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Existence of Pseudo Almost Automorphic Solutions for the Heat Equation with
-Pseudo Almost Automorphic Coefficients
Boundary Value Problems volume 2009, Article number: 182527 (2009)
Abstract
We obtain the existence of pseudo almost automorphic solutions to the -dimensional heat equation with
-pseudo almost automorphic coefficients.
1. Introduction
Let be an open bounded subset with
boundary
, and let
be the space square integrable functions equipped with its natural
topology. Of concern is the study of pseudo almost automorphic solutions to the
-dimensional heat equation with divergence terms

where the symbols and
stand, respectively, for the first- and second-order differential operators defined by

and the coefficients are
-pseudo almost automorphic.
To analyze (1.1), our strategy will consist of studying the existence of pseudo almost automorphic solutions to the class of partial hyperbolic differential equations

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
Remark 2.2.
-
(i)
A function
,
,
, is the Bochner transform of a certain function
,
if and only if
for all
,
and
.
-
(ii)
Note that if
, then
. Moreover,
for each scalar
.
Definition 2.3.
The Bochner transform ,
,
,
of a function
on
, with values in
, is defined by
for each
.
Definition 2.4.
Let . The space
of all Stepanov bounded functions, with the exponent
, consists of all measurable functions
such that
. This is a Banach space with the norm

2.1.
-Pseudo Almost Periodicity
Definition 2.5.
A function is called (Bohr) almost periodic if for each
there exists
such that every interval of length
contains a number
with the property that

The number above is called an
-translation number of
, and the collection of all such functions will be denoted
.
Definition 2.6.
A function is called (Bohr) almost periodic in
uniformly in
where
is any compact subset
if for each
there exists
such that every interval of length
contains a number
with the property that

The collection of those functions is denoted by .
Define the classes of functions and
, respectively, as follows:

and is the collection of all functions
such that

uniformly in .
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
.
Define as the collection of all functions
such that

uniformly in , where
is any bounded subset.
Obviously,

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).
A function is said to be almost automorphic if for every sequence of real numbers
there exists a subsequence
such that

is well defined for each , and

for each .
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.
We will denote by the closed subspace of all functions
with
. Equivalently,
if and only if
is almost automorphic, and the convergences in Definition 2.12 are uniform on compact intervals, that is, in the Fréchet space
. Indeed, if
is almost automorphic, then its range is relatively compact. Obviously, the following inclusions hold:

Definition 2.14 (see [22]).
The space of Stepanov-like almost automorphic functions (or
-almost automorphic) consists of all
such that
. That is, a function
is said to be
-almost automorphic if its Bochner transform
is almost automorphic in the sense that for every sequence of real numbers
there exists a subsequence
and a function
such that

as pointwise on
.
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.
A function with
for each
, is said to be
-almost automorphic in
uniformly in
if
is
-almost automorphic for each
; that is, for every sequence of real numbers
, there exists a subsequence
and a function
such that

as pointwise on
for each
.
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
.
Obviously, the following inclusions hold:

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]).
A function is called
-pseudo almost automorphic (or Stepanov-like pseudo almost automorphic) if it can be expressed as

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,
.
Obviously, the following inclusions hold:

Theorem 3.3 (see [7]).
The space equipped with the norm
is a Banach space.
Definition 3.4.
A function with
for each
, is said to be
-pseudo almost automorphic if there exists two functions
such that

where and
. The collection of those
-pseudo almost automorphic functions will be denoted by
.
We have the following composition theorems.
Theorem 3.5.
Let be a
-pseudo almost automorphic function. Suppose that
is Lipschitzian in
uniformly in
; that is there exists
such

for all .
If , then
defined by
belongs to
.
Proof.
Let , where
and
. Similarly, let
, where
and
, that is,

for all .
It is obvious to see that . Now decompose
as follows:

Using the theorem of composition of almost automorphic functions, it is easy to see that . Now, set

Clearly, . Indeed, we have

and hence for ,

Now using (3.5), it follows that

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
.
It is obvious to see that . Now decompose
as follows:

Using the theorem of composition of almost automorphic functions, it is easy to see that . Now, set

We claim that . First of all, note that the uniformly continuity of
on bounded subsets
yields the uniform continuity of its Bohr transform
on bounded subsets of
. Since both
are bounded functions, it follows that there exists
a bounded subset such that
for each
. Now from the uniform continuity of
on bounded subsets of
, it obviously follows that
is uniformly continuous on
uniformly for each
. Therefore for every
there exists
such that for all
with
yield

Using the proof of the composition theorem [2, Theorem 2.4], (applied to ) it follows

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.
A linear operator (not necessarily densely defined) is said to be sectorial if the following holds: there exist constants
,
, and
such that
,

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.
Define the linear operator as follows:

It can be checked that the operator is sectorial on
.
It is wellknown that [14] if is sectorial, then it generates an analytic semigroup
, which maps
into
and such that there exist
with


Throughout the rest of the paper, we suppose that the semigroup is hyperbolic; that is, there exist a projection
and constants
such that
commutes with
,
is invariant with respect to
,
is invertible, and the following hold:


where and, for
,
.
Recall that the analytic semigroup associated with
is hyperbolic if and only if

see details in [23, Proposition 1.15, page 305]
Definition 4.3.
Let . A Banach space
is said to be an intermediate space between
and
, or a space of class
, if
, and there is a constant
such that

where is the graph norm of
.
Concrete examples of include
for
, the domains of the fractional powers of
, the real interpolation spaces
,
, defined as the space of all
such

with the norm

the abstract Hölder spaces as well as the complex interpolation spaces
; see Lunardi [14] for details.
For a hyperbolic analytic semigroup , one can easily check that similar estimations as both (4.5) and (4.6) still hold with the
-norms
. In fact, as the part of
in
is bounded, it follows from (4.6) that

Hence, from (4.8) there exists a constant such that

In addition to the above, the following holds:

and hence from (4.5), one obtains

where depends on
. For
, by (4.4) and (4.8),

Hence, there exist constants and
such that

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 . A bounded continuous function
is said to be a mild solution to (1.3) provided that the function
is integrable on
,
is integrable on
for each
and

for all .
Throughout the rest of the paper we denote by and
the nonlinear integral operators defined by

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.
(H.2)Let ,
be an 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

for all and
.
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]).
Let . Then


The proof for the pseudo almost automorphy of is similar to that of
and hence will be omitted.
Lemma 5.3.
Under assumptions (H.1)-(H.2), consider the function , for
, defined by

for each . If

then .
Remark 5.4.
Note that the assumption holds in several case. This is in particular the case when
.
Proof.
Let . Since
, it follows that
. Setting
and using Theorem 3.5 it follows that
. Moreover, using (5.5) it follows that

and hence the function is integrable over
for each
.
Let where
and
. Define, for all
the sequence of integral operators

for each .
Now letting , it follows that

Using Hölder's inequality and the estimate (5.8), it follows that

Using the assumption , we then deduce from the well-known Weirstrass theorem that the series
is uniformly convergent on
. Furthermore,

, and

for each
We claim that . Indeed, let
be a sequence of real numbers. Since
, there exists a subsequence
of
and a function
such that

Define

Set for
. Then using both Hölder's inequality and (5.5), we obtain

where , as
.
Obviously,

Similarly, we can prove that

Therefore the sequence for each
, and hence its uniform limit
.
Let us show that each . Indeed,

and hence , as
. Furthermore, using the assumption
, we then deduce from the well-known Weirstrass theorem that the series

is uniformly convergent on . Moreover,

, and

for each
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.
Under assumptions (H.1)-(H.2), consider the function , for
, defined by

for each .
If , then
.
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).
Throughout the rest of the paper, the constant denotes the bound of the embedding
, that is,

Theorem 5.6.
Under the previous assumptions and if assumptions (H.1)-(H.2) hold, then the evolution equation (1.3) has a unique pseudo almost automorphic solution whenever is small enough, that is,

where .
Proof.
In , define the operator
by setting

for each .
As we have previously seen, for every ,
. From previous assumptions one can easily see that
is well defined and continuous. Moreover, from Theorem 3.5, Lemma 5.3, and Lemma 5.5 we infer that
maps
into
. In particular,
maps
into
. To complete the proof one has to show that
has a unique fixedpoint. Let
. It is routine to see that

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
.
Define the linear operator appearing in (1.3) as follows:

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.
(H.3)Let , let
be an
-pseudo almost automorphic function in
uniformly in
, and let
be
-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

for all and
.
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.
Classical examples of the above-mentioned functions are given as follows:

where the functions are
-pseudo almost automorphic.
In this particular case, the corresponding heat equation, that is,

has a unique pseudo almost automorphic solution whenever
is small enough.
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.2007.09.065
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-y
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.52.4.907
Zhang CY: Pseudo-almost-periodic solutions of some differential equations. Journal of Mathematical Analysis and Applications 1994, 151: 62–76.
Zhang CY: Pseudo almost periodic solutions of some differential equations. II. Journal of Mathematical Analysis and Applications 1995, 192: 543–561. 10.1006/jmaa.1995.1189
Zhang CY: Integration of vector-valued pseudo-almost periodic functions. Proceedings of the American Mathematical Society 1994, 121: 167–174. 10.1090/S0002-9939-1994-1186140-8
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.2008.07.034
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.2008.10.100
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.2008.03.061
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-y
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. 10.1080/00036810600708499
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.
Diagana T: Pseudo Almost Periodic Functions in Banach Spaces. Nova Science, New York, NY, USA; 2007.
Lunardi A: Analytic Semigroups and Optimal Regularity in Parabolic Problems, Progress in Nonlinear Differential Equations and Their Applications. Volume 16. Birkhäuser, Basel, Switzerland; 1995.
Diagana T, HenrÃquez N, Hernà ndez 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.2007.06.048
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.2006.05.015
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.2005.11.031
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.
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.
Diagana T: Stepanov-like pseudo almost periodic functions and their applications to differential equations. Communications in Mathematical Analysis 2007, 3(1):9–18.
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.
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.2007.02.012
Engel KJ, Nagel R: One-Parameter Semigroups for Linear Evolution Equations, Graduate Texts in Mathematics. Volume 194. Springer, New York, NY, USA; 2000.
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Diagana, T., Agarwal, R. Existence of Pseudo Almost Automorphic Solutions for the Heat Equation with -Pseudo Almost Automorphic Coefficients.
Bound Value Probl 2009, 182527 (2009). https://doi.org/10.1155/2009/182527
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DOI: https://doi.org/10.1155/2009/182527
Keywords
- Fractional Power
- Bounded Subset
- Interpolation Space
- Analytic Semigroup
- Uniform Continuity