Integral manifolds for partial functional differential equations in admissible spaces on a half line

Integral manifolds for partial functional differential equations in admissible spaces on a half line tài liệu, giáo án,...

JID:YJMAA AID:17976 /FLA [m3G; v 1.114; Prn:23/10/2013; 10:53] P.1 (1-13) J Math Anal Appl ••• (••••) •••–••• Contents lists available at ScienceDirect Journal of Mathematical Analysis and Applications www.elsevier.com/locate/jmaa Integral manifolds for partial functional differential equations in admissible spaces on a half-line ✩ Nguyen Thieu Huy a,∗ , Trinh Viet Duoc b a b School of Applied Mathematics and Informatics, Hanoi University of Science and Technology, Dai Co Viet, Hanoi, Viet Nam Faculty of Mathematics, Mechanics, and Informatics, Hanoi University of Science, 334 Nguyen Trai, Hanoi, Viet Nam a r t i c l e i n f o Article history: Received December 2012 Available online xxxx Submitted by C.E Wayne Keywords: Exponential dichotomy and trichotomy Partial functional differential equations Stable and center-stable manifolds Admissibility of function spaces a b s t r a c t In this paper we investigate the existence of stable and center-stable manifolds for solutions to partial functional differential equations of the form u˙ (t ) = A (t )u (t ) + f (t , ut ), 0, when its linear part, the family of operators ( A (t ))t , generates the evolution family t (U (t , s))t s having an exponential dichotomy or trichotomy on the half-line and the nonlinear forcing term f satisfies the ϕ -Lipschitz condition, i.e., f (t , ut ) − f (t , v t ) ϕ (t ) ut − v t C where ut , v t ∈ C := C ([−r , 0], X ), and ϕ (t ) belongs to some admissible function space on the half-line Our main methods invoke Lyapunov–Perron methods and the use of admissible function spaces © 2013 Elsevier Inc All rights reserved Introduction Consider the partial functional differential equation du dt = A (t )u (t ) + f (t , ut ), t ∈ [0, +∞), (1.1) where A (t ) is a (possibly unbounded) linear operator on a Banach space X for every fixed t; f : R+ × C → X is a continuous nonlinear operator with C := C ([−r , 0], X ), and ut is the history function defined by ut (θ) := u (t + θ) for θ ∈ [−r , 0] When the family of operators ( A (t ))t generates the evolution family having an exponential dichotomy (or trichotomy), one tries to find conditions on the nonlinear forcing term f such that Eq (1.1) has an integral manifold (e.g., a stable, unstable or center manifold) The most popular condition imposed on f is its uniform Lipchitz continuity with a sufficiently small Lipschitz q φ − ψ C for q small enough (see [1,3,14] and references therein) However, for equaconstant, i.e., f (t , φ) − f (t , ψ) tions arising in complicated reaction–diffusion processes, the function f represents the source of material (or population) which, in many contexts, depends on time in diversified manners (see [5, Chapt 11], [6,15]) Therefore, sometimes one may not hope to have the uniform Lipschitz continuity of f Recently, for the case of partial differential equations without delay, we have obtained exciting results in [9], where we have used the Lyapunov–Perron method and the characterization of the exponential dichotomy (obtained in [8]) of evolution equations in admissible function spaces to construct the structures of solutions in mild forms, which belong to some certain classes of admissible spaces on which we could employ some wellknown principles in mathematical analysis such as the contraction mapping principle, the implicit function theorem, etc The use of admissible spaces has helped us to construct the invariant manifolds without using the smallness of Lipschitz constants of nonlinear forcing terms in classical sense (see [9,10]) ✩ * This research is funded by Vietnam National Foundation for Science and Technology Development (NAFOSTED) under grant number 101.01-2011.25 Corresponding author E-mail addresses: huy.nguyenthieu@hust.vn (T.H Nguyen), tvduoc@gmail.com (V.D Trinh) 0022-247X/$ – see front matter © 2013 Elsevier Inc All rights reserved http://dx.doi.org/10.1016/j.jmaa.2013.10.027 JID:YJMAA AID:17976 /FLA [m3G; v 1.114; Prn:23/10/2013; 10:53] P.2 (1-13) T.H Nguyen, V.D Trinh / J Math Anal Appl ••• (••••) •••–••• Another point we would like to mention is that in some applications the partial differential operator A (t ) is defined only for t (see e.g., [4,11,13] and references therein) Therefore, the evolution family generated by ( A (t ))t is defined only on a half-line The purpose of the present paper is to prove the existence of stable and center-stable manifolds for Eq (1.1) when its linear