Russian Journal of Mathematical Physics, Vol. 8, No. 3, 2001, pp. 311–330. Copyright \begin{matrix}c \\ ◯ \end{matrix} 2001 by MAIK “Nauka/Interperiodica” (Russia).

Dedicated to M. S. Agranovich on the occasion of his 70th anniversary

Global Attractor and Its Perturbations for a Dissipative Hyperbolic Equation

V. V. Chepyzhov and M. I. Vishik

Institute for Information Transmission Problems, Russian Academy of Sciences Bol’shoi Karetnyi per. 19, 103051, Moscow, Russia

Received June 8, 2001

Abstract. We consider a dissipative hyperbolic equation with nonlinear term having high degree of polynomial growth. In this case, the uniqueness theorem for the corresponding Cauchy problem is not proved. For this equation we construct the global attractor, study its properties, and prove that the global attractors of the Galerkin approximation systems converge from below to the global attractor of the original hyperbolic equation. We prove that the dependence of global attractors of hyperbolic equations with rapidly oscillating spatial terms depend on the global attractor of the averaged equation is upper semicontinuous as the oscillation frequency tends to infinity. Stronger results are obtained for hyperbolic equations with moderate growth of nonlinear terms.

We study a dissipative wave equation of the form

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

where γ > 0, g(x) ∈ L_{2}(Ω), and Ω ⋐ ℝ^{n}. In the first four sections of the paper we do not assume that a solution of the Cauchy problem for (1) is unique. As is well known (see, e.g., [1]), the uniqueness theorem for (1) is proved at present only for the case in which the function f(u) is of polynomial growth of degree less than or equal to 1 + 2/(n − 2) for n ⩾ 3. At the same time, the existence theorem for a global weak solution u(x, t), t ⩾ 0, of (1) supplemented with the corresponding initial data is proved without restrictions to the degree of the function f(u). The problem to describe the limit behavior of these solutions u(x, t) as time t goes to infinity naturally arises. To solve this problem, in [2], a trajectory attractor was constructed for (1). The trajectory attractor 𝔄 consists of all weak global solutions (trajectories) (u(x, t), ∂_{t}u(x, t)), t ⩾ 0, of (1) that are bounded with respect to the energy norm and have prolongation to the entire time axis −∞ < t < ∞ as a weak solution bounded with respect to the energy norm. The construction and the main properties of this trajectory attractor 𝔄 are given in Section 1.

In Section 2 we construct the global attractor A for the same equation. We prove that weak solutions (u(x, t), ∂_{t}u(x, t)), t ⩾ 0, of (1) belong to the space C (ℝ_{+}; H^{1−δ}(Ω) × H^{−δ}(Ω)), where δ > 0 is arbitrarily small. In particular, the trajectory attractor 𝔄 belongs to this space formed by continuous functions with values in H^{1−δ}(Ω) × H^{−δ}(Ω). Therefore, a section of the trajectory attractor is well defined for any specified instant t,

𝔄(t) = {(u(·, t), ∂_{t}u(·, t)) | (u, ∂_{t}u) ∈ 𝔄} .

Clearly, the set 𝔄(t) belongs to the space H^{1−δ}(Ω) × H^{−δ}(Ω) and does not depend on t because equation (1) is autonomous, and the trajectory attractor is strictly invariant with respect to the time translations (see Section 1). The section of the trajectory attractor at time t = 0 is called the global attractor A of equation (1):

A =^{def} 𝔄(0).

The research was supported in part by the Russian Foundation of Basic Researches under grant no. 99-01-00304 and by INTAS grant.

The properties of the global attractor A are similar to those of the ordinary global attractor of the semigroup {S(t)} corresponding to (1) for the case in which the uniqueness theorem holds. In particular, as t → +∞, the global attractor A attracts any section B(t) for any bounded set B of trajectories of (1), i.e.,

dist_{H^{1−δ}×H^{−δ}} (B(t), A) → 0 (t → +∞).

(Here dist_{M} (A, B) stands for the Hausdorff distance from a set A to a set B in a metric space M.) Recall that we do not assume that the Cauchy problem for (1) is uniquely solvable.

In Section 3 we prove that the dependence of the global attractors A_{m} of the Galerkin approximation systems of order m for equation (1) on m is upper semicontinuous, i.e.,

dist_{H^{1−δ}×H^{−δ}} (A_{m}, A) → 0 (ε → +0),

where A is the global attractor of (1). Here A_{m} are the conventional global attractors known from the usual theory of semigroups of equations with uniqueness property. As above, the uniqueness for (1) is not assumed.

In Section 4 we study hyperbolic equations similar to (1) that contain rapidly oscillating terms γ(x, x/ε)∂_{t}u, b(x, x/ε)f(u), and g(x, x/ε). We assume that the functions γ(x, x/ε), b(x, x/ε), and g(x, x/ε) weakly converge as ε → 0+ to the corresponding averaged functions γ̄(x), b̄(x), and ḡ(x), respectively. Along with a hyperbolic equation containing rapidly oscillating terms, we consider the averaged hyperbolic equation with averaged terms. As usual, we do not assume that, for any of these equations, the solutions of the corresponding Cauchy problem are unique. Nevertheless, the existence theorems hold, and we can construct the corresponding trajectory attractors 𝔄_{ε} and \overline{𝔄}. Using sections of these trajectory attractors, we also construct the global attractors A_{ε} and \overline{A}. In Section 4 we prove that the trajectory attractor 𝔄_{ε} converges to \overline{𝔄} as ε → 0+ and that the global attractor A_{ε} converges to \overline{A} as ε → 0+ (in the corresponding spaces). To be more precise, we prove that

dist_{H^{1−δ}×H^{−δ}} (A_{ε}, \overline{A}) → 0 (ε → +0).

In Section 5 we prove explicit estimates for the distance between the global attractor A_{ε} of (1) perturbed with terms of the form εf_{1}(u) and εg_{1}(x) and the global attractor A_{0} (ε = 0) of the initial equation (1). We now assume that the Cauchy problem for these equations is uniquely solvable. We prove the following estimate:

dist_{H^{1}×H^{0}} (A_{ε}, A_{0}) ⩽ Cε^{η}, where η = \frac{1}{(1 + \frac{ρ}{ν} )}.

Here ν is the exponent of attraction of the solutions of equation (1) to A_{0} as t → +∞, and the constant ρ is the factor in the exponent of the deviation of a solution u_{ε}(x, t) of the perturbed equation (1) from the solution u_{0}(x, t) of the nonperturbed equation (1) provided that the initial conditions are the same, u_{ε}(·, 0) = u_{0}(·, 0), ∂_{t}u_{ε}(·, 0) = ∂_{t}u_{0}(·, 0), namely,

∥u_{ε}(·, t) − u_{0}(·, t)∥ ⩽ C_{1}e^{ρt}, t ⩾ 0.

1. TRAJECTORY ATTRACTORS OF HYPERBOLIC EQUATIONS

Let us recall the definition of the trajectory attractor for a dissipative hyperbolic equation

∂_{t}^{2}u + γ∂_{t}u = Δu − f(u) + g (x) ,  u|_{∂Ω} = 0,  x ∈ Ω ⋐ ℝ^{n}.  (2)

We assume that g(x) ∈ L_{2}(Ω) and that the nonlinear term f(v) ∈ C(ℝ; ℝ) satisfies the inequality

|f(v)| ⩽ γ_{0}(|v|^{p−1} + 1),  p > 1, γ_{0} > 0.  (3)

We denote by

F (v) = ∫_{0}^{v} f(w) dw

the standard antiderivative of f(v). We also assume that

F (v) ⩾ γ_{1}|v|^{p} − C_{1},  (4)
f(v)v ⩾ γ_{2}F (v) − C_{2} for any
quadv ∈ ℝ,  (5)

where γ_{i} > 0 and C_{i} > 0 for i = 1, 2.

Remark 1.1. Equations of this type occur in relativistic quantum mechanics (see [1, 5, 6] and the references therein), where nonlinear term is of the form

f(u) = |u|^{p−2}u, p ⩾ 2.

