Existence and Uniqueness of Pseudo Almost Automorphic Mild Solutions to Some Classes of Partial Hyperbolic Evolution Equations

Abstract

We will establish an existence and uniqueness theorem of pseudo almost automorphic mild solutions to the following partial hyperbolic evolution equation (𝑑/𝑑𝑡)[𝑢(𝑡)+𝑓(𝑡,𝐵𝑢(𝑡))]=𝐴𝑢(𝑡)+𝑔(𝑡,𝐶𝑢(𝑡)),??𝑡∈ℝ, under some assumptions. To illustrate our abstract result, a concrete example is given.

1. Introduction

Let (𝕏,‖⋅‖) be a Banach space, and let (𝕏_𝛼,‖⋅‖_𝛼) for 0<𝛼<1 be an arbitrary abstract intermediate Banach space between 𝐷(𝐴) (domain of the operator 𝐴) and 𝕏. In this paper, we deal with sufficient conditions for the existence and uniqueness of pseudo almost automorphic mild solutions to equation𝑑𝑑𝑡[𝑢(𝑡)+𝑓(𝑡,𝐵𝑢(𝑡))]=𝐴𝑢(𝑡)+𝑔(𝑡,𝐶𝑢(𝑡)),𝑡∈ℝ, (1.1)where 𝐴 is a sectorial linear operator on a Banach space 𝕏 and 𝜎(𝐴)∩𝑖ℝ=∅, 𝐵,𝐶 are bounded linear operators on 𝕏_𝛼, 𝑓∶ℝ×𝕏↦𝕏_𝛽, 𝑔∶ℝ×𝕏↦𝕏 are jointly continuous functions. This turns out to be a nontrivial problem due to the complexity and importance of pseudo almost automorphic functions (e.g., see [1–8] and the references therein).

Upon making some additional assumptions, it will be shown that (1.1) admits a unique 𝕏_𝛼-valued pseudo almost automorphic mild solution. Applications include the study of pseudo almost automorphic mild solutions to some nonlinear heat equation:𝑢_𝑡(𝑡,𝑥)=𝑢_𝑥𝑥(𝑡,𝑥)+𝑎𝑢(𝑡,𝑥)+𝑔(𝑡,𝑞(𝑥)𝑢(𝑡,𝑥)),𝑡∈ℝ,𝑥∈[0,𝜋],𝑢(𝑡,0)=𝑢(𝑡,𝜋)=0,𝑡∈ℝ, (1.2) where 𝑎∈ℝ is a constant, and 𝑔,𝑞 are continuous functions.

As a natural and important generalization of almost automorphy as well as pseudo almost periodicity, pseudo almost automorphy has recently been investigated (for more details, see [1–3]). Moreover, Liang et al. in [2] established a composition theorem about pseudo almost automorphic functions; Xiao et al. in [3] obtained sufficient conditions for the existence and uniqueness of pseudo almost automorphic mild solutions to semilinear differential equations in Banach spaces. In particular, they proved in [3] that the space (PAA(ℝ,𝕏),‖⋅‖_∞) of pseudo almost automorphic functions, endowed with sup norm, is a Banach space, thus pushed the door open to study pseudo almost automorphic mild solutions to various differential equations.

The method used here was firstly developed in [9], in which Diagana established an existence and uniqueness theorem of pseudo almost periodic mild solutions to (1.1) under some similar assumptions. Combining bounded invariance theorem of pseudo almost automorphic functions established in [6] with composition theorem of pseudo almost automorphic functions and completeness of the space (PAA(ℝ,𝕏),‖⋅‖_∞), this paper extends the results in [9], since such function as 𝑔(𝑡,𝑢)=𝑢sin(1/(2+cos𝑡+cos√2𝑡))+max_𝑘∈ℤ{𝑒^{−(𝑡±𝑘2)2}}cos𝑢,𝑡∈ℝ,𝑢∈𝕏 is not pseudo almost periodic, while they are pseudo almost automorphic.

It should be pointed out that one can also make use of (1.1) to study several types of evolution equations including partial functional differential equations, integro-differential equations, and reaction diffusion equations.

2. Preliminaries

In this section, we collect some preliminary facts from [9–11] that will be used later. Throughout this paper, ℕ, ℤ, ℝ, and ℂ stand for the sets of positive integer, integer, real and complex numbers; (𝕏,‖⋅‖), (𝕐,‖⋅‖_𝕐) stand for Banach spaces. If 𝐴 is a linear operator on 𝕏, then 𝜌(𝐴), 𝜎(𝐴), 𝐷(𝐴), ker(𝐴), 𝑅(𝐴) stand for the resolvent set, spectrum, domain, kernel, and range of 𝐴. The space 𝐵(𝕏,𝕐) denotes the Banach space of all bounded linear operators from 𝕏 into 𝕐 equipped with natural norm ‖𝐴‖_{𝐵(𝕏,𝕐)}=sup_{𝑥∈𝕏,𝑥≠0}(‖𝐴𝑥‖_𝕐/‖𝑥‖). If 𝕐=𝕏, it is simply denoted by 𝐵(𝕏) with ‖𝐴‖_𝐵(𝕏)=sup_{𝑥∈𝕏,𝑥≠0}(‖𝐴𝑥‖/‖𝑥‖).

