Uniform Attractor for the Fractional Nonautonomous Long-Short Wave Equations

Definition 1. Suppose X is a Banach space, $$ is a function, and T(·) is the translation operator. The hull $$ of f is defined by

• (i)

  f  is said to be translation bounded in $$ if $$ is bounded in which

  $$ ()
  Then $$ consists of all the translation bounded functions in $$.

• (ii)

  f is called translation compact in $$ if $$ is compact in $$, where the convergence is taken in the sense of local quadratic mean convergence topology of $$. The collection of all the translation compact functions in $$ is denoted by $$.

Definition 2. Let Σ be a parameter set. $$, σ ∈ Σ is said to be a family of processes in X, if, for each σ ∈ Σ, U_σ(t, τ) is a process; that is, the two-parameter mapping U_σ(t, τ) from X to X satisfies

• (i)

  $$,

• (ii)

  $$,

where Σ is called the symbol space and σ ∈ Σ is called the symbol.

Definition 3. A closed set $$ is called the uniform attractor of the family of processes {U_σ(t, τ)} _σ∈Σ if it is uniformly attracting (attracting property) and it is contained in any closed uniformly attracting set $$ of the family of processes $$ (minimality property).

Definition 4. {U_σ∈Σ(t, τ)}, a family of processes in X, is said to be (X × Σ, X)-continuous, if, for any fixed   T and τ, T⩾τ, projection (u_τ, σ) → U_σ(T, τ)u_τ is continuous from X × Σ to X.

Definition 5. The space L^p(0, T; X) denotes all measurable functions f : [0, T] → X with the norm

for 1 ⩽ p < ∞, and

Lemma 6. Let ∑  be a compact metric space and suppose {T(h)∣h⩾0} is a family of operators defined on ∑ , satisfying

• (i)
  $$ ()

• (ii)

  translation identity:

  $$ ()

where U_σ(T, τ) is an arbitrary process in compact metric space E. Note that if the family of processes {U_σ∈∑(T, τ)} is (E × ∑ , E) continuous and it has a uniform compact attracting set, then the skew product flow corresponding to it has a global attractor $$ on E × ∑ . And the projection of $$ on ∑ , $$, is the compact uniform attractor of {U_σ∈∑(T, τ)}.

Remark 7. Assumption (11) holds if the system has a unique solution.

Lemma 8. Let (X, ∥·∥_X) be a uniform convex Banach space (particularly, a Hilbert space), and let $$ be a sequence in X. If x_k⇀x₀ and ∥x_k∥_X → ∥x₀∥_X, then x_k → x₀.

Lemma 9. Let $$ be a sequence in B^* space X. If x_k⇀x₀, then

Since the solution u(x, t), if it exists, is a 2π-periodic function, we have the Fourier expansion:

where $$. Therefore, and  (−Δ)^αu is defined by Since the following definitions make sense. Let and let H^2α be a complete space of the set A under the norm: Then we can easily check that H^2α is a Banach space and that, for ∀u ∈ H^2α, u is space-periodic with the period 2π and its 2α order derivatives are in $$. And for ∀u, v ∈ H^α, when α₁ + α₂ = α, 0 ⩽ α₁, α₂ ⩽ α. H^2α is a Hilbert space with the inner product

For brevity, we introduce W(x, t) = (u(x, t), n(x, t)) and Y(x, t) = (f(x, t), g(x, t)). We denote the space of functions W(x, t) = (u(x, t), n(x, t)) by $$ with norm

Similarly, we denote the space of Y(x, t) by ∑₀  with norm

Assumption 10. Suppose that the symbol Y(x, t) belongs to the symbol space ∑ , defined by

where $$ and the closure is taken in the sense of local quadratic mean convergence topology in the topological space $$. Moreover, we assume $$.

Remark 11. Due to the conception of translation compact/boundedness, we remark that

• (i)

  ∀Y₁ ∈ ∑ , $$;

• (ii)

  $$, where T(t)φ(s) = φ(s + t) is a translation operator.

Lemma 12. Let $$. Then for 0 ⩽ j ⩽ m, j/m ⩽ a ⩽ 1, there exists a constant C such that

where 1/p = j/n + a(1/r − m/n) + (1 − a)(1/q).