Recall that a function u(x, t), x ∈ Ω, t ⩾ 0 is called a solution of equation (2) in the weak sense (or a weak solution) if u(x, t) ∈ L_{∞}^{loc}(ℝ_{+}; L_{p}(Ω)) ∩ L_{∞}^{loc}(ℝ_{+}; H_{0}^{1}(Ω)), ∂_{t}u(x, t) ∈ L_{∞}^{loc}(ℝ_{+}; L_{2}(Ω)), and u(x, t) satisfies equation (2) in the sense of distributions for the space D′(ℝ_{+}; H^{−r}(Ω)), where r ≡ max {1, n(1/q − 1/2)} and 1/q+1/p = 1 (see [1]). According to the Sobolev embedding theorem, the above exponent r is chosen in such a way that Δu(x, t) ∈ L_{∞}^{loc}(ℝ_{+}; H^{−1}(Ω)) ⊂ L_{∞}^{loc}(ℝ_{+}; H^{−r}(Ω)) and f(u(x, t)) ∈ L_{∞}^{loc}(ℝ_{+}; L_{q}(Ω)) ⊂ L_{∞}^{loc}(ℝ_{+}; H^{−r}(Ω)). Therefore, all terms in (2) belong to the space L_{∞}^{loc}(ℝ_{+}; H^{−r}(Ω)) and can be regarded as distributions belonging to D′(ℝ_{+}; H^{−r}(Ω)) (for more detail, see [2, 3]).

It follows from the above properties of weak solutions that

u ∈ C_{w} (ℝ_{+}; H_{0}^{1}(Ω)) ,  u ∈ C_{w}(ℝ_{+}; L_{p}(Ω)),  ∂_{t}u ∈ C_{w}(ℝ_{+}; L_{2}(Ω)),  (see [3])

for any weak solution u(x, t) of (2). Therefore, for any instant t, t ⩾ 0, the values u(·, t) and ∂_{t}u(·, t) are uniquely defined. Moreover,

u(·, t) ∈ H_{0}^{1}(Ω) ∩ L_{p}(Ω), and ∂_{t}u(·, t) ∈ L_{2}(Ω).

For a given initial data

u|_{t=0} = u_{0}(x) ∈ H_{0}^{1}(Ω) ∩ L_{p}(Ω),  ∂_{t}u|_{t=0} = u_{1}(x) ∈ L_{2}(Ω),  (6)

the Cauchy problem (2), (6) has at least one weak solution u(x, t) [2, 4]. Thus, the set of weak solutions of (2) is very large. For this reason we are interested in the study of the behavior of weak solutions of (2) as t → +∞.

Let v(x) ∈ H_{0}^{1}(Ω) ∩ L_{p}(Ω) and v_{1}(x) ∈ L_{2}(Ω). Consider the nonlinear functional

J_{α}(v, v_{1}) = ∫_{Ω} (|∇v(x)|^{2} + |v_{1}(x) + αv(x)|^{2} + 2F (v(x))) dx,  (7)

where α is positive and satisfies the inequality

0 < α(γ − α) < λ_{1}.  (8)

Here λ_{1} is the first eigenvalue of the Laplace operator {−Δu, u|_{∂Ω} = 0}.

Let u(t) be a weak solution of (2). Write

z(t) = z_{α}(t) =^{def} J_{α}(u(t), ∂_{t}u(t)).  (9)

In view of (3) and (4),

∥v∥^{2} + |v_{1} + αv|^{2} + 2γ_{1}∥v∥_{L_{p}(Ω)}^{p} − C′_{1} ⩽ J_{α}(v, v_{1}) ⩽ ∥v∥^{2} + |v_{1} + αv|^{2} + 2γ′′_{0}∥v∥_{L_{p}(Ω)}^{p} + C′_{0}. (10)

It is proved formally in [2] (i.e., under regularity assumptions for u(x, t)) that the function z(t) satisfies the following differential inequality

\frac{d}{dt}z(t) + δ_{α}z(t) ⩽ \frac{2}{γ − α}|g|^{2} + ρ_{α} for t ⩾ 0,  (11)

where the constants δ_{α} > 0 and ρ_{α} > 0 depend explicitly on the data of the problem (see [3]). It follows from (11) that

z(t) ⩽ R_{α} + z(0)e^{−δ_{α}t} for t ⩾ 0,  (12)

where R_{α} = R_{α} = (ρ_{α} + 2(γ − α)^{−1}|g|^{2}) δ^{−1}.

We now choose a number N > 0 and denote by K^{+}(N) the set of all weak solutions of (2) such that the following inequality holds for every positive α satisfying (8):

z_{α}(t) ⩽ R_{α} + Ne^{−δ_{α}t} for t ⩾ 0.  (13)

Note that we simply replaced the value z(0) in (12) by N. Thus, if z(0) = J_{α}(u(0), ∂_{t}u(0))) ⩽ N, then u(·) ∈ K^{+}(N). The set K^{+}(N) is called the trajectory space of (2) and the elements of K^{+}(N) are called trajectories of (2).

As was shown in [2],

(i) the set K^{+}(N) is nonempty for large N; (ii) the set K^{+}(N) is translation invariant, i.e., if u(·, t) ∈ K^{+}(N), then u(·, t + h) ∈ K^{+}(N) for

all h ⩾ 0. Property (ii) follows directly from (13) if we replace t by t + h. We now define the set of translation operators {T (h), h ⩾ 0} that act on any function w(x, t)

by the formula

T (h)w(x, t) = w(x, t + h), h ⩾ 0.

Property (ii) of the trajectory space K^{+}(N) can be represented as follows:

T (h)K^{+}(N) ⊆ K^{+}(N) for any h ⩾ 0.  (14)

Clearly, the operators {T (h), h ⩾ 0} form a semigroup, which is called the translation semigroup. Consider the following space:

F_{+}^{loc} = L_{∞}^{loc}(ℝ_{+}; L_{p}(Ω) ∩ H_{0}^{1}(Ω)) ∩ { ∂_{t}v ∈ L_{∞}^{loc}(ℝ_{+}; L_{2}(Ω))}.  (15)

Recall that the space L_{∞}^{loc}(ℝ_{+}; Y), where Y is a Banach space, is the inductive limit of the spaces L_{∞}(0, M; Y) as M → +∞. We also introduce the Banach space

F_{+}^{b} = L_{∞}(ℝ_{+}; L_{p}(Ω) ∩ H_{0}^{1}(Ω)) ∩ { ∂_{t}v ∈ L_{∞}(ℝ_{+}; L_{2}(Ω))}

endowed with the norm

∥w(·)∥_{F_{+}^{b}} = ess sup_{t⩾0} ∥w(·, t)∥_{L_{p}∩H_{0}^{1}} + ess sup_{t⩾0} ∥∂_{t}w(·, t)∥_{L_{2}}.  (16)

The following embedding result was proved in [2]:

K^{+}(N) ⊂ F_{+}^{b}.

In other words, for any weak solution of (2) satisfying (13), the norm (16) is finite. In fact, one can prove that the space K^{+}(N) is a bounded subset of F_{+}^{b} (see (10)).

We now define a topological space Θ_{+}^{loc} whose underlying set coincides with that of the Fréchet space F_{+}^{loc} and endow it with the local ∗-weak convergence topology. More precisely, by definition, a sequence of functions {w_{m}(x, t)} ⊂ F_{+}^{loc} converges to a function w(x, t) as m → ∞ in Θ_{+}^{loc} if, for any M > 0, one has

u_{m}(·, t) ⇁ u(·, t)  (m → ∞)  ∗ -weakly in L_{∞} (0, M; H_{0}^{1}(Ω) ∩ L_{p}(Ω))
∂_{t}u_{m}(·, t) ⇁ ∂_{t}u(·, t)  (m → ∞)  ∗ -weakly in L_{∞} (0, M; L_{2}(Ω)) .

