Continuous Dependence on a Parameter of Exponential Attractors for Nonclassical Diffusion Equations

In this paper, a new abstract result is given to verify the continuity of exponential attractors with respect to a parameter for the underlying semigroup. We do not impose any compact embedding on the main assumptions in the abstract result which is different from the corresponding result established by Efendiev et al. in 2004. Consequently, it can be used for equations whose solutions have no higher regularity. As an application, we prove the continuity of exponential attractors in H0 for a class of nonclassical diffusion equations with initial datum in H0.


Introduction
In this paper, we study the existence and robustness of exponential attractors for the following nonclassical diffusion equation: with the initial-boundary value conditions where ϵ ∈ [0, 1] and Ω ⊂ R N (N ≥ 3) is a bounded open set with smooth boundary zΩ. When ϵ � 0, it turns out to be the classical reaction-diffusion equation. We assume that g ∈ L 2 (Ω) and the nonlinearity f ∈ C 1 (R, R) satisfies the following (see, e.g., [1]): Nonclassical diffusion equations appear in fluid mechanics, soil mechanics, and heat conduction theory (see, e.g., [2]). e long-time behavior of solutions to nonclassical diffusion equations has been extensively studied by many authors for both autonomous and nonautonomous cases [3][4][5][6][7][8][9]. e global attractor plays an important role in the study of long-time behavior of infinite dimension systems arising from physics and mechanics. It is a compact invariant set and attracts uniformly the bounded sets of the phase space. However, the rate of attraction may be arbitrary, and it may be sensible to perturbations. ese drawbacks can be overcome by creating the notion of the exponential attractor [10], which is a compact, positively invariant set of finite dimension and exponentially attracts each bounded set. e existence of the exponential attractor has been extensively studied since 1994, see e.g., [5,[10][11][12][13][14][15][16][17].
As discussed in [12], exponential attractors are to be more robust objects under perturbations than global attractors. In general, global attractors are only upper semicontinuous with respect to perturbations, and the lower semicontinuity property is much more delicate and can be established only for some particular cases. However, one can prove the continuity of exponential attractors under perturbations in many cases [5,13]. In particular, for problems (1) and (2), the existence of a pullback attractor was shown by Anh and Bao in [3] for the subcritical case in H 1 0 (Ω). ey also proved the upper semicontinuity of the pullback attractors. However, this upper semicontinuity of the pullback attractor was established only in L 2 (Ω), and the upper semicontinuity in H 1 0 (Ω) remains an open problem. In this paper, we not only prove the upper and lower semicontinuity of the exponential attractor but also show these continuities in a stronger space, i.e., H 1 0 (Ω) when the initial value only belongs to H 1 0 (Ω). In [12], see also [17], Efendiev et al. gave an abstract result about the robustness of exponential attractors ( eorem 4.4 in [12]). A main assumption called "compact Lipschitz condition" was proposed in that theorem. e main difficulty when we apply this result to problems (1) and (2) is that the solution to problems (1) and (2) has no regularity as ϵ > 0 [3]. For example, if the initial datum u 0 belongs to H 1 0 (Ω), the solution with initial u(0) � u 0 is always in H 1 0 (Ω) and has no higher regularity. us, it is impossible to verify the "compact Lipschitz condition" when we want to prove the continuity of exponential attractors in H 1 0 (Ω). Motivated by [18], we modify the result in [12] to adapt it to our case. Moreover, some of the coefficients are allowed to be dependent on the parameter ϵ which relax the conditions, see eorem 1 in the following. e rest of this paper is organized as follows. In Section 2, we formulate and prove the main abstract result, i.e., eorem 1. In Section 3, we apply eorem 1 to the dynamical system generated by (1) and (2) to prove the continuity of the exponential attractors, and we consider two cases according to the constant ] in this section. roughout this paper, we denote by ‖ · ‖ X the norm of a Banach space X. e inner product and norm of L 2 (R n ) are written as (·, ·) and ‖·‖, respectively. We also use ‖u‖ r to denote the norm of u ∈ L r (R n ) (r ≥ 1, r ≠ 2) and |u| to denote the modular of u. Letter c is a generic positive constant independent of ϵ which may change its values from line to line even in the same line (sometimes for the special case, we also denote different positive constants by c i (i � 1, 2, . . .)).

