Russian Journal of Mathematical Physics, Vol. 12, No. 1, 2005, pp. 17–39. Copyright \begin{matrix}c \\ ◯ \end{matrix} 2005 by MAIK “Nauka/Interperiodica” (Russia).

To Leonid Volevich on the occasion of his 70th birthday

Integral Manifolds and Attractors with Exponential Rate for Nonautonomous Hyperbolic Equations with Dissipation

V. V. Chepyzhov*, A. Yu. Goritsky**, and M. I. Vishik*

*Institute for Information Transmission Problems, Russian Academy of Sciences Bol’shoi Karetnyi per. 19, Moscow 111994, Russia E-mail: chep@iitp.ru, vishik@iitp.ru **Department of Mechanics and Mathematics, Moscow State University Vorob’evy Gory, Moscow 119992, Russia E-mail: goritsky@mech.math.msu.su

Received October 24, 2004

Abstract. Integral manifolds and global attractors for nonautonomous hyperbolic quasilinear partial differential equations with dissipation in a bounded domain under the Neumann boundary conditions are constructed and their properties are studied. Special attention is focused on the case in which the forcing terms rapidly oscillate in time.

1. INTRODUCTION

In the theory of dynamical systems corresponding to nonlinear problems of mathematical physics, great attention is paid to methods of reducing infinite-dimensional dynamics to dynamics on a finitedimensional integral manifold. These integral manifolds are invariant with respect to the action of the original infinite-dimensional dynamical system and attract (e.g., with exponential rate) the solutions of the equations in question. Usually, these integral manifolds contain global attractors of the dynamical systems under consideration. In the theory of autonomous equations, these objects are referred to as inertial manifolds. Inertial manifolds were introduced and studied in [1–4] (see also [5–10]). Mostly in these papers, inertial manifolds were constructed for various quasilinear parabolic equations and systems with a linear self-adjoint leading operator term on the right-hand side of the equation. The crucial assumption for the construction of inertial manifolds is the socalled gap property, i.e., the existence of large gaps in the spectrum of the above self-adjoint linear operator. The widths of these gaps must be (several times) greater than the Lipschitz constant in the global Lipschitz condition satisfied by the nonlinear operator on the right-hand side of the equation.

Inertial manifolds for damped hyperbolic equations were constructed, e.g., in [11, 12, 7, 13] (see also [37]). In this case, the corresponding leading linear operator is non-self-adjoint, and this leads to significant difficulties when formulating appropriate gap properties.

In [14, 15], local finite-dimensional integral manifolds with exponential tracking for nonautonomous equations in a Hilbert space were constructed (the term “local” means that these manifolds are in a small neighborhood of a stationary point). In these works, the results obtained in this way are applied to nonautonomous parabolic equations. It should be mentioned that no gap property is assumed there. Sometimes it is possible to combine these local integral manifolds into a single global exponentially attracting set. This possibility is realized for the case in which the nonautonomous equation under consideration is a small perturbation of an autonomous equation possessing a global Lyapunov function. Therefore, there exists a so-called regular attractor (see [30]) for the autonomous equation. A similar apparatus was used in [16] to study global attractors of some nonautonomous hyperbolic equations with dissipation.

The work is partially supported by the RFBR (Grant no. 02-01-00277), by Russian Science Support Foundation, and by the Alexander von Humboldt Foundation.

In the present paper, we study nonautonomous sine–Gordon type equations with dissipation in a bounded domain Ω ⋐ ℝ^{n} with Neumann boundary conditions:

∂_{t}^{2}u + 2γ∂_{t}u = Δu − f(u, t) + g(x, t),  \frac{∂u}{∂n}|_{∂Ω} = 0,  (1)

where γ > 0 is a dissipation coefficient and the nonlinear function f(u, t) satisfies the inequality

|\frac{∂f}{∂u} (u, t)| ⩽ l  ∀u ∈ ℝ, t ∈ ℝ.  (2)

The forcing term g(x, t) is bounded in the space L_{2}^{b}(ℝ; L_{2}(Ω)) (see formula (12)). Denote by H = H^{1}(Ω) × L_{2}(Ω) the energy space of the hyperbolic equation.

In Sections 2 and 3, using the methods developed in [9, 10, 13], we construct (N +2)-dimensional integral manifolds M with exponential tracking in the extended phase space H ×ℝ_{t} for the problem in question under the following assumptions:

λ_{N+1} − λ_{N} > 4l,  γ^{2} > λ_{N+1} + \frac{λ_{N+1} − λ_{N}}{4R(R − 1)} , where R = \frac{λ_{N+1} − λ_{N}}{4l} > 1.  (3)

Here the symbols λ_{n}, n = 0, 1, …, stand for the eigenvalues of the minus Laplacian with Neumann boundary conditions. Recall that λ_{0} = 0.

We note that inequalities (3) imply the corresponding gap property for equation (1) in the space H endowed with an appropriate inner product (the choice of this inner product is the most important part in the construction of the integral manifold M). These inequalities seem to be the best up to date conditions ensuring the existence of the inertial (integral) manifold for a dissipative hyperbolic equation.

In Section 4, we study the dependence of integral manifolds on the nonlinearity of the nonautonomous equation in the general setting in a Hilbert space, assuming that an appropriate gap property holds. We prove that the integral manifold depends Lipschitz-continuously on the nonlinear function in the right-hand side of the equation. As a corollary, we prove that some properties of nonlinearity (like almost periodicity or quasiperiodicity in time) yield the same properties for the integral manifold thus constructed. In that section, we also show a simple but useful trick which enables us to obtain some properties of integral manifolds for nonlinear equations by studying appropriate linear nonhomogeneous equations.

Section 5 is devoted the sine–Gordon-type equations whose external forcing terms are rapidly oscillating (in time). We study the relationship between the integral manifolds and, in Section 6, between the global attractors of oscillatory and averaged equations.

The averaging of global attractors and local invariant manifolds of nonautonomous partial differential equations with almost periodic terms rapidly oscillating in time was studied in [17–19] from the standpoint of the Bogolyubov averaging principle [20] (see also [21]).

The behavior of global attractors of damped wave equations and their averaging were studied in [22] (for the temporal oscillation of terms) and in [23] (for the spatial oscillation of terms). Explicit estimates were proved for the deviation of the global attractors of oscillatory wave equations from the global attractor of the corresponding averaged equations. For parabolic equations and systems with oscillating coefficients, similar problems were treated in [24, 25] (see also [26, 27]).

In [28], the dissipative wave equation (1) was studied with zero Dirichlet boundary conditions. It is proved there that, if the Lipschitz constant l (see (2)) is less than the first eigenvalue of the minus Laplacian with Dirichlet boundary conditions and the dissipation coefficient is sufficiently large, then the global attractor A has a simple structure, namely, it is the closure of the values of a unique bounded complete trajectory of the wave equation. (A complete trajectory of an equation is a solution defined for all values t ∈ ℝ.) This complete trajectory attracts any solution of the wave equation with exponential rate as t → +∞, and hence the global attractor is exponential. For the special case in which the external force g(x, t) is a quasiperiodic function with k rationally independent frequencies (see page 29), it was proved that the global attractor A is a Lipschitzcontinuous image of a k-dimensional torus in the space H.

In [28], the dissipative wave equation with rapidly oscillating external force g^{ε}(x, t) = g(x, t, t/ε) having the uniform average g^{0}(x, t) as ε → 0+ was studied. It was assumed that the function g(x, t, ζ) − g^{0}(x, t) has a bounded antiderivative with respect to ζ. Under this assumption, it was proved that the Hausdorff distance between the global attractor A_{ε} of the original oscillatory equation and the global attractor A_{0} of the averaged equation is less than O(ε^{1/2}).

In the present paper, in Section 5, we consider the hyperbolic equation (1) with Neumann boundary conditions having the rapidly oscillating external force term g^{ε}(x, t) = g^{0}(x, t)+g(x, t/ε). We assume that the antiderivative J(x, ζ) of the function g(x, ζ) with respect to ζ is bounded in H^{1}(Ω),

\frac{∂J}{∂ζ} = g(x, ζ);  ∥J(·, ζ)∥_{H^{1}(Ω)} ⩽ K  ∀ζ ∈ ℝ.  (4)

Under assumptions (3) and (4), we prove the following estimate for the deviation of the integral manifold M_{ε} of the equation with external force g^{ε}(x, t) from the integral manifold M_{0} of the averaged equation with external force g^{0}(x, t):

dist(M_{ε}, M_{0}) ⩽ CKε = O(ε)  (C > 0).  (5)

The distance between the integral manifolds is measured in the uniform metric (see Section 4). The proof of this assertion is based on a construction of a special bounded solution of the corresponding linear nonhomogeneous dissipative wave equation.

In Section 6, we apply the results thus obtained to the study of an important case of the dissipative hyperbolic equation (1), namely, to the damped sine–Gordon equation with nonlinear function f(u) = l sin u. We assume that inequalities (3) hold for N = 0, that is,

λ_{1} > 4l,  γ^{2} > λ_{1} + \frac{λ_{1}}{4R(R − 1)}, where R = \frac{λ_{1}}{4l} > 1.  (6)

(Recall that λ_{0} = 0.) Then the sine–Gordon equation clearly has a 2-dimensional integral manifold M in the extended phase space H × ℝ_{t}.

Note that the sine–Gordon equation with Neumann boundary conditions has the following translation property: if u(t) is a solution of this equation, then u(t) + 2πk is also a solution for each k ∈ ℤ. In particular, this means that the corresponding dynamical system (or the process) acting on the phase space H has no bounded absorbing set, and one cannot construct a (bounded or compact) global attractor in the space H for this system. Nevertheless, the above translation property enables us to construct a dynamical system acting on the factorized phase space H^{F} = (H^{1}(Ω) mod 2π)) × L_{2}(Ω). The factorized dynamical system does have a bounded absorbing set in H^{F}.