2.1. Sectorial Linear Operators and Analytic Semigroups

Definition 2.1. A linear operator 𝐴∶𝐷(𝐴)⊂𝕏↦𝕏 (not necessarily densely defined) is said to be sectorial if the following hold. There exist constants 𝜔∈ℝ, 𝜃∈(𝜋/2,𝜋), and 𝑀>0 such that𝜌(𝐴)⊃𝑆_𝜃,𝜔𝑀∶={𝜆∈ℂ∶𝜆≠𝜔,|arg(𝜆−𝜔)|<𝜃},‖𝑅(𝜆,𝐴)‖≤|𝜆−𝜔|,𝜆∈𝑆_𝜃,𝜔, (2.1)where 𝑅(𝜆,𝐴)=(𝜆𝐼−𝐴)⁻¹ for each 𝜆∈𝜌(𝐴).Remark 2.2. If 𝐴 is sectorial, then it generates an analytic semigroup (𝑇(𝑡))_𝑡≥0, which maps (0,∞) into 𝐵(𝕏) such that there exist 𝑀₀,𝑀₁>0 with‖𝑇(𝑡)‖≤𝑀₀𝑒^𝜔𝑡,𝑡>0,‖𝑡(𝐴−𝜔𝐼)𝑇(𝑡)‖≤𝑀₁𝑒^𝜔𝑡,𝑡>0. (2.2)Definition 2.3. A semigroup (𝑇(𝑡))_𝑡≥0 is said to be hyperbolic, if there exist a projection 𝑃 and constants 𝑀,𝛿>0 such that each 𝑇(𝑡) commutes with 𝑃, ker(𝑃) is invariant with respect to 𝑇(𝑡), 𝑇(𝑡)∶𝑅(𝑄)→𝑅(𝑄) is invertible and‖𝑇(𝑡)𝑃𝑥‖≤𝑀𝑒^−𝛿𝑡‖𝑥‖for𝑡≥0,‖𝑇(𝑡)𝑄𝑥‖≤𝑀𝑒^𝛿𝑡‖𝑥‖for𝑡≤0, (2.3)where 𝑄∶=𝐼−𝑃 and 𝑇(𝑡)∶=(𝑇(−𝑡))⁻¹ for 𝑡≤0.Recall that if a semigroup (𝑇(𝑡))_𝑡≥0 is analytic, then (𝑇(𝑡))_𝑡≥0 is hyperbolic if and only if 𝜎(𝐴)∩𝑖ℝ=∅ (see, e.g., [11, Definition 1.14 and Proposition 1.15, page 305]).

2.2. Intermediate Banach Spaces

Definition 2.4. Let 𝛼∈(0,1). A Banach space (𝕏_𝛼,‖⋅‖_𝛼) is said to be an intermediate space between 𝐷(𝐴) and 𝕏, if 𝐷(𝐴)⊂𝕏_𝛼⊂𝕏 and there is a constant 𝐶>0 such that‖𝑥‖_𝛼≤𝐶‖𝑥‖^1−𝛼‖𝑥‖^𝛼_𝐴,𝑥∈𝐷(𝐴), (2.4)where ‖⋅‖_𝐴 is the graph norm of 𝐴.

Concrete examples of 𝕏_𝛼 include 𝐷(𝐴^𝛼) for 𝛼∈(0,1), the domains of the fractional powers of 𝐴, the real interpolation spaces 𝐷_𝐴(𝛼,∞), 𝛼∈(0,1), defined as follows:𝐷_𝐴(𝛼,∞)∶={𝑥∈𝕏∶[𝑥]_𝛼=sup_0<𝑡≤1‖𝑡^1−𝛼(𝐴−𝜔𝐼)𝑒^−𝜔𝑡𝑇(𝑡)𝑥‖<∞},‖𝑥‖_𝛼=‖𝑥‖+[𝑥]_𝛼, (2.5)and the abstract Hölder spaces 𝐷_𝐴(𝛼)∶=𝐷(𝐴)^{‖⋅‖𝛼}.Lemma 2.5. For the hyperbolic analytic semigroup (𝑇(𝑡))_𝑡≥0, there exist constants 𝐶(𝛼)>0, 𝛿>0,𝑀(𝛼)>0, and 𝛾>0 such that‖𝑇(𝑡)𝑄𝑥‖_𝛼≤𝐶(𝛼)𝑒^𝛿𝑡‖𝑥‖for𝑡≤0,‖𝑇(𝑡)𝑃𝑥‖_𝛼≤𝑀(𝛼)𝑡^−𝛼𝑒^−𝛾𝑡‖𝑥‖for𝑡>0. (2.6)Lemma 2.6 (see [9]). Let 0<𝛼<𝛽<1. For the hyperbolic analytic semigroup (𝑇(𝑡))_𝑡≥0, there exist constants 𝑐>0,𝛿>0, and 𝛾>0 such that‖𝐴𝑇(𝑡)𝑄𝑥‖_𝛼≤𝑐𝑒^𝛿𝑡‖𝑥‖_𝛽for𝑡≤0,‖𝐴𝑇(𝑡)𝑃𝑥‖_𝛼≤𝑐𝑡^{𝛽−𝛼−1}𝑒^−𝛾𝑡‖𝑥‖_𝛽for𝑡>0. (2.7)

