Almost Periodic ( Type ) Solutions to Parabolic Cauchy Inverse Problems

and Applied Analysis 3 2. Solutions of Parabolic Equations Lemma 2.1. Let T > 0. If φ ∈ APT R × R and


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 3-26 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 27-32 and references therein , but little has been done about the inverse problems except for our work in 33-36 .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 J ∈ {R, R n }.Let C J resp., C J × Ω , where Ω ⊂ R m denote the C * -algebra of bounded continuous complex-valued functions on J resp.J × Ω with the supremum norm.For f ∈ C J resp., C J ×Ω and s ∈ J, the translation of f by s is the function R s f t f t s resp., R s f t, Z f t s, Z , t, Z ∈ J × Ω . is relatively dense in J. Denote by AP J the set of all such functions.The number vector τ is called -translation number vector of f. 2 A function f ∈ C J × Ω is said to be almost periodic in t ∈ J and uniform on compact subsets of Ω if f •, Z ∈ AP J for each Z ∈ Ω and is uniformly continuous on J × K for any compact subset K ⊂ Ω. Denote by AP J × Ω the set of all such functions.For convenience, such functions are also called uniformly almost periodic. 3 uniformly with respect to Z ∈ K, where K is any compact subset of Ω. Denote by PAP J PAP J × Ω the set of all such functions.Set

1.4
Members of APT J APT J × Ω are called almost periodic type.We will use the notations throughout the paper: 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.

Solutions of Parabolic Equations
Proof.First consider the case that ϕ ∈ AP R n × R m .Let τ ∈ R n be an -translation vector of ϕ: where 0 ∈ R n m is the zero vector.Note that R n m Z 0, t; ξ, s dξ 1, we get where t ∈ s, T and uniformly with respect to x 2 , t ∈ Ω, here Ω is any compact subset of R m × s, T .Since ϕ ∈ PAP 0 R n × R m , for > 0 there exist positive numbers A and r 0 such that, when r ≥ r 0 for all ξ 1 ∈ −A, A n and ξ 2 ∈ Ω ∩ R m , one has

2.5
Therefore, uniformly with respect to x 2 , t ∈ Ω, where by The proof is similar to that of Lemma 2.1, so we omit it.

2.10
If Proof.Problem 2.10 has the standard solution see 37 : where and L is the parabolic operator Abstract and Applied Analysis f ξ, s LZ l y, η; ξ, s dξ.

2.17
By the induction assumption and Lemma 2.3, we have The proof is complete.

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:

3.1
The following estimates are easily obtained: where m T are positive and increasing for T ≥ 0 and m T → 0 as T → 0.
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 where K T 1 T q T e T q T .
One sees that K T depends on q T only and is bounded near zero.
Proof.The solution u can been written as u X, t By 3.13 and 3.14 , one sees that h x, 0 ϕ x, 0 .We have the following additional problem. 3.17 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 V X, t u x n X, t .Then, V X, t satisfies

3.20
So, if Problem 1 has a solution u, q , then Problems 3.18 -3.20 have the solution V, q with V X, t u x n X, t .Obviously V X, t On the other hand, if V X, t ∈ APT R n−1 × R T and q x, t ∈ APT R n−1 T satisfy 3.18 -3.20 , then we will show that Problem 1 has a unique solution u, q and u X, t ∈ APT R n−1 × R T .
The uniqueness comes from the uniqueness of Cauchy Problem 1 -2 .For the existence, note the fact that if u, q is a solution of 3.12 -3.14 , then V u x n .Thus, we define u X, t x n 0 V x, y, t dy Φ x, t .

3.23
Thus, u satisfies 3.12 and u, q is a unique solution of Problem 1.
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 V, q is a solution of 3.18 -3.20 , let W X, t V x n X, t .Then one can directly calculate that W, q is a solution of 3.15 -3.17 and W X, t ∈ APT R n−1 × R T .
On the other hand, if W, q is a solution of 3.15 -3.17 , let V X, t x n 0 W x, y, t dy Φ x, t , 3.24 where Φ is the solution of the Cauchy problem , 0, t , and this shows that V satisfies 3.20 .V satisfies 3.19 because V X, 0

