# Existence of Pseudo Almost Automorphic Solutions for the Heat Equation with -Pseudo Almost Automorphic Coefficients

- Toka Diagana
^{1}and - RaviP Agarwal
^{2}Email author

**2009**:182527

**DOI: **10.1155/2009/182527

© T. Diagana and R. P. Agarwal. 2009

**Received: **12 March 2009

**Accepted: **3 July 2009

**Published: **17 August 2009

## Abstract

## 1. Introduction

and the coefficients are -pseudo almost automorphic.

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.

Indeed, letting for all , for all and and , one can readily see that (1.1) is a particular case of (1.3).

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 [3] 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 [7], 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 [13], 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 [14], 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.

## 2. Preliminaries

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 ).

The notation stands for the Banach space of bounded linear operators from into equipped with its natural topology; in particular, this is simply denoted whenever .

Definition 2.1 (see [19]).

The Bochner transform , , of a function is defined by

- (i)
- (ii)

Definition 2.3.

The Bochner transform , , , of a function on , with values in , is defined by for each .

Definition 2.4.

### 2.1. -Pseudo Almost Periodicity

Definition 2.5.

The number
above is called an
*-translation* number of
, and the collection of all such functions will be denoted
.

Definition 2.6.

The collection of those functions is denoted by .

Definition 2.7 (see [13]).

A function is called pseudo almost periodic if it can be expressed as where and . The collection of such functions will be denoted by .

Definition 2.8 (see [13]).

A function is said to be pseudo almost periodic if it can be expressed as where and . The collection of such functions will be denoted by .

uniformly in , where is any bounded subset.

A weaker version of Definition 2.8 is the following.

Definition 2.9.

A function is said to be B-pseudo almost periodic if it can be expressed as where and . The collection of such functions will be denoted by .

Definition 2.10 (see [20, 21]).

A function is called -pseudo almost periodic (or Stepanov-like pseudo almost periodic) if it can be expressed as where and . The collection of such functions will be denoted by .

In other words, a function is said to be -pseudo almost periodic if its Bochner transform is pseudo almost periodic in the sense that there exist two functions such that , where and .

To define the notion of -pseudo almost automorphy for functions of the form , we need to define the -pseudo almost periodicity for these functions as follows.

Definition 2.11.

A function with for each , is said to be -pseudo almost periodic if there exist two functions such that , where and .

The collection of those -pseudo almost periodic functions will be denoted .

### 2.2. -Almost Automorphy

The notion of -almost automorphy is a new notion due to N'Guérékata and Pankov [22].

Definition 2.12 (Bochner).

Remark 2.13.

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 [22]).

Remark 2.15.

It is clear that if and is -almost automorphic, then is -almost automorphic. Also if , then is -almost automorphic for any . Moreover, it is clear that if and only if . Thus, can be considered as .

Definition 2.16.

The collection of those -almost automorphic functions will be denoted by .

### 2.3. Pseudo Almost Automorphy

The notion of pseudo almost automorphy is a new notion due to Liang et al. [2, 9].

Definition 2.17.

A function is called pseudo almost automorphic if it can be expressed as where and . The collection of such functions will be denoted by .

Definition 2.18.

A function is said to be pseudo almost automorphic if it can be expressed as where and . The collection of such functions will be denoted by .

A substantial result is the next theorem, which is due to Liang et al. [2].

Theorem 2.19 (see [2]).

The space equipped with the sup norm is a Banach space.

We also have the following composition result.

Theorem 2.20 (see [2]).

If belongs to and if is uniformly continuous on any bounded subset of for each , then the function defined by belongs to provided .

## 3. -Pseudo Almost Automorphy

This section is devoted to the notion of -pseudo almost automorphy. Such a concept is completely new and is due to Diagana [7].

Definition 3.1 (see [7]).

where and . The collection of such functions will be denoted by .

Clearly, a function is said to be -pseudo almost automorphic if its Bochner transform is pseudo almost automorphic in the sense that there exist two functions such that , where and

Theorem 3.2 (see [7]).

If , then for each . In other words, .

Theorem 3.3 (see [7]).

The space equipped with the norm is a Banach space.

Definition 3.4.

where and . The collection of those -pseudo almost automorphic functions will be denoted by .

We have the following composition theorems.

Theorem 3.5.

If , then defined by belongs to .

Proof.

Using the theorem of composition of functions of (see [13]) it is easy to see that .

Theorem 3.6.

