INTEGRAL MANIFOLDS OF NONAUTONOMOUS BOUNDARY CAUCHY PROBLEMS

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1 Journal of Nonlinear Evolution Equations and Applications ISSN Volume 212, Number 1, pp (January 212) INTEGRAL MANIFOLDS OF NONAUTONOMOUS BOUNDARY CAUCHY PROBLEMS T. S. DOAN Department of Mathematics, Center for Dynamics, TU Dresden, 162 Dresden, Germany & Institute of Mathematics, Vietnam Academy of Science and Technology, 18 Hoang Quoc Viet Road, Hanoi, Viet Nam M. MOUSSI Department of Mathematics and Informatics, Faculty of Sciences, University Mohammed I, 6 Oujda, Morocco S. SIEGMUND Department of Mathematics, Center for Dynamics, TU Dresden, 162 Dresden, Germany Received August 22, 211 Accepted November 22, 211 Communicated by Khalil Ezzinbi Abstract. The existence of integral manifolds for nonlinear boundary Cauchy problems is established using an extension of the variation of constants formula recently established in [4]. Examples include nonautonomous structured population equations and nonautonomous retarded differential equations. Keywords: nonautonomous boundary Cauchy problem; integral manifold; structured population equation; retarded differential equation. 21 Mathematics Subject Classification: 34K1, 34K19. address: dtson@tu-dresden.de address: m.moussi@fso.ump.ma address: stefan.siegmund@tu-dresden.de c 212 Journal of Nonlinear Evolution Equations and Applications, JNEEA.com

2 2 T. S. Doan, M. Moussi and S. Siegmund, J. Nonl. Evol. Equ. Appl. 212 (212) Introduction Consider the nonlinear boundary Cauchy problem for arbitrary R + = [, ) d dt u(t) = A max(t)u(t), t [, ), L(t)u(t) = f(t, u(t)), t [, ), u() = x, where A max (t) is a closed operator on a Banach space X endowed with a maximal domain D(A max (t)), and L(t) : D(A max (t)) X, with a boundary space X and a function f : R + X X, the solution u : [, ) X takes the initial value x X at time. This type of equation has recently been suggested and investigated as a model class with various applications like population equations, retarded differential (difference) equations, heat equations and boundary control problems (see e.g. [1, 3] and the references therein). The corresponding linear boundary Cauchy problem of (1.1) is given by d dt u(t) = A max(t)u(t), L(t)u(t) =, u() = x. t [, ), t [, ), In the autonomous case these abstract Cauchy problems were first studied by Greiner [6, 7, 8] and Thieme [13], e.g. by using perturbation results for the domains of semigroups. The homogeneous boundary Cauchy problem (1.2) has been investigated by Kellermann [9] and Nguyen Lan [1]. In these papers, the authors proved the existence of solutions to these problems and generation of an evolution family. In [4] the authors have studied the boundary Cauchy problem in the case that the first equation in (1.1) is replaced by an inhomogeneous equation d dt u(t) = A max(t)u(t) + g(t) and f in the second equation is replaced by f(t, u(t)) f(t). For this type of equation they established a variation of constants formula which can be easily extended to a variation of constants formula for (1.1) using the contraction fixed point theorem. Utilizing the variation of constants formula we will follow the Lyapunov-Perron approach to develop an invariant manifold theory for the class of equations (1.1). The structure of the paper is as follows: In Section 2 we list natural assumptions for wellposedness of equation (1.1), the concepts of mild solution and exponential splitting. Moreover, we cite two examples illustrating our abstract problem and general assumptions. Section 3 is devoted to an invariant manifold theorem for (1.1) which yields sufficient conditions for the existence of e.g. a stable or unstable manifold. To conclude the introductory section, we collect notation used in this paper. For Banach spaces X, Y, let L(X, Y ) denote the space of all linear bounded operators from X to Y, define L(X) := L(X, X). We denote by id X the identity map defined on X. By C b (R +, X) we denote the space of all continuous and bounded functions from R + into X. Let A : D(A) X X be a closed linear operator, we denote by ρ(a) := {λ C λ id X A : D(A) X is bijective the resolvent set of A. For λ ρ(a), the operator R(λ, A) := (λ id X A) 1 is called the resolvent of A. (1.1) (1.2)