It is proved in [2] that the trajectory space K^{+}(N) is closed in Θ_{+}^{loc}. Moreover, it was shown in [3] that

K^{+}(N) ⋐ C^{loc} (ℝ_{+}; H^{1−δ}(Ω)) ∩ C^{1,loc} (ℝ_{+}; H^{−δ}(Ω)) for 0 < δ ⩽ 1.  (17)

It readily follows that the translation semigroup {T (h)} is continuous on K^{+}(N) with respect to the topology Θ_{+}^{loc}.

From now on, it is convenient for our purposes to regard elements of K^{+}(N) as pairs of the form (u(x, t), ∂_{t}u(x, t)), where u(x, t) is a weak solution of (2) satisfying (13) for every α in (8). We also regard elements of the spaces F_{+}^{loc}, F_{+}^{b}, and Θ_{+}^{loc} as pairs (w(x, t), ∂_{t}w(x, t)).

We claim that the translation semigroup {T (h)} acting on K^{+}(N) has an absorbing set P that is bounded in F_{+}^{b} and does not depend on N. Indeed, let us choose a number α = α_{0} satisfying inequality (8). Then it follows from (13) that the set

P = \{ \begin{matrix}(w(x, t), ∂_{t}w(x, t)) ∈ F_{+}^{b} | z_{α_{0}}(t) = J_{α_{0}}(w(t), ∂_{t}w(t)) ⩽ 2R_{α_{0}} \\ and w(x, t) is a weak solution of (2) \end{matrix} \}  (18)

is absorbing for every K^{+}(N). Let B ⊂ K^{+}(N) be a set of trajectories of (2). Take an h_{0} such that Ne^{−δ_{α_{0}} h} ⩽ R_{α_{0}} and note that T (h)B ⊆ P for any h ⩾ h_{0}. Hence, P is absorbing for K^{+}(N). By (10), the set P is bounded in F_{+}^{b}.

Note that the set P has the following important property. Since the set P is bounded in F_{+}^{b}, it follows that P is compact in the topological space Θ_{+}^{loc}. This is a consequence of the following theorem: if a space X^{∗} is dual to a separable Banach space X, then any bounded sequence in X^{∗} contains a ∗-weakly convergent subsequence. Similarly to (17), we see that

P ⋐ C^{loc} (ℝ_{+}; H^{1−δ}(Ω)) ∩ C^{1,loc} (ℝ_{+}; H^{−δ}(Ω)) for 0 < δ ⩽ 1.  (19)

Thus, we have proved that

(i) the translation semigroup {T (h)} is continuous on K^{+}(N) with respect to Θ_{+}^{loc}; (ii) the space K^{+}(N) is closed in Θ_{+}^{loc}; (iii) there is a Θ_{+}^{loc}-compact absorbing set P that does not depend on N. It follows from these facts that the translation semigroup {T (h)} acting on K^{+}(N) has the global

attractor 𝔄. As is well known, 𝔄 coincides with the ω-limit set of P in Θ_{+}^{loc} : 𝔄 = ω(P) [2]. The set 𝔄 does not depend on N.

The global attractor 𝔄 thus constructed is called the trajectory attractor of equation (2) in K^{+}(N). The trajectory attractor has the following properties.

Theorem 1.1.

(i) the set 𝔄 is compact both in Θ_{+}^{loc} and in C^{loc} (ℝ_{+}; H^{1−δ}(Ω)) ∩ C^{1,loc} (ℝ_{+}; H^{−δ}(Ω)) for 0 < δ ⩽ 1;

(ii) the set 𝔄 is strictly invariant with respect to {T (h)}, i.e.,

T (h)𝔄 = 𝔄 for any h ⩾ 0;  (20)

(iii) the set 𝔄 attracts any bounded set B ⊆ K^{+}(N) with respect to the translation semigroup {T (h)}, i.e., for any neighborhood O(𝔄) of the set 𝔄 in the topological space Θ_{+}^{loc}, there exists an h_{1} = h_{1}(B, O) such that

T (h)B ⊆ O(𝔄) for any h ⩾ h_{1}.

(iv) the set 𝔄 does not depend on N.

It follows from property (iii) and from the embedding (17) that

dist_{C([0,M];H^{1−δ})∩C^{1}([0,M];H^{−δ})} (Π_{M}T (h)B, Π_{M}𝔄) → 0 (h → +∞), (0 < δ ⩽ 1), (21)

for any set B ⊆ K^{+}(N) and for every M ⩾ 0. Here Π_{M} stands for the operator of restricting the functions of t ∈ ℝ_{+} to the interval [0, M].

Property (21) means that, taking the time shift of any trajectory u(x, t) by a sufficiently large h, one obtains a solution u(x, t + h) whose restriction to the interval t ∈ [0, M] lies in an arbitrarily small neighborhood, taken with respect to the metric of C (ℝ_{+}; H^{1−δ}) ∩ C^{1} (ℝ_{+}; H^{−δ}), of the trajectory attractor 𝔄. The trajectory attractor itself is a set of weak solutions u(x, t) belonging to P.

To describe the structure of the trajectory attractor, we introduce the notion of kernel of equation (2). Similarly to the spaces F_{+}^{loc} and F_{+}^{b}, we consider the following spaces F^{loc} and F^{b} defined on the entire time axis ℝ :

F^{loc} = L_{∞}^{loc}(ℝ; L_{p}(Ω) ∩ H_{0}^{1}(Ω)) ∩ { ∂_{t}v ∈ L_{∞}^{loc}(ℝ; L_{2}(Ω))},
F^{b} = L_{∞}(ℝ; L_{p}(Ω) ∩ H_{0}^{1}(Ω)) ∩ { ∂_{t}v ∈ L_{∞}(ℝ; L_{2}(Ω))}.

By definition, the kernel K of equation (2) is the union of all weak solutions (u(x, t), ∂_{t}u(x, t)), t ∈ ℝ of (2) defined on the time axis −∞ < t < +∞ and satisfying the inequality

z_{α}(t) = J_{α}(u(t), ∂_{t}u(t)) ⩽ R_{α} for any t ∈ ℝ  (22)

for any α in (8). Denote by Π_{+} the operator of restricting to the time semi-axis ℝ_{+}, i.e., if f(t) is a function of t ∈ ℝ, then Π_{+}f(t) is a function of t ∈ ℝ_{+} such that Π_{+}f(t) ≡ f(t) for t ∈ ℝ_{+}.

Remark 1.2. Since equation (2) is autonomous, it follows from the definition of the kernel K that K is translation invariant with respect to the translation group {T (h), h ∈ ℝ}, that is, if u(t) ∈ K, then T (h)u(t) = u(t + h) ∈ K for any h ∈ ℝ.

We have the following corollary.

Corollary 1.1. The trajectory attractor 𝔄 of equation (2) coincides with the restriction of the kernel K of equation (2) to the time semi-axis ℝ_{+} :

𝔄 = Π_{+}K.  (23)

The set K is bounded in F^{b} and Θ^{loc}-compact. At the same time it is a compact set in the space C^{loc} (ℝ; H^{1−δ}(Ω)) ∩ C^{1,loc} (ℝ; H^{−δ}(Ω)) for 0 < δ ⩽ 1.

Identity (23) follows from the invariance property (20) of the trajectory attractor 𝔄.

2. GLOBAL ATTRACTORS OF HYPERBOLIC EQUATIONS

In Section 1 we treated weak solutions u(x, t) of equation (2) and defined the trajectory space K^{+}(N) of (2). Write y(t) = (u(x, t), ∂_{t}u(x, t)). It follows from the definition of u(x, t) that y(t) ∈ L_{∞}(ℝ_{+}; H_{0}^{1}(Ω) × L_{2}(Ω)). It was also shown that any trajectory y(t) = (u(x, t), ∂_{t}u(x, t)) ∈ K^{+}(N) has the following properties:

u(x, t) ∈ C_{w} (ℝ_{+}; H_{0}^{1}(Ω)) , ∂_{t}u(x, t) ∈ C_{w}(ℝ_{+}; L_{2}(Ω)) for t ⩾ 0.