part ( A (t ))t generates the evolution family having an exponential dichotomy or trichotomy on the half-line under more general conditions on the nonlinear forcing term f , that is the ϕ -Lipschitz continuity of f , i.e., f (t , φ) − f (t , ψ) ϕ (t ) φ − ψ C where φ, ψ ∈ C , and ϕ (t ) is a real and positive function which belongs to admissible function spaces defined in Definition 2.3 below We will extend the methods in [9] to the case of partial functional differential equations (PFDE) The main difficulties that we face when passing to the case of PFDE are the following two features: Firstly, since the nonlinear operator f is ϕ -Lipschitz, the existence and uniqueness theorem for solutions to (1.1) is not available Secondly, the evolution family generated by ( A (t ))t is defined only on a half-line R+ and doesn’t act on the same Banach space as that the surfaces of the integral manifold belong to (in fact, the former acts on X , and the latter belongs to C ) Therefore, the standard methods of nonlinear perturbations of an evolutionary process using graph transforms as formulated in [1,3] cannot be applied here To overcome such difficulties, we reformulate the definition of invariant manifolds such that it contains the existence and uniqueness theorem as a property of the manifold (see Definition 3.3 below) Furthermore, we construct the structure of the mild solutions to (1.1) using the Lyapunov–Perron equation (see Eq (3.5)) in such a way that it allows to combine the exponential dichotomy of the linear part of Eq (1.1) with the existence and uniqueness of its bounded solutions in the case of ϕ -Lipschitz nonlinear forcing terms Then, we use the admissible spaces to construct the integral manifolds for Eq (1.1) in the case of dichotomic linear part without using the smallness of Lipschitz constants of the nonlinear terms in t +1 ϕ (τ ) dτ Consequently, classical sense Instead, the “smallness” is now understood as the sufficient smallness of supt t we obtain the existence of invariant stable manifolds for the case of dichotomic linear parts under very general conditions on the nonlinear term f (t , ut ) Moreover, using these results and rescaling procedures we prove the existence of center-stable manifolds for the mild solutions to Eq (1.1) in the case of trichotomic linear parts under the same conditions on the nonlinear term f as in the dichotomic case Our main results are contained in Theorems 3.7, 4.2 We now recall some notions For a Banach space X (with a norm · ) and a given r > we denote by C := C ([−r , 0], X ) the Banach space of all continuous functions from [−r , 0] into X , equipped with the norm φ C = supθ∈[−r ,0] φ(θ) for φ ∈ C For a continuous function v : [−r , ∞) → X the history function v t ∈ C is defined by v t (θ) = v (t + θ) for all θ ∈ [−r , 0] Definition 1.1 A family of bounded linear operators {U (t , s)}t bounded) evolution family if s on a Banach space X is a (strongly continuous, exponential (i) U (t , t ) = Id and U (t , r )U (r , s) = U (t , s) for all t r s 0, (ii) the map (t , s) → U (t , s)x is continuous for every x ∈ X , K e c (t −s) x for all t (iii) there are constants K , c such that U (t , s)x s and x ∈ X The notion of an evolution family arises naturally from the theory of non-autonomous evolution equations which are well-posed Meanwhile, if the abstract Cauchy problem ⎧ ⎨ du ⎩ = A (t )u (t ), dt u ( s ) = xs ∈ X t s 0, (1.2) is well-posed, there exists an evolution family {U (t , s)}t s such that the solution of the Cauchy problem (1.2) is given by u (t ) = U (t , s)u (s) For more details on the notion of evolution families, conditions for the existence of such families and applications to partial differential equations we refer the readers to Pazy [11] (see also Nagel and Nickel [7] for a detailed discussion of well-posedness for non-autonomous abstract Cauchy problems on the whole line R) Function spaces and admissibility We recall some notions on function spaces and refer to Massera and Schäffer [2], Räbiger and Schnaubelt [12] for concrete applications Denote by B the Borel algebra and by λ the Lebesgue measure on R+ The space L 1,loc (R+ ) of real-valued locally integrable functions on R+ (modulo λ-nullfunctions) becomes a Fréchet space for the seminorms pn ( f ) := J | f (t )| dt, n where J n = [n, n + 1] for each n ∈ N (see [2, Chapt 2, §20]) We can now define Banach function spaces as follows Definition 2.1 A vector space E of real-valued Borel-measurable functions on R+ (modulo λ-nullfunctions) is called a Banach function space (over (R+ , B , λ)) if JID:YJMAA AID:17976 /FLA [m3G; v 1.114; Prn:23/10/2013; 10:53] P.3 (1-13) T.H Nguyen, V.D Trinh / J Math Anal Appl ••• (••••) •••–••• 1) E is Banach lattice with respect to a norm · E , i.e., ( E , · E ) is a Banach space, and if ϕ ∈ E and ψ is a real-valued Borel-measurable function such that |ψ(·)| |ϕ (·)|, λ-a.e., then ψ ∈ E and ψ E ϕ E, 2) the characteristic functions χ A belong to E for all A ∈ B of finite measure, and supt χ[t ,t +1] E < ∞ and inft χ[t ,t +1] E > 0, 3) E → L 1,loc (R+ ), i.e., for each seminorm pn of L 1,loc (R+ ) there exists a number β pn > such that pn ( f ) β pn f E for all f ∈ E We then define Banach spaces of vector-valued functions corresponding to Banach function spaces as follows Definition 2.2 Let E be a Banach function space and X be a Banach space endowed with the norm · We set E := E (R+ , X ) = f : R+ → X: f is strongly measurable and f (·) ∈ E (modulo λ-nullfunctions) endowed with the norm f E = f (·) it the Banach space corresponding to the Banach function space E E One can easily see that E is a Banach space We call We now introduce the notion of admissibility in the following definition Definition 2.3 The Banach function space E is called admissible if (i) there is a constant M b such that for every compact interval [a, b] ∈ R+ we have M (b − a) ϕ (t ) dt χ[a,b] a ϕ E, E t +1 (ii) for ϕ ∈ E the function Λ1 ϕ defined by Λ1 ϕ (t ) = t ϕ (τ ) dτ belongs to E, (iii) E is T τ+ -invariant and T τ− -invariant, where T τ+ and T τ− are defined for τ ∈ R+ by T τ+ ϕ (t ) = ϕ (t − τ ) for t τ 0, t τ, T τ− ϕ (t ) = ϕ (t + τ ) for for t Moreover, there are constants N , N such that T τ+ Example 2.4 Besides the spaces L p (R+ ), p N , T τ− N for all τ ∈ R+ ∞, and the space t +1 M(R+ ) := f ∈ L 1,loc (R+ ): sup t f (τ ) dτ < ∞ t t +1 endowed with the norm f M := supt t | f (τ )| dτ , many other function spaces occurring in interpolation theory, e.g., the Lorentz spaces L p ,q , < p < ∞, < q < ∞ are admissible Remark 2.5 One can easily see that if E is an admissible Banach function space then E → M(R+ ) We now collect some properties of admissible Banach function spaces in the following proposition (see [9, Proposition 2.6]) Proposition 2.6 Let E be an admissible Banach function space Then the following assertions hold (a) Let ϕ ∈ L 1,loc (R+ ) such that ϕ and Λσ ϕ by and Λ1 ϕ ∈ E, where Λ1 is defined as in Definition 2.3(ii) For σ > we define functions Λσ ϕ t e −σ (t −s) ϕ (s) ds, Λσ ϕ (t ) = ∞ e −σ (s−t ) ϕ (s) ds Λσ ϕ (t ) = t JID:YJMAA AID:17976 /FLA [m3G; v 1.114; Prn:23/10/2013; 10:53] P.4 (1-13) T.H Nguyen, V.D Trinh / J Math Anal Appl ••• (••••) •••–••• Then, Λσ ϕ and Λσ ϕ belong to E In particular, if supt then Λσ ϕ and Λσ ϕ are bounded Moreover, denoted by Λσ ϕ N1 ∞ − e −σ Λ1 T 1+ ϕ and ∞ t +1 | t · ϕ (τ )| dτ < ∞ (this will be satisfied if ϕ ∈ E (see Remark 2.5)) ∞ for ess sup-norm, we have N2 Λσ ϕ ∞ − e −σ Λ1 ϕ ∞ (b) E contains exponentially decaying functions ψ(t ) = e −αt for t and any fixed constant α > (c) E does not contain exponentially growing functions f (t ) = ebt for t and any constant b > Exponential dichotomy and stable manifolds In this section, we will find condition for the existence of stable manifolds Throughout this section we assume that the evolution family {U (t , s)}t s has an exponential dichotomy on R+ We now make precisely the notion of exponential dichotomies in the following definition Definition 3.1 An evolution family {U (t , s)}t s if there exist bounded linear projections P (t ), t on the Banach space X is said to have an exponential dichotomy on [0, ∞) 0, on X and positive constants N , ν such that (a) U (t , s) P (s) = P (t )U (t , s), t s 0, (b) the restriction U (t , s)| : Ker P (s) → Ker P (t ), t s 0, is an isomorphism, and we denote its inverse by U (s, t )| := (U (t , s)| )−1 , s t, (c) U (t , s)x Ne −ν (t −s) x for x ∈ P (s) X , t s 0, (d) U (s, t )| x Ne −ν (t −s) x for x ∈ Ker P (t ), t s The projections P (t ), t 0, are called the dichotomy projections, and the constants N , ν – the dichotomy constants Using the projections ( P (t ))t on X , we can define the family of operators ( P (t ))t on C as follows P (t ) : C → C , P (t )φ (θ) = U (t − θ, t ) P (t )φ(0) for all θ ∈ [−r , 0] (3.1) Then, we have that ( P (t ))2 = P (t ), and therefore the operators P (t ), t 0, are projections on C Moreover, Im P (t ) = {φ ∈ C : φ(θ) = U (t − θ, t )ν0 ∀θ ∈ [−r , 0] for some ν0 ∈ Im P (t )} To obtain the existence of stable manifolds we need the following notion of the ϕ -Lipschitz of the nonlinear term f Definition 3.2 Let E be an admissible Banach function space and f : [0, ∞) × C → X is said to be ϕ -Lipschitz if f satisfies (i) (ii) ϕ be a positive function