In Section 6, we prove that the closure (in H^{F}) of the values of all complete trajectories of the sine–Gordon equation lying in the constructed two-dimensional integral manifold M forms the global attractor A of the factorized dynamical system. Moreover, this attractor attracts all solutions of the equation as t → +∞ with exponential rate. If the external force g(x, t) is a quasiperiodic function with k rationally independent frequencies, then the global attractor A is a Lipschitzcontinuous image of a (k + 1)-dimensional torus in the space H^{F}. Finally, if the external force is of the form g^{ε}(x, t) = g^{0}(x, t) + g(x, t/ε), then (provided that the gap condition (6) and inequality (4) hold) there exist global attractors A_{ε} and A_{0} for both (oscillating and averaged) nonautonomous problems, and

dist_{symm}(A_{ε}, A_{0}) = O(ε).  (7)

In Section 7, we consider the dissipative hyperbolic equations with a singularly oscillating forcing term and present some new results concerning their global attractors. We also formulate some open problems.

2. INTEGRAL MANIFOLDS FOR SINE–GORDON TYPE EQUATIONS

Nonautonomous equation and the corresponding process. We consider the following problem for a dissipative nonautonomous hyperbolic equation in a bounded domain Ω:

∂_{t}^{2}u + 2γ∂_{t}u = Δu − f(u, t) + g(x, t),  \frac{∂u}{∂n}|_{∂Ω} = 0,  (8)
u|_{t=τ} = u_{τ} (x) ∈ H^{1}(Ω),  (9)
∂_{t}u|_{t=τ} = p_{τ} (x) ∈ L_{2}(Ω).  (10)

Here γ stands for a positive dissipation coefficient, the nonlinear function f(u, t) is continuously differentiable, and

|\frac{∂f}{∂u} (u, t)| ⩽ l  ∀u ∈ ℝ, t ∈ ℝ.  (11)

It is also assumed that the forcing term g(x, t) satisfies the condition

sup_{t∈ℝ} ∫_{t}^{t+1}∫_{Ω} g^{2}(x, s) dxds < +∞.  (12)

In the particular case f(u, t) ≡ l sin u, equation (8) is the so-called sine–Gordon equation with dissipation.

Remark 1. Along with the Neumann boundary conditions, we can consider periodic boundary conditions as well. These problems are more interesting and more complicated than those related to the Dirichlet boundary conditions (treated in [28, 29]), because of the existence of an unstable (zero) eigenvalue of the Laplace operator.

The problem (8)–(10) treated here has a unique solution u ∈ C([τ, T]; H^{1}(Ω)) with the time derivative ∂_{t}u ∈ C([τ, T]; L_{2}(Ω)) for any T ⩾ τ (see [30–33]). We define a family of operators {Y (t, τ)}, Y (t, τ): H → H, where t ⩾ τ, τ ∈ ℝ, acting on the phase space H = H^{1}(Ω)×L_{2}(Ω) by the formula

Y (t, τ) (u_{τ} (x), p_{τ} (x)) = y(t) ≡ (u(t, x), p(t, x)) ∈ H^{1}(Ω) × L_{2}(Ω) = H;

here u(t, x) stands for a solution of problem (8)–(10) and p(t, x) = ∂_{t}u(t, x) is the derivative of the solution with respect to t, y = (u, p) ∈ H. The two-parameter family {Y (t, τ)} is referred to as a process corresponding to problem (8)–(10). The process {Y (t, τ)} is continuous with respect to (y_{τ} , t) (see [30, 31, 33]).

We rewrite the problem in question as a first-order differential equation for a vector function y = (u, p),

\frac{d}{dt}y(t) + Ay = F (y, t),  Ay = − ( \begin{matrix}0 & 1 \\ Δ & −2γ \end{matrix} ) y, F (y, t) = ( \begin{matrix}0 \\ −f(u, t) + g \end{matrix} ) .  (13)

Gap property. Let e_{k}(x) and λ_{k} be the eigenfunctions and eigenvalues of the operator −Δ in the domain Ω with Neumann boundary conditions,

Δe_{k}(x) = −λ_{k}e_{k},  \frac{∂e_{k}}{∂n} |_{∂Ω} = 0,  ∥e_{k}∥^{2} = (e_{k}, e_{k}) = 1,
0 = λ_{0} < λ_{1} ⩽ λ_{2} ⩽ ⋯,  lim_{k→+∞} λ_{k} = +∞.

Here the symbols ∥ · ∥ and (· , ·) stand for the norm and the inner product in the space L_{2}(Ω). The standard norm in the Hilbert space H = H^{1}(Ω) × L_{2}(Ω) is defined by

∥y∥_{H}^{2} = ∥(u, p)∥_{H}^{2} = u_{0}^{2} + p_{0}^{2} + ∑_{k=1}^{∞} (λ_{k}u_{k}^{2} + p_{k}^{2}) ,

where u_{k} = (u, e_{k}) and p_{k} = (p, e_{k}).

Theorem 1. Let the inequalities

λ_{N+1} − λ_{N} > 4l  (14)

and

γ^{2} > λ_{N+1}

hold for some N. Then we can equip the space H with an inner product [· , ·] equivalent to the initial one and find orthogonal projections P and Q (with respect to the new inner product) such that

1. dim P (H) = N + 1, Q = Id −P, and Q(H)⊥P (H);

2. the linear operator A satisfies the inequalities

[Ay, y] ⩽ μ_{N}[y, y]  ∀y ∈ P (H),  (15)
[Ay, y] ⩾ μ_{N+1}[y, y]  ∀y ∈ Q(H) ∩ D(A),  (16)

where μ_{N} = γ − \sqrt{γ^{2} − λ_{N}} and μ_{N+1} = γ − \sqrt{γ^{2} − λ_{N+1}} (note that μ_{N} < μ_{N+1});

3. in the new norm |||y||| = \sqrt{[y, y]} , the nonlinear operator F (y, t) satisfies the Lipschitz condition with respect to y,

|||F (y_{1}, t) − F (y_{2}, t)||| ⩽ L|||y_{1} − y_{2}|||,  L = \frac{l}{\sqrt{γ^{2} − λ_{N+1}}} ;  (17)

4. if γ is sufficiently large, namely,

γ^{2} > λ_{N+1} + \frac{λ_{N+1} − λ_{N}}{4R(R − 1)} = λ_{N+1} + \frac{l}{R − 1}, where R = \frac{λ_{N+1} − λ_{N}}{4l} > 1  (18)

(see (14)), then the following gap property holds :

μ_{N+1} − μ_{N} > 2L.  (19)

The proof is given in Section 3 below.

Integral manifolds. Our initial problem (8)–(10) can now be represented as an abstract equation of the form

\frac{dy}{dt} + Ay = F (y, t)  (20)

in a Hilbert space H with the initial data

y|_{t=τ} = y_{τ} ∈ H.  (21)

Here A is a linear operator (see (13)) which is semibounded below and has dense domain D(A) ⊂ H and F is a nonlinear operator. Below, we deal with the new norm |||y||| and new inner product [y, y] constructed in Theorem 1 but use the notation |y| and (y, y). We set

θ = \frac{μ_{N+1} + μ_{N}}{2} ⩾ 0, δ = \frac{μ_{N+1} − μ_{N}}{2} > 0 ⟺ μ_{N+1} = θ + δ, μ_{N} = θ − δ.

Theorem 1 claims the existence of a specific direct sum decomposition H = P (H) ⊕ Q(H) saisfying (15)–(16). Under the new notation, inequalities (15)–(16) become

(Ap, p) ⩽ (θ − δ)|p|^{2}  ∀p ∈ P (H),  (22)
(Aq, q) ⩾ (θ + δ)|q|^{2}  ∀q ∈ Q(H) ∩ D(A),  (23)

where P (H) and Q(H) are invariant subspaces of the operator A, P (H)⊥Q(H), and P (H) is finitedimensional. The nonlinear operator F (v, t) satisfies the global Lipschitz condition with respect to the space variable v (see (17)),

|F (v_{1}, t) − F (v_{2}, t)| ⩽ L |v_{1} − v_{2}|  ∀v_{1}, v_{2} ∈ H, t ∈ ℝ,  (24)

and, since μ_{N+1} − μ_{N} = 2δ, assumption (19) becomes

L < δ.  (25)

Now we are going to define the main object of our investigation, the integral manifold M of a nonautonomous equation.

We determine an invariant set M in an extended phase space H × ℝ_{t} as the set-theoretic union of integral curves (y(t), t) defined for all values t ∈ ℝ_{t} and satisfying the asymptotic property |y(t)| = O(e^{−θt}) as t → −∞. It can be shown that the set M defined in such a way is exponentially attracting. Moreover, M has a one-to-one projection to the space P (H) × ℝ_{t}, and can thus be represented as the graph of a function q = Φ(p, t).

To be precise, the following theorem holds.

Theorem 2. Let conditions (22)–(25) hold. Suppose that equation (20) has at least one solution ȳ(t) defined for all values t ∈ ℝ_{t} and satisfying the asymptotic property O(e^{−θt}) as t → −∞. Then the following assertions hold:

1. the set M is the graph of a function Φ(p, t), Φ: P (H) × ℝ_{t} → Q(H), which is Lipschitz continuous with respect to p, i.e.,

|Φ(p_{1}, t) − Φ(p_{2}, t)| ⩽ l^{∗} |p_{1} − p_{2}|  ∀p_{1}, p_{2} ∈ P (H), t ∈ ℝ,  (26)

and the Lipschitz constant l^{∗} is less than 1;

2. every solution y(t) of equation (20) admits a trace ỹ(t), t ∈ ℝ_{t}, on the integral manifold M, i.e., ỹ(t) is also a solution of (20), (ỹ(t), t) ∈ M for all t ∈ ℝ_{t}, and

|y(t) − ỹ(t)| ⩽ c |y(τ) − ỹ(τ)| e^{−(θ+μ)(t−τ)} ∀t ⩾ τ,

where μ = δ − L > 0 (see (25)) and c is a positive constant.

The complete proof of the theorem is given in [9, Th. 2.1] under the additional assumption that the right-hand side of equation (20) vanishes for y = 0. To satisfy the last property, we can rewrite equation (20) for the difference y − ȳ. Note that the Lipschitz condition with respect to y remains the same under this substitution.

Remark 2. Below, in Section 6, for the sine–Gordon equation, we state the existence of a solution ȳ(t), t ∈ ℝ_{t}, that has at most linear growth as t → ±∞.

The research methods of [9] (as well as of [10, 13–15]) are based on differential inequalities, and the same approach is used in the present paper in the proof of Theorem 3 below. It should be noted that there are other methods to prove the existence of integral (inertial) manifolds (see, e.g., [6, 8]) when assuming the gap property (25) (or, equivalently, (19)).

3. PROOF OF THEOREM 1

New norm. We decompose the whole space H into the direct sum of N + 2 subspaces H = H_{0} ⊕ H_{1} ⊕ ⋯ ⊕ H_{N} ⊕ H_{∞} orthogonal with respect to the initial inner product. Here every twodimensional space H_{k}, k = 0, 1, …, N, corresponds to an eigenvector e_{k} with respect to u and to p, i.e., H_{k} = {y = (u, p) = (u_{k}e_{k}, p_{k}e_{k}) = (u_{k}, p_{k})e_{k} | (u_{k}, p_{k}) ∈ ℝ^{2}}, and H_{∞} = (H_{0}⊕H_{1}⊕⋯⊕H_{N})^{⊥} is a subspace of codimension 2(N + 1) corresponding to the eigenvectors e_{N+1}, e_{N+2}, … of the Laplace operator. We note that the spaces H_{k} are invariant with respect to the linear operator A.

Introducing a new inner product [· , ·], we retain the pairwise orthogonality of the spaces H_{k}, for k = 0, 1, …, N, ∞, but we change the inner product in each of these spaces. For any vector y = (u, p) ∈ H, denote its orthogonal projections to H_{k}, k = 0, 1, …, N, ∞, by y_{k} = (u_{k}e_{k}, p_{k}e_{k}) ∈ H_{k}, k = 0, 1, …, N, and y_{∞} = y − ∑_{k=0}^{N} y_{k} ∈ H_{∞}. Thus, the new norm and the new inner product in H are defined as follows:

|||y|||^{2} = ∑_{k=0}^{N} |||y_{k}|||_{k}^{2} + |||y_{∞}|||_{∞}^{2}, [y, ỹ] = ∑_{k=0}^{N}[y_{k}, ỹ_{k}]_{k} + [y_{∞}, ỹ_{∞}]_{∞},

where ||| · |||_{k}^{2} and [· , ·]_{k} are the new norm and the new inner product in H_{k}, k = 0, 1, …, N, ∞, which are defined below.

Norm in H_{∞}. The initial energy norm in H_{∞} is

∥y∥_{H}^{2} = ∥∇u∥^{2} + ∥p∥^{2} ∀y = (u, p) ∈ H_{∞}.

Define the new inner product of any two vectors y = (u, p) and ỹ = (ũ, p̃), y, ỹ ∈ H_{∞}, as follows:

[y, ỹ]_{∞} = (∇u, ∇ũ) + (γ^{2} − 2λ_{N+1})(u, ũ) + (γu + p, γũ + p̃).

The corresponding norm is defined by the rule

|||y|||_{∞}^{2} = ∥∇u∥^{2} + (γ^{2} − 2λ_{N+1})∥u∥^{2} + ∥γu + p∥^{2} ∀y = (u, p) ∈ H_{∞}.

Proposition 1. The norms |||y|||_{∞} and ∥y∥_{H} are equivalent on the space H_{∞}.

Proof. For any vector y = (u, p) ∈ H_{∞},

∥∇u∥^{2} ⩾ λ_{N+1}∥u∥^{2},  (27)

and |||y|||_{∞}^{2} is obviously bounded above by (a scalar multiple of) ∥∇u∥^{2} and ∥p∥^{2}.

Let us find a lower bound for |||y|||_{∞}^{2}. Since γ^{2} > λ_{N+1}, we have γ^{2} ⩾ λ_{N+1}(1 + ε) for some ε, 0 < ε < 1. Thus, it follows from (27) that

|||y|||_{∞}^{2} = ∥∇u∥^{2} + (γ^{2} − 2λ_{N+1})∥u∥^{2} + ∥γu + p∥^{2}

= ε∥∇u∥^{2} + (1 − ε)(∥∇u∥^{2} − λ_{N+1}∥u∥^{2}) + (γ^{2} − λ_{N+1}(1 + ε))∥u∥^{2} + ∥γu + p∥^{2}

⩾ ε∥∇u∥^{2} + ∥γu + p∥^{2} ⩾ \frac{ε}{2}∥∇u∥^{2} + \frac{ε}{2λ_{N+1}} ∥u∥^{2} + (γ∥u∥ − ∥p∥)^{2}.

The expression on the right-hand side is a positive-definite quadratic form in ∥∇u∥, ∥u∥, and ∥p∥. Therefore, this form is bounded below by (a scalar multiple of) ∥∇u∥^{2} + ∥p∥^{2}.

Proposition 2. If y = (0, p) ∈ H_{∞}, then |||y|||_{∞} = ∥y∥_{H}.

Proof. If u = 0, then |||y|||_{∞}^{2} = ∥p∥^{2} = ∥y∥_{H}^{2}.

Proposition 3. For every vector y = (u, p) ∈ H_{∞}, we have

|||y|||_{∞} ⩾ \sqrt{γ^{2} − λ_{N+1}} ∥u∥.

Proof. |||y|||_{∞}^{2} ⩾ ∥∇u∥^{2} + (γ^{2} − 2λ_{N+1})∥u∥^{2} ⩾ (γ^{2} − λ_{N+1})∥u∥^{2} by (27).

Proposition 4. For every vector y = (u, p) ∈ H_{∞} ∩ D(A), we have

[Ay, y]_{∞} ⩾ μ_{N+1}[y, y]_{∞}.

Proof.

Ay = (−p, −Δu + 2γp);
[Ay, y]_{∞} = −(∇p, ∇u) − (γ^{2} − 2λ_{N+1})(p, u) + (−γp − Δu + 2γp, γu + p)
= (Δu, p) − (γ^{2} − 2λ_{N+1})(u, p) + (−Δu + γp, γu + p)
= γ∥∇u∥^{2} + γ∥p∥^{2} + 2λ_{N+1}(u, p);
[y, y]_{∞} = ∥∇u∥^{2} + ∥p∥^{2} + 2γ(u, p) + 2(γ^{2} − λ_{N+1})∥u∥^{2};
[Ay, y]_{∞} − γ[y, y]_{∞} = −2γ(γ^{2} − λ_{N+1})∥u∥^{2} − 2(γ^{2} − λ_{N+1})(u, p)
= −2\sqrt{γ^{2} − λ_{N+1}} (\sqrt{γ^{2} − λ_{N+1}} u, γu + p)
⩾ −\sqrt{γ^{2} − λ_{N+1}}((γ^{2} − λ_{N+1})∥u∥^{2} + ∥γu + p∥^{2})
⩾ −\sqrt{γ^{2} − λ_{N+1}} (∥∇u∥^{2} + (γ^{2} − 2λ_{N+1})∥u∥^{2} + ∥γu + p∥^{2})
= −\sqrt{γ^{2} − λ_{N+1}} [y, y]_{∞}.

This proves the proposition because μ_{N+1} = γ − \sqrt{γ^{2} − λ_{N+1}}.

Norm in H_{k}, k = 0, 1, …, N. The initial norm in H_{k} of a vector y = (u_{k}e_{k}, p_{k}e_{k}) ∈ H_{k} is equal to

∥y∥_{H}^{2} = λ_{k}u_{k}^{2} + p_{k}^{2} for k ≠ 0 and ∥y∥_{H}^{2} = u_{0}^{2} + p_{0}^{2} for k = 0.

We take into account the fact that ∥∇u∥^{2} = λ_{k}∥u∥^{2} and proceed as in the space H_{∞}, i.e., for any vector y = (u, p) = (u_{k}e_{k}, p_{k}e_{k}) ∈ H_{k}, we define the new norm as follows:

|||y|||_{k}^{2} = ∥∇u∥^{2} + (γ^{2} − 2λ_{k})∥u∥^{2} + ∥γu + p∥^{2} = (γ^{2} − λ_{k}) u_{k}^{2} + (γu_{k} + p_{k})^{2} = (My, y),

where the matrix M = ( \begin{matrix}2γ^{2} − λ_{k} & γ \\ γ & 1 \end{matrix} ) is positive-definite because γ^{2} > λ_{k}, k = 0, 1, …, N.

The corresponding product of two vectors y = (u, p) and ỹ = (ũ, p̃) is defined by

[y, ỹ]_{k} = (My, ỹ).

Propositions 1–3 have the following obvious analogs.

Proposition 5. The norms |||y|||_{k} and ∥y∥_{H} are equivalent on the space H_{k} (as any two norms in a finite-dimensional space).

Proposition 6. If y = (0, p) ∈ H_{k}, then |||y|||_{k} = ∥y∥_{H}.

Proposition 7. For every vector y = (u, p) ∈ H_{∞}, we have

|||y|||_{k} ⩾ \sqrt{γ^{2} − λ_{k}} |u_{k}|.

The restriction of the linear operator A to the space H_{k} has the matrix ( \begin{matrix}0 & −1 \\ λ_{k} & 2γ \end{matrix} ). The eigen-

values of this restriction are

μ_{k} = γ − \sqrt{γ^{2} − λ_{k}} and ν_{k} = γ + \sqrt{γ^{2} − λ_{k}},

and the corresponding eigenvectors are the vectors

ξ_{k} = (γ + \sqrt{γ^{2} − λ_{k}}, −λ_{k}) and η_{k} = (−1, γ + \sqrt{γ^{2} − λ_{k}}) .

Proposition 8. The vectors ξ_{k} and η_{k} are orthogonal with respect to the new inner product.

Proof.

(Mξ_{k}, η_{k}) = (( \begin{matrix}2γ^{2} − λ_{k} & γ \\ γ & 1 \end{matrix} ) ( \begin{matrix}γ + \sqrt{γ^{2} − λ_{k}} \\ −λ_{k} \end{matrix} ) , ( \begin{matrix}−1 \\ γ + \sqrt{γ^{2} − λ_{k}} \end{matrix} ))

= −(2γ^{2} − λ_{k}) (γ + \sqrt{γ^{2} − λ_{k}}) + γλ_{k} + γ (γ + \sqrt{γ^{2} − λ_{k}})^{2} − λ_{k} (γ + \sqrt{γ^{2} − λ_{k}})

= 0.

Proof of Theorem 1. 1. We denote by H^{I} the set of linear combinations of the vectors ξ_{k}e_{k}, k = 0, 1, …, N, (which are pairwise orthogonal with respect to both the old and the new inner product) and by H^{II} the set of linear combinations of the vectors η_{k}e_{k}. It follows from Proposition 8 that the spaces H^{I} and H^{II} are orthogonal with respect to the new inner product, and both the spaces are orthogonal to H^{∞}.

Let P be an orthogonal projection onto the N-dimensional space H^{I} = P (H). Then Q = Id −P is an orthogonal projection onto H^{II} ⊕ H^{∞} = Q(H). Assertion 1 of Theorem 1 is proved.

2. Since A(ξ_{k}e_{k}) = μ_{k}(ξ_{k}e_{k}) and A(η_{k}e_{k}) = ν_{k}(η_{k}e_{k}), k = 0, 1, …, N, we have

[Ay, y] ⩽ max_{1⩽k⩽N} μ_{k} · [y, y] = μ_{N}[y, y]  ∀y ∈ H^{I},  (28)
[Ay, y] ⩾ min_{1⩽k⩽N} ν_{k} · [y, y] = ν_{N}[y, y]  ∀y ∈ H^{II}.  (29)

Thus, inequality (15) is obtained in (28) and inequality (16) follows from (29) and from Proposition 4 because ν_{N} > γ > μ_{N+1}.

3. The vector F (y, t) has a zero u-component. Thus, Propositions 2 and 6 imply

|||F (y_{1}, t) − F (y_{2}, t)||| = ∥F (y_{1}, t) − F (y_{2}, t)∥_{H} = ∥f(u_{1}, t) − f(u_{2}, t)∥.

It follows from (11) that |f(u_{1}, t) − f(u_{2}, t)| ⩽ l|u_{1} − u_{2}|, and therefore

|||F (y_{1}, t) − F (y_{2}, t)||| ⩽ l∥u_{1} − u_{2}∥.

On the other hand, Propositions 3 and 7 imply

|||y|||^{2} = ∑_{k=1}^{N} |||y_{k}|||_{k}^{2} + |||y_{∞}|||_{∞}^{2} ⩾ ∑_{k=1}^{N}(γ^{2} − λ_{k})|u_{k}|^{2} + (γ^{2} − λ_{N+1})∥u_{∞}∥^{2}

⩾ (γ^{2} − λ_{N+1})( ∑_{k=1}^{N} |u_{k}|^{2} + ∥u_{∞}∥^{2}) = (γ^{2} − λ_{N+1})∥u∥^{2}.

Thus,

|||F (y_{1}, t) − F (y_{2}, t)||| ⩽ \frac{l}{\sqrt{γ^{2} − λ_{N+1}}} |||y_{1} − y_{2}|||,

and inequality (17) is proved.

4. Let us show that inequality (19) is equivalent to estimate (18). We use the formulas for the numbers μ_{N+1}, μ_{N}, and L and rewrite inequity (19) as follows:

μ_{N+1} − μ_{N} = \frac{λ_{N+1} − λ_{N}}{\sqrt{γ^{2} − λ_{N}} + \sqrt{γ^{2} − λ_{N+1}}} > \frac{2l}{\sqrt{γ^{2} − λ_{N+1}}} = 2L.

RUSSIAN JOURNAL OF MATHEMATICAL PHYSICS Vol. 12 No. 1 2005

We solve the last inequality with respect to γ. Making equivalent manipulations, we see that

\sqrt{γ^{2} − λ_{N}} + \sqrt{γ^{2} − λ_{N+1}} < 2R\sqrt{γ^{2} − λ_{N+1}}, where R = \frac{λ_{N+1} − λ_{N}}{4l} > 1,

or

\sqrt{γ^{2} − λ_{N}} < (2R − 1)\sqrt{γ^{2} − λ_{N+1}}.

Squaring the last inequality and reducing similar terms, we obtain

(γ^{2} − λ_{N}) − (γ^{2} − λ_{N+1}) < (4R^{2} − 4R)(γ^{2} − λ_{N+1}),
λ_{N+1} − λ_{N} < 4R(R − 1)(γ^{2} − λ_{N+1}),

which is equivalent to (18). This completes the proof of Theorem 1.

4. DEPENDENCE OF INTEGRAL MANIFOLDS ON NONLINEARITY

The distance between integral manifolds. In this section, we consider two differential equations of type (20),

\frac{dy^{i}}{dt} + Ay^{i} = F^{i}(y^{i}, t),  i = 1, 2,  (30)

and assume that the above properties are satisfied. Theorem 2 states the existence of integral manifolds M^{1} and M^{2} for both equations. These integral manifolds are the graphs of the corresponding Q(H)-valued functions Φ^{1}(p, t) and Φ^{2}(p, t), p ∈ P (H), t ∈ ℝ, which are Lipschitz-continuous with respect to p (see Theorem 2).

We are going to estimate the distance dist(M^{1}, M^{2}) between the integral manifolds M^{1} and M^{2},

dist(M^{1}, M^{2}) = sup_{p∈P (H), t∈ℝ_{t}} |Φ^{1}(p, t) − Φ^{2}(p, t)| ,

by means of the distance between the nonlinear functions F^{i}, provided that the latter distance is finite,

dist(F^{1}, F^{2}) = sup_{y∈H, t∈ℝ_{t}} |F^{1}(y, t) − F^{2}(y, t)| < ∞.  (31)

Recall that | · | here is the norm ∥ · ∥_{H} in the phase space H.

Theorem 3. Let the assumptions of Theorem 2 be satisfied and let (31) hold. Then

dist(M^{1}, M^{2}) ⩽ C dist(F^{1}, F^{2}),  where C = (θ + δ − L)^{−1}.  (32)

Proof. 1. Since L < δ (see (25)), it is possible to choose a positive constant κ so that m := δ − L − κ > 0. Then we introduce the number β, 0 < β < 1, by the formula

\frac{1}{2} (β + \frac{1}{β} ) = 1 + \frac{2m}{L} = \frac{2δ − 2κ − L}{L}.  (33)

Let y^{i}(t), i = 1, 2, be solutions of (30). We define two functions

p(t) = P (y^{1}(t) − y^{2}(t)) , q(t) = Q (y^{1}(t) − y^{2}(t)) .

These functions satisfy the equations

\frac{dp}{dt} + Ap = P (F^{1}(y^{1}, t) − F^{2}(y^{2}, t)) , \frac{dq}{dt} + Aq = Q (F^{1}(y^{1}, t) − F^{2}(y^{2}, t)) .

We take the inner product of the first equation by p and of the other by q,

\frac{1}{2} \frac{d}{dt} |p(t)|^{2} = −(p, Ap) + (p, F^{1}(y^{1}, t) − F^{2}(y^{2}, t)),
\frac{1}{2} \frac{d}{dt} |q(t)|^{2} = −(q, Aq) + (q, F^{1}(y^{1}, t) − F^{2}(y^{2}, t)).

We now multiply the first relation by β and subtract the result from the other relation. Using (22)–(24), we obtain

\frac{1}{2} \frac{d}{dt} (|q(t)|^{2} − β|p(t)|^{2})
= −(q, Aq) + (q, F^{1}(y^{1}, t) − F^{2}(y^{2}, t)) + β(p, Ap) − β(p, F^{1}(y^{1}, t) − F^{2}(y^{2}, t))
= −(q, Aq) + β(p, Ap) + (q − βp, F^{1}(y^{1}, t) − F^{2}(y^{2}, t))
= −(q, Aq) + β(p, Ap) + (q − βp, F^{1}(y^{1}, t) − F^{1}(y^{2}, t)) + (q − βp, F^{1}(y^{2}, t) − F^{2}(y^{2}, t))
⩽ −(θ + δ)|q|^{2} + β(θ − δ)|p|^{2} + |q − βp| L |y^{1} − y^{2}| + |q − βp| dist(F^{1}, F^{2}).

Let us now estimate the last two terms as follows:

|q − βp| L |y^{1} − y^{2}| ⩽ \frac{L}{2} |q − βp|^{2} + \frac{L}{2} |y^{1} − y^{2}|^{2} = \frac{L}{2} (|q|^{2} + β^{2}|p|^{2}) + \frac{L}{2} (|q|^{2} + |p|^{2}) ,
|p − βq| dist(F^{1}, F^{2}) ⩽ κ |p − βq|^{2} + \frac{dist^{2}(F^{1}, F^{2})}{4κ} = κ (|p|^{2} + β^{2}|q|^{2}) + D^{2}

⩽ κ (|p|^{2} + β|q|^{2}) + D^{2}, where D = \frac{dist(F^{1}, F^{2})}{2\sqrt{κ}},

because 0 < β < 1. Recall that m = δ − κ − L and β is defined in (33). Hence,

\frac{1}{2} \frac{d}{dt} (|q(t)|^{2} − β|p(t)|^{2})

⩽ −(θ + δ)|q|^{2}+ β(θ − δ)|p|^{2}+ \frac{L}{2} (|q|^{2} + β^{2}|p|^{2})+ \frac{L}{2} (|q|^{2} + |p|^{2})+κ (|q|^{2} + β|p|^{2})+D^{2}

= − (θ + δ − L − κ) |q|^{2} + β(θ − δ + κ + \frac{L}{2}(β + \frac{1}{β} ))|p|^{2} + D^{2}

= −(θ + m)|q|^{2} + β(θ + m)|p|^{2} + D^{2} = −(θ + m) (|q|^{2} − β|p|^{2}) + D^{2},

or

\frac{1}{2} \frac{d}{dt} (|q(t)|^{2} − β|p(t)|^{2} − D̃^{2}) ⩽ −(θ + m) (|q(t)|^{2} − β|p(t)|^{2} − D̃^{2}) ,

where

D̃ = \frac{D}{\sqrt{θ + m}} = \frac{dist(F^{1}, F^{2})}{2\sqrt{κ(θ + m)}}.

