t) is a local dichotomous solution of the system (4), and z ∈ C 0 ((t 1 , t 2 ); B Zα r ) ∩ C 1 ((t 1 , t 2 ); B Zα r ). Especially, z ∈ C 0 ([t 1 , t 2 ]; B Z r ) ∩ C 1 ((t 1 , t 2 ); B Z r ) is obvious when α = 0. Indeed, for t ∈ (t 1 , t 2 ), each term of (34) and (35) is in D(S) and is also in D(S α ), then operating on both sides of (34) and (35) with S α and adding them we obtain This proves the formula (36).
Concerning the continuity of x with values in B Xα r for α ∈ (0, 1) up to t = t 1 , from [12, Lemma 7.
r , t ≤ t 2 is an infinitely long backward dichotomous solution of system (4) if and only if z(t) satisfies the integral equation Proof. (i) Since z(t), t ≥ t 1 is an infinitely long forward dichotomous solution of (4), by the Definition 2.3, for each t 2 ∈ (t 1 , ∞), we have Let t 2 → ∞, from the estimation (14), we obtain Since then from z(t) = x(t) + y(t), we obtain (38). By Lemma 4.2, it is obvious that Since the right side of (38) is continuously differentiable, differentiate (38) with respect to t, we will have z(t)).
Furthermore, for any z, z ∈ C β ([t 1 , ∞), B Zα r ), we have is a contraction. Hence, from the lemma 4.3, if x 1 ∈ D(S + ) α , the fixed point z(t) of T x1 is the unique infinitely long forward dichotomous solution to system (4) in C β ([t 1 , ∞), B Zα r ) such that P + z(t 1 ) = x 1 . The proof is completed.
Obviously, z C β < ∞ implies z(t) Z α → 0 as t → ∞. In the following we will show that all these inifinitely long forward dichotomous solutions involved in B Zα r lie on the graph of a map h s : R × B Xα r → B Yα r . Prior to this, a generalized Gronwall's inequality with singular kernel will be given, which generalizes the case α = 1/2 in [8, Lemma 6,p.33] to α ∈ [0, 1).
The following lemma gives the existence of local stable integral manifold of system (4). for some positive constants K, K 1 , κ. Then there exists a unique C 0,1 local stable integral manifold for system (4).
Proof. Let κ ∈ (0, 1] be an arbitrary and fixed constant. Define the space Γ(r): Observe that h s Define the Lyapunov-Perron operator L on the Lipschitz function h s in Γ(r) as follows: where x(t) = x(t; t 1 , x 1 , h s ) is the unique solution of the following system Since S + is the infinitesimal generator of strongly continuous and analytic semigroup {T + (t)} t≥0 , from the Hypothesis 3.6 (i), x(t) is well defined for all t ≥ t 1 , and x(t) has the form On the one hand, since T + (t − t 1 )x 1 (44) and (14), we have and note that x(t) X α ≤ r can be verified by x 1 X α ≤ r/(2M + 0 ) and θ + 0,1 < 1/2. Then the integral on the right side of (42) converges and belongs to B Yα r . Indeed, it follows from On the other hand, set (14) and (15), we have Since e β+(t−s) < 1, by the Lemma 4.5, it yields that where K > 1 depends on M + α , L 0,1 (r) and α.
In addition, we use the notation x(t, h s ) to signify the dependence of x(t) on h s . For h s , h s ∈ Γ(r), then by Lemma 4.5, there exists a constant K 1 > 1 that depends on M + α L 0,1 (r)(1 + κ) and α, such that If h s is a fixed point of L in Γ(r), then the graph of h s is the local stable manifold. In the follows, we prove L is a contraction map in Γ(r). First we verify that L(Γ(r)) ⊂ Γ(r).
In the following, we shall focus on the smoothness of h s obtained in Lemma 4.6.
Proof. By Lemma 4.6, we obtain a unique C 0,1 local stable integral manifold characterized by the graph of h s ∈ Γ(r), where h s and Γ(r) are as in to (50) and (41) respectively. We shall proceed to prove that h s is C 1 provided F and G are C 1 in z and (1 + κ)K 2 θ − 1,0 θ + 1,0 < 1 for some positive constant K 2 . Fix t, and defined λ(h s , x 0 ) By Lemma 3.1, h s is C 1 in x if and only if λ(h s , x 0 ) = 0 for every x 0 ∈ D(S + ) α ∩B Xα r . Here we use the notation x(t, x 1 , h s ) to represent the solution of (43). By (44) and the Taylor expansion, we have − h(s, x(s, x 1 , h s )) − h(s, x(s, x 0 + ∆, h s )) + h(s, x(s, x 0 , h s ))] where R 2 (x) represents the sum of higher order Taylor expansions of F in (53) at the point (x(s, x 0 , h s ), h s (s, x(s, x 0 , h s ))). By Lemma 4.5, it follows that as (x 1 , ∆) → (x 0 , 0), where K 2 > 1 which depends on M + α L 1,0 (r) and α. Then, similarly as (53), by (50) and the Taylor expansion, we have where R 2 (x) represents the sum of higher order Taylor expansions of G in (55) at the point (x(s, x 0 , h s ), h s (s, x(s, x 0 , h s ))).