belonging to E A function f (t , 0) ϕ (t ) for all t ∈ R+ , f (t , φ1 ) − f (t , φ2 ) ϕ (t ) φ1 − φ2 C for all t ∈ R+ and all φ1 , φ2 ∈ C ϕ (t )(1 + φ C ) for all φ ∈ C and t Note that if f (t , φ) is ϕ -Lipschitz then f (t , φ) To prove the existence of a stable manifold, instead of Eq (1.1) we consider the following integral equation ⎧ ⎪ ⎪ ⎨ u (t ) = U (t , s)u (s) + ⎪ ⎪ ⎩ t U (t , ξ ) f (ξ, u ξ ) dξ for t s 0, (3.2) s us = φ ∈ C We note that, if the evolution family {U (t , s)}t s arises from the well-posed Cauchy problem (1.2), then the function u : [s − r , ∞) → X , which satisfies (3.2) for some given function f , is called a mild solution of the semilinear problem ⎧ ⎨ du ⎩ = A (t )u (t ) + f (t , ut ), dt us = φ ∈ C t s 0, We refer the reader to J Wu [14] for more detailed treatments on the relations between classical and mild solutions of functional evolution equations We now give the definition of a stable manifold for the solutions of the integral equation (3.2) JID:YJMAA AID:17976 /FLA [m3G; v 1.114; Prn:23/10/2013; 10:53] P.5 (1-13) T.H Nguyen, V.D Trinh / J Math Anal Appl ••• (••••) •••–••• Definition 3.3 A set S ⊂ R+ × C is said to be an invariant stable manifold for the solutions to Eq (3.2) if for every t ∈ R+ the phase space C splits into a direct sum C = X (t ) ⊕ X (t ) with corresponding projections P (t ) (i.e., X (t ) = Im P (t ) and X (t ) = Ker P (t )) such that sup P (t ) < ∞, t and there exists a family of Lipschitz continuous mappings Φt : X (t ) → X (t ), t ∈ R+ with the Lipschitz constants independent of t such that (i) S = {(t , ψ + Φt (ψ)) ∈ R+ × ( X (t ) ⊕ X (t )) | t ∈ R+ , ψ ∈ X (t )}, and we denote S t := ψ + Φt (ψ): t , ψ + Φt (ψ) ∈ S , (ii) S t is homeomorphic to X (t ) for all t 0, (iii) to each φ ∈ S s there corresponds one and only one solution u (t ) to Eq (3.2) on [s − r , ∞) satisfying the conditions that u s = φ and supt s ut C < ∞ Moreover, any two solutions u (t ) and v (t ) of Eq (3.2) corresponding to different functions φ1 , φ2 ∈ S s attract each other exponentially in the sense that, there exist positive constants μ and C μ independent of s such that ut − v t C C μ e −μ(t −s) P (s)φ1 (0) − P (s)φ2 (0) for t s, (3.3) s − r, is a solution to Eq (3.2) satisfying conditions that u s ∈ S s s (iv) S is positively invariant under Eq (3.2), i.e., if u (t ), t and supt s ut C < ∞, then we have ut ∈ S t for all t Note that if we identify X (t ) ⊕ X (t ) with X (t ) × X (t ), then we can write S t = graph(Φt ) Let {U (t , s)}t s have an exponential dichotomy with the dichotomy projections P (t ), t 0, and constants N , ν > Note that the exponential dichotomy of {U (t , s)}t s implies that H := supt P (t ) < ∞ and the map t → P (t ) is strongly continuous (see [4, Lemma 4.2]) We can then define the Green function on the half-line as follows P (t )U (t , τ ) for t > τ 0, −U (t , τ )| ( I − P (τ )) for t < τ G (t , τ ) = It follows from the exponential dichotomy of {U (t , s)}t N (1 + H )e −ν |t −τ | G (t , τ ) (3.4) s that for all t = τ The following lemma gives the form of bounded solutions to Eq (3.2) Lemma 3.4 Let the evolution family {U (t , s)}t s have an exponential dichotomy with the dichotomy projections P (t ), t 0, and constants N , ν > Suppose that ϕ is a positive function which belongs to the admissible space E Let f : R+ × C → X be ϕ -Lipschitz and u (t ) be a solution to Eq (3.2) such that supt s−r u (t ) < ∞ for fixed s Then, for t s we can rewrite u (t ) in the form ⎧ ⎪ ⎪ ⎨ u (t ) = U (t , s)ν + ⎪ ⎪ ⎩ us = φ ∈ C ∞ G (t , τ ) f (τ , u τ ) dτ , (3.5) s for some ν0 ∈ X (s) = P (s) X , where G (t , τ ) is the Green function defined as in (3.4) Proof Put y (t ) = ∞ s G (t , τ ) f (τ , u τ ) dτ We have ∞ N (1 + H )e −ν |t −τ | + u τ C y (t ) ϕ (τ ) dτ s ∞ N (1 + H ) + sup ξ s−r e −ν |t −τ | ϕ (τ ) dτ u (ξ ) Using Proposition 2.6 we obtain y (t ) N (1 + H ) + sup ξ s−r u (ξ ) ( N Λ1 T 1+ ϕ ∞ + N Λ1 ϕ − e −ν ∞) for all t (3.6) JID:YJMAA AID:17976 /FLA [m3G; v 1.114; Prn:23/10/2013; 10:53] P.6 (1-13) T.H Nguyen, V.D Trinh / J Math Anal Appl ••• (••••) •••–••• On the other hand, t U (t , s) y (s) = − U (t , s)U (s, τ )| I − P (τ ) f (τ , u τ ) dτ s ∞ − U (t , s)U (s, τ )| I − P (τ ) f (τ , u τ ) dτ t ∞ t =− U (t , τ ) I − P (τ ) f (τ , u τ ) dτ − s U (t , τ )| I − P (τ ) f (τ , u τ ) dτ t Therefore, t y (t ) = U (t , s) y (s) + U (t , τ ) f (τ , u τ ) dτ s Since u (t ) is a solution of Eq (3.2) we obtain that u (t ) − y (t ) = U (t , s)(u (s) − y (s)) Put now ν0 = u (s) − y (s) The boundedness of u (t ) and y (t ) on [s − r , ∞) implies that ν0 ∈ X (s) and P (s)u (s) = P (s)φ(0) = ν0 Therefore, u (t ) = U (t , s)ν0 + y (t ) for