3.26
Finally, This shows that V satisfies 3.18 .The proof is complete.
By 3.15 -3.16 we have the integral equation about W X, t : where W 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 q x, t ∈ APT R n−1 T , Theorem 2.4 shows the Cauchy problem 3.15 and 3.16 or equivalently 3.28 has a unique solution W ∈ APT R n × R T .Thus, 3.29 does define an operator L. To show that Problem 2 and so Problem 1 has a unique solution, we only need to show that 3.29 has a solution q x, t

3.30
Set B M, T {q x, t ∈ APT R n−1 T : q T ≤ M}.Now, we show that for small T the operator L in 3.29 is a contraction from B M, T into itself.
If q ∈ B M, T , then, according to Theorem 2.4, the function W determined by 3.15 -3.16 and therefore by 3.28 belongs to APT R n−1 × R T .Note that

3.31
where K 0 comes from Lemma 3.3.Noting that m T → 0 ad T → 0, we choose T 1 ≤ T 0 such that when T < T 1 one has So, Lq ∈ B M, T .For q 1 , q 2 ∈ B M, T , by 3.29

3.33
The function V W 1 − W 2 is a solution of the Cauchy problem

3.35
Abstract and Applied Analysis 13 Applying Lemma 3.3 to W 1 , q 1 and W 2 , q 2 , respectively, one gets

3.36
If we choose T 2 ≤ T 1 so that when One sees that for such T , the operator L is a contraction from B M, T into itself and, therefore, has a unique fixed point in B M, T .Thus, we have shown.Proof.We show that the conclusion of Theorem 3.5 can be extended to R n T 0 .Let T sup{s : Problem 1 has solution in R n s }.By Theorem 3.5, T > 0. Suppose that T < T 0 .Consider the problem

3.38
For q we can write the integral equation similar to 3.29 , but this time its domain is x ∈ R n−1 , t ∈ T, T 0 .As the proof above, define the ball B 1 M, S in APT R n−1 × T, T 0 ; then there exists a t 0 > 0 such that the operator L t 0 is a contraction from B 1 M, S into itself.So, 3.29 has a solution for the domain x ∈ R n−1 , t ∈ T, T t 0 .This contradicts the definition of T .We must have T T 0 .
For the stability, we have the following.

3.41
Since the function V W 2 − W 1 is the solution of the Cauchy problem 3.42 one has

3.44
Using Lemma 3.1, we get the estimates desired if we let

Definition 1 . 1 . 1 A
function f ∈ C J is called almost periodic if for every > 0 the set T f, τ ∈ J : R τ f − f < 1.1

T
, and F x n x n ∈ APT R n−1 ×R T .It follows from Lemma 2.3, Theorem 2.4, and 3.29 that Lq ∈ APT R n−1 T and

Theorem 3 . 5 .Theorem 3 . 6 .
If functions F, ϕ, and h satisfy the conditions of Problem 1, M and T are determined by 3.30 and 3.32 , 3.37 respectively, then in R n T , Problem 1 has a unique solution u, q with u ∈ APT R n−1 × R T and q ∈ APT R n−1 T .Furthermore, we have the following.Let F, ϕ, and h be as in Problem 1.Then, there exists an almost periodic type solution for Problem 1 in R n T 0 .
The proof is complete.Let ϕ, ∂ϕ/∂x i ∈ APT R n × R m , and let u be as in Lemma 2.1.Then, ∂u/∂x i ∈ APT R n × R m × s, T .
then u and ∂u x, t /∂x i i 1, 2, . . ., n m are all in of 38 .Let ϕ, φ, and χ be real, continuous functions on 0, T with χ ≥ 0. If Proof.Replacing ϕ s 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 φ, χ 1 , and χ 2 , one gets Let F X, t ∈ C R n T and ϕ ∈ C R n .If u X, t is a solution of the problem t s χ ρ dρ ds t ∈ 0, T .3.4 Lemma 3.2.Let ϕ be a continuous function on 0, T .If φ, χ 1 , and χ 2 are nondecreasing and nonnegative on 0, T and ϕ t ≤ φ t χ 1 t 1 t , ∂ 2 /∂x 2 n ϕ 1 , F 1 t , ∂ 2 /∂x 2n F 1 t , and q i t i 1, 2 only.
If Problem 1 has a solution in R n T 0 , then it has a unique one.