Discrete Dynamics in Nature and Society

Volume 2008 (2008), Article ID 405092, 13 pages

http://dx.doi.org/10.1155/2008/405092

## Existence and Uniqueness of Pseudo Almost Automorphic Mild Solutions to Some Classes of Partial Hyperbolic Evolution Equations

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

Received 1 June 2008; Revised 4 July 2008; Accepted 10 September 2008

Academic Editor: Leonid Shaikhet

Copyright © 2008 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

We will establish an existence and uniqueness theorem of pseudo almost automorphic mild solutions to the following partial hyperbolic evolution equation ?? under some assumptions. To illustrate our abstract result, a concrete example is given.

#### 1. Introduction

Let be a Banach space, and let for be an arbitrary abstract intermediate Banach space between (domain of the operator ) and . In this paper, we deal with sufficient conditions for the existence and uniqueness of pseudo almost automorphic mild solutions to equationwhere is a sectorial linear operator on a Banach space and , are bounded linear operators on , , are jointly continuous functions. This turns out to be a nontrivial problem due to the complexity and importance of pseudo almost automorphic functions (e.g., see [1–8] and the references therein).

Upon making some additional assumptions, it will be shown that (1.1) admits a unique -valued pseudo almost automorphic mild solution. Applications include the study of pseudo almost automorphic mild solutions to some nonlinear heat equation: where is a constant, and are continuous functions.

As a natural and important generalization of almost automorphy as well as pseudo almost periodicity, pseudo almost automorphy has recently been investigated (for more details, see [1–3]). Moreover, Liang et al. in [2] established a composition theorem about pseudo almost automorphic functions; Xiao et al. in [3] obtained sufficient conditions for the existence and uniqueness of pseudo almost automorphic mild solutions to semilinear differential equations in Banach spaces. In particular, they proved in [3] that the space of pseudo almost automorphic functions, endowed with sup norm, is a Banach space, thus pushed the door open to study pseudo almost automorphic mild solutions to various differential equations.

The method used here was firstly developed in [9], in which Diagana established an existence and uniqueness theorem of pseudo almost periodic mild solutions to (1.1) under some similar assumptions. Combining bounded invariance theorem of pseudo almost automorphic functions established in [6] with composition theorem of pseudo almost automorphic functions and completeness of the space , this paper extends the results in [9], since such function as is not pseudo almost periodic, while they are pseudo almost automorphic.

It should be pointed out that one can also make use of (1.1) to study several types of evolution equations including partial functional differential equations, integro-differential equations, and reaction diffusion equations.

#### 2. Preliminaries

In this section, we collect some preliminary facts from [9–11] that will be used later. Throughout this paper, , , , and stand for the sets of positive integer, integer, real and complex numbers; , stand for Banach spaces. If is a linear operator on , then , , , , stand for the resolvent set, spectrum, domain, kernel, and range of . The space denotes the Banach space of all bounded linear operators from into equipped with natural norm . If , it is simply denoted by with .

##### 2.1. Sectorial Linear Operators and Analytic Semigroups

*Definition 2.1. *A linear operator (not necessarily densely defined) is said to
be sectorial if the following hold. There exist constants , ,
and such thatwhere for each *Remark 2.2. *If is sectorial, then it generates an analytic
semigroup ,
which maps into such that there exist with*Definition 2.3. *A semigroup is said to be hyperbolic, if there exist a
projection and constants such that each commutes with , is invariant with respect to , is invertible andwhere and for .Recall that if
a semigroup is analytic, then is hyperbolic if and only if (see, e.g., [11, Definition 1.14 and Proposition 1.15,
page 305]).

##### 2.2. Intermediate Banach Spaces

*Definition 2.4. *Let .
A Banach space is said to be an intermediate space between and ,
if and there is a constant such thatwhere is the graph norm of .