t s ✷ Remark 3.5 Eq (3.5) is called the Lyapunov–Perron equation By computing directly, we can see that the converse of Lemma 3.4 is also true This means that, all solutions of the integral equation (3.5) satisfy Eq (3.2) for t s Theorem 3.6 Let the evolution family {U (t , s)}t s have an exponential dichotomy with the dichotomy projections P (t ), t constants N , ν > Suppose that ϕ is a positive function which belongs to E Let f : R+ × C → X be ϕ -Lipschitz, and let k := e ν r (1 + H ) N ( N Λ1 T 1+ ϕ ∞ + N Λ1 ϕ ∞ ) − e −ν 0, and (3.7) Then, if k < 1, there corresponds to each φ ∈ Im P (s) one and only one solution u (t ) of Eq (3.5) on [s − r , ∞) satisfying the conditions that P (s)u s = φ and supt s ut C < ∞ Moreover, the following estimate is valid for any two solutions u (t ), v (t ) corresponding to different initial functions φ1 , φ2 ∈ Im P (s): ut − v t C C μ e −μ(t −s) φ1 (0) − φ2 (0) for all t s where μ is a positive number satisfying < μ < ν + ln − N (1 + H )e ν r N Λ1 T 1+ ϕ ∞ + N Λ1 ϕ ∞ , Ne ν r C μ := ν r N (1+ H )e − 1−e−(ν −μ) ( N Λ1 T 1+ ϕ ∞ + N Λ1 ϕ ∞ ) and Proof Denote by C b ([s − r , ∞), X ) the Banach space of bounded, continuous and X -valued functions defined on [s − r , ∞), which is endowed with the sup-norm · ∞ Setting ν0 := φ(0) we consider the transformation T defined by ( T u )(t ) = ∞ U (2s − t , s)ν0 + ∞ U (t , s)ν0 + s s G (2s − t , τ ) f (τ , u τ ) dτ G (t , τ ) f (τ , u τ ) dτ for s − r for t t s, s Since ν0 ∈ P (0) X , using the inequality (3.6) we can easily see that T acts from C b ([s − r , ∞), X ) into itself We next prove that, if k < 1, then T is a contraction mapping To this, we estimate ∞ ( T u )(t ) − ( T v )(t ) G (t , τ ) f (τ , u τ ) − f (τ , v τ ) dτ s ∞ e −ν |t −τ | ϕ (τ ) u τ − v τ C dτ N (1 + H ) s ke −ν r sup u (t ) − v (t ) t s−r for t s, JID:YJMAA AID:17976 /FLA [m3G; v 1.114; Prn:23/10/2013; 10:53] P.7 (1-13) T.H Nguyen, V.D Trinh / J Math Anal Appl ••• (••••) •••–••• and ∞ ( T u )(t ) − ( T v )(t ) G (2s − t , τ ) f (τ , u τ ) − f (τ , v τ ) dτ s ∞ e −ν |2s−t −τ | ϕ (τ ) u τ − v τ C dτ N (1 + H ) s ∞ e −ν |s−τ | ϕ (τ ) u τ − v τ C dτ N (1 + H )e ν r s k sup u (t ) − v (t ) s−r t for − r + s t s Therefore, supt s−r ( T u )(t ) − ( T v )(t ) k supt s−r u (t ) − v (t ) Hence, for k < the transformation T : C b ([s − r , ∞), X ) → C b ([s − r , ∞), X ) is a contraction mapping Thus, there exists a unique u (·) ∈ C ([s − r , ∞), X ) such that T u = u This yields that u (t ), t s − r, is the unique solution of Eq (3.5) with ∞ u s (θ) = U (s − θ, s)ν0 + s G (s − θ, τ ) f (τ , u τ ) dτ for all θ ∈ C , and P (s)u (s) = ν0 = φ(0) Therefore, P (s)u s = φ by definition of P (s) (see equality (3.1)) Let u (t ), v (t ) be the two solutions to Eq (3.5) corresponding to different initial functions φ1 , φ2 ∈ Im P (s), respectively Putting ν1 := φ1 (0), ν2 := φ2 (0) we have that Ne −ν (t −s) u (t ) − v (t ) Ne −ν (s−t ) ν1 − ν2 + N (1 + H ) ν1 − ν2 + N (1 + H ) ∞ −ν |t −τ | e ϕ (τ ) u τ ∞ −ν |2s−t −τ | s s − vτ C dτ ϕ (τ ) u τ − v τ e if t s, if s − r C dτ t s Since t + θ ∈ [−r + t , t ] for fixed t ∈ [s, ∞) and θ ∈ [−r , 0], we obtain ∞ ut − v t C Ne ν r e −ν (t −s) e −ν |t −τ | ϕ (τ ) u τ − v τ C dτ , ν1 − ν2 + N (1 + H )eν r t s s Put h(t ) = ut − v t C Then, supt s h (t ) < ∞ and ∞ h(t ) Ne ν r e −ν (t −s) e −ν |t −τ | ϕ (τ )h(τ ) dτ , ν1 − ν2 + N (1 + H )eν r t s (3.8) s We will use the cone inequality theorem (see [9, Theorem 2.8]) applying to Banach space L ∞ [s, ∞) which is the space of real-valued functions defined and essentially bounded on [s, ∞) (endowed with the sup-norm denoted by · ∞ ) with the cone K being the set of all nonnegative functions We then consider the linear operator A defined for g ∈ L ∞ [s, ∞) by ∞ e −ν |t −τ | ϕ (τ ) g (τ ) dτ , ( Ag )(t ) = N (1 + H )eν r t s s By Proposition 2.6 we have that ∞ e −ν |t −τ | ϕ (τ ) g (τ ) dτ sup( Ag )(t ) = sup N (1 + H )e ν r t s t s Therefore, A ∈ L( L ∞ [s, ∞)) and can now be rewritten by h Ah + z k g ∞ s A k < Obviously, the cone K is invariant under the operator A The inequality (3.8) for z(t ) = Ne ν r e −ν (t −s) ν1 − ν2 By the cone inequality theorem [9, Theorem 2.8] we obtain that h g = Ag + z which can be rewritten as g, where g is a solution in L ∞ [s, ∞) of the equation ∞ g (t ) = Ne ν r e −ν (t −s) e −ν |t −τ | ϕ (τ ) g (τ ) dτ , ν1 − ν2 + N (1 + H )eν r s t s JID:YJMAA AID:17976 /FLA [m3G; v 1.114; Prn:23/10/2013; 10:53] P.8 (1-13) T.H Nguyen, V.D Trinh / J Math Anal Appl ••• (••••) •••–••• To estimate g, we put w (t ) = e μ(t −s) g (t ) for t s Then, we obtain that ∞ w (t ) = Ne ν r e −(ν −μ)(t −s) e −ν |t −τ |+μ(t −τ ) ϕ (τ ) w (τ ) dτ ν1 − ν2 + N (1 + H )eν r (3.9) s We next consider the linear operator D defined on L ∞ [s, ∞) as ∞ e −ν |t −τ |+μ(t −τ ) ϕ (τ )φ(τ ) dτ ( D φ)(t ) = N (1 + H )eν r for all t s s One can easily see that D ∈ L( L ∞ [s, ∞)) and for z˜ (t ) = Ne ν r e −(ν −μ)(t −s) w = D w + z˜ N (1+ H )e ν r 1−e −(ν −μ) D ( N Λ1 T 1+ ϕ ∞ + N Λ1 ϕ ∞ ) Eq (3.9) can be rewritten as ν1 − ν2 We have D < if < μ < ν + ln(1 − N (1 + H )e ν r ( N Λ1 T 1+ ϕ ∞ + N Λ1 ϕ ∞ )) Under this condition, the equation w = D w + z˜ has the unique solution w ∈ L ∞ [s, ∞) and w = ( I − D )−1 z˜ Hence, we obtain that Ne ν r w ∞ = ( I − D )−1 z˜ ∞ Ne ν r N (1+ H )e ν r 1−e −(ν −μ) 1− This yields that w (t ) Cμ h(t ) = ut − v t C ν1 − ν2 1− D ν1 − ν2 ( N Λ1 T 1+ ϕ ν1 − ν2 for t g (t ) = e −μ(t −s) ∞ + N Λ1 ϕ ∞) := C μ ν1 − ν2 s Hence, C μ e −μ(t −s) w (t ) ν1 − ν2 for t s ✷ We now prove our main result of this section Theorem 3.7 Let the evolution family {U (t , s)}t s have an exponential dichotomy with the dichotomy projections P (t ), t 0, and constants N , ν > Suppose that ϕ is a positive function which belongs to the admissible space E Let f : R+ × C → X be ϕ -Lipschitz satisfying k < 1+ Ne ν r where k is defined by (3.7) Then, there exists an invariant stable manifold S for the solutions to Eq (3.2) Proof Since {U (t , s)}t s has an exponential dichotomy, we have that for each t the phase space C splits into the direct sum C = Im P (t ) ⊕ Ker P (t ) where the projections P (t ), t 0, are defined as in equality (3.1) Clearly, supt P (t ) < ∞ We now construct a stable manifold S = {(t , S t )}t for the solutions to Eq (3.2) To this, we determine the surface S t for t by the formula S t := φ + Φt (φ): φ ∈ Im P (t ) ⊂ C where the operator Φt0 is defined for each t 0 by ∞ Φt0 (φ)(θ) = G (t − θ, τ ) f (τ , u τ ) dτ for all θ ∈ [−r , 0], t0 here u (·) is the unique solution of Eq (3.2) on [−r + t , ∞) satisfying P (t )ut0 = φ (note that the existence and uniqueness of u (·) is guaranteed by Theorem 3.6) On the other hand, by the definition of the Green function G (see Eq (3.4)) we have that Φt0 (φ) ∈ Ker P (t ) We next show that the stable manifold S satisfies the conditions of Definition 3.3 Firstly, we prove that Φt0 is Lipschitz continuity with Lipschitz constant independent of t Indeed, for φ1 and φ2 belonging to Im P (t ) we have ∞ Φt0 (φ1 )(θ) − Φt0 (φ2 )(θ) e −ν |t0 −θ −τ | ϕ (τ ) u τ − v τ C dτ N (1 + H ) t0 ∞ e −ν |t0 −τ | ϕ (τ ) u τ − v τ C dτ N (1 + H )e ν r t0 ∞ N (1 + H )e ν r sup u τ − v τ C τ t0 e −ν |t0 −τ | ϕ (τ ) dτ t0 N (1 + H )e ν r N Λ1 T 1+ ϕ ∞ + N Λ1 ϕ ∞ sup u τ − v τ C − e −ν τ t0 (3.10) JID:YJMAA AID:17976 /FLA [m3G; v 1.114; Prn:23/10/2013; 10:53] P.9 (1-13) T.H Nguyen, V.D Trinh / J Math Anal Appl ••• (••••) •••–••• Moreover, by the Lyapunov–Perron equation for u (·) and v (·) (see Eq (3.5)) we have N (1 + H )e ν r sup u τ − v τ C Ne ν r φ1 − φ2 C + N Λ1 T 1+ ϕ ∞ + N Λ1 ϕ ∞ sup u τ − v τ C − e −ν τ t0 τ t0 It follows that Ne ν r sup u τ − v τ C φ1 − φ2 1−k τ t0 C Substituting this inequality into (3.10) we obtain that Φt0 (φ1 ) − Φt0 (φ2 ) C = sup θ ∈[−r ,0] Nke ν r Φt0 (φ1 )(θ) − Φt0 (φ2 )(θ) Therefore, Φt0 is Lipschitz continuous with the Lipschitz constant 1−k Nke ν r 1−k φ1 − φ2 C independent of t To show that S t0 is homeomorphic to Im P (t ) We define the transformation D : Im P (t ) → S t0 by Dφ := φ + Φt0 (φ) for all φ ∈ Im P (t ) Then, applying the implicit function theorem for Lipschitz continuous mappings (see [3, Lemma 2.7]) νr < (equivalently k < 1+Ne we have that if the Lipschitz constant Nke ν r ) then D is a homeomorphism Therefore, the 1−k condition (ii) in Definition 3.3 is satisfied The condition (iii) in Definition 3.3 now follows from Theorem 3.6 We now prove that the condition (iv) of Definition 3.3 is satisfied Indeed, let u (·) be solution of Eq (3.2) such that the function u s (θ) ∈ S s Then, by Lemma 3.4, the solution u (t ) for t ∈ [s, ∞) can be rewritten in the form ∞ u (t ) = U (t , s)ν0 + G (t , τ ) f (τ , u τ ) dτ for some ν0 ∈ Im P (s) s Thus, for t s and θ ∈ [−r , 0] we have ∞ u (t − θ) = U (t − θ, s)ν0 + G (t − θ, τ ) f (τ , u τ ) dτ s ∞ t = U (t − θ, s)ν0 + G (t − θ, τ ) f (τ , u τ ) dτ + s G (t − θ, τ ) f (τ , u τ ) dτ t ∞ t = U (t − θ, s)ν0 + U (t − θ, τ ) P (τ ) f (τ , u τ ) dτ + s G (t − θ, τ ) f (τ , u τ ) dτ t ∞ t = U (t − θ, t ) U (t , s)ν0 + U (t , τ ) P (τ ) f (τ , u τ ) dτ s G (t − θ, τ ) f (τ , u τ ) dτ + t t s Put μ0 = U (t , s)ν0 + U (t , τ ) P (τ ) f (τ , u τ ) dτ We have P (t )μ0 = μ0 , hence belongs to Im P (t ) and μ0 ∈ Im P (t ) We thus obtain that U (t − θ, t )μ0 ∞ u (t − θ) = U (t − θ, t )μ0 + G (t − θ, τ ) f (τ , u τ ) dτ t By the uniqueness of