In particular, the function y(t) = (u(x, t), ∂_{t}u(x, t)) is defined for every t ⩾ 0, and the values of y(t) belong to the space H_{0}^{1}(Ω) × L_{2}(Ω) = E.

For a set B of trajectories, B ⊂ K^{+}(N), we define the section of B at time t as follows:

B(t) = {(u(·, t), ∂_{t}u(·, t)) | (u(·), ∂_{t}u(·)) ∈ B} ⊂ E.

Similarly, for the trajectory attractor 𝔄 and for the kernel K of equation (2), we define their sections at time t as follows:

𝔄(t) = {(u(·, t), ∂_{t}u(·, t)) | (u(·), ∂_{t}u(·)) ∈ 𝔄} ⊂ E,  t ⩾ 0,
K(t) = {(u(·, t), ∂_{t}u(·, t)) | (u(·), ∂_{t}u(·)) ∈ K} ⊂ E,  t ∈ ℝ.

The translation invariant property of 𝔄 and K implies that the sets 𝔄(t) and K(t) do not depend on t (see Remark 1.2 and Theorem 1.1). This remark motivates the following definition.

Definition 2.1. The set

A = K(0) = 𝔄(0) ⊂ E

is called the global attractor of equation (2).

Properties (i), (ii), and (iii) of the trajectory attractor 𝔄 in Theorem 1.1 imply the following assertion.

Theorem 2.1. The global attractor A of equation (2) has the following properties:

(i) the set A is bounded and closed in the space E = H_{0}^{1}(Ω) × L_{2}(Ω), and A is compact in H^{1−δ}(Ω) × H^{−δ}(Ω) for 0 < δ ⩽ 1;

(ii) the set A attracts the section B(t) of any bounded set B ⊂ K^{+}(N) for every N > 0, i.e.,

dist_{H^{1−δ}×H^{−δ}} (B(t), A) → 0 (t → +∞);

(iii) the set A is minimal in the following sense: it belongs to any set A′ satisfying (i) and (ii).

Proof. Since the set Π_{M}𝔄 is compact with respect to the topology of Π_{M}Θ_{+}^{loc}, and Π_{M}Θ_{+}^{loc} ⋐ C ([0, M]; H^{1−δ}(Ω)) ∩ C^{1} ([0, M]; H^{−δ}(Ω)) for 0 < δ ⩽ 1 (see [3]), if follows that Π_{0}𝔄 = A ⋐ H^{1−δ}(Ω) × H^{−δ}(Ω) for M = 0, and A is bounded in H_{0}^{1}(Ω) × L_{2}(Ω). This proves assertion (i). To prove (ii), we set M = 0 in (21), which implies

dist_{C([0,0];H^{1−δ})∩C^{1}([0,0];H^{−δ})} (Π_{0}T (h)B, Π_{0}𝔄) = dist_{H^{1−δ}×H^{−δ}} (B(h), A) → 0 (h → +∞).

Finally, let A′ be closed in H_{0}^{1}(Ω) × L_{2}(Ω) and let dist_{H^{1−δ} ×H^{−δ}} (B(t), A′) → 0 (as t → +∞) for any bounded set B ⊂ K^{+}(N). Set B = 𝔄. It is clear that 𝔄 belongs to K^{+}(N). Therefore, dist_{H^{1−δ} ×H^{−δ}} (𝔄(t), A′) → 0 (as t → +∞). However, 𝔄(t) does not depend on t (i.e., 𝔄(t) ≡ 𝔄(0)) because the trajectory attractor 𝔄 is strictly invariant with respect to the translation semigroup {T (h)}, which means that T (h)𝔄 = 𝔄 for h ⩾ 0. Hence, dist_{H^{1−δ} ×H^{−δ}} (𝔄(0), A′) = dist_{H^{1−δ} ×H^{−δ}} (A, A′) = 0. Since A′ is closed in H_{0}^{1}(Ω) × L_{2}(Ω), it follows that A ⊆ A′. Thus, A belongs to any set satisfying (i) and (ii), i.e., A is minimal. This proves assertion (iii). □

We now can formulate another (equivalent) definition of the global attractor for equation (2).

Definition 2.2. A set A ⊂ H_{0}^{1}(Ω) × L_{2}(Ω) is called the global attractor of equation (2) if it satisfies properties (i), (ii), and (iii) in Theorem 2.1.

Remark 2.1. If the nonlinear term f(v) in (2) satisfies conditions that provide the unique solvability of the corresponding Cauchy problem (for example, if f(v) ∈ C^{1}(ℝ) and |f′(v)| ⩽ C (1 + |v|^{p−2}), where 2 ⩽ p < 4 for n = 3, see [5, 6, 3]), then the set A in Definitions 2.1 and 2.2 coincides with the usual global attractor of the semigroup {S(t)} corresponding to equation (2) and acting in E = H_{0}^{1}(Ω) × L_{2}(Ω) by the formula

S(t) (u(x), u_{1}(x)) = (u(x, t), ∂_{t}u(x, t)) t ⩾ 0.

Here u(x, t) is the solution of problem (2), (6).

3. GALERKIN APPROXIMATIONS FOR GLOBAL ATTRACTORS OF HYPERBOLIC EQUATIONS

In this section we study some problems concerning finite-dimensional approximations of the global attractor A of a hyperbolic equation (2). As above, it is assumed that the nonlinear term f(v) can be of arbitrary polynomial growth with respect to v.

We consider a system of Galerkin approximations for equation (2). Let {w_{j}(x)}_{j∈ℕ} be a complete system of functions in H_{0}^{1}(Ω) ∩ L_{p}(Ω). For simplicity, we can assume that {w_{j}(x)}_{j∈ℕ} are the eigenfunctions of the operator \{−Δu, u|_{∂Ω} = 0\}. Each function w_{j}(x), x ∈ \overline{Ω} is bounded. Denote by P_{m} the orthogonal projection of L_{2}(Ω) onto the linear span of the functions {w_{1}(x), w_{2}(x), …, w_{m}(x)}.

The following system of ordinary differential equations is called the Galerkin approximation system of order m for hyperbolic equation (2):

\frac{d^{2}}{dt^{2}} u_{m} + γ \frac{d}{dt}u_{m} = P_{m}Δu_{m} − P_{m}f(u_{m}) + P_{m}g(x),  (24)
u_{m}|_{t=0} = u_{0m}(x),  \frac{d}{dt}u_{m}|_{t=0} = u_{1m}(x).  (25)

Here

u_{m}(x, t) = ∑_{j=1}^{m} a_{j,m}(t)w_{j}(x),

and the coefficients a_{j,m}(t) ∈ C(ℝ; ℝ) are unknown variables. One usually assumes that

u_{0m}(x) = P_{m}u_{0}(x) = ∑_{j=1}^{m} a_{j,m}^{0}w_{j}(x), u_{1m}(x) = P_{m}u_{0}(x) = ∑_{j=1}^{m} a_{j,m}^{1}w_{j}(x),

where u_{0}(x) and u_{1}(x) are the initial data of the original hyperbolic equation (2) (see (6)). Then u_{0m}(x) → u_{0}(x) (as m → ∞) strongly in H_{0}^{1}(Ω)∩L_{p}(Ω), and u_{1m}(x) → u_{1}(x) (as m → ∞) strongly in L_{2}(Ω).

If f(v) ∈ C(ℝ; ℝ), then system (25) satisfies the local Lipschitz condition, and the Cauchy problem (25), (25) has a unique solution {a_{j,m}(t)} for t ⩾ 0. The function u_{m}(x, t) satisfies both the differential inequality (11) and inequality (12) for any α such that (8) holds. It follows from these facts that the semigroup {S_{m}(t)} corresponding to (25), (25) and acting on the finite-dimensional space E_{m} = P_{m} (H_{0}^{1} × L_{2}(Ω)) ⊂ E has an absorbing set B_{m} = {(v_{m}, v_{1m})} ⊂ E_{m} bounded with respect to the norm

∥(v_{m}, v_{1m})∥_{E∩L_{p}} =^{def} ∥v_{m}(·)∥_{H^{1}} + ∥v_{m}(·)∥_{L_{p}} + ∥v_{1m}(·)∥_{H}.