Write w(t) = |q(t)|^{2} − β|p(t)|^{2} − D̃^{2}. Note that ẇ ⩽ −2(θ + m)w. Thus,

w(t) ⩽ w(τ)e^{−2(θ+m)(t−τ)}  ∀t ⩾ τ.  (34)

2. Let us return to the integral manifolds M^{1} and M^{2}. Choose a time T and a point p_{T} ∈ P (H). Since the sets M^{1} and M^{2} are invariant, and these sets are graphs of functions defined on P (H)×ℝ_{t}, there are solutions y^{1}(t) and y^{2}(t) lying on M^{1} and M^{2}, respectively, satisfying the condition P y^{1}(T) = P y^{2}(T) = p_{T}. For these two solutions, we define the functions p(t), q(t), and w(t) as above. Our objective is to estimate the quantity

|q(T)| = |y^{1}(T) − y^{2}(T)| = |Φ^{1}(p_{T} , T) − Φ^{2}(p_{T} , T)| .

Since p(T) = 0, inequality (34) implies

|q(T)|^{2} = D̃^{2} + w(T) ⩽ D̃^{2} + w(τ)e^{−2(θ+m)(T −τ)} ⩽ D̃^{2} + |q(τ)|^{2}e^{−2(θ+m)(T −τ)}  (35)

for any τ ⩽ T. By the definition of M^{i}, we have y^{i}(τ) = O(e^{−θτ}) as τ → −∞, i = 1, 2. Thus,

|q(τ)|^{2} = |Q(y^{1}(τ) − y^{2}(τ))|^{2} = O(e^{−2θτ}),  τ → −∞,
|q(τ)|^{2}e^{−2(θ+m)(T −τ)} = e^{−2θT} e^{−2m(T −τ)}O(1),  τ → −∞.

As the variable τ tends to −∞ in (35) (recall that T is chosen), we obtain

|q(T)| ⩽ D̃ = \frac{dist(F^{1}, F^{2})}{2\sqrt{κ(θ + m)}}.

Since T ∈ ℝ_{t} and p_{T} ∈ P (H) are arbitrary, we obtain (32) for C = (4κ(θ + m))^{−1/2}. One can vary the positive numbers κ and m under the only condition of the form κ + m = δ − L and readily obtain the optimal value of the constant C = (4κ(θ + m))^{−1/2} in (32), which is equal to C = 1/(θ + δ − L). This completes the proof of the theorem.

Properties of integral manifolds. We return to the Cauchy problem in a Hilbert space H of type (20), (21),

\frac{dy}{dt} + Ay = F (y, t),  y|_{t=τ} = y_{τ} ∈ H.  (36)

Suppose that the above conditions are satisfied. Thus, for the process {Y (t, τ)} corresponding to problem (36), there exists an integral manifold M. The set M is the graph of a function Φ(p, t), Φ: P (H) × ℝ_{t} → Q(H). It follows from the definition of M (which is the union of some specific integral curves) that the following assertions hold:

1. if F does not depend on t, then Φ also does not depend on t;

2. if F (u, t) is a periodic function with respect to t, then Φ(p, t) is also periodic in t with the

same period. We claim that some other properties of the nonlinearity F (y, t) (including boundedness, Lipschitz

continuity, almost periodicity, and quasiperiodicity with respect to t) imply the corresponding properties of Φ(p, t).

Proposition 9. If F (y, t) is bounded, then Φ(p, t) is also bounded (i.e., the projection of the integral manifold M onto Q(H) is bounded).

Proof. Consider two equations (30) with F^{1}(y, t) = F (y, t) and F^{2} ≡ 0. Obviously, the integral manifold for the second (linear) problem is the plane q = 0. Thus, we have Φ^{1} ≡ Φ and Φ^{2} ≡ 0. Since

dist(F^{1}, F^{2}) = sup_{y∈H, t∈ℝ_{t}} |F (y, t)| ⩽ C_{0},

it follows from Theorem 3 that

sup_{p∈P (H), t∈ℝ_{t}} |Φ(p, t)| = dist(Φ^{1}, Φ^{2}) ⩽ CC_{0}.

Proposition 10. Let F (y, t) be a Lipschitz-continuous function with respect to t. Then Φ(p, t) is also Lipschitz-continuous with respect to t.

Proof. Choose any T > 0 and consider two equations (30) with F^{1}(y, t) = F (y, t + T) and F^{2}(y, t) = F (y, t). Let d be a Lipschitz constant for F with respect to t. We have

|F^{1}(y, t) − F^{2}(y, t)| = |F (y, t + T) − F (y, t)| ⩽ dT ∀y ∈ H, ∀t ∈ ℝ.

The integral manifolds M^{1} and M^{2} of these two problems are the graphs of the functions Φ^{1}(p, t) ≡ Φ(p, t + T) and Φ^{2}(p, t) ≡ Φ(p, t). Theorem 3 implies that

sup_{p∈P (H), t∈ℝ_{t}} |Φ^{1}(p, t) − Φ^{2}(p, t)| ⩽ C sup_{y∈H, t∈ℝ_{t}} |F^{1}(y, t) − F^{2}(y, t)| ⩽ CdT,

i.e., |Φ(p, t + T) − Φ(p, t)| ⩽ CdT for any t ∈ ℝ, T > 0, and p ∈ P (H). This means that the function Φ(p, t) is Lipschitz continuous with respect to t, and a Lipschitz constant for f can be obtained by multiplying the corresponding constant for F by C.

Proposition 11. Let F (y, t) be almost periodic with respect to time t. Then Φ(p, t) is also almost periodic with respect to t.

Proof. Let a number χ be an ε-period of the function F (y, t), i.e., |F (y, t + χ) − F (y, t)| ⩽ ε for all t ∈ ℝ and y ∈ H. Consider two equations (30) with F^{1}(y, t) = F (y, t+χ) and F^{2}(y, t) = F (y, t). The corresponding processes Y^{i}(t, τ) possess integral manifolds M^{i}, which are the graphs of the functions Φ^{1}(p, t) ≡ Φ(p, t + χ) and Φ^{2}(p, t) ≡ Φ(p, t). According to Theorem 3,

sup_{p∈P (H), t∈ℝ_{t}} |Φ^{1}(p, t) − Φ^{2}(p, t)| ⩽ C sup_{y∈H, t∈ℝ_{t}} |F^{1}(y, t) − F^{2}(y, t)| ⩽ Cε,

i.e., |Φ(p, t + χ) − Φ(p, t)| ⩽ Cε for any t ∈ ℝ and p ∈ P (H). Thus, χ is also a Cε-period of the function Φ(p, t) and Φ(p, t) is almost periodic with respect to t.

Now let the function F (y, t) be quasiperiodic with respect to t, and, more precisely, let

F (y, t) = ϕ(y, α_{1}t, α_{2}t, …, α_{k}t) = ϕ(y, ᾱt),

where ϕ(·, ω) ∈ C^{Lip}(𝕋^{k}; C_{L}), ω̄ = (ω_{1}, …, ω_{k}), and the real numbers (α_{1}, α_{2}, …, α_{k}) = ᾱ are rationally independent. Here the symbol C_{L} stands for the space of bounded functions f: H → H satisfying the Lipschitz condition with the constant L. The space C_{L} is equipped with the C(H; H)-norm.

Proposition 12. Let F (y, t) be quasiperiodic with respect to t. Then the function Φ(p, t) is also quasiperiodic, and, more precisely, there exists a function Ψ(p, ω) ∈ C^{Lip}(P (H) × 𝕋^{k}; Q(H)) such that Φ(p, t) = Ψ(p, ᾱt) and the frequencies ᾱ = (α_{1}, α_{2}, …, α_{k}) coincide with those for the function F (y, t).

Proof. For every ω ∈ 𝕋^{k}, we consider the following equation:

\frac{dy}{dt} + Ay = F_{ω}(y, t), where F_{ω}(y, t) = ϕ(y, ω + ᾱt).

The corresponding process possesses the integral manifold M_{ω} which is the graph of a function Φ_{ω}(p, t). Write Ψ(p, ω) = Φ_{ω}(p, 0). Theorem 2 shows that the function Ψ(p, ω) thus constructed is Lipschitz continuous with respect to p, and Theorem 3 implies that

|Ψ(p, ω_{1}) − Ψ(p, ω_{2})| = |Φ_{ω_{1}}(p, 0) − Φ_{ω_{2}}(p, 0)| ⩽ C sup_{y∈H, t∈ℝ_{t}} |F_{ω_{1}}(y, t) − F_{ω_{2}}(y, t)|

= C sup_{t∈ℝ_{t}} ∥ϕ(·, ω_{1} + ᾱt) − ϕ(·, ω_{2} + ᾱt)∥_{C(H;H)} ⩽ C̃ |ω_{1} − ω_{2}| .

Here we used the assumption on the Lipschitz continuity of the mapping ω → ϕ(·, ω) regarded as a C(H; H)-valued function on 𝕋^{k}.

We must only show that Φ(p, t) ≡ Ψ(p, ᾱt), i.e., Φ_{0}(p, t) ≡ Φ_{ᾱt}(p, 0).

We choose any p ∈ P (H) and h ∈ ℝ and, for the equation with the right-hand side F (y, t) = ϕ(y, ᾱt), find an integral curve (y(t), t) lying on M_{0} such that P y(h) = p. The solution y(t) satisfies the condition y(t) = O(e^{−θt}) as t → −∞. Then the shifted function y_{h}(t) = y(h + t) satisfies equation (20) with the right-hand side F_{ᾱh}(y, t) = ϕ(y, ᾱh+ᾱt), and y_{h}(t) has the same asymptotic property as t → −∞, i.e., (y_{h}(t), t) ∈ M_{ᾱh}. Thus, Φ_{0}(p, h) = Qy(h) = Qy_{h}(0) = Φ_{ᾱh}(p, 0).

Some special cases of nonlinearity. We now consider two equations (30) of the following specific type:

\frac{dy^{i}}{dt} + Ay^{i} = F^{i}(y^{i}, t) ≡ F (y^{i}, t) + g^{i}(t),  i = 1, 2.  (37)

Suppose that the properties (22)–(25) are satisfied. Thus, the integral manifolds M^{1} and M^{2} exist for the processes {Y^{i}(t, τ)} corresponding to our equations. Due to Theorem 3, it is possible to estimate the distance between these manifolds in terms of the distance between g^{1}(t) and g^{2}(t) in C_{B}(ℝ, H). However, this estimate deals with C-norms of g^{i}(t), and thus cannot give good results if, say, g^{1}(t) is a rapidly oscillating function and g^{2}(t) is its averaging. We suggest another approach to estimate dist(M^{1}, M^{2}) in this case. Our main assumption is as follows.

