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MINISTY OF EDUCATION AND TRAINING HANOI NATIONAL UNIVERSITY OF EDUCATION  BUI XUAN QUANG INERTIAL MANIFOLDS FOR CERTAIN CLASSES OF EVOLUTION EQUATIONS Speciality: Integral and Differential Equations Code: 9.46.01.03 SUMMARY OF Ph.D DISSERTATION IN MATHEMATICS Hanoi – 2020 This dissertation has been completed at the Hanoi National University of Education Scientific Advisors: Assoc Prof Dr habil Nguyen Thieu Huy Dr Tran Thi Loan Referee 1: Assoc Prof Dr Khuat Van Ninh Hanoi Pedagogical University Referee 2: Assoc Prof Dr Nguyen Xuan Thao Hanoi University of Science and Technology Referee 3: Assoc Prof Dr Le Van Hien Hanoi National University of Education The dissertation will be presented to the examining committee at the Hanoi National University of Education, 136 Xuan Thuy, Hanoi, Vietnam on at This dissertation is publicly available at HNUE Library Information Centre, the National Library of Vietnam Introduction Motivation Many phenomena in mechanics, physics, ecology, etc can be described by partial differential equations By choosing appropriate function spaces and linear operators, these partial differential equations can be rewritten into semi-linear evolution equations in an infinitedimensional Banach space whose linear part is the generator of a continuous semigroup and the nonlinear term satisfies the Lipschitz condition The investigation of the asymptotic behavior of solutions to partial differential equations in large time is one of the central problems of the theory for infinite dimensional dynamic systems An important tool for such investigation is the concept of inertial manifolds introduced in 1985 by C Foias, G.R Sell & R Temam (1985) when they studied the asymptotic behavior of solutions to Navier-Stokes equations An inertial manifold for an evolution equation is a (Lipschitz) finite-dimensional manifold which is positively invariant and exponentially attracts all other solutions of the equation This fact permits to invoke the reduction principle to study the asymptotic behavior of the solutions to evolution equations in infinite-dimensional spaces by comparing with that of the induced equations in spaces of finite-dimension Nguyen T.H (2012) proved the existence of inertial manifolds for the solutions to the semi-linear parabolic   du + Au = f (t, u), t > s, dt (1)  u(s) = us , when the partial differential operator A is positive definite and self-adjoint with a discrete spectrum having a sufficiently large distance between some two successive points of the spectrum, and the nonlinear forcing term f satisfies the ϕ-Lipschitz conditions These lead us to research the topic “Inertial manifolds for certain classes of evolution equations” 2.1 Overview of the Research Problems Historical Remarks – The existence of inertial manifolds The notion of inertial manifolds has been introduced by C Foias, G.R Sell & R Temam (1985) Inertial manifolds for evolution equations have been systematically studied in many works, for instance, Chow S.N & and Lu K (1988) considered general equations in Banach spaces with the nonlinear term f bounded and of class C , but the exponential attractivity towards the manifold was not proved to be uniform on bounded subsets of the phase space Mallet-Paret J & Sell G.R (1988) introduced the principle of spatial average to prove the existence of inertial manifolds for reaction-diffusion equations when the spectral gap condition is not fully satisfied Also, a more geometric proof was presented by Constantin P et al (1988, 1989) in the Hilbert space case using the concept of spectral barriers in an attempt to