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Abstract and Applied Analysis
Volume 2012 (2012), Article ID 180894, 17 pages
http://dx.doi.org/10.1155/2012/180894
Research Article

Almost Periodic (Type) Solutions to Parabolic Cauchy Inverse Problems

Department of Mathematics, Harbin Institute of Technology, Harbin 150001, China

Received 23 December 2011; Revised 1 March 2012; Accepted 12 March 2012

Academic Editor: István Györi

Copyright © 2012 Fenglin Yang and Chuanyi Zhang. 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 first show the existence and uniqueness of (pseudo) almost periodic solutions of some types of parabolic equations. Then, we apply the results to a type of Cauchy parabolic inverse problems and show the existence, uniqueness, and stability.

1. Introduction

Zhang in [1, 2] defined pseudo almost periodic functions. As almost periodic functions, pseudo almost periodic functions are applied to many mathematical areas, particular to the theory of ordinary differential equations. (e.g., see [326] and references therein). However, the study of the related topic on partial differential equations has only a few important developments. On the other hand, almost periodic functions to various problems have been investigated (e.g., see [2732] and references therein), but little has been done about the inverse problems except for our work in [3336]. In [36], we study pseudo almost periodic solutions to parabolic boundary value inverse problems. In this paper, we devote such solutions to cauchy problems.

To this end, we need first to define the spaces in a more general setting. Let . Let (resp., , where ) denote the algebra of bounded continuous complex-valued functions on (resp. with the supremum norm. For (resp., ) and , the translation of by is the function (resp., .

Definition 1.1. (1) A function is called almost periodic if for every the set is relatively dense in . Denote by the set of all such functions. The number (vector) is called translation number (vector) of .
(2) A function is said to be almost periodic in and uniform on compact subsets of if for each and is uniformly continuous on for any compact subset . Denote by the set of all such functions. For convenience, such functions are also called uniformly almost periodic.
(3) A function is called pseudo almost periodic if where and , uniformly with respect to , where is any compact subset of . Denote by the set of all such functions.
Set

Members of are called almost periodic type.

We will use the notations throughout the paper: . means that is almost periodic type in and uniformly for means that is almost periodic type in and uniformly for .

Let be the fundamental solution of the heat equation [37].

In the next section, we will show the existence and uniqueness of some type of parabolic equations. Sections 3 is devoted to a type of Cauchy Problem respectively.

2. Solutions of Parabolic Equations

Lemma 2.1. Let . If and then for each fixed .

Proof. First consider the case that . Let be an translation vector of : where is the zero vector. Note that, we get where and with a bounded subset of . This shows that .
To show that if , we only need to show that if . That is, uniformly with respect to , here is any compact subset of .
Since , for there exist positive numbers and such that, when for all and , one has Therefore, uniformly with respect to , where by we mean that This shows that . The proof is complete.

Corollary 2.2. Let , and let be as in Lemma 2.1. Then, .

Proof. Note that

By Lemma 2.1 we get the conclusion.

Lemma 2.3. If and then and are all in .

The proof is similar to that of Lemma 2.1, so we omit it.

Theorem 2.4. Consider the heat problem If are in and are in , then (2.10) has a unique solution .

Proof. Problem (2.10) has the standard solution (see [37]): where and is the parabolic operator Now, we show that By Lemmas 2.1 and 2.3, .
Obviously, By Lemmas 2.1 and 2.3 we only need to show that the functions are in . We do this by induction. By Lemmas 2.1 and 2.3 and Corollary 2.2, it is true for the case . Suppose that . Then, By the induction assumption and Lemma 2.3, we have . Similarly one shows that . The proof is complete.

3. Cauchy Problem

Starting this section we will apply the results of the last section to inverse problems of partial differential equations. We will investigate two types of initial value problems in this and the next sections, respectively. We will keep the notation in Section 2 and, at the same time, introduce the following new notation:

The following estimates are easily obtained: where are positive and increasing for and as .

To show the main results of this and the next sections, the following lemmas are needed. The first lemma is the Gronwall-Bellman lemma; the convenient reference should be an ODE text, for instance, it is proved on page 15 of [38].

Lemma 3.1. Let , , and be real, continuous functions on with . If then

Lemma 3.2. Let be a continuous function on . If , , and are nondecreasing and nonnegative on and then where

Proof. Replacing in the two integrals of (*) by the expression on the right-hand side in (26), changing the integral order of the resulting inequality, and making use of the monotonicity of , , and , one gets Using Lemma 3.1 leads to the conclusion.

