Almost Periodic (Type) Solutions to Parabolic Cauchy Inverse Problems

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 [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_𝑠𝑓(𝑡)=𝑓(𝑡+𝑠) (resp., 𝑅_𝑠𝑓(𝑡,𝑍)=𝑓(𝑡+𝑠,𝑍),(𝑡,𝑍)∈𝐽×Ω).

Definition 1.1. (1) A function 𝑓∈𝒞(𝐽) is called almost periodic if for every 𝜖>0 the set 𝑇𝑓,𝜖=𝜏∈𝐽∶𝑅_𝜏𝑓−𝑓<𝜖 (1.1) is relatively dense in 𝐽. Denote by 𝒜𝒫(𝐽) the set of all such functions. The number (vector) 𝜏 is called 𝜖-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) A function 𝑓∈𝒞(𝐽)(𝒞(𝐽×Ω)) is called pseudo almost periodic if 𝑓=𝑔+𝜑, (1.2) where 𝑔∈𝒜𝒫(𝐽)(𝒜𝒫(𝐽×Ω)) and 𝜑∈𝒫𝒜𝒫₀(𝐽)(𝒫𝒜𝒫₀(𝐽×Ω)), 𝒫𝒜𝒫₀(𝐽)=𝜑∈𝒞(𝐽)∶lim_𝑟→∞1(2𝑟)^𝑛_{[−𝑟,𝑟]}^𝑛𝜑(𝑥)𝑑𝑥=0,𝒫𝒜𝒫₀(𝐽×Ω)=𝜑∈𝒞(𝐽×Ω)∶lim_𝑟→∞1(2𝑟)^𝑛_{[−𝑟,𝑟]}^𝑛𝜑(𝑥,𝑍)𝑑𝑥=0, (1.3) 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 (𝐽)∈{(𝐽)(𝐽,()∈{()(.𝒜𝒫𝒯𝒜𝒫,𝒫𝒜𝒫)}𝒜𝒫𝒯𝐽×Ω𝒜𝒫𝐽×Ω,𝒫𝒜𝒫𝐽×Ω)} (1.4)

Members of 𝒜𝒫𝒯(𝐽)(𝒜𝒫𝒯(𝐽×Ω)) are called almost periodic type.

We will use the notations throughout the paper: 𝐑^𝑚_𝑇=𝐑^𝑚×(0,𝑇),‖𝐹‖_𝑇=sup{|𝐹(𝑥,𝑡)|∶𝑥∈𝐑^𝑛,0≤𝑡≤𝑇}. 𝐹∈𝒜𝒫𝒯(𝐑^𝑛×𝐑^𝑚_𝑇) means that 𝐹(𝑥⁽¹⁾,𝑥⁽²⁾,𝑡) is almost periodic type in 𝑥⁽¹⁾∈𝐑^𝑛 and uniformly for (𝑥⁽²⁾,𝑡)∈𝐑^𝑚_𝑇;𝐹∈𝒜𝒫𝒯(𝐑^𝑛×𝐑^𝑚) means that 𝐹(𝑥⁽¹⁾,𝑥⁽²⁾) is almost periodic type in 𝑥⁽¹⁾∈𝐑^𝑛 and uniformly for 𝑥⁽²⁾∈𝐑^𝑚.

Let𝑍()=1𝑥,𝑡;𝜉,𝑠2𝜋(𝑡−𝑠)^𝑛+𝑚−∑𝑥exp^𝑖−𝜉_𝑖²4(),𝑡−𝑠𝑥,𝜉∈𝐑^𝑛+𝑚, (1.5) 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 𝑇>0. If 𝜑∈𝒜𝒫𝒯(𝐑^𝑛×𝐑^𝑚) and 𝑢(𝑥,𝑡;𝑠)=_{𝐑𝑛+𝑚}𝜑(𝜉)𝑍(𝑥,𝑡;𝜉,𝑠)𝑑𝜉, (2.1) then for each fixed 𝑠∈[0,𝑇)𝑢∈𝒜𝒫𝒯(𝐑^𝑛×𝐑^𝑚×[𝑠,𝑇]).

Proof. First consider the case that 𝜑∈𝒜𝒫(𝐑^𝑛×𝐑^𝑚). Let 𝜏∈𝐑^𝑛 be an 𝜖-translation vector of 𝜑: 𝑢𝑥⁽¹⁾+𝜏,𝑥⁽²⁾,𝑡;𝑠−𝑢𝑥⁽¹⁾,𝑥⁽²⁾=,𝑡;𝑠_{𝐑𝑛+𝑚}𝜑𝜉⁽¹⁾,𝜉⁽²⁾𝑍𝑥⁽¹⁾+𝜏,𝑥⁽²⁾,𝑡;𝜉⁽¹⁾,𝜉⁽²⁾,𝑠−𝑍𝑥⁽¹⁾,𝑥⁽²⁾,𝑡;𝜉⁽¹⁾,𝜉⁽²⁾,𝑠𝑑𝜉⁽¹⁾𝑑𝜉⁽²⁾=_{𝐑𝑛+𝑚}𝜑𝑥⁽¹⁾+𝜏+𝜉⁽¹⁾,𝑥⁽²⁾+𝜉⁽²⁾−𝜑𝑥⁽¹⁾+𝜉⁽¹⁾,𝑥⁽²⁾+𝜉⁽²⁾𝑍(0,𝑡;𝜉𝑠)𝑑𝜉, (2.2) where 0∈𝐑^𝑛+𝑚 is the zero vector. