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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 
.
, 
, 
, is the Bochner transform of a certain function 
, 
if and only if 
for all 
, 
and 
.
, 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
-Pseudo Almost PeriodicityDefinition 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
-Almost AutomorphyThe 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
-Pseudo Almost AutomorphyThis 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.
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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