Lemma 3.3. Let and . If is a solution of the problem then where .

One sees that depends on only and is bounded near zero.

Proof. The solution can been written as so, By Lemma 3.1 one gets the desired result. The proof is complete.

Problem 1. Find functions and such that where , and . are constants.

By (3.13) and (3.14), one sees that .

We have the following additional problem.

Problem 2. Find functions and such that
The Cauchy problems with unknown coefficient belong to inverse problems [39]. “In the last two decades, the field of inverse problems has certainly been one of the fastest growing areas in applied mathematics. This growth has largely been driven by the needs of applications both in other sciences and in industry.” [40]. For the two problems above, we have the following.

Lemma 3.4. Problems 1 and 2 are equivalent to each other.

Proof. Let . Then, satisfies So, if Problem 1 has a solution , then Problems (3.18)–(3.20) have the solution with . Obviously if .
On the other hand, if and satisfy (3.18)–(3.20), then we will show that Problem 1 has a unique solution and .
The uniqueness comes from the uniqueness of Cauchy Problem (1)-(2). For the existence, note the fact that if is a solution of (3.12)–(3.14), then . Thus, we define It follows from (3.14) that . Now, satisfies (3.13) because It follows from (3.18), (3.20), and (3.21) that Thus, satisfies (3.12) and is a unique solution of Problem 1.
It follows from , and (3.21) that in (3.21) is in .
Since we have shown that Problem 1 is equivalent to (3.18)–(3.20), to show the lemma we only need to show that Problem 2, equivalent to (3.18)–(3.20) too.
If is a solution of (3.18)–(3.20), let . Then one can directly calculate that is a solution of (3.15)–(3.17) and .
On the other hand, if is a solution of (3.15)–(3.17), let where is the solution of the Cauchy problem Since , . By Theorem 2.4, and so .
Obviously, , and this shows that satisfies (3.20). satisfies (3.19) because Finally, This shows that satisfies (3.18). The proof is complete.

By (3.15)-(3.16) we have the integral equation about : where is determined by (3.28).

It is readily to show that (3.15)–(3.17) are equivalent to (3.28)-(3.29).

Note that, for a given , Theorem 2.4 shows the Cauchy problem (3.15) and (3.16) (or equivalently (3.28)) has a unique solution . Thus, (3.29) does define an operator . To show that Problem 2 (and so Problem 1) has a unique solution, we only need to show that (3.29) has a solution . That is, has a fixed point in . To this end, let Set . Now, we show that for small the operator in (3.29) is a contraction from into itself.

If , then, according to Theorem 2.4, the function determined by (3.15)-(3.16) and therefore by (3.28) belongs to . Note that , , and . It follows from Lemma 2.3, Theorem 2.4, and (3.29) that and where comes from Lemma 3.3. Noting that ad , we choose such that when one has So, . For , by (3.29) The function is a solution of the Cauchy problem Thus, by Lemma 3.3 Applying Lemma 3.3 to and , respectively, one gets If we choose so that when then

One sees that for such , the operator is a contraction from into itself and, therefore, has a unique fixed point in . Thus, we have shown.

Theorem 3.5. If functions , and satisfy the conditions of Problem 1, and are determined by (3.30) and (3.32), (3.37) respectively, then in , Problem 1 has a unique solution with and .

Furthermore, we have the following.

Theorem 3.6. Let be as in Problem 1. Then, there exists an almost periodic type solution for Problem 1 in .

Proof. We show that the conclusion of Theorem 3.5 can be extended to . Let Problem 1 has solution in . By Theorem 3.5, . Suppose that . Consider the problem

For we can write the integral equation similar to (3.29), but this time its domain is . As the proof above, define the ball in ; then there exists a such that the operator is a contraction from into itself. So, (3.29) has a solution for the domain . This contradicts the definition of . We must have .

For the stability, we have the following.

Theorem 3.7. Let functions , and be as in Problem 1. If and are solutions to (3.15)–(3.17), then where depends on , , and only.

Proof. By (3.29), By Lemma 3.3, Since the function is the solution of the Cauchy problem one has Therefore, Using Lemma 3.1, we get the estimates desired if we let where The proof is complete.

Corollary 3.8. If Problem 1 has a solution in , then it has a unique one.

Acknowledgment

The research is supported by the NSF of China (no. 11071048).

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