3 INTEGRAL MANIFOLDS OF NONAUTONOMOUS CAUCHY PROBLEMS 3 Finally, for a measurable set Ω R n and 1 p <, let L p (Ω) denote the space of all measurable functions from Ω to R n satisfying that ( u p := u(x) p dx Ω ) 1 p <. Let L (Ω) denote the space of all essentially bounded measurable functions. The Sobolev space W 1,1 (Ω) is given by W 1,1 (Ω) = { u L 1 (Ω) u L 1 (Ω), where the derivative u is defined in the weak sense. Let W 1,1 (Ω) be endowed with the norm u W 1,1 := u 1 + u 1. 2 Preliminaries In this section we recall some definitions and results, formulate assumptions and discuss some examples. 2.1 Linear nonautonomous boundary Cauchy problems A family of linear (unbounded) operators (A(t)) t T defined on a Banach space X is called a stable family if there are constants M 1 and ω R such that (ω, ) ρ(a(t)) for all t T and k R(λ, A(t i )) M(λ ω) k i=1 for λ > ω and any finite sequence t 1 t k T. Remark 1 In the autonomous case (A(t) = A), suppose that A generates a strongly continuous semigroup. Then, by the Hille-Yosida Theorem (see [5, Theorem II.3.8]) A is stable. A family of linear bounded operators (U(t, s)) t s J, J := R + or R, on a Banach space X is called evolution family if (1) U(t, s) = U(t, r)u(r, s) and U(s, s) = id X for all t r s J, (2) the mapping {(t, s) J J : t s (t, s) U(t, s) L(X) is strongly continuous. The growth bound of (U(t, s)) t s is defined by ω(u) := inf { ω R : M ω 1 with U(t, s) M ω e ω(t s) t s J. The evolution family (U(t, s)) t s is called exponentially bounded provided that ω(u) <. We now turn to the notion of exponential splitting for an evolution family.

4 4 T. S. Doan, M. Moussi and S. Siegmund, J. Nonl. Evol. Equ. Appl. 212 (212) 1 15 Definition 2 (Exponential Splitting) An evolution family (U(t, s)) t s on a Banach space X has an exponential splitting with exponents α < β, if there exist projections P (t), t R +, being uniformly bounded and strongly continuous and a constant N 1 such that (1) P (t)u(t, s) = U(t, s)p (s) for all t s, (2) the restriction U Q (t, s) : Q(s)X Q(t)X is invertible for t s and we set U(s, t) := [U Q (t, s)] 1, where Q(t) := id X P (t). (3) U(t, s)p (s) Ne α(t s) and [U Q (t, s)q(s)] 1 Ne β(t s) for all t s. Let X, D, X be Banach spaces such that D is dense and continuously embedded in X. On these spaces, the operators A max (t) L(D, X), L(t) L(D, X), for t, are supposed to satisfy the following hypotheses: (H1) There are positive constants C 1, C 2 such that for all x D and t ; C 1 x D x + A max (t)x C 2 x D (H2) for each x D the mapping R + t A max (t)x X is continuously differentiable; (H3) the operators L(t) : D X, t, are surjective; (H4) for each x D the mapping R + t L(t)x X is continuously differentiable; (H5) there exist constants γ > and ω R such that L(t)x X γ 1 (λ ω) x X, for x ker(λ id X A max (t)), λ > ω and t ; (H6) the family of operators (A(t)) t, A(t) := A max (t) ker L(t), generates an evolution family (U(t, s)) t s. In the following lemma, we cite consequences of the above assumptions from [6, Lemma 1.2] which will be needed below. Lemma 3 The restriction L(t) ker(λ idx A max(t)) is an isomorphism from ker(λ id X A max (t)) into X and its inverse L λ,t := [L(t) ker(λ idx A max(t))] 1 : X ker(λ id X A max (t)) satisfies L λ,t γ(λ ω) 1 for λ > ω and t. To illustrate sufficient and natural conditions which imply the assumptions (H1) (H6) for application-relevant classes of boundary Cauchy problems we discuss examples of a nonautonomous structured population equation and a nonautonomous functional differential equation.