3. Pseudo Almost Automorphic Functions

In this section, we recall some recent results on almost automorphic functions and pseudo almost automorphic functions. Let 𝐶(ℝ,𝕏) denote the collection of continuous functions from ℝ into 𝕏. Let 𝐵𝐶(ℝ,𝕏) denote the Banach space of all 𝕏-valued bounded continuous functions equipped with the sup norm ‖𝑢‖_∞∶=sup_𝑡∈ℝ‖𝑢(𝑡)‖ for each 𝑢∈𝐵𝐶(ℝ,𝕏). Similarly, 𝐶(ℝ×𝕏,𝕏) denotes the collection of continuous functions from ℝ×𝕏 into 𝕏, 𝐵𝐶(ℝ×𝕏,𝕏) denotes the collection of all bounded continuous functions 𝑓∶ℝ×𝕏↦𝕏.Definition 3.1 ( see[8]). A function 𝑓∈𝐶(ℝ,𝕏) is said to be almost automorphic if for every sequence of real numbers (𝜎_𝑛)_𝑛∈ℕ, there exists a subsequence (𝑠_𝑛)_𝑛∈ℕ such thatlim_𝑚→+∞lim_𝑛→+∞𝑓(𝑡+𝑠_𝑛−𝑠_𝑚)=𝑓(𝑡)foreach𝑡∈ℝ. (3.1)This limit means that𝑔(𝑡)=lim_𝑛→+∞𝑓(𝑡+𝑠_𝑛) (3.2)is well defined for each 𝑡∈ℝ, and𝑓(𝑡)=lim_𝑛→+∞𝑔(𝑡−𝑠_𝑛) (3.3)for each 𝑡∈ℝ. The collection of all such functions will be denoted by AA(ℝ,𝕏).Theorem 3.2 (see [7]). Assume 𝑓,𝑔∶ℝ↦𝕏 are almost automorphic, and 𝜆 is any scalar. Then the following holds true:(1)𝑓+𝑔,𝜆𝑓,𝑓_𝜏(𝑡)∶=𝑓(𝑡+𝜏) and 𝑓(𝑡)∶=𝑓(−𝑡) are almost automorphic;(2)the range 𝑅_𝑓 of 𝑓 is precompact, so 𝑓 is bounded;(3)if {𝑓_𝑛} is a sequence of almost automorphic functions and 𝑓_𝑛→𝑓 uniformly on ℝ, then 𝑓 is almost automorphic.Theorem 3.3 (see [7]). If one equips? AA(ℝ,𝕏) with the sup norm ‖𝑢‖_∞=sup_𝑡∈ℝ‖𝑢(𝑡)‖, then AA(ℝ,𝕏) turns out to be a Banach space.

One setsAA₀(ℝ,𝕏)∶={𝑓∈𝐵𝐶(ℝ,𝕏)∶lim_𝑟→∞12𝑟^𝑟_−𝑟‖𝑓(𝑠)‖𝑑𝑠=0}. (3.4)Definition 3.4 (see [2]). A function 𝑓∈𝐵𝐶(ℝ,𝕏) is said to be pseudo almost automorphic if it can be decomposed as 𝑓=𝑔+𝜙, where 𝑔∈AA(ℝ,𝕏) and 𝜙∈AA₀(ℝ,𝕏). The collection of such functions will be denoted by PAA(ℝ,𝕏).

Theorem 3.5 (see [3]). If one equips PAA(ℝ,𝕏) with the sup norm ‖𝑢‖_∞=sup_𝑡∈ℝ‖𝑢(𝑡)‖, then PAA(ℝ,𝕏) turns out to be a Banach space.Definition 3.6 (see [2]). A function 𝑓∈𝐶(ℝ×𝕏,𝕏) is said to be almost automorphic if 𝑓∈𝐶(ℝ×𝕏,𝕏) is almost automorphic in 𝑡∈ℝ uniformly for all 𝑥∈𝐾, where 𝐾 is any bounded subset of 𝕏. That is to say, for every sequence of real numbers (𝜎_𝑛)_𝑛∈ℕ there exists a subsequence (𝑠_𝑛)_𝑛∈ℕ such that𝑔(𝑡,𝑥)=lim_𝑛→+∞𝑓(𝑡+𝑠_𝑛,𝑥) (3.5)is well defined in 𝑡∈ℝ, for all 𝑥∈𝐾, and𝑓(𝑡,𝑥)=lim_𝑛→+∞𝑔(𝑡−𝑠_𝑛,𝑥) (3.6)for each 𝑡∈ℝ and 𝑥∈𝐾. Denote by AA(ℝ×𝕏,𝕏) the collection of all such functions.One also defines AA₀(ℝ×𝕏,𝕏) as the collection of functions 𝑓∈𝐵𝐶(ℝ×𝕏,𝕏) such thatlim_𝑟→∞12𝑟^𝑟_−𝑟‖𝑓(𝑡,𝑥)‖𝑑𝑡=0 (3.7)uniformly for any bounded subset of 𝕏.Definition 3.7 (see [2]). A function 𝑓∈𝐵𝐶(ℝ×𝕏,𝕏) is said to be pseudo almost automorphic if it can be decomposed as 𝑓=𝑔+𝜙, where 𝑔∈AA(ℝ×𝕏,𝕏) and 𝜙∈AA₀(ℝ×𝕏,𝕏). The collection of such functions will be denoted by PAA(ℝ×𝕏,𝕏).Theorem 3.8 (see [2]). Assume 𝑓=𝑔+𝜙∈PAA(ℝ×𝕏,𝕏) with 𝑔(𝑡,𝑥)∈AA(ℝ×𝕏,𝕏), 𝜙(𝑡,𝑥)∈AA₀(ℝ×𝕏,𝕏), satisfying the following conditions:(1)𝑔(𝑡,𝑥) is uniformly continuous in any bounded subset 𝐾⊂𝕏 uniformly for 𝑡∈ℝ;(2)𝑓(𝑡,𝑥) is uniformly continuous in any bounded subset 𝐾⊂𝕏 uniformly for 𝑡∈ℝ. If 𝑥(⋅)∈PAA(ℝ,𝕏), then 𝑓(⋅,𝑥(⋅))∈PAA(ℝ,𝕏).Remark 3.9. If 𝕏_𝛼 for 𝛼∈(0,1) is an intermediate space between 𝐷(𝐴) and 𝕏, 𝐵𝐶(ℝ,𝕏_𝛼), AA(ℝ,𝕏_𝛼), PAA(ℝ,𝕏_𝛼) are equipped with the 𝛼-sup norm: ‖𝑢‖_∞,𝛼=sup_𝑡∈ℝ‖𝑢(𝑡)‖_𝛼, then they all constitute Banach spaces in view of Definition 2.4, Theorems 3.3 and 3.5.Theorem 3.10 (see [6]). Let 𝑢∈PAA(ℝ,𝕐), 𝐵∈𝐵(𝕐,𝕏). If 𝑣(𝑡)∶=𝐵𝑢(𝑡) for each 𝑡∈ℝ, then 𝑣∈PAA(ℝ,𝕏).