Concrete examples of include for ,
the domains of the fractional powers of ,
the real interpolation spaces , ,
defined as follows:and the abstract Hölder spaces .Lemma 2.5. *For the hyperbolic analytic semigroup ,
there exist constants , ,
and such that*Lemma 2.6 (see [9]). *Let .
For the hyperbolic analytic semigroup ,
there exist constants ,
and such that*

#### 3. Pseudo Almost Automorphic Functions

In this
section, we recall some recent results on almost automorphic functions and
pseudo almost automorphic functions. Let denote the collection of continuous functions
from into .
Let denote the Banach space of all -valued bounded continuous functions equipped
with the sup norm for each .
Similarly, denotes the collection of continuous functions
from into , denotes the collection of all bounded
continuous functions .*Definition 3.1 ( see[8]). *A function is said to be almost automorphic if for every
sequence of real numbers , there exists a subsequence such thatThis limit means thatis well defined for each andfor each . The collection of all such functions will be denoted by Theorem 3.2 (see [7]). *Assume are almost automorphic, and is any scalar. Then the following holds true:*(1)* and are almost automorphic;*(2)

*(3)*

*the range of is precompact, so is bounded;**Theorem 3.3 (see [7]).*

*if is a sequence of almost automorphic functions and uniformly on , then is almost automorphic.**If one equips? with the sup norm , then turns out to be a Banach space.*

One sets*Definition 3.4 (see [2]). *A function is said to be pseudo almost automorphic if it
can be decomposed as ,
where and .
The collection of such functions will be denoted by .

Theorem 3.5 (see [3]). *If one
equips with the sup norm ,
then turns out to be a Banach space.**Definition 3.6 (see [2]). *A function is said to be almost automorphic if is almost automorphic in uniformly for all ,
where is any bounded subset of That is to say, for every sequence of real
numbers there exists a subsequence such thatis well defined in for all ,
andfor each and Denote by the collection of all such functions.One also defines as the collection of functions such thatuniformly for any bounded subset
of .*Definition 3.7 (see [2]). *A function is said to be pseudo almost automorphic if it
can be decomposed as ,
where and .
The collection of such functions will be denoted by .Theorem 3.8 (see [2]). *Assume with , ,
satisfying the following conditions:*(1)* is uniformly continuous in any bounded subset uniformly for *(2)

*is uniformly continuous in any bounded subset uniformly for**If , then .*

*Remark 3.9.*If for is an intermediate space between and , , , are equipped with the -sup norm: , then they all constitute Banach spaces in view of Definition 2.4, Theorems 3.3 and 3.5.Theorem 3.10 (see [6]).

*Let , . If for each , then .*

#### 4. Pseudo Almost Automorphic Mild Solutions

In this section, combining Theorem 3.10 with composition theorem of pseudo almost automorphic functions (Theorem 3.8) and completeness of the space (Theorem 3.5), we will establish the existence and uniqueness theorem of pseudo almost automorphic mild solutions to (1.1) under the following assumptions. (H1)For , are continuously embedded and there exist such that , for each (H2) is a sectorial linear operator on a Banach space and (H3) with , and with , .(H4) The functions are uniformly Lipschitz with respect to the second argument in the sense that: there exists such that for all and .(H5) are uniformly continuous in any bounded subset uniformly for .(H6) The operators and .

*Definition 4.1 (see [9]). *A function is said to be a mild solution to (1.1) if is integrable on , is integrable on and for each .Lemma 4.2. *Let
assumptions (H1)–(H6) hold. Consider the nonlinear operator defined by**Then, maps into itself.**Proof. *From (H1), (H3), (H4), (H5), we deduce that with , ,
satisfying the following conditions.
(1) is uniformly continuous in any bounded subset uniformly for (2) is uniformly Lipschitz with respect to the
second argument in the sense that there exists such thatfor all and .

Let .
Since ,
it follows from Theorem 3.10 that .
Setting and applying Theorem 3.8, we get that Using (H1) and Theorem 3.10, we obtain that Now, write where and ,
thenSet Next, we show that and .

Now, to prove that .
Let us take a sequence and show that there exists a subsequence such thatSince ,
there exists a subsequence such thatOn the other hand, we
havePassing to the norm remembering the triangle inequalities, (2.7), then we obtain thatThus, (4.8) and Lebesgue
dominated convergence theorem lead to (4.7), therefore, to .
To finish the proof, we will prove that .
It is obvious that ,
the left task is to show thatUsing (2.7), we havewhereThen, by the Lebesgue dominated
convergence theorem and the fact that ,
one has Hence, and we end the proof.
Lemma 4.3. *Let assumptions (H2)–(H6) hold. Define the nonlinear operator by**Then, maps into itself.**Proof. *The proof is similar to that of Lemma 4.2, so we omit it.Theorem 4.4. *Under the assumptions (H1)–(H6), partial hyperbolic evolution equation
(1.1) admits a unique pseudo almost automorphic mild solution if**Proof. *Firstly, define the nonlinear operator on byLet ,
then ,
as proved in Lemma 4.2. Together with Lemmas 4.2 and 4.3, it follows that the
operator maps into itself. Secondly, we will show that admits a unique fixed point in

Let ,
then the triangle inequality readsBy (H1), (H5), and (H6), we
obtainBy (H2), Lemma 2.6, (H5), and
(H6), we obtainSimilarly, by (H2), Lemma 2.5,
(H5), and (H6), we obtainCombining the above inequality
together, we obtain ,
whereClearly, if ,
then the operator becomes a strict contraction on .
Remembering that equipped with the -sup norm: is a Banach space by Remark 3.9, the classical Banach fixed-point theorem leads to
the desired conclusion.