The proof is complete.
Remark 1. With the same arguments as Lemma 4.7 and Lemma 3.1, the conclusion of Lemma 4.7 can be improved to the C k (k ≥ 2) case provided in addition to the existence of C k−1,1 local stable integral manifold that F and G are C k in z along with sufficiently small L k,0 (r) and L k,0 (r). While we omit the details and shift attention to the C k,γ (k ≥ 1, γ ∈ (0, 1]) smoothness later.
where Γ(r) refers to (41). By Lemma 4.6, we obtain a unique C 0,1 local stable integral manifold characterized by the graph of h s ∈ Γ, where h s refers to (50). In the following, we shall continue to prove h s is C k,γ in x for γ ∈ (0, 1]. From (42) and Lemma 3.2, it suffices to show L(Γ k (r)) ⊂ Γ k (r) in the C 0 norm.
Step I : Prior to this, we need to prove that x(t) defined by (44) is C k,γ in x 1 and satisfies the estimates on the derivatives up to order k and Hölder derivatives of D k x1 x(t). Now we define the space where M + 0 refers to (14). From the Proposition A2 in [11, p182], we obtain that L β (r) is a complete metric space endowed with the induced metric For any x ∈ L β (r), define [Tx](t, T + (t − s)F (s, x(s, x 1 ), h s (s, x(s, x 1 )))ds. (59) To prove that x(t) defined by (44) is C k,γ in x 1 provided h s ∈ Γ k (r), it suffices to prove T is a contraction map in L β (r). We first prove that T(L β (r)) ⊂ L β (r). Obviously, [Tx](t 1 , x 1 ) = x 1 . For any x ∈ L β (r), [Tx] being C 1 in t and C k,γ in x 1 follow from the fact that, for any l ∈ N and γ ∈ (0, 1], the composition of C l,γ functions is a C l,γ function. Moreover, by (45), we know that [Tx] C 0 (Xα,Xα) ≤ where we use the notation F (s, x 1 ) for F (s, x(s, x 1 ), h s (s, x(s, x 1 ))). Note that the integral in (60) converges. Indeed, since x ∈ L β (r) and h s ∈ Γ k (r), choose L 1,γ (r) where R i (s, x 1 ) is a sum of monomials whose factors are derivatives of F and of h s up to order i and of x up to order i − 1. Note that (61) is well-defined because [Tx] is C k,γ in x 1 and the integral in (61) converges. Moreover, by choosing L i,γ (r) sufficiently small, we can obtain The only thing that remains to ensure that T(L β (r)) ⊂ L β (r) is the estimate on H γ (D k x1 [Tx]). For all x 1 , x 1 ∈ D(S + ) α ∩ B Xα r , x ∈ L β (r) and h s ∈ Γ k (r), we have Since each difference terms in the right side of (62) contain the factors D x1 x(s, x 1 ) and D x1 x(s, x 1 ), we use the triangle inequality to estimate (62) in the C 0 norm and assume β + < (1 + γ)β and L k,γ (r) being sufficiently small, then we can obtain This finishes the verification of T(L β (r)) ⊂ L β (r). In addition, for any x, x ∈ L β (r), we have Since θ + 1,0 ≤ θ + 0,1 1+κ < θ + 0,1 and θ + 0,1 < 1 2 , it follows that d L ([Tx], [T x]) < 1 2 d L (x, x). Thus, the contraction mapping theorem yields that T has a fixed point x on L β (r).
