Asymptotic Periodicity for Strongly Damped Wave Equations

Abstract

This work deals with the existence and uniqueness of asymptotically almost-periodic mild solutions for a class of strongly damped semilinear wave equations.

1. Introduction

Let 𝑋 be a reflexive Banach space and let 𝐴∶𝐷(𝐴)⊆𝑋→𝑋 be a closed densely defined operator and 𝜂>0. Consider the Cauchy problem 𝑢_𝑡𝑡+2𝜂𝐴^1/2𝑢_𝑡+𝐴𝑢=𝑓𝑡,𝑢,𝑢_𝑡𝑢(0),𝑡>0,=𝑢₀∈𝑋^1/2,𝑢_𝑡(0)=𝑣₀∈𝑋, (1) where 𝑋^1/2 is the fractional power space associated with 𝐴 as in [1]. Equations like (1) appear in the literature under the name of strongly damped wave equations. An example of mathematical model represented in form (1) is the wave equation with structural damping (see [2–5]). The strongly damped wave equations has been investigated in several contexts by many authors in the last years, for example, existence [6, 7], global classical solution [6, 8], long-time asymptotic behavior [9–11], attractor [2, 12–16], well-posedness [17], decay estimates [18], blowup [8, 19, 20], controllability [19], bootstrapping, and regularity [21]. Another important aspect of the qualitative study of the solutions of strongly damped wave equations is their asymptotic periodicity. In recent years, the study of periodicity and its various extensions for evolution equations has attracted a great deal of attention of many mathematicians (see [22–32] and references therein).

To the best of our knowledge, the study of the existence of asymptotically almost-periodic solutions for strongly damped wave equations of type (1) is a topic not yet considered in the literature.

Problem (1) can be written as a first order in time Cauchy problem in 𝑌⁰∶=𝑋^1/2×𝑋: 𝑢𝑣_𝑡+𝒜_(1/2)𝑢𝑣𝑢𝑣𝑢(0)𝑣(0)=𝑢=𝐹𝑡,,𝑡>0,₀𝑣₀, (2) where 𝒜_(1/2)=0−𝐼𝐴2𝜂𝐴^1/2∶𝐷𝒜_(1/2)⊆𝑋^1/2×𝑋⟶𝑋^1/2×𝑋 (3) is defined by 𝒜_(1/2)𝜑𝜓=−𝜓𝐴𝜑+2𝜂𝐴^1/2𝜓𝜑𝜓for∈𝐷𝒜_(1/2)=𝑋¹×𝑋^1/2,𝐹𝑢𝑣=0𝑓()𝑡,.𝑡,𝑢,𝑣 (4)

Definition 1. The pair (𝜂,𝐴) is said to be an admissible pair if there exist 𝜓∈(0,𝜋/2) and 𝑀>0 such that (𝜆𝐼−𝐴)⁻¹_ℒ(𝑋)≤𝑀|𝜆|,1+ (5) for all 𝜆 in the sector Σ_𝜓={𝜆∈ℂ∶𝜓≤|arg𝜆|≤𝜋}∪{0} and 𝜋/2>𝜓/2+arg(𝜂+𝜂²−1).
If (𝜂,𝐴) is an admissible pair, by [21, Proposition 2.1] 𝒜_(1/2) is a closed operator with 0∈𝜌(𝒜_(1/2)). Indeed, 𝒜_(1/2) has the inverse expressed in the matrix form as 𝒜₍₎⁻¹_1/2=2𝜂𝐴^−1/2𝐴⁻¹𝑋−𝐼0∈ℒ^1/2×𝑋. (6) Moreover, the operator 𝒜_(1/2) has compact resolvent whenever 𝐴 has compact resolvent. By [21, Theorem 2.3] the operator 𝒜_(1/2) is sectorial in 𝑋^1/2×𝑋. The semigroup {𝑒^{−𝒜(1/2)𝑡}∶𝑡≥0} generated by −𝒜_(1/2) in 𝑋^1/2×𝑋 is exponentially decaying analytic. That is, there are constants 𝐾≥1 and 𝐶>0 such that 𝑒^{−𝒜(1/2)𝑡}_{ℒ(𝑋1/2×𝑋)}≤𝐾𝑒^−𝐶𝑡,𝑡≥0. (7) Throughout this paper, we always assume that (𝜂,𝐴) is an admissible pair.
This paper has four sections. In the next section, we consider some definitions, technical aspects, and basic properties related to asymptotically almost-periodic functions. In the third section, we obtain general results on the existence of asymptotically almost-periodic (mild) solutions to the problem (1). The main abstract results are Theorems 12, 16, 17, and 19. Finally, in the fourth section we consider several applications. In particular, we consider the following class of partial differential equations: 𝑢_𝑡𝑡+𝜈𝑏(𝑡)𝑢_𝑡+Δ²𝑢−𝛿Δ𝑢_𝑡(𝑡)ℎ(𝑢)(𝑢)𝑢()=𝜇𝑎∇⋅𝑢+𝑔Δ𝑢,𝑥∈Ω,𝑡≥0𝑢=Δ𝑢=0,𝑥∈𝜕Ω,𝑡>0,0,𝑥=𝑢₀(𝑥),𝑢_𝑡(0,𝑥)=𝑣₀(𝑥),𝑥∈Ω, (8) in a bounded smooth domain Ω⊆ℝ^𝑁 and where ℎ and 𝑔 satisfy certain growth conditions. We prove that if 𝑎∈𝐴𝑃(ℝ), 𝑏∈𝐶₀(ℝ⁺,ℝ), and |𝜇|+|𝜈| is small enough, then the previous problem has an asymptotically almost periodic mild solution. The same type of conclusion is derived for the wide class 𝑢_𝑡𝑡+Δ²𝑢−𝛿Δ𝑢_𝑡=𝑎(𝑡)_Ω∇⋅𝑢(𝑡,𝑥)𝑑𝑥^𝛽Φ₀,𝑢()𝑥∈Ω,𝑡≥0,𝑢=Δ𝑢=0,𝑥∈𝜕Ω,𝑡>0,0,𝑥=𝑢₀(𝑥),𝑢_𝑡(0,𝑥)=𝑣₀(𝑥),𝑥∈Ω, (9) where Φ₀∈𝐿^𝑝(Ω) and 𝑝>𝑁/2.

2. Preliminaries

In this section, we present some concepts and properties needed to develop the following sections. Let 𝑋 be a reflexive Banach space. For an interval 𝐼⊆ℝ, 𝐶_𝑏(𝐼,𝑋) denotes the space formed by the bounded continuous functions from 𝐼 into 𝑋, endowed with the norm of uniform convergence. When 𝑋=ℝ, we denote 𝐶_𝑏(𝐼) instead of 𝐶_𝑏(𝐼,ℝ). The notation 𝐶₀(ℝ⁺,𝑋) stands for the subspace of 𝐶_𝑏(ℝ⁺,𝑋) consisting of functions that vanish at infinity. We denote by ℒ(𝑋) the Banach algebra of bounded linear operators defined on 𝑋. For 𝑟>0, the notation 𝐵_𝑟(𝑋) stands for the closed ball {𝑥∈𝑋∶‖𝑥‖≤𝑟}. For a linear operator 𝐴 with domain 𝐷(𝐴) and range ℛ(𝐴) in 𝑋, we represent by 𝜎(𝐴) (resp., 𝜌(𝐴)) the spectrum (resp., the resolvent set) of 𝐴. For 𝜆∈𝜌(𝐴), we denote by 𝑅(𝜆,𝐴)=(𝜆𝐼−𝐴)⁻¹ the resolvent operator of 𝐴. When 𝐴 is closed, we denote by [𝐷(𝐴)] the domain of 𝐴 endowed with the graph norm ‖𝑥‖_𝐴=‖𝑥‖+‖𝐴𝑥‖.

Next, we present a brief summary of the main properties of asymptotically almost-periodic functions.

Definition 2 (see [33]). A continuous function 𝑓∶ℝ→𝑋 is called almost-periodic if for each 𝜀>0 there exists 𝑙(𝜀)>0 such that for every interval of length 𝑙(𝜀) it contains a number 𝜏 with the property that ‖𝑓(𝑡+𝜏)−𝑓(𝑡)‖≤𝜀 for each 𝑡∈ℝ.
The previous number 𝜏 is called an 𝜀-translation number for 𝑓. We denote by 𝐴𝑃(𝑋) the space formed by the almost periodic functions 𝑓∶ℝ⁺→𝑋. We note that each almost-periodic function is bounded and uniformly continuous. It is well known that the range ℛ(𝑓)={𝑓(𝑡)∶𝑡∈ℝ} of an almost periodic function 𝑓 is relatively compact. The space 𝐴𝑃(𝑋) is a Banach space endowed with the norm of uniform convergence.
Let 𝑌 be a Banach space. We have the following concept of parameter-dependent almost-periodic function.

Definition 3. A continuous function 𝑓∶ℝ×𝑌→𝑋 is called almost-periodic in 𝑡 uniformly for 𝑦 in compact subsets of 𝑌 if for every compact subset 𝐾 of 𝑌 and each 𝜀>0 there exists 𝑙(𝜀)>0 such that every interval 𝐼 of length 𝑙(𝜀) contains a number 𝜏 with the property that ‖𝑓(𝑡+𝜏,𝑦)−𝑓(𝑡,𝑦)‖≤𝜀 for all 𝑡∈ℝ, 𝑦∈𝐾.
Henceforth, we abbreviate the terminology by calling almost-periodic from ℝ×𝑌 into 𝑋 to those functions that are almost-periodic in 𝑡 uniformly for 𝑦 in compact subsets of 𝑌, and we denote by 𝐴𝑃(ℝ×𝑌,𝑋) the set formed by the almost-periodic functions from ℝ×𝑌 into 𝑋.
The proof of the following result is similar to the proof of [22, Lemma 2.12] and therefore omitted.

Lemma 4. If ℎ∶ℝ→𝑋^1/2×𝑋 is an almost-periodic function and 𝑢(𝑡)=^𝑡_−∞𝑒^{−𝒜(1/2)(𝑡−𝑠)}ℎ(𝑠)𝑑𝑠,𝑡∈ℝ, (10) then 𝑢 is almost-periodic.

It is well known that the study of composition of two functions with special properties is important and basic for deep investigations. The following result has been established in [34].

Lemma 5. Let 𝑓∶ℝ×𝑌→𝑋 be an almost-periodic function and let 𝑢∶ℝ→𝑌 be an almost-periodic function. Then the function ℝ→𝑋,𝑡→𝑓(𝑡,𝑢(𝑡)) is almost-periodic.

We will need the following definition.

Definition 6. A continuous function 𝑓∶ℝ⁺→𝑋 is called asymptotically almost-periodic if there exist two functions 𝑓_𝑎𝑝∈𝐴𝑃(𝑋) and 𝜑_𝑓∈𝐶₀(ℝ⁺,𝑋) such that 𝑓(𝑡)=𝑓_𝑎𝑝(𝑡)+𝜑_𝑓(𝑡),𝑡∈ℝ⁺. (11) The function 𝑓_𝑎𝑝 is called the almost-periodic part of 𝑓. We denote by 𝐴𝐴𝑃(𝑋) the space formed by the asymptotically almost-periodic functions 𝑓∶ℝ⁺→𝑋. The space 𝐴𝐴𝑃(𝑋) is a Banach space endowed with the norm of uniform convergence. Furthermore, 𝐴𝐴𝑃(𝑋)=𝐴𝑃(𝑋)⊕𝐶₀(ℝ⁺,𝑋).
In what follows 𝐶₀(ℝ⁺×𝑌,𝑋) denotes the space consisting of continuous functions 𝑓∶ℝ⁺×𝑌→𝑋 such that lim_𝑡→∞𝑓(𝑡,𝑦)=0 uniformly for 𝑦 in compact subsets of 𝑌.

Definition 7. A continuous function 𝑓∶ℝ⁺×𝑌→𝑋 is called asymptotically almost-periodic if there are two functions 𝑔∈𝐴𝑃(ℝ×𝑌,𝑋) and 𝜑∈𝐶₀(ℝ⁺×𝑌,𝑋) such that 𝑓𝑡,𝑦=𝑔𝑡,𝑦+𝜑𝑡,𝑦,𝑦∈𝑌,𝑡∈ℝ⁺. (12) We denote by 𝐴𝐴𝑃(ℝ⁺×𝑌,𝑋) the set consisting of all asymptotically almost-periodic functions from ℝ⁺×𝑌 into 𝑋.
Let 𝐼⊆ℝ be an interval. We have the following concept of function uniformly continuous on compacts sets.

Definition 8. A continuous function 𝑓∶𝐼×𝑌→𝑋 is called uniformly continuous on compact sets if for all compact set 𝐾⊆𝑌 and all 𝜀>0 there is 𝛿_𝜀,𝐾>0 such that ‖𝑓(𝑡,𝑦₁)−𝑓(𝑡,𝑦₂)‖≤𝜀 for all 𝑡∈𝐼 and 𝑦₁,𝑦₂∈𝐾 with ‖𝑦₁−𝑦₂‖≤𝛿_𝜀,𝐾.

Lemma 9 (see [22]). Let 𝑓∶ℝ⁺×𝑌→𝑋 be an asymptotically almost-periodic and uniformly continuous on compact sets function. Let 𝑢∶ℝ⁺→𝑌 be an asymptotically almost-periodic function. Then the function ℝ⁺→𝑋, 𝑡→𝑓(𝑡,𝑢(𝑡)) is asymptotically almost-periodic.

The proof of the following result is similar to the proof of [22, Lemma 2.13]. Therefore, we will omit it.

Lemma 10. If ℎ∶ℝ⁺→𝑋^1/2×𝑋 is an asymptotically almost-periodic function and 𝑢(𝑡)=^𝑡₀𝑒^{−𝒜(1/2)(𝑡−𝑠)}ℎ(𝑠)𝑑𝑠,𝑡≥0. (13) Then, 𝑢 is asymptotically almost-periodic.

3. Asymptotically Almost-Periodic Mild Solutions

We recall the following definition that will be essential for us.

Definition 11. Let 𝑢₀𝑣₀ be in 𝑋^1/2×𝑋. We say that ^𝑢(⋅)_𝑣(⋅)∶ℝ⁺→𝑋^1/2×𝑋 is a mild solution to (2) (or to (1)) if it satisfies the Cauchy integral formula: 𝑢(𝑡)𝑣(𝑡)=𝑒^{−𝒜(1/2)𝑡}𝑢₀𝑣₀+^𝑡₀𝑒^{−𝒜(1/2)(𝑡−𝑠)}𝐹𝑢(𝑠)𝑣(𝑠)𝑠,𝑑𝑠,𝑡≥0. (14)

Theorem 12. Let 𝑓∶ℝ⁺×𝑋^1/2×𝑋→𝑋 be an asymptotically almost-periodic function and assume that there exists a locally integrable function 𝐿_𝑓∶ℝ→ℝ satisfying 𝑓𝑡,𝑢₁,𝑣₁−𝑓𝑡,𝑢₂,𝑣_2𝑋≤𝐿_𝑓(𝑡)𝑢₁−𝑢_2𝑋1/2+𝑣₁−𝑣_2𝑋, (15) for all ^𝑢𝑖_𝑣𝑖∈𝑋^1/2×𝑋, 𝑖=1,2 and each 𝑡≥0. If 𝐾sup_𝑡≥0^𝑡₀𝑒^{−𝐶(𝑡−𝑠)}𝐿_𝑓(𝑠)𝑑𝑠<1, (16) where 𝐾 and 𝐶 are given in (7). Then, (2) has a unique asymptotically almost-periodic mild solution.