#### 5. Application

*Example 5.1. *Take equipped with the sup norm. Define the
operator bywhere and is a constant.

Clearly, is sectorial, and hence is the generator of an
analytic semigroup. If ,
then generates a hyperbolic analytic semigroup on .

Letwhere (see [10] for more
details).

Define the operator bywhere ,
then and

Letthen , is Lipschitz with respect to the second
argument with is uniformly continuous in any bounded subset uniformly for .

Now the above discussion can be formulated as
follows.Proposition 5.2. *Let Suppose that the constant for .
Then, nonlinear heat equation (1.2) admits a unique -valued pseudo almost automorphic mild solution
if*

#### Appendix

*Proof. *Write where and ,
thenSetWe are going to prove that and .

Let us take a sequence and show that there exists a subsequence such thatSince ,
there exists a subsequence such thatOn the other hand, we haveThus, (A.4) leads to (A.3),
therefore, to .

To complete the proof, we will prove that .
In fact, it is easy to check that andHence, and we end the proof.

#### Acknowledgments

The authors would like to thank referees for their helpful suggestions. This work is supported by the National Science Foundation of China (60771026), the Special Scientific Research Foundation for the Subjects of Doctors in University (20060110005), and the Program for New Century Excellent Talents in University (NCET050271).

#### References

- T. Diagana, “Existence of $p$-almost automorphic mild solution to some abstract differential equations,”
*International Journal of Evolution Equations*, vol. 1, no. 1, pp. 57–67, 2005. View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - J. Liang, J. Zhang, and T.-J. Xiao, “Composition of pseudo almost automorphic and asymptotically almost automorphic functions,”
*Journal of Mathematical Analysis and Applications*, vol. 340, no. 2, pp. 1493–1499, 2008. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - T.-J. Xiao, J. Liang, and J. Zhang, “Pseudo almost automorphic solutions to semilinear differential equations in Banach spaces,”
*Semigroup Forum*, vol. 76, no. 3, pp. 518–524, 2008. View at Publisher · View at Google Scholar · View at MathSciNet - T. Diagana, G. M. N'Guérékata, and N. Van Minh, “Almost automorphic solutions of evolution equations,”
*Proceedings of the American Mathematical Society*, vol. 132, no. 11, pp. 3289–3298, 2004. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - R. A. Johnson, “A linear, almost periodic equation with an almost automorphic solution,”
*Proceedings of the American Mathematical Society*, vol. 82, no. 2, pp. 199–205, 1981. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - Z. R. Hu and Z. Jin, “Pseudo almost automorphic mild solutions to some classes of partial evolution equations,” submitted to
*Nonlinear Analysis: Real World Applications*. - G. M. N'Guérékata,
*Almost Automorphic and Almost Periodic Functions in Abstract Spaces*, Kluwer Academic/Plenum, New York, NY, USA, 2001. View at Zentralblatt MATH · View at MathSciNet - G. M. N'Guérékata,
*Topics in Almost Automorphy*, Springer, New York, NY, USA, 2005. View at Zentralblatt MATH · View at MathSciNet - T. Diagana, “Existence of pseudo almost periodic solutions to some classes of partial hyperbolic evolution equations,”
*Electronic Journal of Qualitative Theory of Differential Equations*, no. 3, pp. 1–12, 2007. View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - A. Lunardi,
*Analytic Semigroups and Optimal Regularity in Parabolic Problems*, vol. 16 of*Progress in Nonlinear Differential Equations and Their Applications*, Birkhäuser, Basel, Switzerland, 1995. View at Zentralblatt MATH · View at MathSciNet - K.-J. Engel and R. Nagel,
*One-Parameter Semigroups for Linear Evolution Equations*, vol. 194 of*Graduate Texts in Mathematics*, Springer, New York, NY, USA, 2000. View at Zentralblatt MATH · View at MathSciNet