The Abstract Result and Its Proof
In this section, we modify eorem 4.4 in [12] to adapt to problems (1) and (2). We start with the definition of exponential attractors. Definition 1. Let E be a metric space, B is a bounded set in E, and let S: B ⟶ B be a map. We define a discrete semigroup S n , n ∈ Z + { } by S n x: � S∘ · · · ∘S (n times). A set M ⊂ B is an exponential attractor for the map if the following properties hold: (1) e set M is compact in E and has finite fractal dimension.
where dist E (C 1 , C 2 ) denotes the Hausdorff semidistance between C 1 and C 2 in E given by Definition 2. Let X be a complete metric space endowed with the metric d and M be a bounded closed set in X.
Assume that ϱ is a pseudometric [19] defined on M. Let B ⊂ M and ε > 0.
is finite for every ε > 0. (iii) For any r > 0, we define a local (r, ε, ϱ)-capacity of set M by the formula We now state and prove the main abstract theorem. e proof is essentially a combination of that in [18] and that of [12]. Theorem 1. Let X be a Banach space and B be a bounded set in X. We assume that there exists a family of operators S ϵ : B ⟶ B, ϵ ∈ [0, ϵ 0 ], which satisfy, for any ϵ ∈ [0, ϵ 0 ], (i) S ϵ is Lipschitz on B, i.e., there exists L ϵ > 0 such that (ii) ere exist constants η and K ϵ and compact seminorms n ϵ 1 (x) and n ϵ 2 (x) on B such that for any x 1 , x 2 ∈ B, where 0 < η < 1 is independent of ϵ and K ϵ > 0 is a constant which may be dependent on ϵ(seminorm n(x) on B is said to be compact iff for where κ > 0 and α ∈ (0, 1) are constants independent of ϵ and for mapping V, V i � V∘ · · · ∘V (i times).
Then, ∀ϵ ∈ [0, ϵ 0 ], and the discrete dynamical system generated by the iterations of S ϵ possesses an exponential attractor M ϵ on B such that We assume diam B � 2R. Let δ � 1 − η/2 and x i 1 : i 1 � 1, . . . , n 1 be a maximal (δR, ϱ ϵ )-distinguishable subset of B. en, from (14), we have erefore, If y 1 , y 2 ∈ B i 1 , then from (7), we have Next, for any fixed i 1 ∈ 1, . . . , n i 1 , we assume that If y 1 , y 2 ∈ B i 1 ,i 2 , then from (7), we have We set E 2 ϵ � S ϵ E 1 ϵ ∪ S ϵ x i 1 ,i 2 , and then we have By the induction procedure, we can find sets E i ϵ , i ∈ N, enjoy the following properties: We now define the exponential attractor for the map S 0 : B ⟶ B as follows: Discrete Dynamics in Nature and Society en, from (24)-(27), we see that M 0 is indeed an exponential attractor for the map S 0 : B ⟶ B (see [12]). For ϵ ∈ (0, ϵ 0 ], one can also construct exponential attractors for S ϵ : B ⟶ B as above. However, they are not the ones we needed. Totally similar to [12], one can construct exponential attractors M ϵ based on E i 0 . We note that the only difference between our construction procedure and that of in [12] is that the number P ϵ here may be dependent on ϵ. However, (26) only contributes to the fractal dimension of M ϵ . erefore, (11) and (12) in eorem 1 hold true. Finally, assertion (9) is a direct result of [18]. e proof is completed.
Remark 1. Similar to [12], we can give an explicit expression for c 4 , i.e., c 4 � −α ln q/ln κ − ln q. If L ϵ , K ϵ , n ϵ 1 (x), and n ϵ 2 (x) are all independent ϵ, we can obtain the uniform bound with respect to ϵ of the fractal dimension, and we will apply this abstract result in the last section.
Lemma 3 (see [1]). Assume (F1)-(F4) hold and g ∈ L 2 (Ω). en, for any bounded set D ⊂ H 1 0 (Ω), there exist positive constants E(D), c 0 , and T 1 (D) such that, for any solution u ϵ of problems (1) and (2), where c 0 is the constant in Lemma 3; then, B is a uniformly (with respect to ϵ) bounded absorbing set for S ϵ (t) in H 1 0 (Ω). We note that the absorbing time is independent of ϵ, and we can choose The following lemma gives several priori estimates for the derivative u ϵ t (t) of the solution to (1) and (2).

Lemma 4. Under the assumptions (F1)-(F4), for any
for any u 0 ∈ D and any t ≥ T 2 (D), where c is a constant independent of ϵ.
Proof. We first take the inner product of (1) with u ϵ (t) in By condition (F4), for any θ > 0, there exists c θ > 0 satisfying (36) By condition (F3), for any c > 0, there exists c c > 0 satisfying Combining (36) and (37), we get d dt that is, 4 Discrete Dynamics in Nature and Society We choose α 1 > 0 small enough such that α 1 λ 1 < 1 and 1 − α 1 > 1/2, and apply Poincare inequality in (39) to get d dt Choosing θ, c small enough and using H€ older inequality, we can get from (40) that We set σ � α 1 λ 1 /2; then, Integrating the above inequality from 0 to t, it yields that Now, we consider (36) again. If 0 < κ 2 ≤ 1, we can get from (36) that d dt Using a similar process as (39)-(42), we deduce that, for θ small enough, If κ 2 > 1, we can get from (36) that Applying (37) to the above, we obtain d dt Following a similar process as (39)-(42), we deduce that, for θ, c small enough, From (47) and (51), we see that, for any where c 5 � κ 2 when 0 < κ 2 ≤ 1 and c 5 � 1 when κ 2 > 1. For any t > 0, we integrate (52) over [t, t + 1] to get t+1 t e σs ∇u ϵ (s) Putting (44) into (53), we have, for any t > 0, t+1 t e σs ∇u ϵ (s) Next, we multiply (1) with u t and integrate it in Ω to get Applying H€ older inequality to the above we get us, Discrete Dynamics in Nature and Society (57) Dropping the first term in the left-hand side of (57), from (54) and using uniform Gronwall inequality, we conclude that We now differentiate equation (1) with respect to t to get Taking the inner product of (59) with u ϵ (61) Integrating (57) over (t, t + 1) and using (58), we deduce that t+1 t e σs u ϵ t (s)
Proof. By (F2), we have where l(t, ϵ 1 , From (31) and the Sobolev embedding 6 Discrete Dynamics in Nature and Society e proof is completed.

e Main Result for a General
Case: ] ∈ R + . Now, we verify the conditions in eorem 1 for S ϵ (t) in this case. We first verify condition (i), i.e., the Lipschitz continuity in H 1 0 (actually uniform Lipschtz continuity in H 1 0 since the coefficient is independent of ϵ ). (F1)-(F4), we have, for any ϵ ∈ [0, 1] and any

Lemma 6. Under assumptions
where c 6 is the constant in Lemma 5.