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

We obtain the existence of pseudo almost automorphic solutions to the -dimensional heat equation with -pseudo almost automorphic coefficients.

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.

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 .

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 .

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

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).

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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