Let the linear equation

\frac{dw}{dt} + Aw = g^{1}(t) − g^{2}(t)

have a bounded solution w(t) defined for all values t ∈ ℝ.

Remark 3. If A is a positive-definite self-adjoint operator and g^{i}(t), i = 1, 2, are bounded functions, then the solution w(t) is defined by the formula

w(t) = ∫_{−∞}^{t} e^{−A(t−s)} (g^{1}(s) − g^{2}(s)) ds = ∫_{0}^{+∞} e^{−As} (g^{1}(t − s) − g^{2}(t − s)) ds.

If the spectrum of the self-adjoint operator A does not contain zero, then the solution is

w(t) = ∫_{−∞}^{t} e^{−A(t−s)}Π_{+} (g^{1}(s) − g^{2}(s)) ds − ∫_{t}^{+∞} e^{A(s−t)}Π_{−} (g^{1}(s) − g^{2}(s)) ds,

where Π_{+} and Π_{−} are orthogonal projections onto invariant subspaces of A corresponding to the positive and negative parts of the spectrum σ(A).

Theorem 4. There exists a constant C^{∗} such that

dist(M^{1}, M^{2}) ⩽ C^{∗} sup_{t∈ℝ} |w(t)|.  (38)

Proof. The difference y^{3}(t) = y^{1}(t) − w(t) satisfies the equation

\frac{dy^{3}}{dt} + Ay^{3} = F^{3}(y^{3}, t) ≡ F (y^{3} + w(t), t) + g^{2}(t).

The process Y^{3}(t, τ) acts by the formula Y^{3}(t, τ)y_{τ} = Y^{1}(t, τ)(y_{τ} + w(τ)) − w(t) and possesses the integral manifold M^{3}, which can readily be obtained from M^{1}. Namely, M^{3} is the graph of a function Φ^{3} : P (H) × ℝ_{t} → Q(H) given by

Φ^{3}(p, t) = Φ^{1}(p + P w(t), t) − Qw(t).

(As above, Φ^{1} is the function defining the integral manifold M^{1}.)

Theorem 2 states that the function Φ^{1}(p, t) is Lipschitz continuous with respect to p. Thus,

dist(M^{1}, M^{3}) = sup_{p∈P (H), t∈ℝ} |Φ^{3}(p, t) − Φ^{1}(p, t)| ⩽ sup_{t∈ℝ} (l|P w(t)| + |Qw(t)|) ⩽ \sqrt{1 + l^{∗}^{2}} sup_{t∈ℝ} |w(t)|,

(39)

where l^{∗}, l^{∗} < 1, is a Lipschitz constant (with respect to p) for the function Φ^{1}(p, t). On the other hand,

dist(F^{3}, F^{2}) = sup_{y∈H, t∈ℝ} |F^{3}(y, t) − F^{2}(y, t)| = sup_{y∈H, t∈ℝ} |F (y + w(t), t) − F (y, t)| ⩽ L sup_{t∈ℝ} |w(t)|,

where L is the Lipschitz constant for F (see (24)), and it follows from Theorem 3 that

dist(M^{3}, M^{2}) ⩽ CL sup_{t∈ℝ} |w(t)|.  (40)

Inequalities (39) and (40) imply inequality (38) for C^{∗} = CL + \sqrt{1 + l^{∗}^{2}}.

5. HYPERBOLIC EQUATION WITH RAPIDLY OSCILLATING FORCE

The following problem for a dissipative nonautonomous hyperbolic equation in a bounded domain Ω is considered:

∂_{t}^{2}u + 2γ∂_{t}u = Δu − f(u, t) + g^{0}(x, t) + g(x, \frac{t}{ε}),  \frac{∂u}{∂n}|_{∂Ω} = 0,  (41)
u|_{t=τ} = u_{τ} (x) ∈ H^{1}(Ω),  (42)
∂_{t}u|_{t=τ} = p|_{t=τ} = p_{τ} (x) ∈ L_{2}(Ω).  (43)

Here γ > 0 stands for the dissipation coefficient, the nonlinearity f(u, t) is C^{1}-smooth, |∂f/∂u| ⩽ l, g^{0}(x, t) ∈ C_{B}(ℝ, L_{2}(Ω)), the rapidly oscillating forcing term g(x, t/ε) is of class C_{B}(ℝ, H^{1}(Ω)), and 0 < ε ⩽ ε_{0}. We assume that the oscillating term g(x, ζ) has zero average with respect to the fast time variable ζ. Theorem 2 is applicable to this case.

Theorem 5. Let conditions (14) and (18) hold. Then there exists an (invariant, finite-dimen-sional, and Lipschitz-continuous) integral manifold M_{ε} that exponentially attracts all solutions of problem (41)–(43) as t → +∞.

We also consider the averaged equation

∂_{t}^{2}u + 2γ∂_{t}u = Δu − f(u, t) + g^{0}(x, t).  (44)

By Theorem 2, this equation also has an exponentially attracting integral manifold M_{0}.

Let J(x, ζ) be an antiderivative of g(x, ζ) with respect to the fast time variable ζ, that is, \frac{∂J}{∂ζ} (x, ζ) = g(x, ζ).

Theorem 6. Let J(·, ζ) be bounded in H^{1}(Ω),

∥J(·, ζ)∥_{H^{1}(Ω)} ⩽ K  ∀ζ ∈ ℝ.  (45)

Then there exists a constant C such that

dist(M_{ε}, M_{0}) ⩽ CKε.  (46)

Remark 4. The boundedness condition for J(·, ζ) with respect to the norm of L_{2}(Ω) is insufficient. Representing solutions in the form of Fourier series, one can readily construct a corresponding counterexample for the linear equation (i.e., with f ≡ 0) with a special inhomogeneity g(x, t/ε).

Proof of Theorem 6. Let us apply Theorem 4 to the case in question. Thus, we must construct a bounded solution w(t), t ∈ ℝ, of the linear equation

∂_{t}^{2}w + 2γ∂_{t}w = Δw + g(x, t/ε), \frac{∂w}{∂n} |_{∂Ω} = 0,

and estimate the C-norm of this solution.

We obtain the desired solution in the form of a Fourier series

w(t, x) = ∑_{k=0}^{∞} w_{k}(t)e_{k}(x).

Here the symbols e_{k}(x), k = 0, 1, 2, …, stand for the eigenvectors of the operator −Δ in the domain Ω under the Neumann boundary conditions, and w_{k}(t) is a solution of the ordinary differential equation

ẅ_{k} + 2γẇ_{k} + λ_{k}w_{k} = g_{k}(t/ε), g_{k}(ζ) = (g(x, ζ), e_{k}(x)) .

Denote by J_{k}(ζ) = (J(x, ζ), e_{k}(x)) the Fourier coefficients of the functions J(x, ζ). Inequality (45) yields

∑_{k=0}^{∞} λ_{k}J_{k}^{2}(ζ) ⩽ K^{2},  ∑_{k=0}^{∞} J_{k}^{2}(ζ) ⩽ K^{2}  ∀ζ ∈ ℝ.  (47)

We must estimate w(t) in H^{1}(Ω) and ∂_{t}w(t) in L_{2}(Ω), i.e., we must estimate the sum

∥w(t)∥_{H^{1}(Ω)}^{2} + ∥∂_{t}w(t)∥_{L_{2}(Ω)}^{2} = w_{0}^{2} + ẇ_{0}^{2} + ∑_{k=1}^{∞} (λ_{k}w_{k}^{2} + ẇ_{k}^{2}) = ∑_{k=0}^{∞} (v_{k}^{2} + p_{k}^{2}) ,

where the vector function (v_{k}(t), p_{k}(t)) = (\sqrt{λ_{k}}w_{k}, ẇ_{k}), k ⩾ 1, is a solution of the system

\frac{d}{dt}( \begin{matrix}v_{k} \\ p_{k} \end{matrix} ) + A_{k}( \begin{matrix}v_{k} \\ p_{k} \end{matrix} ) = ( \begin{matrix}0 \\ g_{k}(t) \end{matrix} ),  A_{k} = ( \begin{matrix}0 & −\sqrt{λ_{k}} \\ \sqrt{λ_{k}} & 2γ \end{matrix} ) ,  (48)

and the pair (v_{0}, p_{0}) = (w_{0}, ẇ_{0}) satisfies the equation

v̈_{0} + 2γv̇_{0} = g_{0}(t/ε),  p_{0} = v̇_{0}.  (49)

We seek bounded solutions of (48) and (49). Let us begin with v_{0}(t). The function

v_{0}(t) = ε ∫_{−∞}^{t} e^{−2γ(t−s)}J_{0}(s/ε) ds

is a bounded solution of the equation

v̇_{0} + 2γv_{0} = εJ_{0}(t/ε), where \frac{d}{dt}(εJ_{0}(t/ε)) = g_{0}(t/ε).

Thus, the function v_{0}(t) satisfies (49), and

|v_{0}(t)| ⩽ \frac{Kε}{2γ},  |p_{0}(t)| = |εJ_{0}(t/ε) − 2γv_{0}(t)| ⩽ 2Kε.  (50)

Now let us consider equation (48). Both eigenvalues μ_{k} = γ − \sqrt{γ^{2} − λ_{k}} and ν_{k} = γ + \sqrt{γ^{2} − λ_{k}} of the 2×2-matrix A_{k} have positive real parts. Hence, the desired solution which is bounded for all values of t is

( \begin{matrix}v_{k}(t) \\ p_{k}(t) \end{matrix} ) = ∫_{−∞}^{t} exp(−A_{k}(t − s)) ( \begin{matrix}0 \\ g_{k}(s) \end{matrix} )ds = ε ∫_{−∞}^{t} exp(−A_{k}(t − s)) ( \begin{matrix}0 \\ \frac{d}{ds}J_{k}(s/ε) \end{matrix} ) ds

= ε[( \begin{matrix}0 \\ J_{k}(0) \end{matrix} ) − ∫_{−∞}^{t} A_{k} exp(−A_{k}(t − s)) ( \begin{matrix}0 \\ J_{k}(s/ε) \end{matrix} ) ds].  (51)

Therefore,

\frac{1}{ε^{2}} ∑_{k=1}^{∞} (v_{k}^{2} + p_{k}^{2}) = ∑_{k=1}^{∞} |( \begin{matrix}0 \\ J_{k}(0) \end{matrix} ) − ∫_{−∞}^{t} A_{k} exp(−A_{k}(t − s)) ( \begin{matrix}0 \\ J_{k}(s/ε) \end{matrix} ) ds|^{2}

⩽ 2 ∑_{k=1}^{∞} J_{k}^{2}(0, 0) + 2 ∑_{k=1}^{∞} |∫_{−∞}^{t} A_{k} exp(−A_{k}(t − s)) ( \begin{matrix}0 \\ J_{k}(s/ε) \end{matrix} )ds|^{2} . (52)

Obviously, the first term on the right-hand side is less than 2K^{2}. To estimate the integral, we need the following auxiliary statements.

