© T. Diagana and R. P. Agarwal. 2009
Received: 12 March 2009
Accepted: 3 July 2009
Published: 17 August 2009
where is a sectorial linear operator on a Banach space whose corresponding analytic semigroup is hyperbolic; that is, the operator are arbitrary linear (possibly unbounded) operators on , and are -pseudo almost automorphic for and jointly continuous functions.
The concept of pseudo almost automorphy, which is the central tool here, was recently introduced in literature by Liang et al. [1, 2]. The pseudo almost automorphy is a generalization of both the classical almost automorphy due to Bochner  and that of pseudo almost periodicity due to Zhang [4–6]. It has recently generated several developments and extensions. For the most recent developments, we refer the reader to [1, 2, 7–9]. More recently, in Diagana , the concept of -pseudo almost automorphy (or Stepanov-like pseudo almost automorphy) was introduced. It should be mentioned that the -pseudo almost automorphy is a natural generalization of the notion of pseudo almost automorphy.
In this paper, we will make extensive use of the concept of -pseudo almost automorphy combined with the techniques of hyperbolic semigroups to study the existence of pseudo almost automorphic solutions to the class of partial hyperbolic differential equations appearing in (1.3) and then to the -dimensional heat equation (1.1).
In this paper, as in the recent papers [10–12], we consider a general intermediate space between and . In contrast with the fractional power spaces considered in some recent papers by Diagana , the interpolation and Hölder spaces, for instance, depend only on and and can be explicitly expressed in many concrete cases. Literature related to those intermediate spaces is very extensive; in particular, we refer the reader to the excellent book by Lunardi , which contains a comprehensive presentation on this topic and related issues.
Existence results related to pseudo almost periodic and almost automorphic solutions to the partial hyperbolic differential equations of the form (1.3) have recently been established in [12, 15–18], respectively. Though to the best of our knowledge, the existence of pseudo almost automorphic solutions to the heat equation (1.1) in the case when the coefficients are -pseudo almost automorphic is an untreated original problem and constitutes the main motivation of the present paper.
Let be two Banach spaces. Let (resp., ) denote the collection of all -valued bounded continuous functions (resp., the class of jointly bounded continuous functions ). The space equipped with the sup norm is a Banach space. Furthermore, (resp., ) denotes the class of continuous functions from into (resp., the class of jointly continuous functions ).
Definition 2.1 (see ).
Definition 2.7 (see ).
Definition 2.8 (see ).
A weaker version of Definition 2.8 is the following.
The notion of -almost automorphy is a new notion due to N'Guérékata and Pankov .
Definition 2.12 (Bochner).
The function in Definition 2.12 is measurable but not necessarily continuous. Moreover, if is continuous, then is uniformly continuous. If the convergence above is uniform in , then is almost periodic. Denote by the collection of all almost automorphic functions . Note that equipped with the sup norm, , turns out to be a Banach space.
Definition 2.14 (see ).
2.3. Pseudo Almost Automorphy
A substantial result is the next theorem, which is due to Liang et al. .
Theorem 2.19 (see ).
We also have the following composition result.
Theorem 2.20 (see ).
This section is devoted to the notion of -pseudo almost automorphy. Such a concept is completely new and is due to Diagana .
Definition 3.1 (see ).
Theorem 3.2 (see ).
Theorem 3.3 (see ).
We have the following composition theorems.
Using the theorem of composition of functions of (see ) it is easy to see that .
Using the theorem of composition [2, Theorem 2.4] for functions of it is easy to see that .
4. Sectorial Linear Operators
The class of sectorial operators is very rich and contains most of classical operators encountered in literature.
see details in [23, Proposition 1.15, page 305]
the abstract Hölder spaces as well as the complex interpolation spaces ; see Lunardi  for details.
5. Existence of Pseudo Almost Automorphic Solutions
This section is devoted to the search of an almost automorphic solution to the partial hyperbolic differential equation (1.3).
In order to show that and are well defined, we need the next lemma whose proof can be found in Diagana .
Lemma 5.2 (see ).
Consequently the uniform limit ; see [21, Lemma 2.5] . Therefore, is pseudo almost automorphic.
The proof is similar to that of Lemma 5.3 and hence omitted, though here we make use of the approximation (5.4) rather than (5.5).
We require the following assumption.
We have the following.
Under the previous assumptions including (H.3), then the -dimensional heat equation (1.1) has a unique pseudo almost automorphic solution whenever is small enough.
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