Proof. We define the map 𝔉 on the space 𝐴𝐴𝑃(𝑋^1/2×𝑋) by the expression 𝔉𝑢𝑣(𝑡)=𝑒^{−𝒜(1/2)𝑡}𝑢₀𝑣₀+^𝑡₀𝑒^{−𝒜(1/2)(𝑡−𝑠)}𝐹𝑢(𝑠)𝑣(𝑠)𝑠,𝑑𝑠,𝑡∈ℝ⁺, (17) where 𝑢(⋅)𝑣(⋅)=𝑢_𝑎𝑝(⋅)𝑣_𝑎𝑝(⋅)+𝜑_𝑢(⋅)𝜑_𝑣(⋅) (18) is an asymptotically almost-periodic function; that is, 𝑢_𝑎𝑝(⋅)𝑣_𝑎𝑝(⋅)∈𝐴𝑃𝑋^1/2,𝜑×𝑋_𝑢(⋅)𝜑_𝑣(⋅)∈𝐶₀ℝ⁺,𝑋^1/2×𝑋. (19) Since 𝑓∈𝐴𝐴𝑃(ℝ⁺×𝑋^1/2×𝑋,𝑋); then there are two functions Φ∈𝐴𝑃(ℝ⁺×𝑋^1/2×𝑋,𝑋) and Ψ∈𝐶₀(ℝ⁺×𝑋^1/2×𝑋,𝑋) so that 𝑓=Φ+Ψ.
We set for [^𝑎_𝑏]∈𝑋^1/2×𝑋Φ𝑎𝑏=0Φ(),Ψ𝑎𝑏=0Ψ()𝑠,𝑠,𝑎,𝑏𝑠,𝑠,𝑎,𝑏. (20) For 𝑡∈ℝ⁺, we have the following decomposition: 𝔉𝑢𝑣(𝑡)=^𝑡₀𝑒^{−𝒜(1/2)(𝑡−𝑠)}Φ𝑢(𝑠)𝑣(𝑠)−Φ𝑢𝑠,𝑠,^𝑎𝑝(𝑠)𝑣_𝑎𝑝(𝑠)+𝑑𝑠_𝑡−∞𝑒^{−𝒜(1/2)(𝑡−𝑠)}Φ𝑢𝑠,^𝑎𝑝(𝑠)𝑣_𝑎𝑝(𝑠)−𝑑𝑠_0−∞𝑒^{−𝒜(1/2)(𝑡−𝑠)}Φ𝑢𝑠,^𝑎𝑝(𝑠)𝑣_𝑎𝑝(𝑠)𝑑𝑠+𝑒^{−𝒜(1/2)𝑡}𝑢₀𝑣₀+^𝑡₀𝑒^{−𝒜(1/2)(𝑡−𝑠)}Ψ𝑢(𝑠)𝑣(𝑠)(𝑡)(𝑡),𝑠,𝑑𝑠∶=𝒢+ℋ (21) where 𝒢(𝑡)=^𝑡_−∞𝑒^{−𝒜(1/2)(𝑡−𝑠)}Φ𝑢𝑠,^𝑎𝑝(𝑠)𝑣_𝑎𝑝(𝑠)𝑑𝑠 (22) and ℋ(𝑡) denotes the remained terms of the previous decomposition.
Next, let us show that ℋ∈𝐶₀(ℝ⁺,𝑋^1/2×𝑋). By (7) we have that 𝑡⟶𝑒^{−𝒜(1/2)𝑡}𝑢₀𝑣₀∈𝐶₀ℝ⁺,𝑋^1/2×𝑋. (23) We observe that 𝑢(𝑡)𝑣(𝑡)ℬ∶=∶𝑡≥0,ℬ_𝑎𝑝𝑢∶=^𝑎𝑝(𝑡)𝑣_𝑎𝑝(𝑡)∶𝑡∈ℝ (24) are relatively compact in 𝑋^1/2×𝑋. Since Ψ∈𝐶₀(ℝ⁺×𝑋^1/2×𝑋,𝑋), for each 𝜀>0 there exists a constant 𝑇>0 such that Ψ(𝑠,𝑧)_{𝑋1/2×𝑋}≤𝜀 (25) for all 𝑡≥2𝑇,𝑧∈ℬ.
We deduce ^𝑡₀𝑒^{−𝒜(1/2)(𝑡−𝑠)}Ψ𝑢(𝑠)𝑣(𝑠)𝑠,𝑑𝑠_{𝑋1/2×𝑋}≤𝐾𝐶𝑒^{−𝐶𝑡/2}sup_{𝑡∈ℝ+,𝑧∈ℬ}Ψ(𝑠,𝑧)_{𝑋1/2×𝑋}+𝐾𝐶𝜀. (26) On the other hand, since Φ∈𝐴𝑃(ℝ⁺×𝑋^1/2×𝑋,𝑋) by [35, Appendix] we get ⁰_−∞𝑒^{−𝒜(1/2)(𝑡−𝑠)}Φ𝑢𝑠,^𝑎𝑝(𝑠)𝑣_𝑎𝑝(𝑠)𝑑𝑠_{𝑋1/2×𝑋}≤𝐾𝐶𝑒^−𝐶𝑡sup_{𝑡∈ℝ,𝑧∈ℬ𝑎𝑝}‖Φ(𝑠,𝑧)‖_𝑋,for𝑡∈ℝ⁺. (27) Next, we estimate the first term of (21). For 𝜀>0, we choose 𝑇>0 big enough so that 𝜑_𝑢(𝑠)_𝑋1/2+𝜑_𝑣(𝑠)_𝑋≤𝜀, (28) for all 𝑠≥𝑇, and ‖Ψ(𝑠,𝑧)‖_𝑋^1/2_×𝑋≤𝜀 for all 𝑠≥𝑇 and all 𝑧∈ℬ∪ℬ_𝑎𝑝. We have the following estimates: ^𝑡₀𝑒^{−𝒜(1/2)(𝑡−𝑠)}Φ𝑢(𝑠)𝑣(𝑠)−Φ𝑢𝑠,𝑠,^𝑎𝑝(𝑠)𝑣_𝑎𝑝(𝑠)𝑑𝑠_{𝑋1/2×𝑋}≤𝐾^𝑡₀𝑒^{−𝐶(𝑡−𝑠)}𝐿_𝑓(𝑠)×𝑢(𝑠)−𝑢_𝑎𝑝(𝑠)_𝑋1/2+𝑣(𝑠)−𝑣_𝑎𝑝(𝑠)_𝑋𝑑𝑠+𝐾^𝑡₀𝑒^{−𝐶(𝑡−𝑠)}Ψ𝑢(𝑠)𝑣(𝑠)−Ψ𝑢𝑠,𝑠,^𝑎𝑝(𝑠)𝑣_𝑎𝑝(𝑠)_{𝑋1/2×𝑋}𝑑𝑠≤𝐾^𝑇₀𝑒^{−𝐶(𝑡−𝑠)}𝐿_𝑓(𝑠)𝜑_𝑢(𝑠)_𝑋1/2+𝜑_𝑣(𝑠)_𝑋𝑑𝑠+𝐾^𝑡_𝑇𝑒^{−𝐶(𝑡−𝑠)}𝐿_𝑓(𝑠)𝜑_𝑢(𝑠)_𝑋1/2+𝜑_𝑣(𝑠)_𝑋+𝑑𝑠𝐾^𝑇₀𝑒^{−𝐶(𝑡−𝑠)}×𝑑𝑠sup_{𝜉∈ℝ+,𝑧∈ℬ}Ψ(𝜉,𝑧)_{𝑋1/2×𝑋}+sup_{𝜉∈ℝ+,𝑧∈ℬ𝑎𝑝}Ψ(𝜉,𝑧)_{𝑋1/2×𝑋}+𝐾^𝑡_𝑇𝑒^{−𝐶(𝑡−𝑠)}×Ψ𝑢(𝑠)𝑣(𝑠)𝑠,_{𝑋1/2×𝑋}+Ψ𝑢𝑠,^𝑎𝑝(𝑠)𝑣_𝑎𝑝(𝑠)_{𝑋1/2×𝑋}𝑑𝑠≤𝐾𝑒^{−𝐶𝑡𝑇}₀𝑒^𝐶𝑠𝐿_𝑓(𝑠)𝑑𝑠sup_𝜉∈ℝ+𝜑_𝑢(𝜉)𝜑_𝑣(𝜉)_{𝑋1/2×𝑋}+𝐾+𝜀𝐾𝐶𝑒^𝐶𝑇𝑒^−𝐶𝑡×sup_{𝜉∈ℝ+,𝑧∈ℬ}Ψ(𝜉,𝑧)_{𝑋1/2×𝑋}+sup_{𝜉∈ℝ+,𝑧∈ℬ𝑎𝑝}Ψ(𝜉,𝑧)_{𝑋1/2×𝑋}𝐾+2𝜀𝐶. (29) From (23) to (29), we deduce that lim_𝑡→∞ℋ(𝑡)=0; that is, ℋ∈𝐶₀(ℝ⁺,𝑋^1/2×𝑋).
Since Φ∈𝐴𝑃(ℝ⁺×𝑋^1/2×𝑋,𝑋^1/2×𝑋) and ^𝑢𝑎𝑝_𝑣𝑎𝑝∈𝐴𝑃(𝑋^1/2×𝑋), we get from Lemma 5 that Φ𝑢⋅,^𝑎𝑝(⋅)𝑣_𝑎𝑝(⋅)∈𝐴𝑃𝑋^1/2×𝑋. (30) Now, by Lemma 4, we obtain that 𝒢∈𝐴𝑃(𝑋^1/2×𝑋), and hence 𝔉 is well defined. It suffices to show that the operator 𝔉 has a unique fixed point in 𝐴𝐴𝑃(𝑋^1/2×𝑋).
For this, we consider that ^𝑢1_𝑣1,^𝑢2_𝑣2∈𝐴𝐴𝑃(𝑋^1/2×𝑋). We can deduce that 𝔉𝑢₁𝑣₁𝑢−𝔉²𝑣_2∞=sup_𝑡≥0𝔉𝑢₁𝑣₁𝑡)𝑢(−𝔉²𝑣₂(𝑡)_{𝑋1/2×𝑋}≤𝐾sup_𝑡≥0^𝑡₀𝑒^{−𝐶(𝑡−𝑠)}×𝑓𝑠,𝑢₁(𝑠),𝑣₁(𝑠)−𝑓𝑠,𝑢₂(𝑠),𝑣₂(𝑠)_𝑋𝑑𝑠≤𝐾sup_𝑡≥0^𝑡₀𝑒^{−𝐶(𝑡−𝑠)}𝐿_𝑓(𝑠)×𝑢₁(𝑠)−𝑢₂(𝑠)_𝑋1/2+𝑣₁(𝑠)−𝑣₂(𝑠)_𝑋≤𝑑𝑠𝐾sup_𝑡≥0^𝑡₀𝑒^{−𝐶(𝑡−𝑠)}𝐿_𝑓(𝑠)𝑢𝑑𝑠¹𝑣₁−𝑢₂𝑣_2∞, (31) by the contraction principle, 𝔉 has a unique fixed point in 𝐴𝐴𝑃(𝑋^1/2×𝑋). This completes the proof.

Remark 13. We wish to emphasize that condition (15) is optimal in the sense that the function 𝐿_𝑓(⋅) is locally integrable. This is the largest possible class of Lipschitz constant 𝐿_𝑓(⋅) for which the conclusion of Theorem 12 holds true. However, this condition in 𝐿_𝑓(⋅) makes our analysis much more harder, because to prove Theorem 12 we cannot use the standard composition lemma for asymptotically almost-periodic functions (see Lemma 9). To overcome this difficulty we need to use a suitable decomposition for the natural operator associated with the mild solution (see (14) and (21)). In contrast, we note that in the more restrictive case of 𝐿_𝑓(⋅) to be an integrable bounded function we can use Lemma 9 directly.

Corollary 14. Let 𝑓∶ℝ⁺×𝑋^1/2×𝑋→𝑋 be an asymptotically almost-periodic function that satisfies the Lipschitz condition (15) with 𝐿_𝑓(⋅)≡𝐿. If 𝐾𝐿𝐶<1, (32) then problem (2) has a unique asymptotically almost-periodic mild solution.

Remark 15. Let 𝑓∶ℝ⁺×𝑋^1/2×𝑋→𝑋 be an asymptotically almost-periodic function that satisfies the Lipschitz condition (15). We can avoid the condition (32) by using the fixed-point iteration method. Indeed, we consider two cases.

Case 1. 𝐿_𝑓(⋅)≡𝐿. We consider the following space 𝐴𝐴𝑃⁰𝑋^1/2𝑢𝑣𝑋×𝑋∶=∈𝐴𝐴𝑃^1/2×𝑋∶𝑢(0)=0,𝑣(0)=0 (33) endowed with the norm of the uniform convergence. We define the map Γ on the space 𝐴𝐴𝑃⁰(𝑋^1/2×𝑋) by Γ𝑢𝑣(𝑡)=^𝑡₀𝑒^{−𝒜(1/2)(𝑡−𝑠)}𝑢(𝑠)𝑣(𝑠)×𝐹𝑠,+𝑒^{−𝒜(1/2)𝑠}𝑢₀𝑣₀𝑑𝑠. (34) Combining Lemmas 9 and 10, we know that Γ is a continuous function from 𝐴𝐴𝑃⁰(𝑋^1/2×𝑋) into 𝐴𝐴𝑃⁰(𝑋^1/2×𝑋). Moreover, for ^𝑢𝑖_𝑣𝑖∈𝐴𝐴𝑃⁰(𝑋^1/2×𝑋), 𝑖=1,2 we have Γ𝑢₁𝑣₁(𝑡)𝑢−Γ²𝑣₂(𝑡)_{𝑋1/2×𝑋}≤𝐾𝐿^𝑡₀𝑒^{−𝐶(𝑡−𝑠)}sup_{0≤𝜉≤𝑠}𝑢₁(𝜉)−𝑢₂(𝜉)_𝑋1/2+𝑣₁(𝜉)−𝑣₂(𝜉)_𝑋𝑑𝑠. (35) With the notation Φ(𝑠)=𝐾𝐿𝑒^−𝐶𝑠 and 𝛼(𝑠)=sup_{0≤𝜉≤𝑠}𝑢₁(𝜉)−𝑢₂(𝜉)_𝑋1/2+𝑣₁(𝜉)−𝑣₂(𝜉)_𝑋, (36) the previous estimate yields Γ𝑢₁𝑣₁(𝑡)𝑢−Γ²𝑣₂(𝑡)_{𝑋1/2×𝑋}≤(Φ∗𝛼)(𝑡),𝑡≥0. (37) Since Φ∗𝛼 is a nondecreasing function, proceeding inductively, we can show that Γ^𝑛𝑢₁𝑣₁(𝑡)−Γ^𝑛𝑢₂𝑣₂(𝑡)_{𝑋1/2×𝑋}≤Φ^∗𝑛∗𝛼(𝑡),𝑡≥0, (38) where Φ^∗𝑛 denotes the 𝑛-fold convolution of Φ with itself.
On the other hand, the map 𝒮∶𝐶_𝑏(ℝ⁺)→𝐶_𝑏(ℝ⁺), given by (𝒮𝛼)(𝑡)=^𝑡₀Φ(𝑡−𝑠)𝛼(𝑠)𝑑𝑠,𝑡≥0, (39) is a bounded linear map. Moreover, it follows from [36, Theorem2.3.5] that 𝜎(𝒮)={0}, which implies that ‖𝒮^𝑛‖=‖Φ^∗𝑛‖₁→0 as 𝑛→∞. This shows that Γ^𝑛 is a contraction for 𝑛 sufficiently large. As a consequence Γ has a fixed point [^𝑢_𝑣] in 𝐴𝐴𝑃⁰(𝑋^1/2×𝑋).
We note that the function 𝑥(𝑡)𝑦(𝑡)=𝑢(𝑡)𝑣(𝑡)+𝑒^{−𝒜(1/2)𝑡}𝑢₀𝑣₀ (40) is an asymptotically almost-periodic mild solution to problem (2).