5 INTEGRAL MANIFOLDS OF NONAUTONOMOUS CAUCHY PROBLEMS 5 Example 4 (Nonautonomous Structured Population Equation) Consider a nonautonomous population equation t u(t, a, x) = u(t, a, x) µ(a, x)u(t, a, x) + A(a, x)u(t, a, x), a t s, a, x Ω, (2.1) u(t,, x) = β(t, a, x)u(t, a, x) da, t s, x Ω, u(s, a, x) = f(a, x), a, x Ω. Here Ω is a bounded domain in R n, the function u(t, a, x) represents the density of individuals of the population of age a and size x at time t. The functions µ and β correspond to the aging and the birth rates, respectively. Finally, we note that this equation is a special case of the very general nonautonomous population equation with diffusion treated by Rhandi and Schnaubelt in [12]. We impose the following conditions: (i) A(a, ) L (Ω) for all a and A(, ) C b (R +, L (Ω)). Moreover, (A(a, )) a is a family of operators generating an exponentially bounded evolution family U(a, r) a r on the Banach space L 1 (Ω). (ii) µ C b (R +, L (Ω)). (iii) β C 1 (R +, L (R + Ω) L 1 (R + Ω)) the space of continuously differentiable functions from R + into L (R + Ω) L 1 (R + Ω). Our aim is to write equation (2.1) as a boundary Cauchy problem of the form (1.2) satisfying the hypotheses (H1) (H6). For this purpose, we define the Banach spaces X := L 1 (Ω), X := L 1 (R +, X) L 1 (R + Ω) and D := W 1,1 (R +, X), and for each t the operator A max (t) : X X by D(A max (t)) = D and for all ϕ D, where For each t, we define L(t) : D X by where Φ(t) : X X given by (A max (t)ϕ)(a) = ϕ(a, ) + B(a, )ϕ(a, ) (2.2) a B(a, )ϕ(a, ) := A(a, )ϕ(a, ) µ(a, )ϕ(a, ). (2.3) L(t)ϕ = ϕ(, ) Φ(t)ϕ for all ϕ D, (2.4) Φ(t)ϕ := β(t, a, )ϕ(a, ) da. It is obvious to see that Φ(t) L(X, X). We show now that the hypotheses (H1) (H6) are satisfied:

6 6 T. S. Doan, M. Moussi and S. Siegmund, J. Nonl. Evol. Equ. Appl. 212 (212) 1 15 Verification of (H1): Since A( ) C b (R +, L (Ω)) and µ( ) C b (R +, L (Ω)) it follows that A := sup a A(a) < and µ := sup µ(a) <. a Let ϕ D be arbitrary. From the definition of D, we have ϕ D = On the other hand, ϕ(a) 1 da + ϕ X + A max (t)ϕ X + ϕ (a) 1 da + (1 + A + µ )( ϕ X + A max (t)ϕ X ). A(a)ϕ(a) µ(a)ϕ(a) 1 da ϕ X + A max (t)ϕ X = ϕ X + ϕ + A( )ϕ µ( )ϕ X (1 + A + µ ) ϕ D. This shows the assumption (H1) with C 1 = (1 + A + µ ) 1 and C 2 = (1 + A + µ ). Verification of (H2): From (2.2), we derive that A max (t) is independent of t. Therefore, the map t A max (t)ϕ is continuously differentiable for each fixed ϕ X. Verification of (H3): See Appendix, Lemma 11. Verification of (H4): From continuous differentiability of β, we derive that for each ϕ D the mapping from R + X, t L(t)ϕ defined as in (2.4) is also continuously differentiable. Verification of (H5): Define a family of linear operators (C(a)) a on X by C(a)ϕ := µ(a, )ϕ. From (ii), we have µ C b (R +, L (Ω)) and therefore sup C(a) 1 sup µ(a, ) <, a R + a R + which together with (i) implies that the family of operators (B(a, )) a, given by (2.4), generates on X an exponentially bounded evolution family (V (t, s)) t s given by V (t, s)ϕ = U(t, s)ϕ + t s U(t, σ)c(σ)v (σ, s) dσ, for all t s and ϕ X. Then (H5) is an application of Appendix, Lemma 12. Verification of (H6): The corresponding evolution semigroup (T (t)) t of the evolution family (V (a, r)) a r is given by { V (a, a t)ϕ(a t) a t, (T (t)ϕ)(a) = (2.5), a < t, for ϕ X. One can show that (T (t)) t is a strongly continuous semigroup. Its generator denoted by A is a restriction of A max (t) with D(A ) = {ϕ D ϕ() =.