Moreover,

∥B_{m}∥_{E∩L_{p}} ⩽ R for m ∈ ℕ,

and the constant R does not depend on m. Hence, the global attractor A_{m} (for the semigroup {S_{m}(t)}) exists, A_{m} ⊂ B_{m}, and

∥A_{m}∥_{E∩L_{p}} ⩽ R for m ∈ ℕ.  (26)

We can similarly prove the existence of the trajectory attractor 𝔄_{m} for the Galerkin system (25) and the property

∥𝔄_{m}∥_{L_{∞}(ℝ_{+};E∩L_{p})} ⩽ R_{1} < +∞ for m ∈ ℕ.  (27)

In particular, this means that

∥(u_{m}(·), ∂_{t}u_{m}(·))∥_{L_{∞}(ℝ_{+};E∩L_{p})} ⩽ R_{1}

for every u_{m}(·) ∈ 𝔄_{m}. Relation (25) implies the inequality

∥(∂_{t}^{2}u_{m}(·))∥_{L_{∞}(ℝ_{+};H^{−r})} ⩽ R_{2} < +∞.  (28)

Indeed, to prove (28), we can estimate the norm ∥(∂_{t}^{2}u_{m}(t))∥_{H^{−r}} using (25) together with the relation

∥P_{m}f(u_{m})∥_{H^{−r}} ⩽ c∥f(u_{m})∥_{H^{−r}} ⩽ c′∥f(u_{m})∥_{L_{q}} ⩽ c_{1} (∫_{Ω} |u_{m}(x)|^{p}dx + 1)^{1/q}⩽ c_{2} (∥u_{m}∥_{L_{p}}^{p−1} + 1) .

We obtain the following result.

Theorem 3.1. The global attractor A_{m} of the Galerkin approximation system (25) of the hyperbolic equation (2) converges from below, as m → ∞, to the global attractor A of equation (2) in the following sense:

dist_{H^{1−δ}×H^{−δ}} (A_{m}, A) → 0 (as m → +∞) for 0 < δ ⩽ 1.  (29)

Proof. As was proved in [2], the trajectory attractor 𝔄_{m} of system (25) converges from below, as m → ∞, to the trajectory attractor 𝔄 of equation (2) in the weak topology Θ_{+}^{loc} (see Section 1). Moreover, 𝔄_{m} ⊂ P for all m ⩾ 1 (see (18)). Using the compact embedding (19), we see that

dist_{C([0,M];H^{1−δ})∩C^{1}([0,M];H^{−δ})} (Π_{M}𝔄_{m}, Π_{M}𝔄) → 0  (as m → +∞)  (30)

for every nonnegative M. Setting M = 0 in (30), we obtain

dist_{H^{1−δ}×H^{−δ}} (𝔄_{m}(0), 𝔄(0)) ≡ dist_{H^{1−δ}×H^{−δ}} (A_{m}, A) → 0 (m → +∞). □

4. GLOBAL ATTRACTOR OF A HYPERBOLIC EQUATION WITH RAPIDLY OSCILLATING TERMS

We consider a dissipative hyperbolic equation of the form

∂_{t}^{2}u + γ (x, \frac{x}{ε}) ∂_{t}u = Δu − b (x, \frac{x}{ε}) f(u) + g (x, \frac{x}{ε}) ,  u|_{∂Ω} = 0,  (31)

where x ∈ Ω ⋐ ℝ^{n}. A similar hyperbolic equation with constant coefficients γ and b was treated in the previous sections. The construction of the trajectory attractor and the global attractor is also applicable to equation (31) without significant modifications.

We assume that γ (x, z) , b (x, z) ∈ C_{b}(\overline{Ω} × ℝ^{n}) and

0 < γ̄_{1} ⩽ γ(x, z) ⩽ γ̄_{2},  (32)
0 < β̄_{1} ⩽ b(x, z) ⩽ β̄_{2}, for any x ∈ \overline{Ω} × ℝ^{n}.  (33)

Suppose that the function f(v) ∈ C(ℝ) satisfies inequalities (3)–(5). The degree p can be arbitrarily large. Note that, in this case, it is not known whether the solutions of the Cauchy problem for equation (31) are unique.

Assume that the functions γ (x, \frac{x}{ε}) and b (x, \frac{x}{ε}) have averages γ̄ (x) and b̄ (x) as ε → 0+ in the space L_{∞,∗w}(Ω), i.e.,

∫_{Ω} γ (x, \frac{x}{ε}) φ(x) dx → ∫_{Ω} γ̄ (x) φ(x) dx,

∫_{Ω} b (x, \frac{x}{ε}) φ(x) dx → ∫_{Ω} b̄ (x) φ(x) dx  (as ε → 0+)  (34)

for every function φ ∈ L_{1}(Ω). Finally, assume that the function g (x, \frac{x}{ε}) has average ḡ (x) in L_{2,w}(Ω) as ε → 0+,

∫_{Ω} g (x, \frac{x}{ε}) φ(x) dx → ∫_{Ω} ḡ (x) φ(x) dx  (ε → 0+) for any φ ∈ L_{2}(Ω).  (35)

Remark 4.1. In particular, conditions (34) and (35) hold if the functions γ (x, z), b (x, z), and g (x, z) are periodic, quasiperiodic, or at least almost periodic with respect to any variable z_{i}, i = 1, …, n.

Remark 4.2. One can readily generalize the below results to the case in which g = g (x, \frac{x}{ε}, t, \frac{t}{ε}). We can also assume that the nonlinear interaction term is of the form ∑_{j=1}^{N} β_{j} (x, \frac{x}{ε}) f_{j}(v) or is a of sums of this kind as N tends to infinity.

We also introduce the averaged equation

∂_{t}^{2}ū + γ̄ (x) ∂_{t}ū = Δū − b̄ (x) f(ū) + ḡ (x) ,  ū|_{∂Ω} = 0.  (36)

It is clear that the functions γ̄ (x) and b̄ (x) satisfy inequalities (32) and (33) as well.

It follows from assumptions (32)–(35) that both equations (31) for ε > 0 and equation (36) have the trajectory attractors, which we denote by 𝔄_{ε} and \overline{𝔄}, respectively. The construction and the properties of these trajectory attractors were described in Section 1. The corresponding spaces F_{+}^{loc}, Θ_{+}^{loc} and F_{+}^{b} do not depend on ε.

Similarly to (7), we define the functional

J_{ε,α}(v, v_{1}) = ∫_{Ω} (|∇v(x)|^{2} + |v_{1}(x) + αv(x)|^{2} + 2b (x, \frac{x}{ε}) F (v(x))) dx.  (37)

Let u(x, t) be a weak solution of (31). Set

z(t) = J_{ε,α}(u(t), ∂_{t}u(t)).

The trajectory space K_{ε}^{+}(N), N ⩾ 0, of equation (31) consists of the weak solutions of (31) for which the relation

z(t) ⩽ R_{α} + N exp (−δ_{α}t) for any t ⩾ 0  (38)

holds for every α such that

0 < α(γ̄_{1} − α) < λ_{1}, 0 < α < γ̄_{1}.

Here positive constants R_{α} and δ_{α} can be defined explicitly for given equation (31) and do not depend on ε (see [3, 4]). We can similarly define the trajectory space \overline{K}^{+}(N) for equation (36). The translation semigroup \{T (h)\} acting on the spaces K_{ε}^{+}(N) and \overline{K}^{+}(N) has the trajectory attractors 𝔄_{ε} and \overline{𝔄}, respectively.