Case 2. 𝐿_𝑓(⋅) in (15) is an integrable bounded function. By Lemmas 9 and 10 the space 𝐴𝐴𝑃(𝑋^1/2×𝑋) is invariant under 𝔉 (see (17)). The fixed point iteration method and the following estimate 𝔉^𝑛𝑢₁𝑣₁−𝔉^𝑛𝑢₂𝑣_2∞≤𝐾‖𝐿_𝑓‖₁^𝑛𝑢𝑛!¹𝑣₁−𝑢₂𝑣_2∞ (41) are responsible for the fact that 𝔉 has a unique fixed point in 𝐴𝐴𝑃(𝑋^1/2×𝑋). This concludes the discussion of Remark 15.
We have the following results of the existence of local type.

Theorem 16. Let 𝑓∶ℝ⁺×𝑋^1/2×𝑋→𝑋 be an asymptotically almost-periodic function that satisfies the Lipschitz condition 𝑓𝑡,𝑢₁,𝑣₁−𝑓𝑡,𝑢₂,𝑣_2𝑋≤𝐿_𝑓(𝑟)𝑢₁−𝑢_2𝑋1/2+𝑣₁−𝑣_2𝑋, (42) for each 𝑡≥0 and ^𝑢𝑖_𝑣𝑖∈𝑋^1/2×𝑋 such that ‖𝑢_𝑖‖_𝑋1/2+‖𝑣_𝑖‖_𝑋≤𝑟, 𝑖=1,2 where 𝐿_𝑓∶ℝ⁺→ℝ⁺ is nondecreasing continuous function such that 𝐿_𝑓(0)=0 and 𝑓(𝑡,0,0)=0 for all 𝑡≥0. Then, there exists 𝜀>0 such that, for each ^𝑢0_𝑣0∈𝑋^1/2×𝑋 satisfying ‖𝑢₀‖_𝑋1/2+‖𝑣₀‖_𝑋≤𝜀, there is a unique asymptotically almost-periodic mild solution of problem (2).

Proof. We choose 𝜆,𝑟>0 small enough such that 𝜆<1 and 𝐾(𝜆+(1/𝐶)𝐿_𝑓(𝑟))<1. Assume that ‖𝑢₀‖_𝑋1/2+‖𝑣₀‖_𝑋≤𝜆𝑟. We consider the space 𝐴𝐴𝑃_𝑟𝑢𝑣∶=∈𝐴𝐴𝑃𝑋^1/2×𝑋∶𝑢(0)=𝑢₀,𝑣(0)=𝑣₀,‖𝑢(𝑡)‖_𝑋^1/2+‖𝑣(𝑡)‖_𝑋≤𝑟,𝑡≥0, (43) endowed with the norm of the uniform convergence. We consider the map 𝔉 given by (17) on 𝐴𝐴𝑃_𝑟. By Lemma 9𝔉𝐴𝐴𝑃_𝑟⊆𝐴𝐴𝑃𝑋^1/2×𝑋. (44) We next prove that 𝔉(𝐴𝐴𝑃_𝑟)⊆𝐴𝐴𝑃_𝑟.
In fact, if [^𝑢_𝑣]∈𝐴𝐴𝑃_𝑟, we have 𝔉𝑢𝑣(𝑡)_{𝑋1/2×𝑋}≤𝐾𝑒^−𝐶𝑡𝑢_0𝑋1/2+𝑣_0𝑋+𝐾^𝑡₀𝑒^{−𝐶(𝑡−𝑠)}𝐿_𝑓(𝑟)×‖𝑢(𝑠)‖_𝑋^1/2+‖𝑣(𝑠)‖_𝑋1𝑑𝑠≤𝐾𝜆+𝐶𝐿_𝑓(𝑟)𝑟≤𝑟, (45) which permit us to infer that 𝔉(𝐴𝐴𝑃_𝑟)⊆𝐴𝐴𝑃_𝑟.
On the other hand, for ^𝑢𝑖_𝑣𝑖∈𝑋^1/2×𝑋,𝑖=1,2, we have that 𝔉𝑢₁𝑣₁(𝑡)𝑢−𝔉²𝑣₂(𝑡)_{𝑋1/2×𝑋}≤𝐾𝐿_𝑓(𝑟)^𝑡₀𝑒^{−𝐶(𝑡−𝑠)}×𝑢₁(𝑠)−𝑢₂(𝑠)_𝑋1/2+𝑣₁(𝑠)−𝑣₂(𝑠)_𝑋≤𝐾𝑑𝑠𝐶𝐿_𝑓(𝑟)𝑢₁𝑣₁−𝑢₂𝑣_2∞, (46) which shows that 𝔉 is a contraction from 𝐴𝐴𝑃_𝑟 into itself. Therefore, the assertion holds for 𝜀=𝜆𝑟.

Theorem 17. Let 𝑓∈𝐴𝐴𝑃(ℝ⁺×𝑋^1/2×𝑋,𝑋) be a function that satisfies the local Lipschitz condition (42) with 𝐿_𝑓(⋅) a nondecreasing function. Assume that there is a constant 𝑟>0 such that 𝐾𝐶𝐿_𝑓𝑟+𝐶_#+1𝑟𝐿_𝑓𝐶_#𝐶_#+sup_𝑡≥0𝑓(𝑡,0,0)<1, (47) where 𝐶_#=𝐾[‖𝑢₀‖_𝑋1/2+‖𝑣₀‖_𝑋], 𝐾, and 𝐶 are the constant given in (7). Then, there is an asymptotically almost-periodic mild solution of problem (2).

Proof. We define the map Γ on the closed ball ℬ_𝑟=𝑢𝑣∈𝐴𝐴𝑃⁰𝑋^1/2×𝑋∶‖𝑢(𝑡)‖_𝑋^1/2+‖𝑣(𝑡)‖_𝑋≤𝑟,𝑡≥0 (48) by means of the expression (34).
If [^𝑢_𝑣]∈ℬ_𝑟, we have the estimate Γ𝑢𝑣(𝑡)_{𝑋1/2×𝑋}≤Γ𝑢𝑣(𝑡)00−Γ(𝑡)_{𝑋1/2×𝑋}+Γ00(𝑡)_{𝑋1/2×𝑋}≤𝐾𝐿_𝑓𝑟+𝐶_#^𝑡₀𝑒^{−𝐶(𝑡−𝑠)}×‖𝑢(𝑠)‖_𝑋^1/2+‖𝑣(𝑠)‖_𝑋𝑑𝑠+𝐾^𝑡₀𝑒^{−𝐶(𝑡−𝑠)}×𝐹𝑠,𝑒^{−𝒜(1/2)𝑠}𝑢₀𝑣_{0𝑋1/2×𝑋}≤𝐾𝑑𝑠𝐶𝐿_𝑓𝑟+𝐶_#𝑟+𝐾𝐶𝐿_𝑓𝐶_#𝐶_#+sup_𝑡≥0𝑓(≤𝐾𝑡,0,0)𝐶𝐿_𝑓𝑟+𝐶_#+1𝑟𝐿_𝑓𝐶_#𝐶_#+sup_𝑡≥0𝑓(𝑡,0,0)𝑟≤𝑟. (49)
Moreover, for ^𝑢𝑖_𝑣𝑖∈ℬ_𝑟, 𝑖=1,2, we have Γ𝑢₁𝑣₁(𝑡)𝑢−Γ²𝑣₂(𝑡)_{𝑋1/2×𝑋}≤𝐾𝐿_𝑓𝑟+𝐶_#^𝑡₀𝑒^{−𝐶(𝑡−𝑠)}×𝑢₁(𝑠)−𝑢₂(𝑠)_𝑋1/2+𝑣₁(𝑠)−𝑣₂(𝑠)_𝑋𝑑𝑠≤𝐾𝐿_𝑓𝑟+𝐶_#𝑢₁𝑣₁−𝑢₂𝑣_2∞. (50)
Using (50), we get that Γ is a contraction on ℬ_𝑟.

In many concrete situations, the operator 𝐴 has compact resolvent, which in turn implies that the semigroup 𝑒^{−𝒜(1/2)𝑡} generated by the operator −𝒜_(1/2) is compact for 𝑡>0. To exploit this property of compactness, we need to introduce some preliminaries.

Let 𝑌 be an arbitrary Banach space and let ℎ∶[0,∞)→[1,∞) be a nondecreasing continuous function such that ℎ(𝑡)→∞ when 𝑡→∞. In next, we denote by 𝐶_ℎ(𝑌) the space 𝐶_ℎ(𝑌)=𝑢∈𝐶ℝ⁺,𝑌∶lim_𝑡→0𝑢(𝑡)ℎ(𝑡)=0 (51) endowed with the norm ‖𝑢‖_ℎ=sup_𝑡≥0‖𝑢(𝑡)‖_𝑌ℎ(𝑡). (52)

For reference purposes, we state the following property.

Lemma 18 (see [26]). A set 𝒦⊆𝐶_ℎ(𝑌) is relatively compact in 𝐶_ℎ(𝑌) if the following conditions are fulfilled: (C1) For all 𝑏>0, the set 𝒦_𝑏={𝑢|_[0,𝑏]∶𝑢∈𝒦} is relatively compact in 𝐶([0,𝑏],𝑌). (C2)lim_𝑡→∞(‖𝑢(𝑡)‖_𝑌/ℎ(𝑡))=0 uniformly for 𝑢∈𝒦.

To establish our next result we introduce the following condition.

(𝑊) There is a continuous nondecreasing function 𝑊_𝑓∶ℝ⁺→ℝ⁺ such that 𝑓(𝑡,𝑢,𝑣)_𝑋≤𝑊_𝑓‖𝑢‖_𝑋^1/2+‖𝑣‖_𝑋, (53) for all 𝑡≥0 and all [^𝑢_𝑣]∈𝑋^1/2×𝑋.