7 INTEGRAL MANIFOLDS OF NONAUTONOMOUS CAUCHY PROBLEMS 7 Thus, according to Remark 1 we obtain that A is stable. Since A(t) := A max (t) ker L(t) is a bounded perturbation of A with A(t)ϕ = A ϕ, D(A(t)) = {ϕ D ϕ() = Φ(t)ϕ it follows together with [1, Theorem 2.3] that A(t) generates an evolution family. Hence, (H6) is satisfied. Example 5 (Nonautonomous functional differential equation) { d dtx(t) = B(t)x(t), t s, x s = ϕ C([ r, ], E). (2.6) Here B(t) is defined on a Banach space E. Furthermore, r, ϕ C([ r, ], E) and the retarded function x s is defined as x s () := x(s + ) for [ r, ]. We assume the following conditions: (i) The family of linear operators B(t), t, is stable and generates an evolution family (U(t, s)) t s satisfying U(t, s) Me ω(t s), t s ; (ii) the domain D(B(t)) := D B is independent of t, B() is a closed operator in E and the function t B(t)x is continuously differentiable for all x E. Define the Banach spaces X := C([ r, ], E), X := E and D := { ϕ C 1 ([ r, ], E) such that ϕ() D B endowed with the norm ϕ := ϕ C 1 ([ r,],e]) + B()ϕ(). For each t, we define the operator A max (t) : X X with D(A max ) = D by and the operator L(t) : D(A max (t)) X by A max (t)ϕ = ϕ for all ϕ D, x L(t)ϕ = ϕ () B(t)ϕ() for all ϕ D. Then, the above retarded differential equation (2.6) can be written as a linear boundary Cauchy problem (1.2). For more details, we refer the reader to [1].