4. Pseudo Almost Automorphic Mild Solutions

In this section, combining Theorem 3.10 with composition theorem of pseudo almost automorphic functions (Theorem 3.8) and completeness of the space (PAA(ℝ,𝕏),‖⋅‖_∞) (Theorem 3.5), we will establish the existence and uniqueness theorem of pseudo almost automorphic mild solutions to (1.1) under the following assumptions. (H1)For 0<𝛼<𝛽<1, 𝕏_𝛽↪𝕏_𝛼↪𝕏 are continuously embedded and there exist 𝑘₀>0,𝑘(𝛼)>0 such that ‖𝑢‖≤𝑘₀‖𝑢‖_𝛽, ‖𝑢‖_𝛼≤𝑘(𝛼)‖𝑢‖_𝛽 for each 𝑢∈𝕏_𝛽.(H2)𝐴 is a sectorial linear operator on a Banach space 𝕏 and 𝜎(𝐴)∩𝑖ℝ=∅.(H3)𝑓=ℳ+𝜑∈PAA(ℝ×𝕏,𝕏_𝛽) with ℳ∈AA(ℝ×𝕏,𝕏_𝛽), 𝜑∈AA₀(ℝ×𝕏,𝕏_𝛽) and 𝑔=𝒩+𝜓∈PAA(ℝ×𝕏,𝕏) with 𝒩∈AA(ℝ×𝕏,𝕏), 𝜓∈AA₀(ℝ×𝕏,𝕏).(H4) The functions 𝑓,𝑔 are uniformly Lipschitz with respect to the second argument in the sense that: there exists 𝐿>0 such that‖𝑓(𝑡,𝑥)−𝑓(𝑡,𝑦)‖_𝛽≤𝐿‖𝑥−𝑦‖,‖𝑔(𝑡,𝑥)−𝑔(𝑡,𝑦)‖≤𝐿‖𝑥−𝑦‖, (4.1) for all 𝑡∈ℝ and 𝑥,𝑦∈𝕏.(H5)ℳ,𝒩 are uniformly continuous in any bounded subset 𝐾⊂𝕏 uniformly for 𝑡∈ℝ.(H6) The operators 𝐵,𝐶∈𝐵(𝕏_𝛼,𝕏) and 𝜔=max(‖𝐵‖_{𝐵(𝕏𝛼,𝕏)},‖𝐶‖_{𝐵(𝕏𝛼,𝕏)}).

Definition 4.1 (see [9]). A function 𝑢∈𝐵𝐶(ℝ,𝕏_𝛼) is said to be a mild solution to (1.1) if 𝑠→𝐴𝑇(𝑡−𝑠)𝑃𝑓(𝑠,𝐵𝑢(𝑠)) is integrable on (−∞,𝑡), 𝑠→𝐴𝑇(𝑡−𝑠)𝑄𝑓(𝑠,𝐵𝑢(𝑠)) is integrable on (𝑡,+∞) and𝑢(𝑡)=−𝑓(𝑡,𝐵𝑢(𝑡))−^𝑡_−∞𝐴𝑇(𝑡−𝑠)𝑃𝑓(𝑠,𝐵𝑢(𝑠))𝑑𝑠+_𝑡^+∞+𝐴𝑇(𝑡−𝑠)𝑄𝑓(𝑠,𝐵𝑢(𝑠))𝑑𝑠_𝑡−∞𝑇(𝑡−𝑠)𝑃𝑔(𝑠,𝐶𝑢(𝑠))𝑑𝑠−_𝑡^+∞𝑇(𝑡−𝑠)𝑄𝑔(𝑠,𝐶𝑢(𝑠))𝑑𝑠 (4.2) for each 𝑡∈ℝ.Lemma 4.2. Let assumptions (H1)–(H6) hold. Consider the nonlinear operator Λ₁ defined by(Λ₁𝑢)(𝑡)=^𝑡_−∞𝐴𝑇(𝑡−𝑠)𝑃𝑓(𝑠,𝐵𝑢(𝑠))𝑑𝑠−_𝑡^+∞𝐴𝑇(𝑡−𝑠)𝑄𝑓(𝑠,𝐵𝑢(𝑠))𝑑𝑠. (4.3)Then, Λ₁ maps PAA(ℝ,𝕏_𝛼) into itself.Proof. From (H1), (H3), (H4), (H5), we deduce that 𝑓=ℳ+𝜑∈PAA(ℝ×𝕏,𝕏) with ℳ∈AA(ℝ×𝕏,𝕏), 𝜑∈AA₀(ℝ×𝕏,𝕏), satisfying the following conditions. (1)ℳ(𝑡,𝑥) is uniformly continuous in any bounded subset 𝐾⊂𝕏 uniformly for 𝑡∈ℝ.(2)𝑓(𝑡,𝑥) is uniformly Lipschitz with respect to the second argument in the sense that there exists 𝑘₀𝐿>0 such that‖𝑓(𝑡,𝑥)−𝑓(𝑡,𝑦)‖≤𝑘₀𝐿‖𝑥−𝑦‖, (4.4)for all 𝑡∈ℝ and 𝑥,𝑦∈𝕏.
Let 𝑢∈PAA(ℝ,𝕏_𝛼). Since 𝐵∈𝐵(𝕏_𝛼,𝕏), it follows from Theorem 3.10 that 𝐵𝑢(⋅)∈PAA(ℝ,𝕏). Setting ℎ(⋅)=𝑓(⋅,𝐵𝑢(⋅)) and applying Theorem 3.8, we get that ℎ∈PAA(ℝ,𝕏). Using (H1) and Theorem 3.10, we obtain that ℎ∈PAA(ℝ,𝕏_𝛽). Now, write ℎ=𝜉+𝜂 where 𝜉∈AA(ℝ,𝕏_𝛽) and 𝜂∈AA₀(ℝ,𝕏_𝛽), then(Λ₁𝑢)(𝑡)=^𝑡_−∞𝐴𝑇(𝑡−𝑠)𝑃𝜉(𝑠)𝑑𝑠+^𝑡_−∞+𝐴𝑇(𝑡−𝑠)𝑃𝜂(𝑠)𝑑𝑠_𝑡+∞𝐴𝑇(𝑡−𝑠)𝑄𝜉(𝑠)𝑑𝑠+^𝑡_+∞𝐴𝑇(𝑡−𝑠)𝑄𝜂(𝑠)𝑑𝑠. (4.5)SetΞ(𝑡)=^𝑡_−∞𝐴𝑇(𝑡−𝑠)𝑃𝜉(𝑠)𝑑𝑠+^𝑡_+∞𝐴𝑇(𝑡−𝑠)𝑄𝜉(𝑠)𝑑𝑠,Π(𝑡)=^𝑡_−∞𝐴𝑇(𝑡−𝑠)𝑃𝜂(𝑠)𝑑𝑠+^𝑡_+∞𝐴𝑇(𝑡−𝑠)𝑄𝜂(𝑠)𝑑𝑠. (4.6) Next, we show that Ξ∈AA(ℝ,𝕏_𝛼) and Π∈AA₀(ℝ,𝕏_𝛼).
Now, to prove that Ξ∈AA(ℝ,𝕏_𝛼). Let us take a sequence (𝜎_𝑛)_𝑛∈ℕ and show that there exists a subsequence (𝑠_𝑛)_𝑛∈ℕ such thatlim_𝑚→+∞lim_𝑛→+∞‖Ξ(𝑡+𝑠_𝑛−𝑠_𝑚)−Ξ(𝑡)‖_𝛼=0foreach𝑡∈ℝ. (4.7)Since 𝜉∈AA(ℝ,𝕏_𝛽), there exists a subsequence (𝑠_𝑛)_𝑛∈ℕ such thatlim_𝑚→+∞lim_𝑛→+∞‖𝜉(𝑡+𝑠_𝑛−𝑠_𝑚)−𝜉(𝑡)‖_𝛽=0foreach𝑡∈ℝ. (4.8)On the other