refine the spectral gap condition Demengel E & Ghidaglia J.M (1991) studied the Hilbert space case with A self-adjoint obtaining the first proof for the case where f is not bounded Debussche A & Temam R (1993) gave another proof for the case where f is not necessarily bounded, but now in the more general case of Banach spaces, and f is assumed to be of class C Other proofs for the nonself-adjoint case in Hilbert spaces were given by Debussche A & Temam R (1991) and Sell G.R & You Y (1992) A nice study of the role of the cone and strong squeezing conditions in the construction of inertial manifolds in Hilbert spaces was made by Robinson J.C (1993) Mora X (1989) considered damped semilinear wave equations The notion of inertial manifolds has been translated and extended to more general classes of differential equations in applications, for example stochastic inertial manifolds Bensoussan A & Landoli F (1995), the existence of inertial manifolds for non-autonomous evolution equations Koksch N & Siegmund S (2011), or for retarded partial differential equations (1998, 2001) In of all the above publications, the nonlinear term is assumed to be Lipschitz continuity However, for equations arising in complicated reaction- diffusion processes, the Lipschitz coefficients may depend on time In 2012, Nguyen T.H considered the parabolic equations (1) and proved the existence of inertial manifolds when the nonlinear term f (t, u) is ϕ-Lipschitz, i.e., i.e., f (t, x) ϕ(t) + Aβ x and f (t, x) − f (t, y) ϕ(t) Aβ (x − y) where ϕ belongs to one of admissible function spaces containing wide classes of function spaces like Lp -spaces, the Lorentz spaces Lp,q and many other function spaces occurring in interpolation theory – Generalizations of inertial manifolds The notion of inertial manifolds has also been extended to various concepts such as slow manifolds motivated by phenomena in meteorology, to inertial manifolds for equations with delay Moreover, the existence of a new type of inertial manifolds, namely the admissibly inertial manifolds of E-class, has been proved by Nguyen T.H (2013) Such manifolds were consisted of solution trajectories belonging to a Banach space E, which could be the Lp -space, Lorentz space Lp,q or many other function spaces occurring in interpolation theory – Applications of inertial manifolds Besides the existence of inertial manifolds for specific partial differential equations, inertial manifolds have found numerous useful applications in other branches of mathematics These include the connection of inertial manifold with the multigrid methods of numerical analysis or an attempt of inertial manifold to describe the turbulence of fluid mechanics This dissertation will emphasize the applications of inertial manifolds in control theory 2.2 Classes of evolution equations in this dissertation A Parabolic equations du(t) + Au(t) = f (t, u(t)) dt (2) B Partial functional differential equations (with finite delay) du(t) + Au(t) = L(t)ut + g(t, ut ) dt (3) C Partial neutral functional differential equations ∂ F ut + AF ut = Φ(t, ut ) ∂t (4) Purpose, Objects and Scope of the Dissertation Purpose We study the existence of inertial manifolds and the asymptotic behavior of solutions to certain classes of evolution equations in an infinite-dimensional Banach space The evolution equations considered with the linear parts is the generator of a semigroup and the Lipschitz coefficient of the nonlinear term may depend on time and belongs to admissible function spaces which contain wide classes of function spaces like Lp -spaces, the Lorentz spaces Lp,q and many other function spaces occurring in interpolation