We next denote 𝛽(𝐿)=sup_𝑡≥0𝐾ℎ(𝑡)^𝑡₀𝑒^{−𝐶(𝑡−𝑠)}𝑊_𝑓(𝐿ℎ(𝑠))𝑑𝑠. (54) We have the following result.

Theorem 19. Assume that the operator 𝐴 in (1) has compact resolvent. Suppose, in addition, that the following conditions are fulfilled:(a) The function 𝑓∈𝐴𝐴𝑃(ℝ⁺×𝑋^1/2×𝑋,𝑋) is uniformly continuous on compact sets and satisfies the condition (𝑊).(b) For each 𝑎≥0, 𝑟>0, and 0≤𝜉≤𝑎, 𝑒^{−𝒜(1/2)(𝑎+𝑠−𝜉)}−𝑒^{−𝒜(1/2)(𝑎−𝜉)}𝐹𝑢𝑣𝜉,⟶0 (55)
as 𝑠→∞ uniformly for all [^𝑢_𝑣] so that ‖𝑢‖_𝑋^1/2+‖𝑣‖_𝑋≤𝑟.(c) For each 𝐿>0, lim_𝑡→∞1ℎ(𝑡)^𝑡₀𝑒^{−𝐶(𝑡−𝑠)}𝑊_𝑓(𝐿ℎ(𝑠))𝑑𝑠=0. (56)(d) For each 𝜀>0, there is 𝛿>0 such that for every ^𝑢𝑖_𝑣𝑖∈𝐶_ℎ(𝑋^1/2×𝑋), 𝑖=1,2 with 1ℎ(𝑡)𝑢₁(𝑡)−𝑢₂(𝑡)_𝑋1/2+𝑣₁(𝑡)−𝑣₂(𝑡)_𝑋≤𝛿,𝑡≥0 (57)
implies that ^𝑡₀𝑒^{−𝐶(𝑡−𝑠)}𝑓𝑠,𝑢₁(𝑠),𝑣₁(𝑠)−𝑓𝑠,𝑢₂(𝑠),𝑣₂(𝑠)_𝑋𝑑𝑠≤𝜀, (58)
for each 𝑡≥0.(e)liminf_𝜉→∞𝛽(𝜉)/𝜉<1.
Then, problem (2) has an asymptotically almost-periodic mild solution.

Proof. Let 𝐶⁰_ℎ(𝑋^1/2×𝑋) be the space consisting of the functions 𝑢(⋅)𝑣(⋅) in 𝐶_ℎ(𝑋^1/2×𝑋) such that 𝑢(0)=0, 𝑣(0)=0. It is clear that 𝐶⁰_ℎ(𝑋^1/2×𝑋) is a closed subspace of 𝐶_ℎ(𝑋^1/2×𝑋). We define the operator Γ on 𝐶⁰_ℎ(𝑋^1/2×𝑋) by (34). It follows from conditions (a) and (c) that 1ℎ(𝑡)Γ𝑢𝑣(𝑡)_{𝑋1/2×𝑋}≤𝐾ℎ(𝑡)^𝑡₀𝑒^{−𝐶(𝑡−𝑠)}𝑊_𝑓(𝐿ℎ(𝑠))𝑑𝑠⟶0,𝑡⟶∞, (59) where 𝑢𝑣𝐿=_ℎ+𝐾𝑢_0𝑋1/2+𝑣_0𝑋. (60) Thus, we have that Γ∶𝐶⁰_ℎ(𝑋^1/2×𝑋)→𝐶⁰_ℎ(𝑋^1/2×𝑋).
We divide the rest of the proof into several steps.(i) The map Γ is continuous. For each 𝜀>0 there is 𝛿>0 such that for ^𝑢𝑖_𝑣𝑖∈𝐶⁰_ℎ(𝑋^1/2×𝑋), 𝑖=1,2 with 𝑢₁−𝑢₂𝑣₁−𝑣_2ℎ≤𝛿 (61) implies that 1ℎ(𝑡)Γ𝑢₁𝑣₁(𝑡)𝑢−Γ²𝑣₂(𝑡)_{𝑋1/2×𝑋}≤1ℎ(𝑡)𝜀≤𝜀, (62) which shows the assertion.(ii) The map Γ is completely continuous. We take 𝑟>0 and we set 𝑉=Γ(𝐵_𝑟(𝐶⁰_ℎ(𝑋^1/2×𝑋))). For 𝑡≥0, we set 𝑉(𝑡)𝑢(𝑡)𝑣(𝑡)∶𝑢𝑣∶=∈𝑉. (63) We first show that 𝑉(𝑡) is a relatively compact set in 𝑋^1/2×𝑋 for each 𝑡≥0. It follows from the mean value theorem that 𝑉(𝑡)∈𝑡𝑐(𝒦₀), where 𝑐(𝒦₀) denotes the convex hull of 𝒦₀=𝑒^{−𝒜(1/2)(𝑡−𝑠)}𝐹𝑢𝑣‖𝑢‖𝑠,∶0≤𝑠≤𝑡,_𝑋1/2+‖𝑣‖_𝑋≤𝜌, (64) where 𝜌=ℎ(𝑡)(𝑟+𝐾(‖𝑢₀‖_𝑋1/2+‖𝑣₀‖_𝑋)), and 𝐾 is a constant given in (7). Since 𝐹𝑢𝑣sup𝑠,_{𝑋1/2×𝑋}‖𝑢‖∶0≤𝑠≤𝑡,_𝑋1/2+‖𝑣‖_𝑋≤𝜌≤𝑊_𝑓𝜌, (65) and taking into account that 𝑒^{−𝒜(1/2)𝑡} is compact for 𝑡>0, we infer that 𝒦₀ is a relatively compact set in 𝑋^1/2×𝑋 and consequently 𝑉(𝑡) is also relatively compact. Let now 𝑏>0 fixed and 𝑉_𝑏 the set formed by the functions [^𝑢_𝑣]∈𝑉 restricted to the interval [0,𝑏]. We affirm that the set 𝑉_𝑏 is equicontinuous. In fact, if 𝑢𝑣𝑢𝑣̃𝑢𝑣̃=Γ,with∈𝐵_𝑟𝐶⁰_ℎ𝑋^1/2×𝑋. (66) For 𝑠≥0, we obtain the following estimate: 𝑢()(𝑡)𝑣()(𝑡)𝑡+𝑠−𝑢𝑡+𝑠−𝑣_{𝑋1/2×𝑋}≤𝐾_𝑡^𝑡+𝑠𝑒^{−𝐶(𝑡+𝑠−𝜉)}𝑊_𝑓((𝜉+𝐿ℎ))𝑑𝜉_𝑡0𝑒^{−𝒜(1/2)(𝑡+𝑠−𝜉)}−𝑒^{−𝒜(1/2)(𝑡−𝜉)}𝑢(𝜉)𝑣(̃𝜉)×𝐹𝜉,+𝑒^{−𝒜(1/2)𝜉}𝑢₀𝑣_{0𝑋1/2×𝑋}𝑑𝜉, (67) where 𝐿=𝑟+𝐾(‖𝑢₀‖_𝑋1/2+‖𝑣₀‖_𝑋). It is immediate that the first term on the right hand side converges to zero when 𝑠→0 and, using condition (𝑏) we obtain the second term on the right-hand side also converges to zero when 𝑠→0 and the convergence is independent of the function 𝑢_(⋅)𝑣̃_(⋅). We now show that lim_𝑡→∞1ℎ(𝑡)𝑢(𝑡)𝑣(𝑡)=0, (68) independent of 𝑢𝑣̃∈𝐵_𝑟(𝐶⁰_ℎ(𝑋^1/2×𝑋)). This assertion is a direct consequence of the following estimate and the condition (c)∶1ℎ(𝑡)𝑢(𝑡)𝑣(𝑡)_{𝑋1/2×𝑋}≤𝐾ℎ(𝑡)^𝑡₀𝑒^{−𝐶(𝑡−𝑠)}𝑊_𝑓(𝐿ℎ(𝑠))𝑑𝑠. (69) Combining these assertions with Lemma 18, we get that 𝑉 is a relatively compact set in 𝐶⁰_ℎ(𝑋^1/2×𝑋). Since 𝑟 was chosen arbitrary, this proves that Γ is completely continuous.(iii) There is 𝑟₀>0 such that Γ(𝐵_𝑟0(𝐶⁰_ℎ(𝑋^1/2×𝑋)))⊆𝐵_𝑟0(𝐶⁰_ℎ(𝑋^1/2×𝑋)). In fact, if we assume that the assertion is false; then for all 𝑟>0 we can choose ^𝑢𝑟_𝑣𝑟∈𝐵_𝑟0(𝐶⁰_ℎ(𝑋^1/2×𝑋)) such that 1ℎ(𝑡)Γ𝑢^𝑟𝑣^𝑟(𝑡)_{𝑋1/2×𝑋}>𝑟 (70) for all 𝑡∈ℝ⁺. Then 𝐾𝑢1≤1+^0𝑋1/2+𝑣_0𝑋𝑟×𝛽𝑟+𝐾𝑢_0𝑋1/2+𝑣_0𝑋𝑟+𝐾𝑢_0𝑋1/2+𝑣_0𝑋,1≤liminf_𝜉→∞𝛽(𝜉)𝜉, (71)(72) which contradicts condition (e) and establishes the assertion.(iv) If [^𝑢_𝑣]∈𝐴𝐴𝑃⁰(𝑋^1/2×𝑋), then the function ℝ⁺→𝑋^1/2×𝑋 given by 𝑢(𝑡)𝑣(𝑡)𝑡⟶+𝑒^{−𝒜(1/2)𝑡}𝑢₀𝑣₀ (73) is in 𝐴𝐴𝑃(𝑋^1/2×𝑋). Since 𝑓∈𝐴𝐴𝑃(ℝ⁺×𝑋^1/2×𝑋,𝑋) is uniformly continuous on compact sets, we have from Lemma 9 that 𝑢(𝑡)𝑣(𝑡)𝑡⟶𝐹𝑡,+𝑒^{−𝒜(1/2)𝑡}𝑢₀𝑣₀∈𝐴𝐴𝑃𝑋^1/2×𝑋. (74) Applying Lemma 10, we obtain that Γ(𝐴𝐴𝑃⁰(𝑋^1/2×𝑋))⊆𝐴𝐴𝑃⁰(𝑋^1/2×𝑋). Consequently, combining with (iii) we infer that Γ𝐵_𝑟0𝐶⁰_ℎ𝑋^1/2×𝑋∩𝐴𝐴𝑃⁰𝑋^1/2×𝑋^ℎ⊆𝐵_𝑟0𝐶⁰_ℎ𝑋^1/2×𝑋∩𝐴𝐴𝑃⁰𝑋^1/2×𝑋^ℎ, (75) where 𝒮^ℎ denotes the closure of 𝒮 in 𝐶_ℎ(𝑋^1/2×𝑋). Using Schauder fixed point theorem, we deduce that Γ has a fixed point [^𝑢_𝑣]∈𝐵_𝑟0(𝐶⁰_ℎ(𝑋^1/2×𝑋))∩𝐴𝐴𝑃⁰(𝑋^1/2×𝑋)^ℎ.(v) Finally, we show that [^𝑢_𝑣]∈𝐴𝐴𝑃⁰(𝑋^1/2×𝑋). Let (^𝑢𝑛_𝑣𝑛)_𝑛 be a sequence in 𝐵_𝑟0(𝐶⁰_ℎ(𝑋^1/2×𝑋))∩𝐴𝐴𝑃⁰(𝑋^1/2×𝑋) that converges to [^𝑢_𝑣] for the topology in 𝐶_ℎ(𝑋^1/2×𝑋). For 𝜀>0, let 𝛿>0 be the constant in condition (𝑑); there is 𝑛₀∈ℕ so that 1ℎ(𝑡)𝑢^𝑛(𝑡)(𝑡)𝑣−𝑢_𝑛(𝑡)−𝑣(𝑡)_{𝑋1/2×𝑋}≤𝛿, (76) for all 𝑡≥0 and all 𝑛≥𝑛₀. Therefore, for 𝑛≥𝑛₀sup_𝑡≥0Γ𝑢^𝑛𝑣^𝑛(𝑡)𝑢𝑣−Γ(𝑡)_{𝑋1/2×𝑋}≤𝜀. (77) Hence lim_𝑛→∞Γ𝑢^𝑛𝑣^𝑛−𝑢𝑣_∞=0. (78) Since Γ^𝑢𝑛_𝑣𝑛∈𝐴𝐴𝑃⁰(𝑋^1/2×𝑋) we get that [^𝑢_𝑣]∈𝐴𝐴𝑃⁰(𝑋^1/2×𝑋) and completes the proof.

