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Discrete Dynamics in Nature and Society

Volume 2013 (2013), Article ID 183420, 10 pages

http://dx.doi.org/10.1155/2013/183420

## Almost Automorphic Mild Solutions to Neutral Parabolic Nonautonomous Evolution Equations with Nondense Domain

^{1}Department of Mathematics, North University of China, Taiyuan 030051, China^{2}School of Mechatronic Engineering, North University of China, Taiyuan 030051, China

Received 5 March 2013; Accepted 22 April 2013

Academic Editor: Yonghui Xia

Copyright © 2013 Zhanrong Hu and Zhen Jin. 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.

#### Abstract

Combining the exponential dichotomy of evolution family, composition theorems for almost automorphic functions with Banach fixed point theorem, we establish new existence and uniqueness theorems for almost automorphic mild solutions to neutral parabolic nonautonomous evolution equations with nondense domain. A unified framework is set up to investigate the existence and uniqueness of almost automorphic mild solutions to some classes of parabolic partial differential equations and neutral functional differential equations.

#### 1. Introduction

In this paper, we are interested in the existence and uniqueness of almost automorphic mild solutions to the following neutral parabolic evolution equations in Banach space : where sectorial operators have a domain not necessarily dense in and satisfy “Acquistapace-Terreni" conditions, , are almost automorphic in the first argument and Lipschitz in the second argument, and , are bounded linear operators.

Bochner has shown in the seminal work [1] that in certain situations it is possible to establish the almost periodicity of an object by first establishing its almost automorphy and then invoking auxiliary conditions which, when coupled with almost automorphy, give almost periodicity. From then on, automorphy has been widely investigated. Fundamental properties of almost automorphic functions on groups and abstract almost automorphic minimal flows were studied by Veech [2, 3] and others. Afterwards, Zaki [4] extended the notion of scalar-valued almost automorphy to the one of vector-valued almost automorphic functions, paving the road to many applications to differential equations and dynamical systems. Among other things, Shen and Yi [5] showed that almost automorphy is essential and fundamental in the qualitative study of almost periodic differential equations in the sense that almost automorphic solutions are the right class for almost periodic systems. We refer the readers to the monographs [6, 7] by N’Guérékata for more information on this topic.

In the autonomous case, namely , the existence and uniqueness of almost automorphic mild solutions to evolution equation (1) with have been successfully investigated in [6–13] in the framework of semigroups of bounded linear operators. In [13], N’Guérékata studied the existence and uniqueness of almost automorphic solutions for semilinear evolution equation where generates an exponentially stable semigroup on Banach space and is almost automorphic. The author proved that the unique bounded mild solution of (3) is almost automorphic. In [8], Boulite et al. studied the existence and uniqueness of almost automorphic solutions for evolution equation (3), assuming that generates a hyperbolic semigroup on Banach space and is almost automorphic, where is an intermediate space between and . The authors proved that the unique bounded mild solution of (3) is almost automorphic. Cieutat and Ezzinbi [9] studied the existence of bounded and compact almost automorphic solutions for semilinear evolution equation (3). The main methods are through the minimizing of some subvariant functionals. They gave sufficient conditions ensuring the existence of an almost automorphic mild solution when there is at least one bounded mild solution on .

In the nonautonomous case, almost automorphic mild solutions to evolution equation (1) with have been successfully investigated in [14–16] in the framework of evolution family. Among others, Baroun et al. [14] generalized the main results of [8] to the nonautonomous case. The authors proved that the unique bounded mild solution of the semilinear evolution equation is almost automorphic, assuming that the evolution family generated by has an exponential dichotomy and is almost automorphic. Ding et al. [15] established the existence and uniqueness theorem of almost automorphic mild solutions to the evolution equation (4), where the evolution family generated by has an exponential dichotomy and is almost automorphic. Liu and Song [16] proved the existence and uniqueness of an almost automorphic or a weighted pseudo almost automorphic mild solution to (4), assuming that the evolution family generated by is exponentially stable and is almost automorphic or weighted pseudo almost automorphic.