theory Objects Inertial manifolds and finite-dimensional feedback control for evolution equations (3), (2) and (4) in admissible spaces Scope of the Dissertation The scope of the dissertation is defined by the following contents ◦ Content Study the existence of inertial manifolds for du(t) +Au(t) = f (t, u(t)) under dt the conditions that the partial differential operator A is positive such that −A is sectorial with a sufficiently large gap in its spectrum and f is a nonlinear operator satisfying ϕ-Lipschitz condition ◦ Content Study the regularity of the inertial manifolds for du(t) + Au(t) = f (t, u(t)) dt and using the theory of inertial manifolds for non-autonomous semi-linear evolution equations, we construct a feedback controller for a class of control problems for the one-dimensional reaction-diffusion equations with the Lipschitz coefficient of the nonlinear term may depends on time and belongs to an admissible space ◦ Content Study the existence of inertial manifolds for partial functional differential equation du(t) + Au(t) = L(t)ut + g(t, ut ) under the conditions that the partial dt differential operator A is positive such that −A is sectorial with a sufficiently large gap in its spectrum; the operator L(t) is linear, and g is a nonlinear operator satisfying ϕ-Lipschitz condition for ϕ belonging to an admissible function space ◦ Content Study the existence of inertial manifolds for partial neutral functional dif∂ ferential equation ∂t F ut + AF ut = Φ(t, ut ), where the partial differential operator A is positive definite and self-adjoint with a discrete spectrum having a sufficiently large gap; the difference operator F : Cβ → X is bounded linear operator, and the nonlinear delay operator Φ satisfies the ϕ-Lipschitz condition Research Methods The dissertation uses the tools of functional analysis, fixed point theorem, semigroup theory to study the contents Moreover, we use some special techniques to get our purpose: • Linear operators: Using semigroup theory, analytic semigroups, the perturbation theory for strongly continuous semigroups • Nonlinear terms: Theory of admissible spaces • To prove existence of inertial manifolds : we use fixed point theory and the LyapunovPerron method Dissertation Outline • Chapter Preliminaries • Chapter Inertial manifolds for a class of parabolic equations and applications • Chapter Inertial manifolds for a class of partial functional differential equations with finite delay • Chapter Inertial manifolds for a class of partial neutral functional differential equations Chapter Preliminaries In this chapter, we present some basic results about linear operators, semigroup theory, and function spaces 1.1 Semigroups In this section we recall the most basic notions of the semigroup theory and their generators The main reference is Engel K.J & Nagel R (2000) (see also C.T Anh & T.D Ke (2016)) 1.2 1.2.1 Linear Operators Positive Operators with Discrete Spectrum Assumption A Let X be a separable Hilbert space and suppose that A is a closed linear operator on X satisfying the following assumption We suppose that A is a positive definite, self-adjoint operator with a discrete spectrum, say < λ1 λ2 · · · λk ··· each with finite multiplicity, and lim λk = ∞ k→∞ We assume that {ek }∞ k=1 is the orthonormal basis of X consisting of the corresponding eigenfunctions of the operator A (i.e., Aek = λk ek ) Let now λN and λN +1 be two successive and different eigenvalues with λN < λN +1 , let further P be the orthogonal projection onto the first N eigenvectors of the operator A 1.2.2 Sectorial Operators and Analytic Semigroups Definition 1.1 Let X be a Banach