Remark 20. Note that in Theorem 19 we do not need to assume that the operator 𝐴 in (1) has compact resolvent if the following condition holds.(f) For all 𝑎≥0 and 𝑟>0 the set {𝑓(𝑡,𝜑₁,𝜑₂)∶0≤𝑡≤𝑎,||𝜑₁||_𝑋^1/2+||𝜑₂||_𝑋≤𝑟} is relatively compact in 𝑋.

4. Applications

Suppose that ℎ,𝑔∈𝐶(ℝ,ℝ), 𝑎∶ℝ→ℝ (𝑏∶ℝ⁺→ℝ) is a bounded continuous function, 𝜈,𝜇∈ℝ, 𝛿>0 and 𝜌₁,𝜌₂∈(1,+∞). In a bounded smooth domain Ω⊆ℝ^𝑁, we consider the following partial differential equation: 𝑢_𝑡𝑡+𝜈𝑏(𝑡)𝑢_𝑡+Δ²𝑢−𝛿Δ𝑢_𝑡(𝑡ℎ(𝑢)(𝑢)𝑢()=𝜇𝑎)∇⋅𝑢+𝑔Δ𝑢𝑥∈Ω,𝑡≥0,𝑢=Δ𝑢=0,𝑥∈𝜕Ω,𝑡>0,0,𝑥=𝑢₀(𝑥),𝑢_𝑡(0,𝑥)=𝑣₀(𝑥),𝑥∈Ω, (79) where ℎ and 𝑔 satisfy the following growth conditions: ℎ𝑠₁−ℎ𝑠₂≤𝐶_ℎ𝑠₁−𝑠₂1+𝑠₁^𝜌1−1+𝑠₂^𝜌1−1,𝑠₁,𝑠₂𝑔𝑠∈ℝ,₁−𝑔𝑠₂≤𝐶_𝑔𝑠₁−𝑠₂1+𝑠₂^𝜌2−1+𝑠₁^𝜌2−1,𝑠₁,𝑠₂∈ℝ, (80) where 𝐶_ℎ and 𝐶_𝑔 are positive constants. Here, we describe the asymptotically almost-periodic behavior of solutions of problem (79) in the 𝐿^𝑝-setting. To model this problem in the abstract form (1) we set that 𝜂=𝛿/2, 𝑝>𝑁/2, the operator 𝐴 is defined in 𝐿^𝑝(Ω) by 𝐴𝑢=Δ²_𝐷𝑢 (Δ_𝐷 is the Dirichlet Laplacian in Ω) on the domain 𝐷Δ²_𝐷=𝜙∈𝐻⁴_𝑝(Ω)∶𝜙=Δ𝜙=0on𝜕Ω, (81) where 𝐻⁴_𝑝(Ω)=𝑊^4,𝑝(Ω) is the standard Sobolev space (see [37]). With this specification, problem (79) will fall into the abstract formulation (1). Since 𝐴^1/2=−Δ_𝐷, we can choose the angle 𝜓 for the sector Σ_𝜓/2=𝜓𝜆∈ℂ∶2≤𝜋arg𝜆≤𝜋with𝜓∈0,2 (82) as small as needed and therefor (see [21, Example 4.3]) (𝜂,𝐴) will be an admissible pair for any 𝜂>0. From [21, Section 3] we get that 𝐿^𝑝(Ω)^1/2=𝜙∈𝐻²_𝑝(Ω)∶𝜙=Δ𝜙=0on𝜕Ω. (83) We define 𝑓∶ℝ⁺×[𝐿^𝑝(Ω)]^1/2×𝐿^𝑝(Ω)→𝐿^𝑝(Ω) by 𝑓𝑡,𝜑₁,𝜑₂=𝜇𝑎(𝑡)ℎ^𝑒𝜑₁∇⋅𝜑₁+𝑔^𝑒𝜑₁Δ𝜑₁−𝜈𝑏(𝑡)𝜑₂,𝑡∈ℝ⁺,𝜑₁∈𝐿^𝑝(Ω)^1/2,𝜑₂∈𝐿^𝑝(Ω), (84) where 𝜃^𝑒 is the Nemytskiĭ operator associated with 𝜃, and ∇𝜑 represents the divergence of 𝜑. For 𝜑₁∈[𝐿^𝑝(Ω)]^1/2 and 𝜑₂∈𝐿^𝑝(Ω). Using Minkowski's inequality and Sobolev embedding, we have the estimate 𝑓𝑡,𝜑₁,𝜑_{2𝐿𝑝(Ω)}≤𝜇|𝑎(𝑡)|ℎ^𝑒𝜑_1𝐿∞(Ω)+𝑔^𝑒𝜑_1𝐿∞(Ω)×𝜑_{1𝐻2𝑝(Ω)}+|𝜈||𝑏(𝑡)|𝜑_{2𝐿𝑝(Ω)}. (85) Whence 𝑓 is well defined. We claim that 𝑓 satisfies (42) with 𝐿_𝑓(𝑟)=𝐶𝜇+|𝜈|‖𝑎‖_∞+‖𝑏‖_∞×1+2𝑟+3𝑟^𝜌1+𝑟^𝜌2. (86) Indeed, it is an easy consequence of the following estimates (here 𝐶 will stand for some positive constant independent of 𝜑₁ and 𝜑₂ with 𝜑₁,𝜑₂∈[𝐿^𝑝(Ω)]^1/2). ℎ^𝑒𝜑₁−ℎ^𝑒𝜑₂∇⋅𝜑_{1𝐿𝑝(Ω)}≤𝐶𝜑₁−𝜑_{2𝐻2𝑝(Ω)}×1+𝜑₁^𝜌1_𝐻⁻¹_2𝑝(Ω)+𝜑₂^𝜌1_𝐻⁻¹_2𝑝(Ω)𝜑_{1𝐻2𝑝(Ω)},ℎ^𝑒𝜑₂∇⋅𝜑₁−𝜑_{2𝐿𝑝(Ω)}≤𝐶1+𝜑_{2𝐻2𝑝(Ω)}+𝜑₂^𝜌1_{𝐻2𝑝(Ω)}𝜑₁−𝜑_{2𝐻2𝑝(Ω)},𝑔^𝑒𝜑₁Δ𝜑₁−𝜑_{2𝐿𝑝(Ω)}≤𝐶1+𝜑_{1𝐻2𝑝(Ω)}+𝜑₁^𝜌2_{𝐻2𝑝(Ω)}𝜑_{2𝐻2𝑝(Ω)},𝑔^𝑒𝜑₁−𝑔^𝑒𝜑₂Δ𝜑_{2𝐿𝑝(Ω)}≤𝐶𝜑₁−𝜑_{2𝐻2𝑝(Ω)}×1+𝜑₁^𝜌2_𝐻⁻¹_2𝑝(Ω)+𝜑₂^𝜌2_𝐻⁻¹_2𝑝(Ω)𝜑_{2𝐻2𝑝(Ω)}. (87) Let 𝐾 be a compact subset of [𝐿^𝑝(Ω)]^1/2×𝐿^𝑝(Ω). For 𝜑₁∈[𝐿^𝑝(Ω)]^1/2 and 𝜑₂∈𝐿^𝑝(Ω), we set that 𝑓_𝑎𝑝𝑡,𝜑₁,𝜑₂=𝜇𝑎(𝑡)ℎ^𝑒𝜑₁∇⋅𝜑₁+𝑔^𝑒𝜑₁Δ𝜑₁,Φ_𝑓𝑡,𝜑₁,𝜑₂=𝜈𝑏(𝑡)𝜑₂,𝑀_𝐾𝜑=sup¹𝜑_{2[𝐿𝑝(Ω)]1/2×𝐿𝑝(Ω)}∶𝜑₁𝜑₂∈𝐾. (88) We have the following estimates: Φ_𝑓𝑡,𝜑₁,𝜑_{2𝐿𝑝(Ω)}≤|𝜈||𝑏(𝑡)|𝑀_𝐾,𝑓_𝑎𝑝𝑡+𝜏,𝜑₁,𝜑₂−𝑓_𝑎𝑝𝑡,𝜑₁,𝜑_{2𝐿𝑝(Ω)}≤𝜇𝑎()(𝑡×ℎ|𝑡+𝜏−𝑎)|_𝑒𝜑_1𝐿∞(Ω)+𝑔^𝑒𝜑_1𝐿∞(Ω)𝜑_{1𝐻2𝑝(Ω)}≤𝐶𝜇𝑎()(𝑡×𝜑|𝑡+𝜏−𝑎)|1+_{1𝐻2𝑝(Ω)}+𝜑₁^𝜌1_{𝐻2𝑝(Ω)}+𝜑₁^𝜌2_{𝐻2𝑝(Ω)}𝜑_{1𝐻2𝑝(Ω)}≤𝐶(𝐾)|𝑎(𝑡+𝜏)−𝑎(𝑡)|, (89)(90) where 𝐶(𝐾) is a constant depending on 𝐾.