Lemma 1. The following inequality holds:

∑_{k=1}^{∞} (∫ a(t)b_{k}(t) dt)^{2} ⩽ (∫ |a(t)| dt)^{2} sup_{t} (∑_{k=1}^{∞} b_{k}^{2}(t))  (53)

provided that the terms on the right-hand side are finite.

Proof. Write ψ_{k}(t) = b_{k}(t)/ sup_{t} \sqrt{∑_{k=1}^{∞} b_{k}^{2}(t)} and represent inequality (53) in the form

∑_{k=1}^{∞} (∫ a(t)ψ_{k}(t) dt)^{2} ⩽ (∫ |a(t)| dt)^{2}  if  |ψ⃗(t)|^{2} ≡ ∑_{k=1}^{∞} ψ_{k}^{2}(t) ⩽ 1 ∀t.  (54)

Below we prove (54) by using the following discrete arithmetic statement:

∑_{k} ( ∑_{l} A_{l}Ψ_{k}^{(l)})( ∑_{l} B_{l}Ψ_{k}^{(l)}) ⩽ ( ∑_{l} |A_{l}|)( ∑_{l} |B_{l}|) if |Ψ⃗^{(l)}| ≡ \sqrt{∑_{k} |Ψ_{k}^{(l)}|^{2}} ⩽ 1 ∀l. (55)

In order to verify (55), one can readily remove the parentheses in this inequality, collect the terms containing A_{l}B_{m}, and obtain

A_{l}B_{m} ∑_{k} (Ψ_{k}^{(l)}Ψ_{k}^{(m)}) ⩽ |A_{l}||B_{m}||Ψ⃗^{(l)}||Ψ⃗^{(m)}| ⩽ |A_{l}||B_{m}| ∀l, m.

To complete the proof of the lemma, we note that, when using the notation A_{l} = B_{l} = a(t_{l})Δt_{l} and Ψ_{k}^{(l)} = ψ_{k}(t_{l}), the terms in inequality (55) become integral sums for the integrals in (54).

Proposition 13. In ℝ^{2}, consider the linear operator A_{k}, k ⩾ 1, defined in (48). The following estimates hold (here the symbol ∥·∥ stands for the standard norm of the linear operator acting from ℝ^{2} to ℝ^{2}) :

1.

∥A_{k}∥ ⩽ \sqrt{λ_{k}} + 2γ,  (56)

2.

∥exp(−A_{k}s)∥ ⩽ C exp(−μ_{1}s)  ∀s ⩾ 0,  (57)

where μ_{1} = γ − \sqrt{γ^{2} − λ_{1}} > 0 and the constant C does not depend on k.

Proof. The first estimate is evident:

∥A_{k}∥ ⩽ ∥( \begin{matrix}0 & −\sqrt{λ_{k}} \\ \sqrt{λ_{k}} & 0 \end{matrix} )∥ + ∥( \begin{matrix}0 & 0 \\ 0 & 2γ \end{matrix} )∥ = \sqrt{λ_{k}} + 2γ.

We readily obtain inequality (57) with C depending on k, i.e., C = C_{k}. In fact, if γ^{2} > λ_{k}, then both eigenvalues μ_{k} = γ −\sqrt{γ^{2} − λ_{k}} and ν_{k} = γ +\sqrt{γ^{2} − λ_{k}} of the matrix A_{k} are positive. We pass to an eigenbasis of the operator A_{k} and, since 0 < μ_{1} ⩽ μ_{k} < ν_{k}, obtain the estimate

∥exp(−A_{k}s)∥ ⩽ exp(−μ_{k}s) ⩽ exp(−μ_{1}s) ∀s ⩾ 0,

where the norm of the operator is evaluated in this very basis. Coming back to the initial basis, we only obtain some constant C_{k} in the inequality.

If γ^{2} < λ_{k}, then the eigenvalues μ_{k} and ν_{k} of the operator A_{k} are complex conjugate, and their real parts are equal to γ. We take the real and imaginary parts of the eigenvectors as a new basis and obtain (in this basis)

∥exp(−A_{k}s)∥ ⩽ exp(−γs) ⩽ exp(−μ_{1}s) ∀s ⩾ 0

since μ_{1} < γ. The constants C_{k} (appearing when returning to the initial basis) tend to 1 as k → ∞ because the eigenvectors of the matrix A_{k} tend to (1; ±i) as \sqrt{λ_{k}} → +∞.

We have thus proved (57) with C = max C_{k}.

Remark 5. The degenerate situation γ^{2} = λ_{k} (where both eigenvalues of A_{k} are equal to γ) is not excluded from our consideration either. As is well known, in this case, the estimate

∥exp(−A_{k}s)∥ ⩽ exp(−(γ − ε)s) ∀s ⩾ 0

with an arbitrary ε > 0 can be proved by choosing a proper basis.

Now we return to inequality (52). Using Lemma 1, Proposition 13, and (47), we see that

\frac{1}{ε^{2}} ∑_{k=1}^{∞} (v_{k}^{2} + p_{k}^{2}) ⩽ 2K^{2} + ∑_{k=1}^{∞} |∫_{−∞}^{t} A_{k} exp(−A_{k}(t − s)) ( \begin{matrix}0 \\ J_{k}(s/ε) \end{matrix} )ds|^{2}

⩽ 2K^{2} + (∫_{0}^{+∞} ∥exp(−A_{k}s)∥ ds)^{2} sup_{s} (∑_{k=1}^{∞} ∥A_{k}∥^{2} J_{k}^{2}(s/ε))

⩽ 2K^{2} + (\frac{C}{μ_{1}} )^{2} sup_{ζ} (∑_{k=1}^{∞}(\sqrt{λ_{k}} + 2γ)^{2}J_{k}^{2}(ζ))

⩽ 2K^{2} + 2(\frac{C}{μ_{1}} )^{2} [sup_{ζ} (∑_{k=1}^{∞} λ_{k}J_{k}^{2}(ζ)) + 4γ^{2} sup_{ζ} (∑_{k=1}^{∞} J_{k}^{2}(ζ))]

⩽ 2K^{2} + 2(\frac{C}{μ_{1}} K)^{2} [1 + 4γ^{2}] = C_{2}K^{2}.

Together with (50), this gives

sup_{t∈ℝ} (∥w(t)∥_{H^{1}(Ω)}^{2} + ∥∂_{t}w(t)∥_{L_{2}(Ω)}^{2}) = sup_{t∈ℝ} ∑_{k=0}^{∞} (v_{k}^{2}(t) + p_{k}^{2}(t)) ⩽ C_{3}K^{2}ε^{2},

where C_{3} depends only on Ω and γ. It follows from the last inequality and Theorem 4 that inequality (46) holds. This completes the proof of Theorem 6.

6. GLOBAL ATTRACTOR FOR THE DAMPED NONAUTONOMOUS SINE–GORDON EQUATION

The following problem in a bounded domain Ω is considered:

∂_{t}^{2}u + 2γ∂_{t}u = Δu − l sin u + g(x, t),  \frac{∂u}{∂ν}|_{∂Ω} = 0,  (58)
u|_{t=τ} = u_{τ} (x) ∈ H^{1}(Ω),  (59)
∂_{t}u|_{t=τ} = p|_{t=τ} = p_{τ} (x) ∈ L_{2}(Ω).  (60)

Here γ > 0 stands for the dissipation coefficient and g(·, t) ∈ C_{B}(ℝ, L_{2}(Ω)) is an external force.

Remark 6. All the statements in this section remain valid if the term l sin u in equation (58) is replaced by a function f(u, t) that is continuous, bounded, 2π-periodic with respect to u, and satisfies the condition |∂f/∂u| ⩽ l.

Problem (58)–(60) generates the process Y (t, τ)(u_{τ} , p_{τ}) = (u(t, ·), ∂_{t}u(t, ·)). Along with any solution u(t, x), equation (58) has also the solutions u(t, x) + 2πk, k ∈ ℤ. Thus, we can define the process {Y^{F}(t, τ)} acting on the factorized phase space

H^{F} = (H^{1}(Ω) mod 2π) × L_{2}(Ω) = (H^{1}(Ω) × L_{2}(Ω))/(2πℤ, 0).

The nonlinearity in equation (58) is bounded, and the linear part of this equation is stable on the subspace of codimension 1 and has a single zero eigenvalue (the corresponding eigenspace coincides with the space of constants with respect to which the process is factorized). Thus, the process {Y^{F}(t, τ)} possesses a bounded absorbing set. The standard apparatus for hyperbolic equations enables us to construct at least one bounded complete trajectory y^{F}(t) of this process, i.e., y^{F}(t) is defined for all values t ∈ ℝ and y^{F}(t) = Y^{F}(t, τ)y^{F}(τ) for any t ⩾ τ (see [30, 33, 34, 31]). The corresponding solution y(t) = (u(t, ·), ∂_{t}u(t, ·)) = Y (t, τ)y(τ) of the initial equation can be unbounded in the direction of constants ū(t) = ∫_{Ω} u(t, x)dx; however, the growth of this solution is at most linear because the time derivative ∂_{t}ū(t) = ∂_{t}ū^{F}(t) is bounded. Recall that in the proof of Theorem 2, we needed a solution y(t) of this kind to construct an integral manifold.