hand, we haveΞ(𝑡+𝑠_𝑛−𝑠_𝑚)−Ξ(𝑡)=^{(𝑡+𝑠𝑛−𝑠𝑚)}_−∞𝐴𝑇(𝑡+𝑠_𝑛−𝑠_𝑚−𝑠)𝑃𝜉(𝑠)𝑑𝑠−^𝑡_−∞𝐴𝑇(𝑡−𝑠)𝑃𝜉(𝑠)𝑑𝑠+^{𝑡+𝑠𝑛−𝑠𝑚}_+∞𝐴𝑇(𝑡+𝑠_𝑛−𝑠_𝑚−𝑠)𝑄𝜉(𝑠)𝑑𝑠−^𝑡_+∞=𝐴𝑇(𝑡−𝑠)𝑄𝜉(𝑠)𝑑𝑠_0+∞𝐴𝑇(𝑠)𝑃𝜉(𝑡+𝑠_𝑛−𝑠_𝑚−𝑠)𝑑𝑠−₀^+∞𝐴𝑇(𝑠)𝑃𝜉(𝑡−𝑠)𝑑𝑠−(⁰_−∞𝐴𝑇(𝑠)𝑄𝜉(𝑡+𝑠_𝑛−𝑠_𝑚−𝑠)𝑑𝑠−⁰_−∞=𝐴𝑇(𝑠)𝑄𝜉(𝑡−𝑠)𝑑𝑠)_0+∞𝐴𝑇(𝑠)𝑃[𝜉(𝑡+𝑠_𝑛−𝑠_𝑚−−𝑠)−𝜉(𝑡−𝑠)]𝑑𝑠_0−∞𝐴𝑇(𝑠)𝑄[𝜉(𝑡+𝑠_𝑛−𝑠_𝑚−𝑠)−𝜉(𝑡−𝑠)]𝑑𝑠. (4.9)Passing to the norm ‖⋅‖_𝛼,0<𝛼<1, remembering the triangle inequalities, (2.7), then we obtain that‖Ξ(𝑡+𝑠_𝑛−𝑠_𝑚)−Ξ(𝑡)‖_𝛼≤₀^+∞‖𝐴𝑇(𝑠)𝑃[𝜉(𝑡+𝑠_𝑛−𝑠_𝑚−𝑠)−𝜉(𝑡−𝑠)]‖_𝛼+𝑑𝑠_0−∞‖𝐴𝑇(𝑠)𝑄[𝜉(𝑡+𝑠_𝑛−𝑠_𝑚−𝑠)−𝜉(𝑡−𝑠)]‖_𝛼𝑑𝑠≤𝑐₀^+∞𝑠^{𝛽−𝛼−1}𝑒^−𝛾𝑠‖𝜉(𝑡+𝑠_𝑛−𝑠_𝑚−𝑠)−𝜉(𝑡−𝑠)‖_𝛽𝑑𝑠+𝑐⁰_−∞𝑒^𝛿𝑠‖𝜉(𝑡+𝑠_𝑛−𝑠_𝑚−𝑠)−𝜉(𝑡−𝑠)‖_𝛽𝑑𝑠. (4.10)Thus, (4.8) and Lebesgue dominated convergence theorem lead to (4.7), therefore, to Ξ∈AA(ℝ,𝕏_𝛼). To finish the proof, we will prove that Π∈AA₀(ℝ,𝕏_𝛼). It is obvious that Π∈𝐵𝐶(ℝ,𝕏_𝛼), the left task is to show thatlim_𝑟→∞12𝑟^𝑟_−𝑟‖Π(𝑡)‖_𝛼𝑑𝑡=0. (4.11)Using (2.7), we have0≤lim_𝑟→∞12𝑟^𝑟_−𝑟‖Π(𝑡)‖_𝛼𝑑𝑡≤𝐼+𝐽, (4.12)where𝐼∶=lim_𝑟→∞𝑐2𝑟^𝑟_−𝑟𝑑𝑡^𝑡_−∞(𝑡−𝑠)^{𝛽−𝛼−1}𝑒^{−𝛾(𝑡−𝑠)}‖𝜂(𝑠)‖_𝛽𝑑𝑠=lim_𝑟→∞𝑐2𝑟^𝑟_−𝑟𝑑𝑡₀^+∞𝜎^{𝛽−𝛼−1}𝑒^−𝛾𝜎‖𝜂(𝑡−𝜎)‖_𝛽𝑑𝜎=lim_𝑟→∞𝑐₀^+∞𝜎^{𝛽−𝛼−1}𝑒^−𝛾𝜎[12𝑟^𝑟_−𝑟‖𝜂(𝑡−𝜎)‖_𝛽𝑑𝑡]𝑑𝜎,𝐽∶=lim_𝑟→∞𝑐2𝑟^𝑟_−𝑟𝑑𝑡_𝑡^+∞𝑒^{𝛿(𝑡−𝑠)}‖𝜂(𝑠)‖_𝛽𝑑𝑠=lim_𝑟→∞𝑐2𝑟^𝑟_−𝑟𝑑𝑡⁰_−∞𝑒^𝛿𝜎‖𝜂(𝑡−𝜎)‖_𝛽𝑑𝜎=lim_𝑟→∞𝑐⁰_−∞𝑒^𝛿𝜎[12𝑟^𝑟_−𝑟‖𝜂(𝑡−𝜎)‖_𝛽𝑑𝑡]𝑑𝜎. (4.13)Then, by the Lebesgue dominated convergence theorem and the fact that 𝜂∈AA₀(ℝ,𝕏_𝛽), one has 𝐼=𝐽=0. Hence, Π∈AA₀(ℝ,𝕏_𝛼) and we end the proof. Lemma 4.3. Let assumptions (H2)–(H6) hold. Define the nonlinear operator Λ₂ by(Λ₂𝑢)(𝑡)=^𝑡_−∞𝑇(𝑡−𝑠)𝑃𝑔(𝑠,𝐶𝑢(𝑠))𝑑𝑠−_𝑡^+∞𝑇(𝑡−𝑠)𝑄𝑔(𝑠,𝐶𝑢(𝑠))𝑑𝑠. (4.14)Then, Λ₂ maps PAA(ℝ,𝕏_𝛼) into itself.Proof. The proof is similar to that of Lemma 4.2, so we omit it.Theorem 4.4. Under the assumptions (H1)–(H6), partial hyperbolic evolution equation (1.1) admits a unique pseudo almost automorphic mild