Lemma 13. Let a, b > 0. Then for each (p, q) satisfying 1 < p, q < ∞, 1/p + 1/q = 1, it holds that

Lemma 14. Assume that

• (i)

  Y(x, t) satisfy Assumption 10;

• (ii)

  u_τ ∈ L²(Ω) and u(t) solves problem (1)–(4).

Then there exist positive constants $$ and $$ such that

Proof. Taking the inner product of (1) with u, we have

Taking the imaginary part of (27), we get By (28) and Remark 11, we have By Gronwall inequality we get the lemma.

Lemma 15. Assume that

• (i)

  α⩾1, and Y(x, t) satisfy Assumption 10;

• (ii)

  W(τ) ∈ H^α × H and W(t) solves problem (1)–(4).

Then there exist positive constants $$ and $$ such that

Proof. Taking the inner product of (1) with u_t, we have

Taking the real part of (31), we have where So we have Taking the inner product of (1) with δu, we have Taking the real part of (35), we have By (34) and (36), we have By Lemmas 12 and 13 and the condition α⩾1, we have So we reduce that Similarly, we also derive that Taking the inner product of (2) with n, we have By (1), we have where By (41)~(43), we get By Lemmas 12~14 and the condition α⩾1, we have So we obtain that Similarly, we can also get that Set So by (39) and (46), we get By (40) and (47), we also get Setting γ = min⁡⁡{δ, β}, ρ ⩽ min⁡⁡{δ/4, β/2}, then we deduce that By Gronwall inequality, we have Obviously for any t⩾t₀, we have So by (52)~(54), there exists a $$ such that for any t⩾t₂.

By (48),(53), and (55), we get

Then setting ρ = min⁡⁡{δ/4, β/2, 1/2}, we get By using Lemma 14, we conclude the lemma.

Lemma 16. Assume that

• (i)

  α⩾1, and Y(x, t) satisfy Assumption 10;

• (ii)

  W(τ) ∈ E₀ and W(t) solves problem (1)–(4).

Then there exist positive constants $$ and $$ such that

Proof. Taking the inner product of (1) with (−Δ) ^αu_t, we have

Taking the real part of (59), we have where, by (1) and (2), we have By Lemmas 12, 14, and 15 and α⩾1, we see that By (60)~(62), Lemmas 12~ 15, and Hölder inequality, we can see that Taking the inner product of (2) with n_xx, we have where, by Lemmas 12~ 15 and α⩾1, we can see that So by (64) and (65), we get Set γ = min⁡⁡{δ, β} and Then by (63) and (66), we can deduce that which has the same form with (51) in the proof of Lemma 15. Similar to the study of (51), there exist positive constants $$ and $$ such that which conclude the proof of Lemma 16.

Theorem 17. Set α⩾1, and Y(x, t) satisfy Assumption 10; for each W_τ ∈ E₀, then system (1)–(4) has a unique global solution W(x, t) ∈ L^∞(τ, T; E₀), ∀T > τ.

Proof. We prove this theorem by two steps.

Step  1. The existence of solution.

By $$’s method, we construct the approximate solution of the periodic initial value problem (1)~(4). We apply the following approximate solution:

to approach W(x, t), the solution of the problem (1)–(4). And for j = 1,2, …, l, W^l(x, t) satisfies We see that system (71) is an initial boundary value problem of ordinary differential equations (ODE). By the standard existence theory for ODE and uniform a priori estimates in Section 3, for any l, there exists a unique solution of (71), such that There is a subsequence $$ of W^l(x, t) and W(x, t) = (u(x, t), n(x, t)) such that Due to the above proof and the continuous extension theorem, W(x, t) is the solution Of (1)~(4).

Step  2. The uniqueness of solution.

Suppose W₁(x, t) = (u₁(x, t), n₁(x, t)), W₂(x, t) = (u₂(x, t), n₂(x, t)) are two solutions of problem (1)–(4). Let W(x, t) = W₁(x, t) − W₂(x, t), and then W(x, t) = (u(x, t), n(x, t)) satisfies

Obviously, $$ is uniformly bounded. Note that $$.

Taking the inner product of (74) with u and taking the imaginary part, we can get

Taking the inner product of (75) with n, we can obtain Taking the inner product of (74) with u_t and taking the real part, we can get Differentiating (74) with respect to t, taking the inner product of with u_t, and taking the imaginary part, we can get Taking the inner product of (75) with n_xx, we have Therefore by (78)~(81), we conclude that From Gronwall inequality and (76), we have Therefore, we complete the proof of the theorem.