8 8 T. S. Doan, M. Moussi and S. Siegmund, J. Nonl. Evol. Equ. Appl. 212 (212) Nonlinear boundary Cauchy problems In case f the boundary Cauchy problem (1.1) reduces to the linear boundary Cauchy problem (1.2) which was studied in the last subsection under the assumptions (H1) (H6). In particular, let (U(t, s)) t s denote the evolution family from (H6). We want to study nonlinear perturbations (1.1) of (1.2) and therefore assume that the nonlinearity f is not too far away from : (H7) The nonlinear part f : R + X X is assumed to be continuous, satisfies that f(t, ) = for all t R + and there exists a positive constant l such that one has the global Lipschitz estimate f(t, x) f(t, x) l x x for all x, x X, t R +. Under the assumptions (H1) (H7) the semilinear boundary Cauchy problem (1.1) admits a unique mild solution. For R +, x X, a function u = u(,, x) : [, ) X is called mild solution of (1.1) if it satisfies the integral equation u(t,, x) = U(t, )x + lim U(t, σ)λl λ,σ f(σ, u(σ,, x)) dσ, t. (2.7) The unique existence follows with the usual contraction arguments (see e.g. [3, 7, 11]) and uses the variation of constants formula from [4] for solutions v : [, ) X of inhomogeneous boundary Cauchy problems, i.e. systems (1.1) with f(t, u(t)) g(t) independent of u(t) v(t) = U(t, )x + lim where L λ,σ is defined as in Lemma 3. U(t, σ)λl λ,σ g(σ) dσ, t, 3 Integral Manifolds of Nonlinear Boundary Cauchy Problems In this section, we consider the following system { d dt u(t) = A max(t)u(t), t [, ), L(t)u(t) = f(t, u(t)), t [, ), where A max (t), L(t), f(t, x) are assumed to satisfy assumptions (H1) (H6). For R + and x X let u(,, x) denote the mild solution of (3.1) satisfying that u() = x. In case the evolution family (U(t, s)) t s of the corresponding linear boundary Cauchy problem has an exponential splitting with exponents α < β, projections P ( ), Q( ) = I P ( ) and constant N then for all ζ (α, β) the two sets { M s ζ = M u ζ = (, ξ) R + X : R +, ξ P ()X { (, ξ) R + X : R +, ξ Q()X are called (pseudo)-stable and (pseudo)-unstable vector bundles or manifolds, they consist of solutions which are exponentially bounded from above and below, respectively, in the sense of Definition 2. In case α < < β they are called stable and unstable, respectively. Our aim in this section is to construct nonlinear analogues of M s ζ and Mu ζ by using the Lyapunov-Perron approach as e.g. in [2]. Since solution curves are sometimes also called integral curves and because M s ζ and Mu ζ are invariant manifolds, i.e. consist of solution curves, they are also called integral manifolds.,, (3.1)

9 INTEGRAL MANIFOLDS OF NONAUTONOMOUS CAUCHY PROBLEMS Pseudo-Stable Manifolds Consider the nonlinear boundary Cauchy problem (3.1) which satisfies additionally assumption (H7). By (H7) equation (3.1) has the zero solution. We show that for each fixed R + the set of mild solutions ϕ C([, ), X) of (3.1) which converge to zero exponentially fast as t forms a so-called stable manifold which is the graph of a Lipschitz-continuous chart. In fact, we prove more generally, that sets of solutions which are exponentially bounded by Ne ζ(t ) for t form manifolds, so-called ζ-pseudo-stable manifolds. To this end choose ζ R, R +. Then the set X +,ζ {ϕ (X) := C([, ), X) : sup e ζ( t) ϕ(t) < t is a Banach space with respect to the norm ϕ,ζ := sup e ζ( t) ϕ(t). t Our overall approach is to characterize stable manifolds as a fixed point problem in X +,ζ (X). Thereto we define the Lyapunov-Perron Operators T + : X +,ζ (X) X X +,ζ (X) by T + (ϕ, x)(t) := U(t, )P ()x + lim where the Green s function G is defined by { U(t, s)p (s) for t s, G(t, s) := U(t, s)q(s) for t s. G(t, σ)λl λ,σ f(σ, ϕ(σ)) dσ, (3.2) We also write T + (t, ϕ, x) = T + (ϕ, x)(t) for all (t, x) [, ) X. Some fundamental properties of the operator T + are established in the following proposition. Proposition 6 Suppose that the evolution family (U(t, s)) t s has an exponential splitting with exponents α < β, projections P ( ) and constant N. Then for all [, ) the following assertions hold: (i) For any ζ (α, β), the Lyapunov-Perron operator T + : X + ζ, (X) X X + ζ, (X) defined as in (3.2) is well-defined. (ii) For any ζ (α, β), let ϕ X +,ζ (X) and ξ im P (). Then, the following statements are equivalent: (a) ϕ is the mild solution of (3.1) with P ()ϕ() = ξ, (b) ϕ is the fixed point of the Lyapunov-Perron operator T + (, ξ) : X +,ζ (X) X +,ζ (X) defined as in (3.2). (iii) Suppose that Nlγ < β α 2 and choose and fix η ( Nlγ 2, β α 2 ). Then, for any ζ [α+η, β η] the Lyapunov-Perron operator is uniformly contractive in the first component. More precisely, for all ϕ 1, ϕ 2 X +,ζ (X) and ξ 1, ξ 2 X we have T + (, ϕ 1, ξ 1 ) T + (, ϕ 2, ξ 2 ),ζ N P ()(ξ 1 ξ 2 ) + 2Nlγ ϕ 1 ϕ 2,ζ. (3.3) η