solution ifΘ=𝐿𝑐𝜔[𝑘(𝛼)+𝛿+𝐶(𝛼)𝛿+𝑐Γ(𝛽−𝛼)𝛾^𝛽−𝛼+𝑀(𝛼)Γ(1−𝛼)𝛾^1−𝛼]<1. (4.15)Proof. Firstly, define the nonlinear operator Λ on 𝐵𝐶(ℝ,𝕏_𝛼) by(Λ𝑢)(𝑡)=−𝑓(𝑡,𝐵𝑢(𝑡))−^𝑡_−∞𝐴𝑇(𝑡−𝑠)𝑃𝑓(𝑠,𝐵𝑢(𝑠))𝑑𝑠+_𝑡^+∞+𝐴𝑇(𝑡−𝑠)𝑄𝑓(𝑠,𝐵𝑢(𝑠))𝑑𝑠_𝑡−∞𝑇(𝑡−𝑠)𝑃𝑔(𝑠,𝐶𝑢(𝑠))𝑑𝑠−_𝑡^+∞𝑇(𝑡−𝑠)𝑄𝑔(𝑠,𝐶𝑢(𝑠))𝑑𝑠=−𝑓(𝑡,𝐵𝑢(𝑡))−(Λ₁𝑢)(𝑡)+(Λ₂𝑢)(𝑡). (4.16)Let 𝑢∈PAA(ℝ,𝕏_𝛼), then 𝑓(⋅,𝐵𝑢(⋅))∈PAA(ℝ,𝕏_𝛽)⊂PAA(ℝ,𝕏_𝛼), as proved in Lemma 4.2. Together with Lemmas 4.2 and 4.3, it follows that the operator Λ maps PAA(ℝ,𝕏_𝛼) into itself. Secondly, we will show that Λ admits a unique fixed point in PAA(ℝ,𝕏_𝛼).
Let 𝑣,𝑤∈PAA(ℝ,𝕏_𝛼), then the triangle inequality reads‖(Λ𝑣)(𝑡)−(Λ𝑤)(𝑡)‖_𝛼≤‖𝑓(𝑡,𝐵𝑣(𝑡))−𝑓(𝑡,𝐵𝑤(𝑡))‖_𝛼+‖(Λ₁𝑣)(𝑡)−(Λ₁𝑤)(𝑡)‖_𝛼+‖(Λ₂𝑣)(𝑡)−(Λ₂𝑤)(𝑡)‖_𝛼. (4.17)By (H1), (H5), and (H6), we obtain‖𝑓(𝑡,𝐵𝑣(𝑡))−𝑓(𝑡,𝐵𝑤(𝑡))‖_𝛼≤𝑘(𝛼)𝐿‖𝐵𝑣(𝑡)−𝐵𝑤(𝑡)‖≤𝑘(𝛼)𝐿𝜔‖𝑣−𝑤‖_∞,𝛼. (4.18)By (H2), Lemma 2.6, (H5), and (H6), we obtain‖(Λ₁𝑣)(𝑡)−(Λ₁𝑤)(𝑡)‖_𝛼≤^𝑡_−∞‖𝐴𝑇(𝑡−𝑠)𝑃[𝑓(𝑠,𝐵𝑣(𝑠))−𝑓(𝑠,𝐵𝑤(𝑠))]‖_𝛼+𝑑𝑠_𝑡+∞‖𝐴𝑇(𝑡−𝑠)𝑄[𝑓(𝑠,𝐵𝑣(𝑠))−𝑓(𝑠,𝐵𝑤(𝑠))]‖_𝛼𝑑𝑠≤𝑐𝐿𝜔‖𝑣−𝑤‖_∞,𝛼^𝑡_−∞(𝑡−𝑠)^{𝛽−𝛼−1}𝑒^{−𝛾(𝑡−𝑠)}𝑑𝑠+𝑐𝐿𝜔‖𝑣−𝑤‖_∞,𝛼𝑡^+∞𝑒^{𝛿(𝑡−𝑠)}𝑑𝑠≤𝑐𝐿𝜔‖𝑣−𝑤‖_∞,𝛼[Γ(𝛽−𝛼)𝛾^𝛽−𝛼+1𝛿]. (4.19)Similarly, by (H2), Lemma 2.5, (H5), and (H6), we obtain‖(Λ₂𝑣)(𝑡)−(Λ₂𝑤)(𝑡)‖_𝛼≤^𝑡_−∞‖𝑇(𝑡−𝑠)𝑃[𝑔(𝑠,𝐶𝑣(𝑠))−𝑔(𝑠,𝐶𝑤(𝑠))]‖_𝛼+𝑑𝑠_𝑡+∞‖𝑇(𝑡−𝑠)𝑄[𝑔(𝑠,𝐶𝑣(𝑠))−𝑔(𝑠,𝐶𝑤(𝑠))]‖_𝛼𝑑𝑠≤𝑀(𝛼)𝐿𝜔‖𝑣−𝑤‖_∞,𝛼^𝑡_−∞(𝑡−𝑠)^−𝛼𝑒^{−𝛾(𝑡−𝑠)}𝑑𝑠+𝐶(𝛼)𝐿𝜔‖𝑣−𝑤‖_∞,𝛼𝑡^+∞𝑒^{𝛿(𝑡−𝑠)}𝑑𝑠≤𝐿𝜔‖𝑣−𝑤‖_∞,𝛼[𝑀(𝛼)Γ(1−𝛼)𝛾^1−𝛼+𝐶(𝛼)𝛿]. (4.20)Combining the above inequality together, we obtain ‖Λ𝑣−Λ𝑤‖_∞,𝛼≤Θ‖𝑣−𝑤‖_∞,𝛼, whereΘ=𝐿𝑐𝜔[𝑘(𝛼)+𝛿+𝐶(𝛼)𝛿+𝑐Γ(𝛽−𝛼)𝛾^𝛽−𝛼+𝑀(𝛼)Γ(1−𝛼)𝛾^1−𝛼]. (4.21)Clearly, if Θ<1, then the operator Λ becomes a strict contraction on PAA(ℝ,𝕏_𝛼). Remembering that PAA(ℝ,𝕏_𝛼) equipped with the 𝛼-sup norm: ‖𝑢‖_∞,𝛼=sup_𝑡∈ℝ‖𝑢(𝑡)‖_𝛼 is a Banach space by Remark 3.9, the classical Banach fixed-point theorem leads to the desired conclusion.