Theorem 18. Under assumptions of Theorem 17, {U_σ∈∑(t, τ)} admits a strong compact uniform attractor $$.

Proof. We prove this theorem by three steps.

Step  1. {U_σ∈∑(t, τ)} possess a bounded uniformly absorbing set in E₀.

Let $$. By Theorem 17, B₀ is a bounded absorbing set of the process $$.

By Assumption 10, we know that, for each Y ∈ ∑ , $$ holds. So the solution of (1)~(4) satisfies

Then we can get that the set B₀ is a bounded uniformly absorbing set of {U_σ∈∑(t, τ)}.

Step  2. we prove the existence of weakly compact uniform attractor $$ in E₀.

From Lemma 6, Theorem 17, and Step 1, we only need to prove that {U_σ∈∑(t, τ)} is (E₀ × ∑ , E₀)-continuous. We denote weak convergence by ⇀ and * weak convergence by $$.

For any fixed  $$, let

If we can deduce that where $$, we will obtain that {U_σ∈∑(t, τ)} is (E₀ × ∑ , E₀)-continuous. By (86) and Theorem 17, we can get that Then by Lemmas 12~16, we can see that Note that and σ_k = (f_k(x, t), g_k(x, t)) ∈ ∑ . By (89) and (90), we find that $$ and Because of Theorem 17 and (93), we easily see that there exist a subsequence $$ of $$ and $$, such that Besides, for any   t₁ ∈ [τ, T], by (89) there exists W⁰≜(u⁰(t₁), n⁰(t₁)) ∈ E₀ such that By (94) and compactness embedding theorem, we can get that

Next, we will obtain that $$ is a solution of problem (1)~(4).

For $$, by (91) we have that

Since by (90), (94), and (97), Then we have And by (94), we have that By using the similar methods to the other terms of (98), we have So, we can get that which shows that ($$ satisfies (1).

For any$$ with ψ(T) = 0, ψ(τ) = 1, by (91) we find that

We know that Assumption (86) implies that Then from (105) and (106), we have while by (104) we know that So by (107) and (108), we have that By (104) and (110), we have For any  $$, with ψ(τ) = 0, ψ(t₁) = 1, then we repeat the procedure of proofs of (105)~(108) by (96) having From (96), (111), and (112), we have that Similarly, we can also derive that From (113) and (114), we deduce (87). We complete the proof of the step.

Step  3. We show the weakly compact uniform attractor $$ is actually the strong one.

From the proof of Lemma 16, we know each solution trajectory for problem (1)–(4) satisfies

where By the uniform boundedness and the compactness embedding, we have that F, G, and G₁ are all weakly continuous in E₀ × Σ.

From Step 2, we can see that the point $$ if and only if there exist two sequences $$ and $$ such that for all σ(t) ∈ Σ, it uniformly satisfies that

where t_k → ∞ as k → ∞. If the weak convergence implies strong one, we obtain $$ is actually the strong compact attractor. For each fixed   h > τ, because of t_k → ∞, we consider it as h < t_k, k ∈ N₊. By Lemma 16 and Theorem 17, $$ is bounded in E₀. Then there exists a subsequence $$ of $$ and a point (v, p) ∈ E₀, such that Let where T(·) is the translation operator on Σ. Since σ(t) is translation compact symbol, there exists a symbol σ^* ∈ Σ such that Then by (118), (119), and the weak (E × Σ)-continuity of U_σ∈Σ(t, τ), we can get that From (119), we can see that the solution trajectory $$ is created by $$ starting at $$. By (115), (119), and (122), we have that Let t = h in (122). Since F and G are weakly continuous in E₀, $$, and the Lebesgue dominated convergence theorem, we can obtain that

Since $$, we can see the solution (w, m) as at h corresponding to the initial data (v, p) and the symbol σ^*. Similarly to (122), we have

Deducting (125) from (124), we can get that As h → ∞, we can get that On the other hand, the weak convergence $$ implies that From the above two inequalities, we get that Similarly to the above arguments, by using (116) we can derive that Then we get that $$ in E₀. We complete the proof of the theorem.