10 1 T. S. Doan, M. Moussi and S. Siegmund, J. Nonl. Evol. Equ. Appl. 212 (212) 1 15 Proof. (i) Let ϕ X +,ζ (X) and ξ X. An elementary computation yields that for all t ( 1 e ζ(t ) T + (t, ϕ, ξ) N P ()ξ + Nlγ ζ α + 1 ) ϕ,ζ, β ζ where we use the fact that lim λl λ,σ γ, see Lemma 3. Therefore, the operator T + is welldefined. (ii) (a) (b): Since ϕ is a mild solution of (3.1) it follows that ϕ(t) = U(t, )ϕ() + lim Together with P ()ϕ() = ξ we get t U(t, σ)λl λ,σ f(σ, ϕ(σ)) dσ. ϕ(t) = U(t, )ξ + lim U(t, σ)p (σ)λl λ,σ f(σ, ϕ(σ)) dσ + ( t ) U(t, )Q() ϕ() + lim U(, σ)q(σ)λl λ,σ f(σ, ϕ(σ)) dσ. Hence, ϕ X +,ζ with ζ (α, β) implies that ϕ() + lim U(, σ)q(σ)λl λ,σ f(σ, ϕ(σ)) dσ =, which concludes that ϕ = T + (ϕ, ξ) and the first implication is proved. (b) (a): Since ϕ is the fixed point of T + (, ξ) it follows that ϕ(t) = U(t, )P ()ξ + lim Replacing t by in the above equality yields that Therefore, we get ϕ() = ξ lim ϕ(t) = U(t, )ϕ() + lim which completes the proof of this part. (iii) From (3.2), we derive that G(t, σ)λl λ,σ f(σ, ϕ(σ)) dσ. U(, σ)q(σ)λl λ,σ f(σ, ϕ(σ)) dσ. t T + (t, ϕ 1, ξ 1 ) T + (t, ϕ 2, ξ 2 ) = U(t, )P ()(ξ 1 ξ 2 ) + + lim U(t, σ)λl λ,σ f(σ, ϕ(σ)) dσ, G(t, σ)λl λ,σ (f(σ, ϕ 1 (σ)) f(σ, ϕ 2 (σ))) dσ. This together with the fact that the evolution family (U(t, s)) t s has an exponential splitting with exponents α < β, projections P ( ) and constant N and the Lipschitz continuity of f implies that e ζ( t) T + (t, ϕ 1, ξ) T + (t, ϕ 2, ξ) N P ()(ξ 1 ξ 2 ) + [ t ] +Nlγ e η(σ t) dσ + e η(t σ) dσ ϕ 1 ϕ 2,ξ. t