A rich source of the literature exists on almost automorphic mild solutions to linear and semilinear evolution equations. However, to the best of our knowledge, there are few results available on the existence and uniqueness of almost automorphic mild solutions to neutral parabolic nonautonomous evolution equations (1) and (2), especially in the case of not necessarily dense domain and bounded perturbations. Nondensity occurs in many situations, from restrictions made on the space where the equation is considered or from boundary conditions. For example, the space of twice continuously differential functions with null value on the boundary is nondense in , the space of continuous functions. One can refer for this to [17–19] or Section 5 for more details. We further remark that our first main result (Theorem 18) recovers partly Theorem 2.2 in [15] and Theorem 3.2 in [16] in the parabolic case. Moreover, a unified framework is set up in the second main result (Theorem 21) to study the existence and uniqueness of almost automorphic mild solutions to some classes of parabolic partial differential equations and neutral functional differential equations. As one will see, the additional neutral term greatly widens the applications of the main result since (2) is general enough to incorporate some classes of parabolic partial differential equations and neutral functional differential equations as special cases.

As a preparation, in Section 2 we fix our notation and collect some basic facts on evolution family and almost automorphy. Section 3 deals with the proof of the existence and uniqueness theorem of almost automorphic mild solutions to evolution equation (1). In Section 4, we study the existence and uniqueness of almost automorphic mild solutions to evolution equation (2) with bounded perturbations. Finally, the abstract results are applied to some classes of parabolic partial differential equations and neutral functional differential equations.

#### 2. Preliminaries

Throughout this paper, , , , and stand for the sets of positive integer, integer, real, and complex numbers, and stands for a Banach space. If is another Banach space, the space denotes the Banach space of all bounded linear operators from into equipped with the uniform operator topology. The resolvent operator is defined by for , the resolvent set of a linear operator .

##### 2.1. Evolution Family and Exponential Dichotomy

*Definition 1 (see [20, 21]). *A family of bounded linear operators on a Banach space is called an evolution family if (1) and for all and ; (2)the map is continuous for all , , and .

*Definition 2 (see [20, 21]). *An evolution family on a Banach space has an exponential dichotomy (or is called hyperbolic) if there exist projections , uniformly bounded and strongly continuous in and constants such that (1) for and ;(2)the restriction of is invertible for (and we set ); (3) for and .Here and below we set .

*Remark 3. *Exponential dichotomy is a classical concept in the study of the long-term behavior of evolution equations, combining forward exponential stability on some subspaces with backward exponential stability on their complements. Its importance relies in particular on the robustness; that is, exponential dichotomy persists under small linear or nonlinear perturbations (see, e.g., [20–24]).

*Definition 4 (see [20, 21]). *Given an evolution family with an exponential dichotomy, one defines its Green’s function by

##### 2.2. Almost Automorphy and Bi-Almost Automorphy

Let denote the collection of continuous functions from into . Let denote the Banach space of all bounded continuous functions from into equipped with the sup norm . Similarly, denotes the collection of all jointly continuous functions from into , and denotes the collection of all bounded and jointly continuous functions .

*Definition 5 (Bochner). *A function is said to be almost automorphic if for any sequence of real numbers , there exists a subsequence such that
pointwise for each . This limit means that
is well defined for each and
for each . The collection of all such functions will be denoted by .

*Example 6 (Levitan). *The function , is almost automorphic but not almost periodic.

*Remark 7. *An almost automorphic function may not be uniformly continuous, while an almost periodic function must be uniformly continuous.

Lemma 8 (see [6, 7]). *Assume that are almost automorphic and is any scalar. Then the following holds true:*(1)* are almost automorphic;*(2)*the range of is precompact, so is bounded;*(3)* defined by , , is almost automorphic.*

Lemma 9 (see [6, 7]). *If is a sequence of almost automorphic functions and uniformly on , then is almost automorphic.*

Lemma 10 (see [6]). *The space equipped with sup norm is a Banach space.*

*Definition 11 (see [25]). *A function is said to be almost automorphic if is almost automorphic in for each . That is to say, for every sequence of real numbers , there exists a subsequence such that
pointwise on for each . Denote by the collection of all such functions.

