Existence of Pseudo Almost Periodic Solutions to Some Classes of Partial Hyperbolic Evolution Equations

The paper examines the existence of pseudo almost periodic solutions to some classes of partial hyperbolic evolution equations. Namely, sufficient conditions for the existence and uniqueness of pseudo almost periodic solutions to those classes of hyperbolic evolution equations are given. Applications include the existence of pseudo almost periodic solutions to the transport and heat equations with delay.


Introduction
Let (X, · ) be a Banach space and let A : D(A) ⊂ X → X be a sectorial linear operator (Definition 2.1). For α ∈ (0, 1), the space X α denotes an abstract intermediate Banach space between D(A) and X. Examples of those X α include, among others, the fractional spaces D((−A) α ) for α ∈ (0, 1), the reel interpolation spaces D A (α, ∞) due to J. L. Lions and J. Peetre, and the Hölder spaces D A (α), which coincide with the continuous interpolation spaces that had been introduced in the literature by G. Da Prato and P. Grisvard.
In [7,11,12,22], some sufficient conditions for the existence and uniqueness of pseudo almost periodic solutions to the abstract (semilinear) differential equations, u ′ (t) + Au(t) = f (t, u(t)), t ∈ R, and (1.1) u ′ (t) + Au(t) = f (t, Bu(t)), t ∈ R, (1.2) where −A is a Hille-Yosida linear operator (respectively, the infinitesimal generator of an analytic semigroup, and the infinitesimal generator of a C 0 -semigroup), B is a densely defined closed linear operator on X, and f : R × X → X is a jointly continuous function, were given. Similarly, in [13], some sufficient conditions for the existence and uniqueness of pseudo almost periodic solutions to the class of partial evolution equations d dt [u(t) + f (t, Bu(t))] = Au(t) + g(t, Cu(t)), t ∈ R (1. 3) where A is the infinitesimal generator of an exponentially stable semigroup acting on X, B, C are arbitrary densely defined closed linear operators on X, and f, g are some jointly continuous functions, were given. Now note the assumptions made in [13] require much more regularity for the operator A, that is, being the infinitesimal generator of an analytic semigroup. In this paper we address such an issue by studying pseudo almost periodic solutions to (1.3) in the case when A is a sectorial operator whose corresponding analytic semigroup (T (t)) t≥0 is hyperbolic, equivalently, Note that (1.3) in the case when A is sectorial corresponds to several interesting situations encountered in the literature. Applications include, among others, the existence and uniqueness of pseudo almost periodic solutions to the hyperbolic transport and heat equations with delay.
As in [5,14] in this paper we consider a general intermediate space X α between D(A) and X. In contrast with the fractional power spaces considered in some recent papers of the author et al. [11,12], the interpolation and Hölder spaces, for instance, depend only on D(A) and X and can be explicitly expressed in many concrete cases. The literature related to those intermediate spaces is very extensive, in particular, we refer the reader to the excellent book by A. Lunardi [23], which contains a comprehensive presentation on this topic and related issues.
The concept of pseudo almost periodicity, which is the central question in this paper was introduced in the literature in the early nineties by C. Zhang [29,30,31] as a natural generalization of the well-known Bohr almost periodicity. Thus this new concept is welcome to implement another existing generalization of almost periodicity, that is, the concept of asymptotically almost periodicity due to Fréchet [6,16].
The existence of almost periodic, asymptotically almost periodic, and pseudo almost periodic solutions is one of the most attractive topics in qualitative theory of differential equations due to their significance and applications in physics, mathematical biology, control theory, physics and others.
Some contributions on almost periodic, asymptotically almost periodic, and pseudo almost periodic solutions to abstract differential and partial differential equations have recently been made in [1,2,3,7,9,11,12,13,22]. However, the existence and uniqueness of pseudo almost periodic solutions to (1.3) in the case when A is sectorial is an important topic with some interesting applications, which is still an untreated question and this is the main motivation of the present paper. Among other things, we will make extensive use of the method of analytic semigroups associated with sectorial operators and the Banach's fixed-point principle to derive sufficient conditions for the existence and uniqueness of a pseudo almost periodic (mild) solution to (1.3).