The sets 𝔄_{ε} and \overline{𝔄} are uniformly bounded in F_{+}^{b}. The proof of these assertions is similar to that of Theorem 1.1. Finally, one can conclude that

𝔄_{ε} = Π_{+}K_{ε} and \overline{𝔄} = Π_{+}\overline{K},  (39)

where K_{ε} and \overline{K} are the kernels of equations (31) and (36), respectively. It readily follows from (38) that K_{ε} consists of all weak solutions u(x, t), t ∈ ℝ of (31) such that

z(t) = J_{ε,α}(u(t), ∂_{t}u(t)) ⩽ R_{α} for any t ∈ ℝ.  (40)

Similarly, the kernel \overline{K} of equation (36) consists of all weak solutions ū(x, s) such that

z̄(t) = J̄_{α}(ū(t), ∂_{t}ū(t)) ⩽ R_{α} for any t ∈ ℝ,  (41)

where J̄_{α}(v, v_{1}) is defined as in (37) in which b (x, \frac{x}{ε}) must be replaced by b̄ (x). Note that the trajectory attractors 𝔄_{ε} and \overline{𝔄} do not depend on the constant N in the definition of the trajectory spaces K_{ε}^{+}(N) and \overline{K}^{+}(N).

Following the scheme in Section 2, we define the global attractors A_{ε} and \overline{A} for equations (31) and (36) as follows:

A_{ε} = 𝔄_{ε}(0) = K_{ε}(0) ⊂ E,  (42)
\overline{A} = \overline{𝔄}(0) = \overline{K}(0) ⊂ E.  (43)

These global attractors have properties (i), (ii), and (iii) of Theorem 3.1.

Theorem 4.1. The trajectory attractors 𝔄_{ε} for equations (31) tend to the trajectory attractor \overline{𝔄} of equation (36) (as ε → 0+) in the space Θ_{+}^{loc},

𝔄_{ε} → \overline{𝔄} as ε → 0 + .  (44)

Theorem 4.2. For every δ ∈]0, 1] we have

dist_{C([0,M];H^{1−δ})∩C^{1}([0,M];H^{−δ})} (Π_{0,M}𝔄_{ε}, Π_{0,M} \overline{𝔄}) → 0 (as ε → 0+) for any M > 0, (45)
dist_{H^{1−δ}×H^{−δ}} (A_{ε}, \overline{A}) → 0 (as ε → 0+).  (46)

Proof of Theorem 4.2. Recall that Theorem 4.1 is proved in [3]. Property (44) together with the embedding 𝔄_{ε} ⊂ P ⋐ C^{loc}(ℝ_{+}; H^{1−δ}(Ω)) ∩ C^{1,loc}(ℝ_{+}; H^{−δ}(Ω)) (for any ε > 0) imply (45). By setting M = 0, we derive (46) from (45). □

In conclusion we briefly consider the averaging of dissipative hyperbolic equations with moderate growth of nonlinear terms. This condition ensures the unique solvability of the corresponding Cauchy problem.

We consider an equation of the form

∂_{t}^{2}u + γ∂_{t}u = Δu − b (x, \frac{x}{ε}) f(u) + g (x, \frac{x}{ε}) ,  u|_{∂Ω} = 0,  ε > 0.  (47)

Here x ∈ Ω ⋐ ℝ^{n} and γ > 0 as usual, and the function f(v) ∈ C^{1}(ℝ) satisfies the conditions (3)–(5). Assume that

|f′(v)| ⩽ C (1 + |v|^{p−2}) , where \{ \begin{matrix}2 ⩽ p < +∞ & for n = 1, 2, \\ 2 ⩽ p < 2 + 2/(n − 2) & for n ⩾ 3, \end{matrix}  (48)

b (x, z) ∈ C_{b}(\overline{Ω} × ℝ^{n}), 0 < β_{1} ⩽ b(x, z) ⩽ β_{2}, and that b (x, \frac{x}{ε}) has average b̄ (x) in L_{∞,∗w}(Ω) as ε → 0+. One has g (x, \frac{x}{ε}) ∈ L_{2}(Ω) for any ε > 0, and this function has the average ḡ (x) in L_{2,w}(Ω) as ε → 0+.

Consider the averaged equation

∂_{t}^{2}ū + γ∂_{t}ū = Δū − b̄ (x) f(ū) + ḡ (x) ,  ū|_{∂Ω} = 0.  (49)

At t = 0 we pose the following initial conditions:

u|_{t=0} = u_{0}(x) ∈ H^{1},  ∂_{t}u|_{t=0} = u_{1}(x) ∈ H,  (50)

where H = L_{2}(Ω) and H^{1} = H_{0}^{1}(Ω).

It is well known that problems (47), (50) and (49), (50) (with u replaced by ū) have unique solutions u(t), t ⩾ 0, and ū(t), t ⩾ 0, such that u, ū ∈ L_{∞}(ℝ_{+}; H^{1}), ∂_{t}u, ∂_{t}ū ∈ L_{∞}(ℝ_{+}; H). In this case it immediately follows from the equations that ∂_{t}^{2}u, ∂_{t}^{2}ū ∈ L_{∞}(ℝ_{+}; H^{−1}), where H^{−1} = H^{−1}(Ω) (see [5, 6, 4]). The results remain valid if there are additional terms of the form b (x, \frac{x}{ε}) and b̄ (x) that depend on x. The trajectory space K_{ε}^{+} consists of all solutions u(x, t) of (47). We can similarly define the trajectory space \overline{K}^{+} of equation (49). Equations (47) and (49) have the trajectory attractors 𝔄_{ε} = Π_{+}K_{ε} and \overline{𝔄} = Π_{+}\overline{K}, where K_{ε} and \overline{K} are the kernels of equations (47) and (49), respectively.

Set H^{σ} = (−Δ)^{−σ/2} H for σ ⩾ 0, where Δ is the Dirichlet Laplacian. In particular, one has H^{0} = H = L_{2}(Ω), H^{1} = H_{0}^{1}(Ω), and H^{2} = H^{2}(Ω) ∩ H_{0}^{1}(Ω). The following lemma is of importance for our purposes.

Lemma 4.1. The translation semigroup \{T (h)\} acting on the trajectory spaces K_{ε}^{+} and \overline{K}^{+} has the attracting set

B_{δ} = {v ∈ F_{+}^{b} | ∥v∥_{C_{b}(ℝ_{+};H^{1+δ})} + ∥∂_{t}v∥_{C_{b}(ℝ_{+};H^{δ})} ⩽ R_{δ}}  (51)

for 0 ⩽ δ < 1. The number R_{δ} does not depend on ε.

The proof repeats that of Lemma 3.4 in [5, Chapter VI].

We also have the following corollary.

Corollary 4.1. The trajectory attractors 𝔄_{ε} (ε > 0) and \overline{𝔄} belong to B_{δ}, i.e.,

∥u(t)∥_{H^{1+δ}} + ∥∂_{t}u(t)∥_{H^{δ}} ⩽ R_{δ},
∥ū(t)∥_{H^{1+δ}} + ∥∂_{t}ū(t)∥_{H^{δ}} ⩽ R_{δ} for t ∈ ℝ,  (52)

for every u ∈ K_{ε} and ū ∈ \overline{K}.

We now consider the semigroups \{S_{ε}(t)\} and \{\overline{S}(t)\} acting on E = H^{1} × H by the formulas

S_{ε}(t) (u_{0}, u_{1}) = (u(t), ∂_{t}u(t)) , \overline{S}(t) (u_{0}, u_{1}) = (ū(t), ∂_{t}ū(t)) ,

where u(t) and ū(t) are solutions of (47), (50) and (49), (50), respectively. These semigroups have the global (E, E)-attractors, which we denote by A_{ε} and \overline{A}, respectively, (see [5, 6]). Recall that

A_{ε} = 𝔄_{ε}(0), \overline{A} = \overline{𝔄}(0).

The estimates (52) imply our main result.

Theorem 4.3. The trajectory attractors 𝔄_{ε} of equations (47) converge to the trajectory attractor \overline{𝔄} of the averaged equation (49) as ε → 0+ in the following sense:

dist_{C([0,M];H^{1+δ})∩C^{1}([0,M];H^{δ})} (Π_{M}𝔄_{ε}, Π_{M} \overline{𝔄}) → 0 (as ε → 0+) for any M > 0,  (53)

where 0 ⩽ δ < 1.