space A closed, linear and densely defined operator B : X ⊃ D(B) → X is called a sectorial operator of (σ, ω)-type if there exist real numbers ω ∈ R, σ ∈ 0, π2 and M such that π ρ(B) ⊃ Σσ (ω) := z ∈ C : | arg(z − ω)| < σ + , z = ω , (1.1) M R(λ, B) for all λ ∈ Σσ (ω) (1.2) |λ − ω| To prove the existence of an inertial manifold, we suppose the following assumption Assumption B Let A be a closed linear operator on a Banach space X such that −A is a sectorial operator of (σ, ω)-type with < σ < π/2 and ω < We suppose that the spectrum σ(−A) of −A can be decomposed as follows: σ(−A) = σu (−A) ∪ σc (−A) ⊂ C− with ωu < ωc < ω < where ωu := sup{Reλ : λ ∈ σu (−A)}, ωc := inf{Reλ : λ ∈ σc (−A)} (1.3) and σc (−A) is compact Assumption B allows us to choose real numbers κ and µ such that ωu < κ < µ < ωc < (1.4) We recall the Riesz projection (or spectral projection) P corresponding to σc (−A), defined by P = 2πi R(λ, −A)dλ, (1.5) + where + is a closed regular curve contained in ρ(−A), surrounding σc (−A) and positively oriented We now recall some properties, called dichotomy estimates, of the analytic semigroup e−tA t Proposition 1.2 Let P be the Riesz projection as mentioned above and choose κ < µ < being the real numbers as in (1.4) For θ > 0, the following dichotomy estimates hold true: e−tA P M1 e−µ|t| for all t ∈ R, (1.6) Aβ e−tA P M2 e−µ|t| for all t ∈ R, (1.7) κt for all t 0, (1.8) for all t > (1.9) −tA e (I − P ) Aβ e−tA (I − P ) Me N κt e tβ Let A satisfy Assumption A or Assumption Assumption1 Then, we can define the Green function as follows: e−(t−τ )A [I − P ] for all t > τ, G(t, τ ) = (1.10) −e−(t−τ )A P for all t τ 1.2.3 Auxiliary Results Theorem 1.3 (Bounded Perturbation Theorem) Let A be the generator of a strongly continuous semigroup etA t on a Banach space X satisfying etA M eωt for all t 0, ω ∈ R and some M If B ∈ L(X) then C := B + A generates a strongly continuous semigroup (S(t))t S(t) D(C) := D(A) with M e(ω+M B )t satisfying for allt Theorem 1.4 Let V be a Banach space Suppose A ∈ L(V ) and G is an open set covering the spectrum σ(A) Then there exists a δ-neighborhood Uδ (A) of A such that σ(X) ⊂ G for all X ∈ Uδ (A) Moreover, for any ε > there exists a δ such that Rλ (X) − Rλ (A) < ε for X ∈ Uδ (A) and λ ∈ / G 1.3 Admissible Spaces Denote by B the Borel algebra and by λ the Lebesgue measure on R The space L1,loc (R) of real-valued locally integrable functions on R (modulo λ-nullfunctions) becomes a Fr´echet space for the seminorms pn (f ) = Jn |g(t)|dt, where Jn = [n, n + 1] for each n ∈ Z We then define Banach function spaces as follows Definition 1.5 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 (1) E is a Banach lattice with respect to the norm · E , i.e., (E, · E ) is a Banach space, and if ϕ ∈ E, ψ is a real-valued Borel-measurable function such that |ϕ(·)| |ψ(·)| (λ-a.e.) then ψ ∈ E and ϕ E ψ E, (2) the characteristic functions χA belongs to E for all A ∈ B of finite measure and sup χ[t,t+1] E < ∞, inf χ[t,t+1] t∈R t∈R E > 0, (3) E → L1,loc (R) We remark that the condition (3) in the above definition means that for each compact interval J ⊂ R, there exists a number βJ such that |g(t)|dt βJ f for all f ∈ E E J We now introduce the notion of admissibility in the following definition Definition 1.6 The Banach function space E is called admissible if it satisfies (1) there is a constant M such that for every compact interval [a, b] ⊂ R we have b M (b − a) ϕ χ[a,b] E |ϕ(t)|dt a E, (1.11) (2) for ϕ ∈ E the function t Λ1 ϕ(t) = ϕ(τ )dτ (1.12) t−1 belong to E, (3) the space E is Tτ+ -invariant and Tτ− -invariant where Tτ+ and Tτ− are defined, for τ ∈ R, by Tτ+ ϕ(t) := ϕ(t − τ ) for t ∈ R, (1.13) Tτ− ϕ(t) := ϕ(t + τ ) for t ∈ R (1.14) Moreover, there are constants N1 and N2 such that Tτ+ N1 Tτ− and N2 for all τ ∈ R Evolution Equations in Admissible Spaces Definition 1.7 (ϕ-Lipschitz functions) Let E be an admissible Banach function space on R and ϕ be a positive function belonging to E Put Xβ := D(Aβ ) for β ∈ [0, 1) and Cβ := C([−h, 0], Xβ ) Then, a function Φ : R × Cβ → X is said to be ϕ-Lipschitz if Φ satisfies (1) Φ(t, ut ) ϕ(t) + |ut | Cβ for a.e t ∈ R and for all ut ∈ Cβ , (2) Φ(t, ut ) − Φ(t, vt ) ϕ(t)|ut − vt | Cβ for a.e t ∈ R and for all ut , ut ∈ Cβ Let χ(s) be an infinitely differentiable function on [0, ∞) such that χ(s) = for s 1, χ(s) = for s 2, χ(s) and |χ (s)| for s ∈ [0, ∞) Define the cut-off mapping by fR (t, u) := χ Aβ u R for all u ∈ D(Aβ ) f (t, u) (1.15) We have Proposition 1.8 If f (t, x) is locally ϕ-Lipschitz in a ball BR , then fR (t, x) is Lipschitz 2R2 +5R+2 ϕ R - Assumption C Let ϕ be a positive function belonging to E such that ϕ(τ ) t−1 (t − τ ) R(ϕ, β) := sup t∈R 2β 1+β 1+β 2β t 1+β dτ n∗ , (2.31) where n∗ ∈ N such that the Theorem 2.2 is satisfied This means that the Theorem 2.2 will be holds for λn∗ and λn∗ +1 Choose then the xj ’s and the xi ’s such that √ ˜ h and h (2.32) 1/2 1/2 4λn0 2λm 14 so that ˜ n − 4hλ and − 2hλm (2.33) We now define g : R × Z2 → Z1 by g(t, y) = (Pn B)−1 APn0 (CPm )−1 y + W t, Pn0 (CPm )−1 y − Pn f t, (CPm )−1 y r (2.34) for all y ∈ Z2 and t ∈ R Thanks to Lemma 2.7, we have g is a globally Lipschitz function with Lip(g) (Pn B)−1 r L(Pn X,Z1 ) ˜ n − 4hλ (CPm )−1 L(Z2 ,X) APn0 L(X) + Lip(W ) + Lip(f ) (λn0 + ς1 (t) + ϕ(t)) − 2hλm (λn0 + ς1 (t) + ϕ(t)) Thus Lip(g) 2.5.5 ξ(t), where ξ(t) := (λn0 + ς1 (t) + ϕ(t)) , for all t ∈ R (2.35) Inertial Manifolds for the Closed-Loop System With g given by (2.34) we can write (2.20) in the closed-loop form du + Au = f (t, u) + Bg(t, Cu) dt (2.36) We shall also consider the following auxiliary parabolic equation dv + Av = Pm f (t, Pm v) + Pm Bg(t, CPm v) dt (2.37) Note that the nonlinear term of both the equations above have Lipschitz coefficient less than or equal to η(t) := ϕ(t) + ξ(t) for t ∈ R We want that, under the suitable conditions, there will be inertial manifolds for parabolic equations (2.36) and (2.37) Applying Theorem 2.2 for the parabolic equations (2.36) and t (2.37), we obtain that, if n∗ is large enough and the norm Λ1 η ∞ = supt∈R t−1 η(τ )dτ is sufficiently small, then have inertial manifolds, M = Mt t∈R and N = Nt t∈R , respectively for (2.36) and (2.37) In more detail, the inertial manifold for the parabolic equation (2.36) is M = Mt t∈R , where Mt = {p + Φt (p) : p ∈ Pn X}, (2.38) here Φt : Pn X → Qn X, defined by Φt0 (p) := Qn x(p)(t0 ) where x(p) is the unique solution in γ,t0 ,θ L∞ to the equation (2.2) satisfying that Pn x(p)(t0 ) = p Similarly, N = Nt t∈R , where Nt = {p + Ψt (p) : p ∈ Pn X}, here Ψt : Pn X → Qn X is the inertial manifold for the auxiliary parabolic equation (2.37) 15 (2.39) When the two evolution equations (2.36) and (2.37) have two inertial manifolds, the corresponding inertial forms on Pn X are and dp + Ap = Pn f (t, p + Φt (p)) + Pn Bg(t, C(p + Φt (p))), dt (2.40) dρ + Aρ = Pn f (t, Pm (ρ + Ψt (ρ))) + Pn Bg(t, CPm (ρ + Ψt (ρ))) dt (2.41) Note that dρ + A(Pn − Pn0 )ρ dt = −APn0 ρ + Pn f (t, Pm (ρ + Ψt (ρ))) +Pn B(Pn B)−1 APn0 (CPm )−1 CPm (ρ + Ψt (ρ)) r +W t, Pn0 (CPm )−1 CPm (ρ + Ψt (ρ)) − Pn f t, (CPm )−1 CPm (ρ + Ψt (ρ) = −APn0 ρ + Pn f t, Pm (ρ + Ψt (ρ)) + APn0 Pm (ρ + Ψt (ρ)) + W t, Pn0 Pm (ρ + Ψt (ρ)) − Pn f t, Pm (ρ + Ψt (ρ)) = W (t, Pn0 ρ) Thus, the inertial form for (2.37) reads dρ + A(Pn − Pn0 )ρ = W (t, Pn0 ρ), dt (2.42) and can be split for ρ = ρ1 + ρ2 , where ρ1 ∈ Pn0 X, ρ2 ∈ (Pn − Pn0 )X as  dρ1   = W (t, ρ1 ), dt   dρ2 + A(Pn − Pn )ρ2 = 0 dt (2.43) The system (2.43) above is now decoupled with (t−s) ρ2 (t) = e−(t−s)A(Pn −Pn0 ) ρ2 (s) = O e−(n0 +1) , as t → ∞ Hence, the long-time dynamics of the inertial form and, hence, of the auxiliary equation (2.37) (t) is given by the system dρdt = W (t, ρ1 ) Concerning the inertial form (2.40), we can write it as dp + A(Pn − Pn0 )p = W (t, Pn0 p) + ε(t, p), dt (2.44) where ε(t, p) is regarded as an error term given by ε(t, p) = Pn f (t, p + Φt (p)) + Pn Bg(t, C(p + Φt (p))) −Pn f (t, p + Pm Ψt (p)) − Pn Bg(t, C(p + Pm Ψt (p))) By using dichotomy estimates and admissibility of function spaces we can obtain ε(t, p) η(t) 1/2 λm for all p ∈ Pn X, (c1 + c2 p ) 16 (2.45) Dε(t, p) L(Pn X) c3 η(t) 1/2 λm c4 + for all p ∈ Pn X, ν/2 λm (2.46) where the ci ’s are constant such that ci = ci (n0 , n, Λ1 ϕ ∞, c4 = c4 (n0 , n, Λ1 ϕ2 ∞, Λ1 ς1 ∞) , Λ1 ς2 for i = 1, 2, 3, ∞ , ν) Thus for each t ∈ R, we have ε(t, p) → and Dε(t, p) L(Pn X) → as m → ∞ We will summarize the above events in the following main results: Theorem 2.8 Consider the open-loop system (2.17) Let non-autonomous ordinary differential equation (2.28) be given with n0 ∈ N and W satisfying (2.25) and (2.26) Suppose that n∗ is the natural number that the conditions in the Theorem 2.2 satisfied with λn∗ and λn∗ +1 , and conditions (2.31) and (2.32) hold If a feedback law g = g(t, y) is given by (2.34), then the closed-loop equation (2.36) has an inertial manifold whose inertial form (2.44) is C -close to (2.42), which has essentially the same dynamics as (2.28), in a weighted metric for the vector fields as estimated in (2.45) and (2.46) Similar to the work R Rosa and R Temam (1997), we state the following result about structural stability of the dynamical systems Theorem 2.9 Assume the hypotheses in Theorem 2.8 hold and the nonlinear funtion W satisfies condition, for some r0 > 0, −α z ((W (t, z), z)) and that the flow induced by dz dt for all z r0 , and for some α > 0, = W (t, z) for z restricted to the ball Brn00 := {z ∈ Pn0 X : z r0 } is structurally stable If feedback law g = g(t, y) is given by (2.34) with m chosen large enough, then the longtime dynamics of the inertial form (2.44) of the closed-loop equation (2.36) is contained in the ball Brn0 = {p ∈ Pn X : p r0 } and the corresponding flow restricted to this ball Brn0 is topologically equivalent to the flow given by (2.42), so that the dynamics of the closed-loop = W (t, z) system is essentially that of dz dt 17 Chapter Inertial Manifolds for a Class of Partial Functional Differential Equations with Finite Delay In this chapter, we prove the existence of inertial manifolds for the partial functional differential equations of the form du(t) + Au(t) = F (t)ut + g(t, ut ) under the conditions that the dt partial differential operator A is positive such that −A is sectorial with a sufficient large enough gap in its spectrum, t → F (t) is an operator-valued function, and g is a nonlinear operator satisfying ϕ-Lipschitz condition, i.e., g(t, ψ) ϕ(t) + |ψ| Cβ and g(t, ψ) − g(t, φ) ϕ(t)|ψ − φ| Cβ for Cβ := C([−h, 0], D(Aβ ) Here, F (·) and ϕ belong to an admissible function space Our main methods are based on Lyapunov-Perron equations combining with analytic semigroups and admissibility of function spaces The content of this chapter is written based on the paper [1] in the List of Publications 3.1 Setting of the Problem We study partial functional differential equation (PFDE) of the form   du(t) + Au(t) = F (t)u + g(t, u ), t > s, t t dt  us = φ ∈ Cβ , s ∈ R = (−∞, +∞), (3.1) where Cβ := C([−h, 0], Xβ ), with Xβ := D(Aβ ), β < 1, being the domain of the fractional power of the positive operator A having the property that −A generates an analytic semigroup {e−tA }t on X, F (t) : Cβ → X is bounded linear operator for each t ∈ R, g : R× Cβ → X is a nonlinear operator, and ut is the history function defined as ut (θ) := u(t+θ) for all θ ∈ [−h, 0] We consider the integral equation t −(t−s)A u(t) = e e−(t−ξ)A F (ξ, u(ξ))dξ u(s) + for a.e t s (3.2) s In this section we will prove the existence of the inertial manifolds for partial functional differential equations with finite delay Then, we state of the main result of this chapter We suppose that A satisfies Assumption B and consider the spectral projection P defined as in (1.5), we define the projector Pˆ in Cβ by Pˆ φ = (Pˆ φ)(θ) := e−θA P φ(0), (3.3) where θ ∈ [−h, 0] and φ = φ(θ) is an element of Cβ Motivated by the definition of inertial manifolds in the case without delay, we then make precisely the notion of inertial manifolds in the following definition 18 Definition 3.1 The inertial manifold of equation (3.1) is a collection of surfaces M = {Mt }t∈R in Cβ of the form Mt = pˆ(θ) + Γt (ˆ p(θ)) : pˆ(θ) ∈ Pˆ Cβ ⊂ Cβ for all t ∈ R (3.4) where Γt (·) is a mapping from P X into (I − Pˆ ) Cβ , possessing the following properties: (1) For every t ∈ R, Mt is the graph of a Lipschitz function with the Lipschitz constants of Γt (·) are independent of t, i.e., there exists a constant C independent of t such that |Γt (p1 ) − Γt (p2 )| Cβ C p1 − p2 β (2) There exists γ > such that to each u0 ∈ Mt0 there corresponds one and only one solution u(t) to (3.2) on (−∞, t0 ] satisfying that u(t0 ) = u0 and sup e−γ(t0 −t) Aβ u(t) < +∞ (3.5) t t0 (3) {Mt }t∈R is invariant under equation (3.2), i.e., if u(t), t ∈ R is a solution to equation (3.2) satisfying conditions that us ∈ Ms and supt s ut Cβ < +∞ for some s ∈ R then we have ut ∈ Mt for all t ∈ R (4) {Mt }t∈R exponentially attracts all the solutions to (3.2), i.e., for any solution u(·) of (3.2) and any fixed s ∈ R, there exists a solution ut ∈ Mt and a positive constant H such that (3.6) |ut − ut | Cβ He−γ(t−s) for t s where γ is the same constant as the one in (3.5) 3.2 Lyapunov-Perron Equation We can now construct the form of the solutions of equation (3.2) which are rescaledly bounded on the half-line (−∞, t0 ] in the following lemma Lemma 3.2 Let the operator A satisfy Assumption B and g : R × Cβ → X be ϕ-Lipschitz for ϕ satisfy Assumption D For fixed t0 ∈ R, let u(t) be a solution of integral equation (3.2) such that u(t) ∈ Xβ for all t ∈ (−∞, t0 ], and sup eγ(t−t0 ) u(t) β < +∞ t∈(−∞,t0 ] Then, for t ∈ (−∞, t0 ], the solution u(t) can be rewritten in the form u(t) = e−(t−t0 )A p + t0 G(t, τ )[F (τ )uτ + g(τ, uτ )]dτ −∞ where p ∈ P X and G(t, τ ) is Green’s function 19 for all t ∈ (−∞, t0 ] (3.7) 3.3 Existence and Uniqueness of Solutions in Weighted Spaces For t0 ∈ R we introduce the space L− γ,t0 := v ∈ C (−∞, t0 ], D(Aβ ) : sup eγ(t−t0 ) v(t) β < +∞ t∈(−∞,t0 ] and put |v|− γ := eγ(t−t0 ) v(t) β sup t∈(−∞,t0 ] − Then the pair (L− γ,t0 , | · |γ ) is Banach space We have the following lemma which describes the existence and uniqueness of certain solutions belonging to weighted spaces Lemma 3.3 Let the operator A satisfy Assumption B and nonlinear term g : R × Cβ → X satisfy Assumption D If < then for any fixed t0 ∈ R and any p ∈ P X, there exists a unique function v(p) ∈ L− γ,t0 satisfying the integral equation (3.7) for all t ∈ (−∞, t0 ] with P v(p)(t0 ) = p Moreover, |v(p)|− γ < +∞, (3.8) M2 1− |v(p) − v(q)|− γ 3.4 p−q β (3.9) The Existence of Inertial Manifolds