Lemma 12 (see [6, Theorem ]). *Assume that and there exists a constant such that for all and ,
**
If , then .*

Corollary 13 (see [6, Corollary ]). *Assume that and . If for each , , then .*

*Definition 14 (see [26]). *A function is called bi-almost automorphic if for every sequence of real numbers , one can extract a subsequence such that
is well defined for each , and
for each . The collection of all such functions will be denoted by .

In other words, a function is said to be bi-almost automorphic if for any sequence of real numbers , there exists a subsequence such that
pointwise for each .

#### 3. Neutral Parabolic Nonautonomous Evolution Equation

In this section, we will establish the existence and uniqueness theorem of almost automorphic mild solutions to neutral parabolic nonautonomous evolution equation (1) under assumptions (H1)–(H5) listed below: (H1) there exist constants , and with such that for ,(H2)the evolution family generated by has an exponential dichotomy with dichotomy constants , dichotomy projections , , and Green’s function , (H3) for each , (H4), and there exists a constant such that for all and , (H5), and there exists a constant such that for all and ,

*Remark 15. *Assumption (H1) is usually called “Acquistapace-Terreni” conditions, which was first introduced in [27] for . If (H1) holds, then there exists a unique evolution family on such that is strongly continuous for , , for . These assertions are established in Theorem 2.3 of [28]. See also [27, 29, 30].

*Definition 16. *A mild solution to (1) is a continuous function satisfying integral equation
for all and all .

Lemma 17. *Assume that (H1)–(H3) and (H5) hold. Define nonlinear operator on by
**
Then maps into itself.*

*Proof. *Combining the ideas from Theorem 4.28 in [22], the technique of exponential dichotomy, composition theorem of almost automorphic functions, and the Lebesgue dominated convergence theorem, we strive for a more self-contained proof. Let . Then it follows from Lemma 12 [6, Theorem ] that , in view of (H5). Hence, can be rewritten as
From triangle inequality and exponential dichotomy of , it follows that
Hence, is well defined for each . To show , we will verify that
By and the strong continuity of , we have for each , , ,
Transforming (19) into another form, we have, for ,
In view of (23), triangle inequality and exponential dichotomy of , we have
Thus, in the Lebesgue dominated convergence theorem, (22) leads to (21) and therefore to .

To show , let us take a sequence of real numbers and show that there exists a subsequence such that
By and (H3), there exists a subsequence such that for each , , ,
Again, in view of (23), triangle inequality and exponential dichotomy of , we obtain
Thus, in the Lebesgue dominated convergence theorem, (26) leads to (25) and therefore to . Here we used the translation invariance of almost automorphic functions, which is collected in Lemma 8(3). The proof is complete.

Now we are in a position to state and prove the first main result of this paper.

Theorem 18. *Suppose that (H1)–(H5) hold. If , then there exists a unique mild solution to (1) such that
*

*Proof. *Firstly, define nonlinear operator on by
Let , then it follows from Lemma 12 [6, Theorem ] that , in view of (H4). Together with Lemma 17, we deduce that the operator is well defined and maps into itself.

Secondly, we will prove that is a strict contraction on . Let . By (H2), (H4), and (H5), we have
Hence,
If , then the operator becomes a strict contraction on . Since the space equipped with sup norm is a Banach space by Lemma 10, an application of Banach fixed point theorem shows that there exists a unique such that (28) holds.

Finally, to prove that satisfies (17) for all , all . For this, we let
Multiplying both sides of (32) by for all , we have
Hence, is a unique mild solution to (1). The proof is complete.

#### 4. Bounded Perturbations

In this section, we consider neutral parabolic nonautonomous evolution equation (2). For this, we need assumptions (H1)–(H5) listed in the previous section and the following assumption: (H6) with .

*Definition 19. *A mild solution to (2) is a continuous function satisfying integral equation
for all and all .