Let be an -pseudo almost automorphic function, where and . Suppose that and are uniformly continuous in every bounded subset uniformly for . If , then defined by belongs to .

Proof.

Let , where and . Similarly, let , where and .

Using the theorem of composition [2, Theorem 2.4] for functions of it is easy to see that .

## 4. Sectorial Linear Operators

Definition 4.1.

The class of sectorial operators is very rich and contains most of classical operators encountered in literature.

Example 4.2.

Let and let be open bounded subset with regular boundary . Let be the Lebesgue space.

It can be checked that the operator is sectorial on .

see details in [23, Proposition 1.15, page 305]

Definition 4.3.

**,**and there is a constant such that

the abstract Hölder spaces as well as the complex interpolation spaces ; see Lunardi [14] 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).

Definition 5.1.

Let and let such that Throughout the rest of the paper, we suppose that the operator is sectorial and generates a hyperbolic (analytic) semigroup and requires the following assumptions.

(H.1)Let . Then , or , or , or . Moreover, we assume that the linear operators are bounded.

*S*

^{ p }-pseudo almost automorphic function in uniformly in , and let be

*S*

^{ p }-pseudo almost automorphic in uniformly in . Moreover, the functions are uniformly Lipschitz with respect to the second argument in the following sense: there exists such that

In order to show that and are well defined, we need the next lemma whose proof can be found in Diagana [12].

Lemma 5.2 (see [12]).

The proof for the pseudo almost automorphy of is similar to that of and hence will be omitted.

Lemma 5.3.

Remark 5.4.

Note that the assumption holds in several case. This is in particular the case when .

Proof.

and hence the function is integrable over for each .

Therefore the sequence for each , and hence its uniform limit .

Consequently the uniform limit ; see [21, Lemma 2.5] . Therefore, is pseudo almost automorphic.

The proof for the almost automorphy of is similar to that of and hence will be omitted.

Lemma 5.5.

Proof.

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).

Theorem 5.6.

Proof.

Therefore, by the Banach fixed-point principle, if , then has a unique fixed-point, which obviously is the only pseudo almost automorphic solution to (1.3).

## 6. Example

Let be an open bounded subset with boundary , and let equipped with its natural topology .

The operator defined above is sectorial and hence is the infinitesimal generator of an analytic semigroup . Moreover, the semigroup is hyperbolic as .

Throughout the rest of the paper, for each , we take equipped with its -norm . Moreover, we let and suppose that . Letting for all , one easily sees that both operators are bounded from into with .

We require the following assumption.

We have the following.

Theorem 6.1.

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.

where the functions are -pseudo almost automorphic.

has a unique pseudo almost automorphic solution whenever is small enough.