If 𝑏∈𝐶₀(ℝ⁺,ℝ) from (89), we get Φ_𝑓∈𝐶₀ℝ⁺×𝐿^𝑝(Ω)^1/2×𝐿^𝑝(Ω),𝐿^𝑝(Ω), (91) and (90) implies that 𝑓_𝑎𝑝∈𝐴𝑃ℝ×𝐿^𝑝(Ω)^1/2×𝐿^𝑝(Ω),𝐿^𝑝(Ω). (92) Hence, 𝑓∈𝐴𝐴𝑃ℝ⁺×𝐿^𝑝(Ω)^1/2×𝐿^𝑝(Ω),𝐿^𝑝(Ω). (93) Applying Theorem 17, we have the following result.

Proposition 21. Under the previous conditions, if 𝑎∈𝐴𝑃(ℝ), 𝑏∈𝐶₀(ℝ⁺,ℝ), and |𝜇|+|𝜈| is small enough, then problem (79) has an asymptotically almost-periodic mild solution.

Example 22. We consider the following partial differential equation 𝑢_𝑡𝑡+𝜈𝑢𝑡+1^𝑡+Δ𝑢²−𝛿Δ𝑢_𝑡=𝜇cos𝑡+cos√2𝑡|𝑢|^𝜌−1(),𝑢()𝑢∇⋅𝑢+𝑢Δ𝑢𝑥∈Ω,𝑡≥0,𝑢=Δ𝑢=0,𝑥∈𝜕Ω,𝑡>0,0,𝑥=𝑢₀(𝑥),𝑢_𝑡(0,𝑥)=𝑣₀(𝑥),𝑥∈Ω. (94) If |𝜇|+|𝜈| is small enough by Proposition 21, problem (94) has an asymptotically almost-periodic mild solution. In fact, take ℎ(𝑢)=𝑔(𝑢)=𝑢|𝑢|^𝜌−1, 𝑎(𝑡)=cos𝑡+cos√2𝑡, and 𝑏(𝑡)=1/(𝑡+1).
The next application is a modification of problem (79). Let 𝑘 be a fixed nonnegative integer and let 𝑐∶ℝ→ℝ⁺ and 𝑏∶ℝ⁺→ℝ be two bounded continuous functions. Suppose that 𝑓∈𝐴𝐴𝑃(ℝ⁺×ℝ^𝑁×ℝ×ℝ,ℝ) satisfies the following growth condition: 𝑓()(𝑤(𝑡𝑤|+|𝑢||𝑢|𝑡,𝑥,𝑢,𝑤−𝑓𝑡,𝑥,𝑢,)≤𝑐)|𝑤−𝑢−1+^𝑘+|𝑢|^𝑘, (95) for each 𝑡≥0, 𝑥∈ℝ^𝑁, 𝑢,𝑢,𝑤,𝑤∈ℝ. We consider the following partial differential equation 𝑢_𝑡𝑡+𝑏(𝑡)𝑢_𝑡+Δ²𝑢−𝛿Δ𝑢_𝑡(),𝑢()=𝑓𝑡,𝑥,𝑢,∇⋅𝑢𝑥∈Ω,𝑡≥0,𝑢=Δ𝑢=0,𝑥∈𝜕Ω,𝑡>0,0,𝑥=𝑢₀(𝑥),𝑢_𝑡(0,𝑥)=𝑣₀(𝑥),𝑥∈Ω. (96)
We model (96) in the abstract form (1) in a similar way to problem (79). That is we set 𝜂=𝛿/2, 𝑝>𝑁 (hence 𝐻²_𝑝(Ω)↪𝐶¹(Ω) which will be useful to get some a priori estimates) and we consider the operator 𝐴 as defined in (81) and (82). Suppose that sup_𝑡≥0Ω𝑓(𝑡,𝑥,0,0)^𝑝𝑑𝑥^1/𝑝<∞. (97) We define 𝐹∶ℝ⁺×[𝐿^𝑝(Ω)]^1/2×𝐿^𝑝(Ω)→𝐿^𝑝(Ω) by 𝐹𝑡,𝜑₁,𝜑₂(𝑥)=𝑓𝑡,𝑥,𝜑₁(𝑥),∇⋅𝜑₁(𝑥)−𝑏(𝑡)𝜑₂(𝑥). (98) We observe that 𝐹 is well defined. In fact, we get the following estimates: ∫_Ω𝑓𝑡,𝑥,𝜑₁(𝑥),∇⋅𝜑₁(𝑥)^𝑝𝑑𝑥^1/𝑝≤𝑐(𝑡)∫_Ω∇⋅𝜑₁(𝑥)^𝑝𝑑𝑥^1/𝑝+∫_Ω𝜑₁(𝑥)^𝑝1+𝜑₁(𝑥)^𝑘𝑝𝑑𝑥^1/𝑝+∫_Ω𝑓(𝑡,𝑥,0,0)^𝑝𝑑𝑥^1/𝑝≤‖𝑐‖_∞∇⋅𝜑_{1𝐿𝑝(Ω)}+1+𝜑₁^𝑘_𝐿∞(Ω)𝜑_{1𝐿𝑝(Ω)}+∫_Ω𝑓(𝑡,𝑥,0,0)^𝑝𝑑𝑥^1/𝑝. (99) On the other hand, for 𝜑₁,𝜑₁∈[𝐿^𝑝(Ω)]^1/2 and 𝜑₂,𝜑₂∈𝐿^𝑝(Ω) we have the following estimate: 𝐹𝑡,𝜑₁,𝜑₂−𝐹𝑡,𝜑₁,𝜑_{2𝐿𝑝(Ω)}≤4max‖𝑐‖_∞,‖𝑏‖_∞1+𝜑₁^𝑘_{𝐻2𝑝(Ω)}+𝜑₁^𝑘_{𝐻2𝑝(Ω)}×𝜑₁−𝜑_{1𝐻2𝑝(Ω)}+𝜑₂−𝜑_{2𝐿𝑝(Ω)}, (100) which means that 𝐹 satisfies (42) with 𝐿𝐹(𝑟)=4max{‖𝑐‖_∞,‖𝑏‖_∞}(1+2𝑟^𝑘).
Since 𝑓∈𝐴𝐴𝑃(ℝ⁺×ℝ^𝑁×ℝ×ℝ,ℝ), there are two functions 𝑓_𝑎𝑝∈𝐴𝑃ℝ×ℝ^𝑁,𝜑×ℝ×ℝ,ℝ_𝑓∈𝐶₀ℝ⁺×ℝ^𝑁×ℝ×ℝ,ℝ, (101) so that 𝑓=𝑓_𝑎𝑝+𝜑_𝑓. Let 𝐾 be a compact subset of [𝐿^𝑝(Ω)]^1/2×𝐿^𝑝(Ω). For 𝜑₁∈[𝐿^𝑝(Ω)]^1/2 and 𝜑₂∈𝐿^𝑝(Ω), we set 𝐹_𝑎𝑝𝑡,𝜑₁,𝜑₂(𝑥)=𝑓_𝑎𝑝𝑡,𝑥,𝜑₁(𝑥),∇⋅𝜑₁(𝑥),Φ𝐹𝑡,𝜑₁,𝜑₂(𝑥)=𝜑_𝑓𝑡,𝑥,𝜑₁(𝑥),∇⋅𝜑₁(𝑥)−𝑏(𝑡)𝜑₂(𝑥). (102) If (𝜑₁,𝜑₂)∈𝐾, then Φ𝐹𝑡,𝜑₁,𝜑_{2𝐿𝑝(Ω)}≤_Ω𝜑_𝑓𝑡,𝑥,𝜑₁(𝑥),∇⋅𝜑₁(𝑥)^𝑝𝑑𝑥^1/𝑝+|𝑏(𝑡)|𝑀_𝐾, (103) where 𝑀_𝐾 is given by (88).
We note that using Sobolev embedding we have that there is a positive constant 𝑐̃>0 depending only on 𝐾 so that sup_𝑥∈Ω𝜑₁(𝑥)≤𝑐,̃sup_𝑥∈Ω∇⋅𝜑₁(𝑥)≤𝑐.̃ (104) Taking into account the fact that 𝑌=Ω×𝐵𝑐(̃ℝ²) is a compact subset of ℝ^𝑁+2, (4.29) and (103). It follows that lim_𝑡→∞Φ𝐹(𝑡,𝜑₁,𝜑₂)=0 uniformly for (𝜑₁,𝜑₂)∈𝐾. Hence, Φ𝐹∈𝐶₀ℝ⁺×𝐿^𝑝(Ω)^1/2×𝐿^𝑝(Ω),𝐿^𝑝(Ω). (105) As an immediate consequence of Definition 3 and using the fact that Ω is bounded we have that 𝐹_𝑎𝑝∈𝐴𝑃ℝ×𝐿^𝑝(Ω)^1/2×𝐿^𝑝(Ω),𝐿^𝑝(Ω). (106) Since 𝐹=𝐹_𝑎𝑝+Φ𝐹, we conclude that 𝐹∈𝐴𝐴𝑃ℝ⁺×𝐿^𝑝(Ω)^1/2×𝐿^𝑝(Ω),𝐿^𝑝(Ω). (107)

Proposition 23. Under the previous conditions, if max{‖𝑐‖_∞,‖𝑏‖_∞} is small enough, then problem (96) has an asymptotically almost-periodic mild solution.

Proof. We argue as follows. Let us choose 𝑟>0 such that 𝐾/𝑟𝐶sup_𝑡≥0(∫_Ω|𝑓(𝑡,𝑥,0,0)|^𝑝𝑑𝑥)^1/𝑝 is small enough, where 𝐾 and 𝐶 are constants given in (7). Then, condition (47) is fulfilled. Now applying Theorem 17 we conclude the proof.