Definition 1 (see, for example, [35, 31] and the references therein). A closed set A is called a uniform (with respect to τ) global attractor of the process {Y (t, τ)} if the following conditions hold:

1. the set A is uniformly (with respect to τ) attracting, i.e., for any bounded set B in H and for any ε > 0, there is a number h = h(B, ε) such that

Y (t, τ)B ⊂ O_{ε}(A) if t − τ ⩾ h;

2. (minimality condition) the set A belongs to any closed uniformly attracting set of the process {Y (t, τ)}.

We need the following simple statement.

Proposition 14. Let y(t), t ∈ ℝ, be a complete trajectory of a process {Y (t, τ)}, i.e., y(t) = Y (t, τ)y(τ) for any t ⩾ τ, t, τ ∈ ℝ, and let the set B = {y(t) | t ⩽ 0} be bounded. Then the set V = {y(s) | s ∈ ℝ} belongs to any closed uniformly attracting set of the process {Y (t, τ)}. Thus, if a uniform attractor A exists, then [V] ⊂ A (the symbol [V] stands for the closure of the set V).

Proof. For every fixed time t_{0}, we have y(t_{0}) = Y (t_{0}, t_{0} − h)y(t_{0} − h) ∈ Y (t_{0}, t_{0} − h)B if h ⩾ max(0, t_{0}). Hence, for any ε > 0,

y(t_{0}) ∈ Y (t_{0}, t_{0} − h)B ⊂ O_{ε}(A)

provided that h = h(ε) is sufficiently large. Thus, any point y(t_{0}) of the set V belongs to O_{ε}(A) for any ε > 0. This implies that y(t_{0}) ∈ [A] = A.

Denote by λ_{1} the first positive eigenvalue of the operator −Δ in the domain Ω with the Neumann boundary conditions.

Theorem 7. Let the following inequalities hold:

λ_{1} > 4l and γ^{2} > \frac{(λ_{1} − 2l)^{2}}{λ_{1} − 4l} .  (61)

Then the extended phase space H × ℝ_{t} contains a two-dimensional integral manifold M which exponentially attracts all solutions of problem (58)–(60) as t → +∞.

Proof. Apply Theorem 1 with N = 0. We have λ_{0} = 0, and thus the inequality λ_{1} > 4l coincides with (14). Inequality (18) becomes

γ^{2} > λ_{1} + \frac{l}{R − 1}, where R = \frac{λ_{1}}{4l} ,

that is, γ^{2} > λ_{1} + 4l^{2}/(λ_{1} − 4l) = (λ_{1} − 2l)^{2}/(λ_{1} − 4l).

We denote by M the set-theoretic union of the integral curves (y(t), t) that are defined for all t ∈ ℝ_{t} and satisfy the condition |y(t)| = O(e^{−θt}) as t → −∞, where θ = \frac{1}{2} (γ − \sqrt{γ^{2} − λ_{1}}) > 0. Due to Theorem 2, this set M has the following properties:

1. the set M is the graph of a function Φ(p, t), Φ: P (H) × ℝ_{t} → Q(H), which is Lipschitzcontinuous with respect to p. Here the space P (H), P (H) ⊂ H ≡ H^{1}(Ω) × L_{2}(Ω), contains the elements of the form (c, 0) ∈ H^{1}(Ω) × L_{2}(Ω), where c is a constant function, and dim P (H) = 1;

2. every solution y(t) = Y (t, τ)y_{τ} of equation (58) admits a trace ỹ(t), t ∈ ℝ, lying on the integral manifold M, i.e., the function ỹ(t) is also a solution of (58), (ỹ(t), t) ∈ M for all t ∈ ℝ, and

|y(t) − ỹ(t)| ⩽ c(|y(τ)|)e^{−(θ+μ)(t−τ)}  ∀t ⩾ τ,  (62)

where μ = θ − l/\sqrt{γ^{2} − λ_{1}} > 0 and c > 0.

Theorem 8. The closure of the projection of M to the phase space H coincides with a uniform attractor A of the process {Y^{F}(t, τ)}. Thus, the global attractor A exists, and it is exponential.

Proof. The nonlinear function f(u) = l sin u is bounded, and the external forcing term g(·, t) is bounded in L_{2}(Ω). It follows from Proposition 9 that Q(M) is also bounded. By definition, the set M is invariant with respect to the translations (u(·), p(·)) → (u(·) + 2πk, p(·)), k ∈ ℤ, and thus the function Φ(p, t) is 2π-periodic with respect to p. Introduce the set

M^{F} = (ΠM)/(2πℤ, 0) ⊂ H^{F},

where Π is the projection operator Π(u, t) = u from the extended phase space H × ℝ_{t} onto the phase space H.

We have the following properties of the set M^{F}.

1. M^{F} is bounded in H^{F}. In fact, Q(M) is bounded, and P (M)/2πℤ = S^{1} is a bounded space.

2. M^{F} is an exponentially attracting set for the process {Y^{F}(t, τ)} by (62). The set [M^{F}] is a closed exponentially attracting set for the process {Y^{F}(t, τ)}. On the other

hand, every point of M^{F} lies on a bounded trajectory y(t) of the process {Y^{F}(t, τ)}. Hence, Proposition 14 implies that [M^{F}] does belong to every closed attracting set of the process {Y^{F}(t, τ)}. Thus, [M^{F}] is a uniform global attractor A, and this attractor is exponential.

Theorem 9. If g(· , t) is a quasiperiodic function with k frequencies, then the global attractor A is a Lipschitz-continuous image of the (k + 1)-dimensional torus 𝕋^{k+1}.

Proof. As was already mentioned in the present section, the function Φ(p, t) specifying the integral manifold M is 2π-periodic with respect to p. On the other hand, Proposition 12 states that Φ(p, t) = Ψ(p, ᾱt), where Ψ(p, ω) ∈ C^{Lip}(P (H) × 𝕋^{k}; Q(H)) and ᾱ = (α_{1}, α_{2}, …, α_{k}) are the frequencies of the quasiperiodic function g(· , t). Thus, the Lipschitz-continuous function Ψ(p, ω) is well defined on the set (P (H) mod 2π) × 𝕋^{k} ≡ 𝕋^{k+1}.

Let us now consider two problems of type (58)–(60) with forcing terms g = g^{1}(x, t) and g = g^{2}(x, t). Suppose that (61) holds. In this case, there exist global attractors A_{1} and A_{2} for both problems. Theorem 3 enables us to estimate the symmetric distance between these attractors,

dist_{symm}(A_{1}, A_{2}) ⩽ C sup_{t∈ℝ_{t}} ∥g^{1}(·, t) − g^{2}(·, t)∥_{L_{2}(Ω)}

provided that the supremum on the right-hand side is finite. By the symmetric distance between two sets X and Y in a space H, we mean the quantity

dist_{symm}(X, Y) = max(dist_{H}(X, Y), dist_{H}(Y, X)), dist_{H}(X, Y) = sup_{x∈X} inf_{y∈Y} ∥x − y∥_{H} .

It is possible to obtain a similar result in the case of rapidly oscillating forcing terms. As in Section 5, we consider problem (41)–(43), i.e., the problem of type (58)–(60) with the external force g = g^{0}(x, t) + g(x, t/ε) and the similar problem for the averaged equation (44). If condition (61) holds, then there exist global attractors A_{ε} and A_{0} for both problems (oscillating and averaged). If the conditions of Theorem 6 are satisfied, then

dist_{symm}(A_{ε}, A_{0}) ⩽ Cε.

7. FURTHER DIRECTIONS OF STUDY

In conclusion, let us present some results concerning global attractors of hyperbolic equations with singularly oscillating external force and formulate open problems on global attractors and analogs of inertial manifolds for these equations.

Consider the following damped hyperbolic equation:

∂_{t}^{2}u + 2γ∂_{t}u = Δu − f(u) + g^{0}(x) + ε^{−α}g(x, \frac{t}{ε})  (63)

with Neumann or Dirichlet boundary conditions:

\frac{∂u}{∂n}|_{∂Ω} = 0  or  u|_{∂Ω} = 0.  (64)

Here it is assumed that γ > 0 and α > 0, g^{0}(·) ∈ L_{2}(Ω), g(·, ζ) ∈ C_{B} (ℝ; H^{1}(Ω)), and |f′(v)| ⩽ l. We also suppose that the function g(x, ζ) has bounded antiderivative with respect to ζ,

\frac{∂J}{∂ζ} (x, ζ) = g(x, ζ);  ∥J(·, ζ)∥_{H^{1}(Ω)} ⩽ K  ∀ζ ∈ ℝ.  (65)

Consider the averaged equation

∂_{t}^{2}u + 2γ∂_{t}u = Δu − f(u) + g^{0}(x),  (66)

with the same boundary conditions (64).

Remark 7. If condition (3) holds, then an analog of Theorem 6, which is proved for equation (41), remains valid for equation (63) as well. Indeed, the crucial point in the proof of Theorem 6 is the estimate of a specific bounded solution w(t) of the linear equation. If the inhomogeneity g(x, t/ε) in this equation is replaced by ε^{−α}g(x, t/ε), then the function w(t) is simply multiplied by ε^{−α}. Thus, in this case, the estimate for the deviation of integral manifolds becomes

dist(M_{ε}, M_{0}) ⩽ CKε^{1−α}, ε > 0.

Under the above assumptions, problems (63), (64) and (66), (64) have the global attractors A_{ε α} and A_{0 0}, respectively. The following estimate for the deviation of these global attractors was proved in [36]:

dist_{H}(A_{ε α}, A_{0 0}) = Cε^{b(α)},  b(α) > 0.  (67)

Here the number α is fixed, 0 < α ⩽ α_{0}, b(α) > 0, and 0 < ε ⩽ ε_{0}(α). The distance in (67) is measured with respect to the norm of the energy space H = H^{1}(Ω) × L_{2}(Ω). The explicit formulas are given for the values α_{0}, b(α), and ε_{0}. The estimate (67) is proved without any spectral gap assumptions concerning equations (63) and (66).

Let us now formulate two open problems for dissipative hyperbolic equations.

1. Is it possible to construct analogs of integral manifolds for equations (63) and (66) and to prove estimates for the distance between these manifolds (estimates similar to (67)) without gap assumptions?

2. Consider the hyperbolic equations of type (63) with nonlinear functions f(v) of superlinear polynomial growth and estimate the distance between the global attractors of singularly oscillating and averaged equations in terms of ε.

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