The global attractors A_{ε} and \overline{A} are compact in E^{δ} = H^{1+δ} × H^{δ} (0 ⩽ δ < 1), and

dist_{E^{δ}} (A_{ε}, \overline{A}) → 0  (ε → 0+).  (54)

Relation (54) follows from (53) for M = 0.

5. PERTURBATION OF GLOBAL ATTRACTORS OF HYPERBOLIC EQUATIONS

We consider the perturbed dissipative hyperbolic equation of the form

∂_{t}^{2}u + γ∂_{t}u = Δu − f_{1}(u) − εf_{2}(u) + g_{1} (x) + εg_{2} (x) ,  (55)
u|_{∂Ω} = 0,  Ω ⋐ ℝ^{n},  0 ⩽ ε ⩽ 1.

We assume that f_{1}(v), f_{2}(v) ∈ C^{1}(ℝ; ℝ) and that these functions satisfy the following conditions:

|f′_{i}(v)| ⩽ C_{0} (1 + |v|^{p_{i}−2}) ,  i = 1, 2,  (56)

where p_{i} ⩾ 2 for n ⩽ 2 and 2 ⩽ p_{i} < 2 + 2/(n − 2) for n ⩾ 3. Moreover, we assume that

F_{i}(v) = ∫_{0}^{v} f_{i}(w) dw ⩾ C_{1}|v|^{p} − C_{2},  (57)
f_{i}(v)v ⩾ cF_{i}(v) − m|v|^{2} − C_{3} for any v ∈ ℝ,  i = 1, 2,  (58)

where m is positive and sufficiently small. Finally, let g_{i} ∈ L_{2}(Ω).

We denote by E the Hilbert space of pairs y(x) = (u(x), u_{1}(x)) ∈ H_{0}^{1}(Ω) × L_{2}(Ω) for which the below norm ∥y∥_{E} is finite,

∥y∥_{E}^{2} = ∫_{Ω} (|∇u(x)|^{2} + |u_{1}(x)|^{2}) dx = ∥u∥^{2} + |u_{1}|^{2} < +∞.  (59)

Here ∥ · ∥ stands for the norm in H_{0}^{1}(Ω) and | · | for the norm in L_{2}(Ω).

As is well known (see, e.g., [6, 5]), if the above assumptions hold, then the Cauchy problem for equation (55) with initial data

u|_{t=0} = u_{0}(x),  ∂_{t}u|_{t=0} = u_{1}(x),  y_{0} = (u_{0}, u_{1}) ∈ E,  (60)

has a unique solution u_{ε}(x, t) such that ∥y_{ε}(t)∥_{E}^{2} = ∥u_{ε}(t)∥^{2} + |∂_{t}u_{ε}(t)|^{2} ⩽ M (∥y_{0}∥_{E}^{2}) for t ⩾ 0. Corresponding to the problem (55), (60) is the semigroup {S_{ε}(t)} acting on E by the formula

S_{ε}(t)(u_{0}, u_{1}) = (u_{ε}(t), ∂_{t}u_{ε}(t)),  t ⩾ 0.  (61)

Moreover, the semigroup {S_{ε}(t)}, S_{ε}(t): E → E, is continuous in E for every ε ∈ [0, 1] and has a uniformly absorbing set B_{0} that is bounded in E and does not depend on ε,

∥B_{0}∥_{E} ⩽ M_{0}.  (62)

The last assertion follows from the inequality

∥S_{ε}(t)y_{0}∥_{E} ⩽ ∥y_{0}∥_{E}e^{−αt} + C′ for t ⩾ 0,  (63)

where C′ does not depend on ε, 0 ⩽ ε ⩽ 1. For the proof, see [5, 6, 4].

Let us prove, using the reasoning applied in these works, that the semigroup {S_{ε}(t)} has a uniform (with respect to ε ∈ [0, 1]) attracting set P ⋐ E that is compact in E, i.e., for any δ > 0 and for any bounded set B (in E), there is a number t(δ, B) such that

dist_{E} (S_{ε}(t)B, P) ⩽ δ for all t ⩾ t(δ, B).  (64)

Note that property (64) was proved in [5, 6] for ε = 0 only. However, all estimates and reasonings hold for ε ∈ [0, 1], and the attracting set P can be chosen to be independent of ε.

Therefore, the semigroup {S_{ε}(t)} has the global attractor A_{ε} ⊆ P. Recall that the global attractor A_{ε} has the following properties:

(i) A_{ε} is compact in E;

(ii) A_{ε} is strictly invariant with respect to {S_{ε}(t)}, i.e.,

S_{ε}(t)A_{ε} = A_{ε} for t ⩾ 0;  (65)

(iii) A_{ε} attracts bounded sets, i.e.,

dist_{E} (S_{ε}(t)B, A_{ε}) → 0 as t → +∞,  (66)

where B is any bounded set in E. Let us now prove the estimate for the deviation of the solution u_{ε}(x, t) of the perturbed equa-

tion (55) for ε > 0 from the solution u_{0}(x, t) of the nonperturbed equation (55) for ε = 0. Let the initial data (u_{0}(x), u_{1}(x)) belong to a chosen bounded set B, for instance,

B = B_{R} = {y_{0} = (u_{0}, u_{1}) ∈ E | ∥y_{0}∥_{E} ⩽ R} .  (67)

Let y_{ε}(x, t) = (u_{ε}(x, t), ∂_{t}u_{ε}(x, t)) be a solution of (55) with initial data

u_{ε}|_{t=0} = u_{0}(x),  ∂_{t}u_{ε}|_{t=0} = u_{1}(x),  (68)

and let y_{0}(x, t) = (u_{0}(x, t), ∂_{t}u_{0}(x, t)) be a solution of (55) (ε = 0) with the same initial data (68). Write

z(x, t) = y_{ε}(x, t) − y_{0}(x, t) = (u_{ε}(x, t) − u_{0}(x, t), ∂_{t}u_{ε}(x, t) − ∂_{t}u_{0}(x, t))
= (w(x, t), ∂_{t}w(x, t)), where w(x, t) = u_{ε}(x, t) − u_{0}(x, t).

Let us subtract equation (55) with ε = 0 from equation (55) with ε > 0. We obtain

∂_{t}^{2}w + γ∂_{t}w = Δw − (f_{1}(u_{ε}) − f_{1}(u_{0})) − εf_{2}(u_{ε}) + εg_{2} (x) ,  (69)
w|_{t=0} = 0,  ∂_{t}w|_{t=0} = 0.  (70)

Theorem 5.1. The following inequality holds:

∥z(·, t)∥_{E}^{2} = ∥w(·, t), ∂_{t}w(·, t))∥_{E}^{2} ⩽ Cε^{2}e^{2ρt} for t ⩾ 0,  (71)

where C = C(R) and ρ = ρ(R).

Proof. Multiplying (69) by ∂_{t}w(x, t) and integrating over Ω, we obtain

\frac{1}{2} \frac{d}{dt} (|∂_{t}w|^{2} + |∇w|^{2}) + γ|∂_{t}w|^{2} + ⟨∫_{0}^{1} f′_{1}(ũ)dsw, ∂_{t}w⟩ = ε ⟨f_{2}(u_{ε}), ∂_{t}w⟩ + ε ⟨g_{2}, ∂_{t}w⟩ , (72)

where ũ = u_{0}(x, t) + s (u_{ε}(x, t) − u_{0}(x, t)). It follows from (56) that

|f′_{1}(ũ)| ⩽ C (|u_{0}(x, t)|^{ρ} + |u_{ε}(x, t)|^{ρ} + 1)  (73)

and

|f_{2}(u_{ε})| ⩽ C (|u_{ε}(x, t)|^{ρ+1} + 1) ⩽ C_{0} (|u_{ε}(x, t)|^{\frac{n}{n−2}} + 1) ,  (74)

where ρ = p − 2 < 2/(n − 2) for n ⩾ 2.