11 Therefore, sup t INTEGRAL MANIFOLDS OF NONAUTONOMOUS CAUCHY PROBLEMS 11 e ζ( t) T + (t, ϕ 1, ξ) T + (t, ϕ 2, ξ) N P ()(ξ 1 ξ 2 ) + 2Nlγ ϕ 1 ϕ 2,ξ, η which completes the proof. Definition 7 (Pseudo-Stable Manifolds) For any ζ R, the ζ-pseudo stable manifold is defined as { Wζ s := (, x) R + X : ϕ X +,ζ. (X) is mild sol. of (3.1) and P ()ϕ() = x We are now ready to state the main result on the existence of pseudo-stable manifolds for nonlinear boundary Cauchy problems. Theorem 8 (Pseudo-stable Manifold Theorem) Assume that (3.1) satisfies the assumptions (H1) (H7) and suppose that the corresponding evolution family (U(t, s)) t s has an exponential splitting with exponents α < β, projections P ( ) and constant N. Furthermore, we assume that Nlγ < β α. Choose and fix η ( Nlγ 2, β α 2 ). Then for any ζ [α + η, β η], the ζ-pseudo stable manifold Wζ s has the following representation Wζ {(, s = ξ + s + (, ξ)) R + X : R +, ξ P ()X, (3.4) with for each R + the uniquely determined continuous mapping s + (, ) : P ()X X given by s + (, ξ) = lim G(, σ)λl λ,σ f(σ, ϕ(σ)) dσ, (3.5) where ϕ is the unique fixed point of T + (, ξ). Furthermore, for each R + the function s + (, ) satisfies s + (, ) and Lip(s + (, )) N 2 lγ η 2Nlγ. Proof. Let (, x) Wζ s, where R + and x X. Define ξ = P ()x. According to Definition 7 and Proposition 6(ii), we obtain that the mild solution ϕ of (3.1) with P ()ϕ() = x is the unique fixed point of the Lyapunov-Perron operator T + (, ξ). Therefore, x = ξ + lim G(, σ)λl λ,σ f(σ, ϕ(σ)) dσ. Then, x = ξ + s + (, ξ). Conversely, let ξ P ()X, where R +. We will show that ξ + s + (, ξ) Wζ s. In light of Proposition 6 (iii), the Lyapunov-Perron operator T+ (, ξ) is contractive and thus has a unique fixed point in X +,ξ (X) denoted by ϕ. This together with Proposition 6 (ii) implies that ϕ is a solution of (3.1) and therefore ϕ() = ξ + lim G(, σ)λl λ,σ f(σ, ϕ(σ)) dσ = ξ + s + (, ξ), which proves (3.4). Since is the fixed point of T + (, ) it follows that s + (, ) = for all R +. To conclude the proof, we prove the Lipschitz continuity of s with respect to the second

12 12 T. S. Doan, M. Moussi and S. Siegmund, J. Nonl. Evol. Equ. Appl. 212 (212) 1 15 argument. Let ξ 1, ξ 2 P ()(X) for a R +. Let ϕ 1, ϕ 2 X +,ζ (X) denote the fixed point of T + (, ξ 1 ), T + (, ξ 2 ), respectively. Using (3.3), we obtain that Therefore, This together with (3.5) implies that ϕ 1 ϕ 2,ζ N ξ 1 ξ 2 + 2Nlγ ϕ 1 ϕ 2,ζ. η ϕ 1 ϕ 2,ζ s + (, ξ 1 ) s + (, ξ 2 ) Nη η 2Nlγ ξ 1 ξ 2. N 2 lγ η 2Nlγ ξ 1 ξ 2, which completes the proof. 3.2 Pseudo-Unstable Manifolds In order to provide the definition of pseudo-unstable manifolds, we introduce the following space: for a given ζ R, R +, the set X,ζ {ϕ (X) := C((, ], X) : sup e ζ( t) ϕ(t) < t is a Banach space with respect to the norm ϕ,ζ := sup e ζ( t) ϕ(t). t Definition 9 (Pseudo-unstable Manifolds) For any ζ R, the ζ-pseudo unstable manifold W u ζ is the set of all (, x) R + X satisfying the following conditions: (i) For any t, there exists a y X (and hence unique due to uniqueness of solution) which is denoted by u(t,, x) such that u(, t, y) = x. (ii) u(,, x) X,ζ (X). The existence of pseudo-unstable manifold for nonlinear boundary Cauchy problem is stated and proved in the following theorem. Theorem 1 (Pseudo-unstable Manifold Theorem) Suppose that the evolution family (U(t, s)) t s associated with the corresponding linear system of (3.1) has an exponential splitting with exponents α < β, projections P ( ) and constant N and the nonlinear part satisfies (H7). Furthermore, we assume that Nlγ < β α. Choose and fix η ( Nlγ 2, β α 2 ). Then for any ζ [α + η, β η], the ζ-pseudo unstable manifold Wζ u has the following presentation Wζ {(, u = ξ + s (, ξ)) R + X : R +, ξ Q()X,