Lemma 20. *Let assumptions (H1)–(H3), (H5), and (H6) hold. Define nonlinear operator on by
**
Then maps into itself.*

*Proof. *Let . By (H6) and Corollary 13, we obtain . Then it follows from Lemma 12 [6, Theorem ] that , in view of (H5). The left is almost same as the proof of Lemma 17, remembering to replace by , and , respectively. This ends the proof.

Now we are in a position to state and prove the second main result of this paper.

Theorem 21. *Suppose that (H1)–(H6) hold. If , then there exists a unique mild solution to (2) such that
*

*Proof. *Firstly, define nonlinear operator on by
Let . By (H6) and Corollary 13, we obtain . Hence, it follows from Lemma 12 [6, Theorem ] that , in view of (H4). Together with Lemma 20, we deduce that the operator is well defined and maps into itself.

Secondly, we will prove that is a strict contraction on and apply Banach fixed point theorem. Let . Then it follows from (H2), (H4), (H5), and (H6) that
Hence,
If , then the operator becomes a strict contraction on . Since the space equipped with sup norm is a Banach space by Lemma 10, an application of Banach fixed point theorem shows that there exists a unique such that (36) holds.

Finally, to prove that satisfies (34) for all , all . For this, we let
Multiplying both sides of (40) by for all , then
Hence, is a unique mild solution to (2). The proof is complete.

#### 5. Applications to Parabolic Partial Differential Equations and Neutral Functional Differential Equations

In this section, two examples are given to illustrate the effectiveness and flexibility of Theorem 21. By a mild solution to a partial or neutral functional differential equation, we mean a mild solution to the corresponding evolution equation.

*Example 22. *Consider the following parabolic partial differential equation:
where is continuous on .

Let denote the space of continuous functions from to equipped with the sup norm and define the operator by
It is known (see [18, Example 14.4]) that is sectorial, is not dense in , and is the generator of an analytic semigroup not strongly continuous at . Since the spectrum of consists of the sequence of eigenvalues , it can be easily checked that for , remembering that the spectral bound of coincides with its growth bound

Define a family of linear operators by
In the case that have a constant domain and , it is known that [31, 32] assumption (H1) can be replaced by the following assumption (ST). (ST) There exist constants , and such that
for .Now, it is not hard to verify that satisfy (H1). By Theorem 2.3 of [28], generate an evolution family that is strongly continuous for . Furthermore,
Hence,
and (H2) is satisfied with .

As for (H3), it is obvious that for each .

To show that for each , let us take a sequence of real numbers and show that there exists a subsequence such that
pointwise for each .

By , there exists a subsequence such that pointwise for each ,
In view of (47) and (50), an application of the Lebesgue dominated convergence theorem shows that
pointwise for each . Hence, (H3) is satisfied.

Define the operators by
then .

In view of the above, (42) can be transformed into the abstract form (2), and assumptions (H1)–(H3) and (H6) are satisfied.

We add the following assumptions: (H4a) is almost automorphic, and there exists a constant such that for all , ,
(H5a) is almost automorphic, and there exists a constant such that for all , ,

Now, the following proposition is an immediate consequence of Theorem 21.

Proposition 23. *Under assumptions (H4a) and (H5a), parabolic partial differential equation (42) admits a unique almost automorphic mild solution if
**
Furthermore, if one takes
**
A simple computation shows that , (H4a), and (H5a) are satisfied with , . By Proposition 23, (42) admits a unique almost automorphic mild solution whenever .*

*Example 24. *Consider neutral functional differential equation
where is a fixed constant.

Take , , , (H4a), and (H5a) as in Example 22. Define the operators by
then .

Now, (57) can be transformed into the abstract form (2) and assumptions (H1)–(H3) and (H6) are satisfied. Hence, Theorem 21 leads also to the following proposition.

Proposition 25. *Under assumptions (H4a) and (H5a), neutral functional differential equation (57) admits a unique almost automorphic mild solution if
**
Furthermore, if one takes
**
A simple computation shows that , (H4a), and (H5a) are satisfied with , . By Proposition 25, (57) admits a unique almost automorphic mild solution.*

#### Acknowledgments

The authors would like to thank the referees for their careful reading of this paper. This work is supported by the National Science Foundation of China (11171314).

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