Using (73), we see from the Holder inequality that

|⟨∫_{0}^{1} f′_{1}(ũ)dsw, ∂_{t}w⟩| ⩽ C (∫_{Ω} (1 + |u_{0}|^{ρ} + |u_{ε}|^{ρ})^{n} dx)^{1/n} ∥w∥_{L_{\frac{2n}{n−2}}} ∥∂_{t}w∥_{L_{2}}
⩽ C_{1} (1 + ∥u_{0}∥^{ρ} + ∥u_{ε}∥^{ρ}) (∥w∥^{2} + |∂_{t}w|^{2}) ⩽ C_{2} (R^{ρ} + 1) ∥z∥_{E}^{2}. (75)

Similarly, relation (74) implies

|ε ⟨f_{2}(u_{ε}), ∂_{t}w⟩| ⩽ εC [∫_{Ω} (1 + |u_{ε}|^{ρ+1})^{2} dx]^{1/2} ∥∂_{t}w∥_{L_{2}}
⩽ εC_{3} (1 + ∥u_{ε}∥^{n/(n−2)}) |∂_{t}w|^{2} ⩽ ε^{2}C_{4} (R^{n/(n−2)} + 1) + ∥z∥_{E}^{2}.  (76)

Finally, for the last term in (72) we obtain

|ε ⟨g_{2}, ∂_{t}w⟩| ⩽ ε^{2}|g_{2}|^{2} + |∂_{t}w|^{2} ⩽ ε^{2}|g_{2}|^{2} + ∥z∥_{E}^{2}.  (77)

Let us apply the estimates (75), (76), and (77) to equality (72). We see that

\frac{d}{dt}∥z(·, t)∥_{E}^{2} ⩽ C_{5}∥z(·, t)∥_{E}^{2} + ε^{2}C_{6},  (78)

where

C_{5} = C_{2} (R^{ρ} + 1) + 2,  C_{6} = C_{4} (R^{n/(n−2)} + 1) + |g_{2}|^{2}.  (79)

Using the Gronwall inequality together with the relation z(x, t)|_{t=0} = 0, we obtain inequality (71) with

C = C_{6} and 2ρ = C_{5}. □  (80)

Recall that the global attractor A_{0} of the limit equation (55) satisfies the exponential attracting property [6]. This means that

dist_{E} (S_{0}(t)B, A_{0}) ⩽ C_{1}e^{−νt} (C_{1} = C_{1}(B)),  ν > 0,  (81)

for any compact set B in E.

We claim that the estimates (71) and (81) imply the following assertion.

Theorem 5.2. The distance in E from the global attractor A_{ε} to the global attractor A_{0} satisfies the estimate

dist_{E} (A_{ε}, A_{0}) ⩽ Cε^{η},  η = \frac{1}{(1 + ρ/ν)},  (82)

where ρ is the exponent in the estimate (71) for the global deviation between the solutions of (71) for ε = 0 and ε > 0 with the same initial data, and ν is the coefficient of exponential attraction of solutions of the limit equation (71) for ε = 0 to the global attractor A_{0} of the same equation.

Proof. Let (u_{0}^{ε}, u_{1}^{ε}) = y^{ε} be an arbitrary point of A_{ε}. Then there is a complete trajectory y_{ε}(t) = (u_{ε}(t), ∂_{t}u_{ε}(t)), t ∈ ℝ, of equation (71) such that y_{ε}(t) ∈ A_{ε} for all t ∈ ℝ and

y_{ε}(0) = y^{ε}.  (83)

Consider the point y_{ε}(−T) (∈ A_{ε}) where T will be specified below. Since A_{ε} ⊂ P and P is compact in E, it follows that the set

B_{ε}^{−} = {y_{ε}(−T) | y^{ε} ∈ A_{ε}}

is precompact in E. Let y_{0}(t) be a solution of (55) for ε = 0 with initial data

y_{0}|_{t=0} = y_{ε}(−T),  y_{ε}(−T) ∈ B_{ε}^{−}.  (84)

Recall that, by (83), the set B_{ε}^{−} is precompact in E. Then the corresponding set of solutions S_{0}(t){y_{ε}(−T)} = S_{0}(t)B_{ε}^{−} converges to A_{0} with exponential rate (see (81)),

dist_{E} (S_{0}(t)B_{ε}^{−}, A_{0}) ⩽ C_{1}e^{−νt}  (C_{1} = C_{1}(B_{ε}^{−}) ⩽ C_{1}(P)),  (85)

where C_{1}(P) is some constant.

The distance from the point S_{0}(T)y_{ε}(−T) := y_{0}(T, y_{ε}(−T)) to A_{0} does not exceed Ce^{−νT} (we use (85) for t = T). Therefore, there is a point y^{0} ∈ A_{0} such that

∥y_{0}(T, y_{ε}(−T)) − y^{0}∥_{E} ⩽ C_{1}e^{−νT} ,  y^{0} ∈ A_{0}.  (86)

Note that the solution y_{0}(t, y_{ε}(−T)) = S_{0}(t)y_{ε}(−T) of equation (55) with ε = 0 and the solution y_{ε}(t − T) = S_{ε}(t)y_{ε}(−T) of (55) with ε > 0 satisfy the same initial data,

y_{ε}(t − T)|_{t=0} = y_{0}(t, y_{ε}(−T))|_{t=0} = y_{ε}(−T).  (87)

Hence, one can apply the estimate (71) with z(x, t) = y_{ε}(t − T) − y_{0}(t, y_{ε}(−T)), which implies

∥z(t)∥_{E} = ∥y_{ε}(t − T) − y_{0}(t, y_{ε}(−T))∥_{E} ⩽ \sqrt{C}εe^{ρt} for t ⩾ 0.  (88)

Set t = T in (88). This yields

∥y_{ε}(0) − y_{0}(T, y_{ε}(−T))∥_{E} ⩽ \sqrt{C}εe^{ρT} .  (89)

It follows from (86) and (89) that

∥y^{ε} − y^{0}∥_{E} = ∥y_{ε}(0) − y^{0}∥_{E} ⩽ ∥y_{ε}(0) − y_{0}(T, y_{ε}(−T))∥_{E} + ∥y_{0}(T, y_{ε}(−T)) − y^{0}∥_{E}
⩽ \sqrt{C}εe^{ρT} + C_{1}e^{−νT} ⩽ C_{0} (εe^{ρT} + e^{−νT} ) .  (90)

(For simplicity we set C_{0} = max \{\sqrt{C}, C_{1}\} in the last estimate.) Let us now choose T such that

εe^{ρT} = e^{−νT} .

It is clear that T = \frac{1}{ρ+ν} ln \frac{1}{ε}. For this value of T, it follows from (90) that

∥y^{ε} − y^{0}∥_{E} ⩽ 2C_{0}e^{(− \frac{ν}{ρ+ν} ln \frac{1}{ε})} = 2C_{0}ε^{\frac{ν}{ρ+ν}} = Cε^{η},

where C = 2C_{0} and η = \frac{1}{1+ρ/ν}. This completes the proof of the theorem. □

REFERENCES

1. Lions, J.-L., Quelques méthodes de résolution des problèmes aux limites non linéaires, Paris: Dunod et Gauthier-Villars, 1969.

2. Chepyzhov, V. V. and Vishik, M. I., Evolution Equations and Their Trajectory Attractors, J. Math. Pures Appl. (9), 1997, vol. 76, no. 10, pp. 913–964.

3. Chepyzhov, V. V. and Vishik, M. I., Averaging of the Trajectory Attractors of Evolution Equations with Rapidly Oscillating Terms, Mat. Sbornik, 2001, vol. 192, no. 1, pp. 16–53; English transl.: Sbornik: Mathematics, 2001, vol. 192, no. 1.

4. Chepyzhov, V. V. and Vishik, M. I., Attractors for Equations of Mathematical Physics, (in press).

5. Temam, R., Infinite-Dimensional Dynamical Systems in Mechanics and Physics, Applied Mathematics Series, 68, New York: Springer, 1988; 1997, 2nd ed..

6. A. V. Babin and M. I. Vishik,, Attractors of Evolution Equations, Moscow: Nauka, 1989; English transl.: Amsterdam: North Holland, 1992.