5. Application

Example 5.1. Take 𝕏∶=𝐶[0,𝜋] equipped with the sup norm. Define the operator 𝐴 by𝐴𝜑(𝜉)∶=𝜑^′′(𝜉)+𝑎𝜑(𝜉),∀𝜉∈(0,𝜋),𝜑∈𝐷(𝐴), (5.1)where 𝐷(𝐴)∶={𝜑∈𝐶²[0,𝜋],𝜑(0)=𝜑(𝜋)=0}⊂𝐶[0,𝜋] and 𝑎∈ℝ is a constant.
Clearly, 𝐴 is sectorial, and hence is the generator of an analytic semigroup. If 𝑎≠𝑛², then 𝐴 generates a hyperbolic analytic semigroup (𝑇(𝑡))_𝑡≥0 on 𝕏.
Let𝕏_𝛼=𝐷_𝐴𝐶(𝛼,∞)=^2𝛼1[0,𝜋],if0<𝛼<2,𝐶_𝜈^2𝛼1[0,𝜋],if2<𝛼<1, (5.2)where 𝐶_𝜈^2𝛼[0,𝜋]={𝜑∈𝐶^2𝛼[0,𝜋],𝜑(0)=𝜑(𝜋)=0} (see [10] for more details).
Define the operator 𝐶 by𝐶𝜑(𝜉)∶=𝑞(𝜉)𝜑(𝜉),∀𝜉∈(0,𝜋),𝜑∈𝐷(𝐶), (5.3)where 𝐷(𝐶)∶={𝜑∈𝐶[0,𝜋],𝜑(0)=𝜑(𝜋)=0}⊂𝐶[0,𝜋], then 𝕏_𝛼⊂𝐷(𝐶) and ‖𝐶‖_{𝐵(𝕏𝛼,𝕏)}=‖𝑞‖_∞=sup_{𝜉∈[0,𝜋]}𝑞(𝜉).
Let1𝑔(𝑡,𝑢)=𝑢sin2+cos𝑡+cos√2𝑡+max_𝑘∈ℤ{𝑒^{−(𝑡±𝑘2)2}}cos𝑢,𝑡∈ℝ,𝑢∈𝕏, (5.4)then 𝑔∈PAA(ℝ×𝕏,𝕏), 𝑔 is Lipschitz with respect to the second argument with 𝐿=2,𝒩(𝑡,𝑢)=𝑢sin(1/(2+cos𝑡+cos√2𝑡)) is uniformly continuous in any bounded subset 𝐾⊂𝕏 uniformly for 𝑡∈ℝ.

Now the above discussion can be formulated as follows.Proposition 5.2. Let 𝑔(𝑡,𝑢)=𝑢sin(1/(2+cos𝑡+cos√2𝑡))+max_𝑘∈ℤ{𝑒^{−(𝑡±𝑘2)2}}cos𝑢,𝑡∈ℝ,𝑢∈𝕏. Suppose that the constant 𝑎≠𝑛² for 𝑛∈ℕ. Then, nonlinear heat equation (1.2) admits a unique 𝕏_𝛼-valued pseudo almost automorphic mild solution if2‖𝑞‖_∞[𝐶(𝛼)𝛿+𝑀(𝛼)Γ(1−𝛼)𝛾^1−𝛼]<1. (5.5)

Appendix

Proof. Write 𝑢=𝜙+𝜓 where 𝜙∈AA(ℝ,𝕐) and 𝜓∈AA₀(ℝ,𝕐), then𝑣(𝑡)=𝐵𝜙(𝑡)+𝐵𝜓(𝑡). (A.1)SetΦ(𝑡)=𝐵𝜙(𝑡),Ψ(𝑡)=𝐵𝜓(𝑡). (A.2)We are going to prove that Φ∈AA(ℝ,𝕏) and Ψ∈AA₀(ℝ,𝕏).
Let us take a sequence (𝜎_𝑛)_𝑛∈ℕ and show that there exists a subsequence (𝑠_𝑛)_𝑛∈ℕ such thatlim_𝑚→+∞lim_𝑛→+∞‖Φ(𝑡+𝑠_𝑛−𝑠_𝑚)−Φ(𝑡)‖=0foreach𝑡∈ℝ. (A.3)Since 𝜙∈AA(ℝ,𝕐), there exists a subsequence (𝑠_𝑛)_𝑛∈ℕ such thatlim_𝑚→+∞lim_𝑛→+∞‖𝜙(𝑡+𝑠_𝑛−𝑠_𝑚)−𝜙(𝑡)‖_𝕐=0foreach𝑡∈ℝ. (A.4)On the other hand, we have0≤‖Φ(𝑡+𝑠_𝑛−𝑠_𝑚)−Φ(𝑡)‖=‖𝐵𝜙(𝑡+𝑠_𝑛−𝑠_𝑚)−𝐵𝜙(𝑡)‖≤‖𝐵‖_{𝐵(𝕐,𝕏)}‖𝜙(𝑡+𝑠_𝑛−𝑠_𝑚)−𝜙(𝑡)‖_𝕐. (A.5)Thus, (A.4) leads to (A.3), therefore, to Φ∈AA(ℝ,𝕏).
To complete the proof, we will prove that Ψ∈AA₀(ℝ,𝕏). In fact, it is easy to check that Ψ∈𝐵𝐶(ℝ,𝕏) andlim_𝑟→∞12𝑟^𝑟_−𝑟‖Ψ(𝑡)‖𝑑𝑡≤‖𝐵‖_{𝐵(𝕐,𝕏)}lim_𝑟→∞12𝑟^𝑟_−𝑟‖𝜓(𝑡)‖_𝕐𝑑𝑡=0. (A.6)Hence, Ψ∈AA₀(ℝ,𝕏) and we end the proof.

Acknowledgments

The authors would like to thank referees for their helpful suggestions. This work is supported by the National Science Foundation of China (60771026), the Special Scientific Research Foundation for the Subjects of Doctors in University (20060110005), and the Program for New Century Excellent Talents in University (NCET050271).