Let 𝑎∶ℝ⁺→ℝ be a bounded continuous function and let 𝛽 be in (0,1) and 𝛿>0. In a bounded smooth domain Ω⊆ℝ^𝑁, we consider the following partial differential equation: 𝑢_𝑡𝑡+Δ²𝑢−𝛿Δ𝑢_𝑡=𝑎(𝑡)_Ω∇⋅𝑢(𝑡,𝑥)𝑑𝑥^𝛽Φ₀,𝑢()𝑥∈Ω,𝑡≥0,𝑢=Δ𝑢=0,𝑥∈𝜕Ω,𝑡>0,0,𝑥=𝑢₀(𝑥),𝑢_𝑡(0,𝑥)=𝑣₀(𝑥),𝑥∈Ω, (108) where Φ₀∈𝐿^𝑝(Ω), 𝑝>𝑁/2. We model (108) in the abstract form (1); that is, we set 𝜂=𝛿/2 and we consider the operator 𝐴 as in (81) and (82). We define 𝑓∶ℝ⁺×[𝐿^𝑝(Ω)]^1/2×𝐿^𝑝(Ω)→𝐿^𝑝(Ω) by 𝑓𝑡,𝜑₁,𝜑₂(𝑥)=𝑎(𝑡)_Ω∇⋅𝜑₁(𝜉)𝑑𝜉^𝛽Φ₀(𝑥). (109) To establish our next result, we assume the following conditions.(H1) Let 𝑎(⋅)=𝑎_𝑎𝑝(⋅)+𝜑_𝑎(⋅) be in 𝐴𝐴𝑃(ℝ⁺), with 𝑎_𝑎𝑝(⋅)∈𝐴𝑃(ℝ) and 𝜑_𝑎(⋅)∈𝐶₀(ℝ⁺).(H2) There is a continuous nondecreasing function ℎ∶[0,∞)→[1,∞) such that ℎ(𝑡)→∞ as 𝑡→∞ and lim_𝑡→∞1ℎ(𝑡)^𝑡₀𝑒^{−𝐶(𝑡−𝑠)}ℎ(𝑠)^𝛽𝑑𝑠=0,sup_𝑡≥0^𝑡₀𝑒^{−𝐶(𝑡−𝑠)}|𝑎(𝑡)|ℎ(𝑠)^𝛽𝑑𝑠<∞. (110)

Proposition 24. Under the previous conditions, problem (108) has an asymptotically almost-periodic mild solution.

Proof. Let 𝑞>0 be given so that (1/𝑞)+(1/𝑝)=1. We can infer the following estimate: 𝑓𝑡,𝜑₁,𝜑_{2𝐿𝑝(Ω)}≤‖𝑎‖_∞Φ_{0𝐿𝑝(Ω)}|Ω|^𝛽/𝑞𝜑₁^𝛽_{𝐻2𝑝(Ω)}. (111) Hence, we can define 𝑊_𝑓 in (𝑊) by 𝑊_𝑓(𝜉)=||𝑎||_∞||Φ₀||_𝐿^𝑝_(Ω)|Ω|^𝛽/𝑞𝜉^𝛽. Let 𝐾 be a compact subset of [𝐿^𝑝(Ω)]^1/2×𝐿^𝑝(Ω) and let 𝑀_𝐾 be as in (88). We set 𝑓_𝑎𝑝𝑡,𝜑₁,𝜑₂(𝑥)=𝑎_𝑎𝑝(𝑡)_Ω∇⋅𝜑₁(𝜉)𝑑𝜉^𝛽Φ₀(𝑥),Φ^#_𝑓𝑡,𝜑₁,𝜑₂(𝑥)=𝜑_𝑎(𝑡)_Ω∇⋅𝜑₁(𝜉)𝑑𝜉^𝛽Φ₀(𝑥),𝜑₁𝜑₂,𝜓₁𝜓₂∈𝐾. (112)
We have the following estimates: 𝑓_𝑎𝑝𝑡+𝜏,𝜑₁,𝜑₂−𝑓_𝑎𝑝𝑡,𝜑₁,𝜑_{2𝐿𝑝(Ω)}≤|Ω|^𝛽/𝑞𝑀^𝛽_𝐾Φ_{0𝐿𝑝(Ω)}𝑎_𝑎𝑝(𝑡+𝜏)−𝑎_𝑎𝑝(𝑡,Φ_#𝑓)𝑡,𝜑₁,𝜑_{2𝐿𝑝(Ω)}≤|Ω|^𝛽/𝑞𝑀^𝛽_𝐾Φ_{0𝐿𝑝(Ω)}𝜑_𝑎(𝑡,𝑓)𝑡,𝜑₁,𝜑₂−𝑓𝑡,𝜓₁,𝜓_{2𝐿𝑝(Ω)}≤|Ω|^𝛽/𝑞‖𝑎‖_∞Φ_{0𝐿𝑝(Ω)}𝜑₁−𝜓₁^𝛽_{𝐻2𝑝(Ω)}. (113)(114)(115) It follows from (113) and (114) that 𝑓∈𝐴𝐴𝑃(ℝ⁺×[𝐿^𝑝(Ω)]^1/2×𝐿^𝑝(Ω),𝐿^𝑝(Ω)). In addition, from the estimate (115), we obtain that 𝑓 is uniformly continuous on compact sets.
From (H₂), for ^𝑢𝑖_𝑣𝑖∈𝐶_ℎ([𝐿^𝑝(Ω)]^1/2×𝐿^𝑝(Ω)),𝑖=1,2, we can infer that 1ℎ(𝑡)^𝑡₀𝑒^{−𝐶(𝑡−𝑠)}𝑊_𝑓((𝑠=‖𝑎‖𝐿ℎ))𝑑𝑠_∞Φ_{0𝐿𝑝(Ω)}|Ω|^𝛽/𝑞𝐿^𝛽1ℎ(𝑡)^𝑡₀𝑒^{−𝐶(𝑡−𝑠)}ℎ(𝑠)^𝛽𝑑𝑠⟶0,𝑡⟶∞,_𝑡0𝑒^{−𝐶(𝑡−𝑠)}𝑓𝑠,𝑢₁(𝑠),𝑣₁(𝑠)−𝑓𝑠,𝑢₂(𝑠),𝑣₂(𝑠)_𝐿𝑝(Ω)≤|Ω|𝑑𝑠_𝛽/𝑞Φ_{0𝐿𝑝(Ω)}^𝑡₀𝑒^{−𝐶(𝑡−𝑠)}|𝑎(𝑠)|𝑢₁(𝑠)−𝑢₂(𝑠)^𝛽_{𝐻2𝑝(Ω)}=|Ω|𝑑𝑠_𝛽/𝑞Φ_{0𝐿𝑝(Ω)}^𝑡₀𝑒^{−𝐶(𝑡−𝑠)}|𝑎(𝑠)|ℎ(𝑠)^𝛽×𝑢₁(𝑠)−𝑢₂(𝑠)_{𝐻2𝑝(Ω)}ℎ(𝑠)^𝛽≤|Ω|𝑑𝑠_𝛽/𝑞Φ_{0𝐿𝑝(Ω)}sup_𝑡≥0^𝑡₀𝑒^{−𝐶(𝑡−𝑠)}|𝑎(𝑡)|ℎ(𝑠)^𝛽×𝑑𝑠sup_𝑡≥01ℎ(𝑡)𝑢₁(𝑡)−𝑢₂(𝑡)_{𝐻2𝑝(Ω)}+𝑣₁(𝑡)−𝑣₂(𝑡)_𝐿𝑝(Ω)^𝛽. (116)(117) Therefore, conditions (c) and (d) of Theorem 19 are satisfied. A straightforward computation shows that (e) holds. We can prove that the set {𝑓(𝑡,𝜑₁,𝜑₂)∶0≤𝑡≤𝑎,||𝜑₁||_𝐻²_𝑝(Ω)+||𝜑₂||_𝐿^𝑝_(Ω)≤𝑟} is relatively compact in 𝐿^𝑝(Ω). In fact, first, we denote by 𝑢 the zero extension of 𝑢 outside Ω; that is, 𝑢(𝑥)=𝑢(𝑥)if𝑥∈Ω,0if𝑥∈ℝ^𝑁⧵Ω. (118) For every number 𝜀>0, there exists a number 𝛿>0 and a subset 𝐺⊂⊂Ω such that for every 𝜚∈ℝ^𝑁 with |𝜚|<𝛿_ΩΦ₀𝑥+𝜚−Φ₀(𝑥)^𝑝𝜀𝑑𝑥<𝑟^𝛽‖𝑎‖_∞|Ω|^{𝛽/𝑞𝑝}_Ω⧵𝐺,Φ₀(𝑥)^𝑝𝜀𝑑𝑥<𝑟^𝛽‖𝑎‖_∞|Ω|^{𝛽/𝑞𝑝}. (119) Hence, _Ω𝑓𝑡,𝜑₁,𝜑₂𝑥+𝜚−𝑓𝑡,𝜑₁,𝜑₂(𝑥)^𝑝≤𝑟𝑑𝑥_𝛽‖𝑎‖_∞|Ω|^{𝛽/𝑞𝑝}×_ΩΦ₀𝑥+𝜌−Φ₀(𝑥)^𝑝𝑑𝑥≤𝜀,_Ω⧵𝐺𝑓𝑡,𝜑₁,𝜑₂(𝑥)^𝑝𝑑𝑥≤𝑟^𝛽‖𝑎‖_∞|Ω|^{𝛽/𝑞𝑝}×_Ω⧵𝐺Φ₀(𝑥)^𝑝𝑑𝑥≤𝜀. (120) From [38, Theorem 2.21] the desired assertion follows. Note finally that using Remark 20 we have that (108) has an asymptotically almost periodic mild solution. This ends the proof.

Acknowledgments

This work was written while the first author was visiting the Universidad de La Frontera (Temuco, Chile). He is grateful to the Applicable Analysis Group and the Department of Mathematics and Statistics, for its generous hospitality and providing a stimulating atmosphere to work. C. Cuevas has been partially supported by Programa Atracción e Inserción (PAI-MEC) Grant 80112008 (CONICYT-CHILE). C. Lizama has been partially supported by Project Anillo ACT 79 and H. Soto has been partially supported by DIUFRO DI 12-0002.