13 INTEGRAL MANIFOLDS OF NONAUTONOMOUS CAUCHY PROBLEMS 13 with for each R + the uniquely determined continuous mapping s (, ) : Q()X X given by s (, ξ) = lim G(, σ)λl λ,σ f(σ, ϕ(σ)) dσ, where ϕ is the unique fixed point of T (, ξ). Furthermore, for each R + the function s (, ) satisfies s (, ) and Lip(s (, )) N 2 lγ η 2Nlγ. Proof. Analog to the proof of Theorem 8. 4 Appendix Let Ω be a bounded set of R n and β : R + Ω R + satisfying that esssup β(a, x) < and β(a, x) da dx <. (4.1) (a,x) R + Ω R + Ω Set X := L 1 (Ω), X := L 1 (R +, X) Define the linear operator L : D X by and D := W 1,1 (R +, X). Lu(x) := u(, x) β(a, x)u(a, x) da for all x Ω. (4.2) In the following lemma, we state and prove some fundamental properties of the operator L. Lemma 11 The operator L defined as in (4.2) is bounded and surjective. Proof. We first show the boundedness of L. Since ϕ(, ) = ϕ(a, ) da for all ϕ D it a follows that L(t)ϕ 1 = ϕ(, ) β(a, )ϕ(a, ) da 1 ϕ(, ) 1 + β(a, )ϕ(a, ) da (1 + β ) ϕ W 1,1. 1 To prove the surjectivity of L, let f X be arbitrary. Define 2f(x) a u(a, x) := 1 + e 2 e 2 β(t,x) dt for all (a, x) R β(t,x) dt + Ω. We have u D. Furthermore, from (4.2) it is easy to see that Lu = f and therefore L is surjective.

14 14 T. S. Doan, M. Moussi and S. Siegmund, J. Nonl. Evol. Equ. Appl. 212 (212) 1 15 Lemma 12 Let (B(a)) a be a family of operators on X which generates an evolution family (V (a, r)) a r with a growth bound ω(v ) < +. Define an operator A max : X X by with the domain (A max ϕ)(a) = ϕ(a) + B(a)ϕ(a) (4.3) a D(A max ) = {ϕ D : ϕ(a) D(B(a)) for a.e. a R +, B( )ϕ( ) X. Then, the following statements hold: (i) For all λ C with Rλ > ω(v ), we have ker(λ id X A max ) = { e λ V (, )f f X. (ii) There exist constants γ > and ω R such that for λ C with Rλ > ω we have Lϕ X γ 1 (Rλ ω) ϕ X for all ϕ ker(λ id X A max ). Proof. (i) See [12]. (ii) Since the evolution family (V (a, r)) a r is exponentially bounded with the growth bound ω(v ), it follows that for each ω > ω(v ) there exists M ω 1 such that V (a, r) M ω e ω(a r) for all a r. (4.4) Take λ C such that Rλ > ω + M ω β and ϕ ker(λ id X A max ). From part (i), we get that ϕ( ) = e λ V (, )ϕ(). This together with (4.4) implies that ϕ X = = e λa V (a, )ϕ() X da M ω e (ω Rλ)a ϕ() X da M ω Rλ ω ϕ() X. (4.5) On the other hand, we have ϕ() X ϕ() β(a, )ϕ(a) da + X Lϕ X + β ϕ X, which together with (4.5) implies that ϕ X M ω Rλ (ω + M ω β ) Lϕ X, β(a, )ϕ(a) da X which proves the lemma with γ = M ω and ω = ω + M ω β.

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