Preliminaries
This section is devoted to some preliminary facts needed in the sequel. Throughout the rest of this paper, (X, · ) stands for a Banach space, A is a sectorial linear operator (Definition 2.1), which is not necessarily densely defined, and B, C are (unbounded) linear operators such that A + B + C is not trivial as each solution to  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 that The class of sectorial operators is very rich and contains most of classical operators encountered in the literature. Two examples of sectorial operators are given as follows: It can be checked that the operator A is sectorial on L p (R).
Define the operator A as follows: It can be checked that the operator A is sectorial on L p (Ω).
It is well-known that [23] if A is sectorial, then it generates an analytic semigroup (T (t)) t≥0 , which maps (0, ∞) into B(X) and such that there exist M 0 , M 1 > 0 with Throughout the rest of the paper, we suppose that the semigroup (T (t)) t≥0 is hyperbolic, that is, there exist a projection P and constants M, δ > 0 such that T (t) commutes with P , N (P ) is invariant with respect to T (t), T (t) : R(Q) → R(Q) is invertible, and the following hold where Q := I − P and, for t ≤ 0, T (t) := (T (−t)) −1 .
Concrete examples of X α include D((−A α )) for α ∈ (0, 1), the domains of the the fractional powers of A, the real interpolation spaces D A (α, ∞), α ∈ (0, 1), defined as follows the abstract Hölder spaces D A (α) := D(A) . α as well as the complex interpolation Lunardi [23] for details. For a hyperbolic analytic semigroup (T (t)) t≥0 , one can easily check that similar estimations as both (2.5) and (2.6) still hold with norms · α . In fact, as the part Hence, from (2.7) there exists a constant c(α) > 0 such that In addition to the above, the following holds and hence from (2.5), one obtains where M ′ depends on α. For t ∈ (0, 1], by (2.4) and (2.7) Hence, there exist constants M (α) > 0 and γ > 0 such that ) denote the collection of all Xvalued bounded continuous functions (respectively, the class of jointly bounded continuous functions F : R × Y → X). The space BC(R, X) equipped with its natural norm, that is, the sup norm defined by is a Banach space. Furthermore, C(R, Y) (respectively, C(R × Y, X)) denotes the class of continuous functions from R into Y (respectively, the class of jointly continuous functions F : R × Y → X).
Definition 2.5. A function f ∈ C(R, X) is called (Bohr) almost periodic if for each ε > 0 there exists l(ε) > 0 such that every interval of length l(ε) contains a number τ with the property that The number τ above is called an ε-translation number of f , and the collection of all such functions will be denoted AP (X).
Definition 2.6. A function F ∈ C(R × Y, X) is called (Bohr) almost periodic in t ∈ R uniformly in y ∈ Y if for each ε > 0 and any compact K ⊂ Y there exists l(ε) such that every interval of length l(ε) contains a number τ with the property that The collection of those functions is denoted by AP (R × Y).  Definition 2.7. A function f ∈ BC(R, X) is called pseudo almost periodic if it can be expressed as f = g + φ, where g ∈ AP (X) and φ ∈ AP 0 (X). The collection of such functions will be denoted by P AP (X).

Set
Remark 2.8. The functions g and φ in Definition 2.7 are respectively called the almost periodic and the ergodic perturbation components of f . Moreover, the decomposition given in Definition 2.7 is unique. Similarly, The collection of such functions will be denoted by P AP (R × Y).

Main results
To study the existence and uniqueness of pseudo almost periodic solutions to (1.3) we need to introduce the notion of mild solution to it.
Definition 3.1. Let α ∈ (0, 1). A bounded continuous function u : R → X α is said to be a mild solution to (1.3) provided that the function s → AT (t−s)P f (s, Bu(s)) is integrable on (−∞, t), s → AT (t − s)Qf (s, Bu(s)) is integrable on (t, ∞) for each t ∈ R, and Throughout the rest of the paper we denote by Γ 1 , Γ 2 , Γ 3 , and Γ 4 , the nonlinear integral operators defined by To study (1.3) we require the following assumptions: (H1) The operator A is sectorial and generates a hyperbolic (analytic) semigroup (T (t)) t≥0 . (H2) Let 0 < α < 1. Then X α = D((−A α )), or X α = D A (α, p), 1 ≤ p ≤ +∞, or X α = D A (α), or X α = [X, D(A)] α . Let B, C : X α −→ X be bounded linear operators. (H3) Let 1 > β > α, and f : R × X −→ X β be a pseudo almost periodic function in t ∈ R uniformly in u ∈ X, g : R × X → X be pseudo almost periodic in t ∈ R uniformly in u ∈ X. (H4) The functions f, g are uniformly Lipschitz with respect to the second argument in the following sense: there exists K > 0 such that for all u, v ∈ X and t ∈ R. In order to show that Γ 1 and Γ 2 are well defined, we need the following estimates.
Throughout the rest of the paper, the constant k(α) denotes the bound of the embedding X β ֒→ X α , that is, and ̟ = max( B B(Xα,X) , C B(Xα,X) ).
Proof. Consider the nonlinear operator M on P AP (X α ) given by AT (t − s)P f (s, Bu(s))ds for each t ∈ R.
As we have previously seen, for every u ∈ P AP (X α ), f (·, Bu(·)) ∈ P AP (X β ) ⊂ P AP (X α ). In view of Lemma 3.3 and Lemma 3.4, it follows that M maps P AP (X α ) into itself. To complete the proof one has to show that M has a unique fixed-point.