The main result of this chapter is stated as follows Theorem 3.4 Let the operator A satisfy Assumption B and nonlinear term g be ϕ-Lipschitz satisfy Assumption D If 20 equation with diffusion t s, t s, x ∈ (0, π), θ ∈ [−1, 0], x ∈ (0, π), (3.11) Chapter Inertial Manifolds for a Class of Partial Neutral Functional Differential Equations In this chapter, we prove the existence of an inertial manifold for the partial neutral func∂ F ut + AF ut = Φ(t, ut ), where the partial differential tional differential equations of the form ∂t operator A is positive definite and self-adjoint with a discrete spectrum having a sufficiently large gap; the difference operator F : Cβ → X is bounded linear operator, and the nonlinear delay operator Φ satisfies the ϕ-Lipschitz condition, i.e., Φ(t, φ) ϕ(t)(1 + |φ| Cβ ) and Φ(t, φ) − Φ(t, ψ) ϕ(t)|φ − ψ| Cβ , where ϕ belongs to an admissible function space defined on R Our main method is based on Lyapunov-Perron’s equations combined with the admissibility of function spaces and the technique of choosing F -induced trajectories The content of this chapter is written based on the paper [2] in the List of Publications 4.1 Setting of the Problem In this section we will prove the existence of the inertial manifolds for partial functional differential equations with finite delay Then, we state of the main result of this chapter Consider the partial neutral functional differential equation   ∂ F u + AF u = Φ(t, u ), t > s, t t t ∂t (4.1)  us = φ, s ∈ R In the case of infinite-dimensional phase spaces, instead of (4.1), we consider the integral equation t F ut = e −(t−s)A e−(t−ξ)A Φ(ξ, uξ )dξ F us + for all λ-a.e t s (4.2) s We suppose that A satisfies Assumption A and consider the orthogonal projection P (defined as in Assumption A), we define the projector Pˆ in Cβ by Pˆ φ = (Pˆ φ)(θ) := e−θA P φ(0), (4.3) where θ ∈ [−h, 0] and φ = φ(θ) is an element of Cβ Definition 4.1 The inertial manifold of equation (4.1) is a collection of surfaces M = {Mt }t∈R in Cβ of the form Mt = pˆ(θ) + Γt (ˆ p(θ)) : pˆ(θ) ∈ Pˆ Cβ ⊂ Cβ for all t ∈ R where Γt (·) is a mapping from P X into (I − Pˆ ) Cβ , possessing the following properties: 21 (4.4) For every t ∈ R, Mt is the graph of a Lipschitz function with the Lipschitz constants of Γt (·) are independent of t, i.e., there exists a constant C independent of t such that |Γt (p1 ) − Γt (p2 )| Cβ C p1 − p2 β There exists γ > such that to each u0 ∈ Mt0 there corresponds one and only one solution u(t) to (4.2) on (−∞, t0 ] satisfying that u(t0 ) = u0 and sup e−γ(t0 −t) Aβ u(t) < ∞ (4.5) t t0 {Mt }t∈R is positively F -invariant under equation (4.2), i.e., if u(t), t ∈ R is a solution to equation (4.2) satisfying conditions that ut0 ∈ Mt0 and supt t0 ut Cβ < ∞ for some t0 ∈ R, then we have ut ∈ Mt for all t ∈ R, where the function ut is defined as in Lemma 4.3 with t0 being replaced by t, i.e., ut (θ) := F ut−θ for all θ ∈ [−h, 0] and t ∈ R {Mt }t∈R exponentially attracts all the solutions to (4.2), i.e., for any solution u(·) of (4.2) and any fixed s ∈ R, there exists a solution ut ∈ Mt and a positive constant H such that |ut − ut | Cβ He−γ(t−s) for t s (4.6) where γ is the same constant as the one in (4.5) 4.2 Lyapunov-Perron Equation We can now construct the form of the solutions of equation (4.2) which are rescaledly bounded on the half-line (−∞, t0 ] in the following lemma Lemma 4.2 Let the operator A satisfy Assumption A and Φ : R × Cβ → X be ϕ-Lipschitz for ϕ satisfy Assumption D For fixed t0 ∈ R, let u(t) be a solution of integral equation (4.2) such that u(t) ∈ Xβ for all t ∈ (−∞, t0 ], and eγ(t−t0 ) u(t) sup t∈(−∞,t0 ] where γ = λN +1 +λN β