u (·) on [s − r , ∞) as in the proof of Theorem 3.6 we have that Eq (3.2) has a unique solution u (·) on [−r + t , ∞) satisfying ( P (t )ut )(θ) = U (t − θ, t )μ0 and ∞ u (ξ ) = U (2t − ξ, t )μ0 + G (2ξ − t , τ ) f (τ , u τ ) dτ t for ξ ∈ [−r + t , t ] Therefore, the history function ut can be viewed as ∞ ut (θ) = u (t + θ) = U (t − θ, t )μ0 + G (t − θ, τ ) f (τ , u τ ) dτ = φ(θ) + Φt (φ)(θ) t Hence, ut ∈ S t for t s ✷ JID:YJMAA AID:17976 /FLA [m3G; v 1.114; Prn:23/10/2013; 10:53] P.10 (1-13) T.H Nguyen, V.D Trinh / J Math Anal Appl ••• (••••) •••–••• 10 Exponential trichotomy and center-stable manifolds In this section, we will generalize Theorem 3.7 to the case that the evolution family {U (t , s)}t s has an exponential trichotomy on R+ and the nonlinear forcing term f is ϕ -Lipschitz In this case, under similar conditions as in the above section we will prove that there exists a center-stable manifold for the solutions to Eq (3.2) We now recall the definition of an exponential trichotomy Definition 4.1 A given evolution family {U (t , s)}t s is said to have an exponential trichotomy on the half-line if there are three families of projections { P j (t )}, t 0, j = 1, 2, 3, and positive constants N , α , β with α < β such that the following conditions are fulfilled: supt P j (t ) < ∞, j = 1, 2, P (t ) + P (t ) + P (t ) = Id for t and P j (t ) P i (t ) = for all j = i P j (t )U (t , s) = U (t , s) P j (s) for t s and j = 1, 2, U (t , s)|Im P j (s) are homeomorphisms from Im P j (s) onto Im P j (t ), for all t denote the inverse of U (t , s)|Im P j (s) by U (s, t )| , s t (v) For all t s and x ∈ X , the following estimates hold: (i) (ii) (iii) (iv) U (t , s) P (s)x Ne −β(t −s) P (s)x , U (s, t )| P (t )x Ne −β(t −s) P (t )x , U (t , s) P (s)x Ne α (t −s) P (s)x The projections { P j (t )}, t constants s and j = 2, 3, respectively; we also 0, j = 1, 2, 3, are called the trichotomy projections, and the constants N , α , β – the trichotomy Using the trichotomy projections we can now construct three families of projections { P j (t )}, t follows: 0, j = 1, 2, 3, on C as P j (t )φ (θ) = U (t − θ, t ) P j (t )φ(0) for all θ ∈ [−r , 0] and φ ∈ C (4.1) We come to our second main result It proves the existence of a center-stable manifold for solutions of Eq (3.2) Theorem 4.2 Let the evolution family {U (t , s)}t s have an exponential trichotomy with the trichotomy projections { P j (t )}t , j = 1, 2, 3, and constants N , α , β given as in Definition 4.1 Suppose that f : R+ × C → X is ϕ -Lipschitz, where ϕ is a positive function which belongs to the admissible space E Set q := sup{ P j (t ) : t 0, j = 1, 3}, N := max{ N , 2Nq}, and k := (1 + H )e ν r N N Λ1 T 1+ ϕ − e −ν ∞ + N Λ1 ϕ ∞ (4.2) Then, if k < 1+ N1 eν r , for each fixed δ > α there exists a center-stable manifold S = {(t , S t )}t which is represented by a family of Lipschitz continuous mappings ⊂ R+ × C for the solutions to Eq (3.2), Φt : Im P (t ) + P (t ) → Im P (t ) with Lipschitz constants being independent of t such that S t = graph(Φt ) has the following properties: (i) S t is homeomorphic to Im( P (t ) + P (t )) for all t (ii) To each φ ∈ S s there corresponds one and only one solution u (t ) to Eq (3.2) on [s − r , ∞) satisfying e −γ (s+θ) u s (θ) = φ(θ) for θ ∈ [−r , 0] and supt s e −γ (t +·) ut (·) C < ∞, where γ = δ+2α Moreover, for any two solutions u (t ) and v (t ) to Eq (3.2) corresponding to different functions φ1 , φ2 ∈ S s we have the estimate ut − v t C C μ e (γ −μ)(t −s) P (s)φ1 (0) − P (s)φ2 (0) for t s (4.3) where μ and C μ are positive constants independent of s, u (·), and v (·) (iii) S is positively invariant under Eq (3.2) in the sense that, if u (t ), t s − r, is the solution to Eq (3.2) satisfying the conditions that the function e −γ (s+·) u s (·) ∈ S s and supt s e −γ (t +·) ut (·) C < ∞, then the function e −γ (t +·) ut (·) ∈ S t for all t s Proof Set P (t ) := P (t ) + P (t ) and Q (t ) := P (t ) = Id − P (t ) for t We have that P (t ) and Q (t ) are projections complemented to each other on X We then define the families of projections { P j (t )}, t 0, j = 1, 2, 3, on C as in equality (4.1) Setting P (t ) = P (t ) + P (t ) and Q (t ) = P (t ), t 0, we obtain that P (t ) and Q (t ) are complemented projections on C for each t We consider the following rescaling evolution family U (t , s) = e −γ (t −s) U (t , s) for all t s JID:YJMAA AID:17976 /FLA [m3G; v 1.114; Prn:23/10/2013; 10:53] P.11 (1-13) T.H Nguyen, V.D Trinh / J Math Anal Appl ••• (••••) •••–••• 11 We now prove that the evolution family U (t , s) has an exponential dichotomy with dichotomy projections P (t ), t Indeed, P (t )U (t , s) = e −γ (t −s) P (t ) + P (t ) U (t , s) = e −γ (t −s) U (t , s) P (s) + P (s) = U (t , s) P (s) Since U (t , s)|Im P (s) is a homeomorphism from Im P (s) onto Im P (t ) and Im P (t ) = Ker P (t ) for all t 0, we have that U (t , s)|Ker P (s) is also a homeomorphism from Ker P (s) onto Ker P (t ), and we denote U (s, t )| := (U (t , s)|Ker P (s) )−1 for s t By the definition of exponential trichotomy we have e −(β+γ )(t −s) Q (t )x U (s, t )| Q (t )x for all t s On the other hand, U (t , s) P (s)x = e −γ (t −s) U (t , s) P (s) + P (s) x Ne −γ (t −s) e −β(t −s) P (s)x + e α (t −s) P (s)x = Ne −γ (t −s) e −β(t −s) P (s) P (s)x + e α (t −s) P (s) P (s)x for all t s and x ∈ X Putting q := sup{ P j (t ) , t 2Nqe − U (t , s) P (s)x 0, j = 1, 3}, we finally get the following estimate (δ−α ) (t −s) P (s)x Therefore, U (t , s) has an exponential dichotomy with the dichotomy projections P (t ), t max{ N , 2Nq}, ν := δ−2α Put x˜ (t ) = e −γ t x(t ), and define the mapping F : R+ × C → X as follows F (t , φ) = e −γ t f t , e γ (t +·) φ(·) 0, and constants N := for (t , φ) ∈ R+ × C ϕ -Lipschitz Thus, we can rewrite Eq (3.2) in the new form ⎧ t ⎪ ⎪ ⎪ ⎨ x˜ (t ) = U (t , s)˜x(s) + U (t , ξ ) F (ξ, x˜ ) dξ for all t s 0, ξ ⎪ s ⎪ ⎪ ⎩ x˜ s (·) = e −γ (s+·) φ(·) ∈ C Obviously, F is also (4.4) Hence, by Theorem 3.7, we obtain that, if k < 1+ N1 eν r , then there exists an invariant stable manifold S for the solutions to Eq (4.4) Returning to Eq (3.2) by using the relation x(t ) := e γ t x˜ (t ) and Theorems 3.6, 3.7, we can easily verify the properties of S which are stated in (i), (ii), (iii) and (iv) Thus, S is a center-stable manifold for the solutions of Eq (3.2) ✷ Example 4.3 We consider the problem ⎧ ⎪ n ⎪ ∂ ∂ ∂ ⎪ −α t ⎪ ⎪ akl (t , x) u (t , x) + δ u (t , x) + bte ln + u (t + θ, x) dθ u (t , x) = ⎪ ⎪ ∂t ∂ xk ∂ xl ⎪ ⎪ k,l=1 ⎨ −r n ⎪ ∂ ⎪ ⎪ nk (x)akl (t , x) u (t , x) = 0, x ∈ ∂Ω, ⎪ ⎪ ∂ xl ⎪ ⎪ k,l=1 ⎪ ⎪ ⎩ u s (θ, x) = u (s + θ, x) = φ(θ, x), θ ∈ [−r , 0], x ∈ Ω for t s 0, x ∈ Ω, (4.5) Here, Ω is a bounded domain in Rn with smooth boundary ∂Ω oriented by outer unit normal vector n(x) The coefficients μ akl (t , x) ∈ C b (R+ , C (Ω)) ∩ C b (R+ , C (Ω)), μ > 12 , are supposed to be real, symmetric, and uniformly elliptic in the sense that n akl (t , x) v k v l η| v |2 , for all x ∈ Ω and for some constant η > k,l=1 Finally, the constants differential operator α > 0, b = and δ > is large enough We now choose the Hilbert space X = L (Ω) and define the n A (t , x, D ) = k,l=1 ∂ ∂ akl (t , x) ∂ xk ∂ xl +δ JID:YJMAA AID:17976 /FLA [m3G; v 1.114; Prn:23/10/2013; 10:53] P.12 (1-13) T.H Nguyen, V.D Trinh / J Math Anal Appl ••• (••••) •••–••• 12 with domain n D A (t ) = f ∈ W 2,2 (Ω): ∂ f (x) = 0, x ∈ ∂Ω ∂ xl nk (x)akl (t , x) k,l=1 Therefore, this problem can rewrite as an abstract Cauchy problem ⎧ ⎨ d ⎩ u (t , ·) = A (t )u (t , ·) + F t , ut (θ, ·) dt u s (θ, ·) = φ(θ, ·) ∈ C for t s 0, for θ ∈ [−r , 0] where F : R+ × C → X is defined by F (t , φ)(x) = bte −α t ln + x ∈ Ω φ(θ) (x) dθ, −r We have F (t , φ)(·) ∈ X because by Minkowski’s inequality it follows that F (t , φ)(x) dx = |b|te −αt ln + Ω φ(θ) (x) dθ −α t 2 ln + −r φ(θ) (x) dx −α t 2 φ(θ) (x) dx −r dθ Ω |b|te dx −r Ω |b|te 2 dθ Ω = |b|te −αt φ(θ) dθ < ∞ −r By Schnaubelt [13, Theorem 3.3, Example 4.2], the family of operators ( A (t ))t generates an evolution family having an exponential dichotomy with the dichotomy constants N , ν provided that the Hölder constants of akl are sufficiently small N Also, the dichotomy projections P (t ), t 0, satisfy supt P (t ) We now check that F is ϕ -Lipschitz with ϕ (t ) = |b|rte −αt ∈ E = L p (R+ ), p Indeed, the condition (i) is evident To verify the condition (ii) we use Minkowski’s inequality and the fact that ln(1 + h) h for all h Then, F (t , φ1 )(x) − F (t , φ2 )(x) = |b|te −αt ln −r Ω |b|te −α t −r ln ln = |b|te −r Ω + |(φ2 (θ))(x)| + |(φ1 (θ))(x)| + |(φ2 (θ))(x)| Ω −α t + |(φ1 (θ))(x)| 2 dθ dx dx dθ |(φ1 (θ))(x)| − |(φ2 (θ))(x)| 1+ dx + |(φ2 (θ))(x)| |b|te −αt φ1 (θ) (x) − φ2 (θ) (x) dx −r Ω = |b|te −αt φ1 (θ) − φ2 (θ) dθ −r |b|rte −αt sup θ ∈[−r ,0] φ1 (θ) − φ2 (θ) dθ dθ JID:YJMAA AID:17976 /FLA [m3G; v 1.114; Prn:23/10/2013; 10:53] P.13 (1-13) T.H Nguyen, V.D Trinh / J Math Anal Appl ••• (••••) •••–••• Hence, F is have 13 ϕ -Lipschitz In the space L p (R+ ), the constants N , N in Definition 2.3 are defined by N = N = Also, we t +1 t ϕ (τ ) dτ and Λ1 T 1+ ϕ (t ) = Λ1 ϕ (t ) = ϕ (τ ) dτ (t −1)+ t where (t − 1)+ = max{0, t − 1} Thus, max Λ1 ϕ ∞, Λ1 T 1+ ϕ ∞ < |b|r (1 + e −1 − e −α ) α2 By Theorem 3.7 we obtain that if |b|r (1 + e 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