Step II : Now we prove that L(Γ k (r)) ⊂ Γ k (r). Since x ∈ L β (r) and h s ∈ Γ k , L(h s ) is C k,γ in x 1 . This is again a result that the composition of C k,γ functions is a C k,γ functions. Differentiable L(h s )(t 1 , x 1 ) with respect to x 1 , we have where we use the notation G(s, x 1 ) for G(x(s, x 1 ), h s (s, x(s, x 1 ))). Choosing L 1,0 (r) R i (x 1 ) is a sum of monomials whose factors are derivatives of G and x up to order i − 1 and h s up to i. Note that all the terms in R i (x 1 ) contain at least one factor which is a derivative of G. Hence, assuming L i,γ (r) is sufficiently small, we can obtain D i x1 L(h s ) ≤ κ. In addition, for Since each difference terms in the right side of (66) contain the factors D x1 x(s, x 1 ) and D x1 x(s, x 1 ), we use the triangle inequality to estimate (66) in the C 0 norm and assume β + < (1 + γ)β and L k,γ (r) being sufficiently small, we obtain that H γ (D k x1 [L(h s )]) ≤ κ. Thus, L(Γ k (r)) ⊂ Γ k (r). By Lemma 3.2, Γ k (r) is a non-empty closed subset of Γ(r)(⊂ C 0 (X α , Y α )) in the C 0 norm, and since L has a fixed point h s in Γ(r), L(Γ k (r)) ⊂ Γ k (r) implies that the fixed point h s of L also lies in Γ k (r) and is therefore of class C k,γ . Thus, W s loc (0) is the unique local stable integral manifold. The proof is complete.

5.
Elliptic equations in infinite cylindrical domain. Consider the following elliptic equation with Dirichlet boundary condition on ∂Ω where Ω is an open and bounded subset of R n with smooth boundary ∂Ω, ∇ y is the gradient in the y-variable and ∆ y is the Laplace operator in the y-variable. The function (x, y, , χ, ω) −→ f (x, y, , χ, ω) is defined in R × Ω × V, and has value in R m , where V ⊂ R m × R m × R mn is an open subset and it is also the higher order term of ( , χ, ω). Assume that f is locally Hölder continuous with respect to x, and locally Lipschitz continuous with respect to ( , χ, ω). Precisely, we assume that for arbitrary finite time interval [t 1 , t 2 ], there exist γ ∈ (0, 1], r > 0 and K 0 = K 0 (r) > 0 such that for x, x ∈ [x 1 , x 2 ] and ( 1 , χ 1 , ω 1 ), ( 2 , χ 2 , ω 2 ) ∈ B R 2m+mn r . Moreover, K 0 → 0 as r → 0.
Based on the idea from Kirchgässner [10] who considers (70) as an evolution equation by treating the unbounded spatial variable x as time variable. We first transfer the problem of elliptic equation (70) to the abstract semilinear problem.
Let A := −∆ y . Then A is a closed operator on X := L 2 (Ω, R m ) with dense domain X 1 := D(A) = H 2 (Ω, R m ) ∩ H 1 0 (Ω, R m ), and it is positive, symmetric with compact inverse. Moreover, σ(A) = {λ n : n ∈ Z + } is a discrete set such that λ n ≥ λ n−1 > 0 and λ n → ∞ as n → ∞. Furthermore, the corresponding eigenfunctions {e n : n ∈ Z + } of A can be chosen to form an orthonormal basis for X, and in terms of this basis, the operator A can be represented by Au = ∞ n=1 λ(u, e n )e n , where (·, ·) be a inner product on X. In particular, −A is sectorial on X.
Then (86) can be transformed into the following abstract linear equation where z = (p, q) T . Moreover, there exists a bounded and boundedly invertible transform Note that u ∈ X 1/2 and v ∈ X signify u(x, ·) ∈ X 1/2 and v(x, ·) ∈ X. Besides, with D(S) = X 1/2 × X 1/2 . Obviously, D(S) = Z and σ(S) = σ(−A 1/2 ) ∪ σ(A 1/2 ). Thus, S is a densely defined and hyperbolic bisectorial operator on Z. Furthermore, we have the following lemma. Based on the Lemma 5.1, we can constrcut the fractional power S α and obtain its domain. The propery of A follows that sectorial operator −S + is also positive, symmetric with compact inverse, and the corresponding eigenfunctions { e n : n ∈ Z + } of −S + can be chosen to form an orthonormal basis for X 1/2 × {0}. In terms of this basis, u = ∞ n=1 ( u, e n ) e n for u ∈ X 1/2 × {0}, and the operator −S + can be represented by −S + u = Since the composition of C 0,1 functions is a C 0,1 function, there exists a local C 0,1 map h s such that h s : R × X 1/2 → X, u x = h s (x, u) = A 1/2 (− h s (x, u) + h s ( h s (x, u))).

Remark 2.
(1) In contrast to the Appendix A in ElBialy [6], under the same Lipschitz condition for the state variables of nonlinear term of elliptic equation in infinite cylinders, we obtain the continuous differentiability of u x additionally.