## Authors’ Affiliations

## References

- Liang J, Zhang J, Xiao T-J: Composition of pseudo almost automorphic and asymptotically almost automorphic functions.
*Journal of Mathematical Analysis and Applications*2008, 340: 1493–1499. 10.1016/j.jmaa.2007.09.065MATHMathSciNetView ArticleGoogle Scholar - Xiao T-J, Liang J, Zhang J: Pseudo almost automorphic solutions to semilinear differential equations in banach spaces.
*Semigroup Forum*2008, 76(3):518–524. 10.1007/s00233-007-9011-yMATHMathSciNetView ArticleGoogle Scholar - Bochner S: Continuous mappings of almost automorphic and almost periodic functions.
*Proceedings of the National Academy of Sciences of the United States of America*1964, 52: 907–910. 10.1073/pnas.52.4.907MATHMathSciNetView ArticleGoogle Scholar - Zhang CY: Pseudo-almost-periodic solutions of some differential equations.
*Journal of Mathematical Analysis and Applications*1994, 151: 62–76.View ArticleGoogle Scholar - Zhang CY: Pseudo almost periodic solutions of some differential equations. II.
*Journal of Mathematical Analysis and Applications*1995, 192: 543–561. 10.1006/jmaa.1995.1189MATHMathSciNetView ArticleGoogle Scholar - Zhang CY: Integration of vector-valued pseudo-almost periodic functions.
*Proceedings of the American Mathematical Society*1994, 121: 167–174. 10.1090/S0002-9939-1994-1186140-8MATHMathSciNetView ArticleGoogle Scholar - Diagana T: Existence of pseudo almost automorphic solutions to some abstract differential equations with -pseudo almost automorphic coefficients.
*Nonlinear Analysis: Theory, Methods & Applications*2009, 70(11):3781–3790. 10.1016/j.na.2008.07.034MATHMathSciNetView ArticleGoogle Scholar - Ezzinbi K, Fatajou S, N'Guérékata GM: Pseudo almost automorphic solutions to some neutral partial functional differential equations in banach space.
*Nonlinear Analysis: Theory, Methods & Applications*2009, 71(1–2):674–684. 10.1016/j.na.2008.10.100View ArticleGoogle Scholar - Liang J, N'Guérékata GM, Xiao T-J, Zhang J: Some properties of pseudo-almost automorphic functions and applications to abstract differential equations.
*Nonlinear Analysis: Theory, Methods & Applications*2009, 70(7):2731–2735. 10.1016/j.na.2008.03.061MATHMathSciNetView ArticleGoogle Scholar - Boulite S, Maniar L, N'Guérékata GM: Almost automorphic solutions for hyperbolic semilinear evolution equations.
*Semigroup Forum*2005, 71: 231–240. 10.1007/s00233-005-0524-yMATHMathSciNetView ArticleGoogle Scholar - Diagana T, N'Guérékata GM: Pseudo almost periodic mild solutions to hyperbolic evolution equations in intermediate banach spaces.
*Applicable Analysis*2006, 85(6–7):769–780. 10.1080/00036810600708499MATHMathSciNetView ArticleGoogle Scholar - Diagana T: Existence of pseudo almost periodic solutions to some classes of partial hyperbolic evolution equations.
*Electronic Journal of Qualitative Theory of Differential Equations*2007, 2007(3):1–12.View ArticleGoogle Scholar - Diagana T:
*Pseudo Almost Periodic Functions in Banach Spaces*. Nova Science, New York, NY, USA; 2007.MATHGoogle Scholar - Lunardi A:
*Analytic Semigroups and Optimal Regularity in Parabolic Problems, Progress in Nonlinear Differential Equations and Their Applications*.*Volume 16*. Birkhäuser, Basel, Switzerland; 1995.Google Scholar - Diagana T, Henríquez N, Hernàndez E: Almost automorphic mild solutions to some partial neutral functional-differential equations and applications.
*Nonlinear Analysis: Theory, Methods & Applications*2008, 69(5):1485–1493. 10.1016/j.na.2007.06.048MATHMathSciNetView ArticleGoogle Scholar - Diagana T, N'Guérékata GM: Almost automorphic solutions to some classes of partial evolution equations.
*Applied Mathematics Letters*2007, 20(4):462–466. 10.1016/j.aml.2006.05.015MATHMathSciNetView ArticleGoogle Scholar - Diagana T: Existence and uniqueness of pseudo-almost periodic solutions to some classes of partial evolution equations.
*Nonlinear Analysis: Theory, Methods & Applications*2007, 66(2):384–395. 10.1016/j.na.2005.11.031MATHMathSciNetView ArticleGoogle Scholar - Diagana T: Existence of almost automorphic solutions to some neutral functional differential equations with infinite delay.
*Electronic Journal of Differential Equations*2008, 2008(129):1–14.MathSciNetGoogle Scholar - Pankov A:
*Bounded and Almost Periodic Solutions of Nonlinear Operator Differential Equations, Mathematics and Its Applications (Soviet Series)*.*Volume 55*. Kluwer Academic Publishers, Dordrecht, The Netherlands; 1990.Google Scholar - Diagana T: Stepanov-like pseudo almost periodic functions and their applications to differential equations.
*Communications in Mathematical Analysis*2007, 3(1):9–18.MATHMathSciNetGoogle Scholar - Diagana T: Stepanov-like pseudo-almost periodicity and its applications to some nonautonomous differential equations.
*Nonlinear Analysis: Theory, Methods & Applications*2009, 69(12):4277–4285.MathSciNetView ArticleGoogle Scholar - N'Guérékata GM, Pankov A: Stepanov-like almost automorphic functions and monotone evolution equations.
*Nonlinear Analysis: Theory, Methods & Applications*2008, 68(9):2658–2667. 10.1016/j.na.2007.02.012MATHMathSciNetView ArticleGoogle Scholar - Engel KJ, Nagel R:
*One-Parameter Semigroups for Linear Evolution Equations, Graduate Texts in Mathematics*.*Volume 194*. Springer, New York, NY, USA; 2000.Google Scholar

## Copyright

This article is published under license to BioMed Central Ltd. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.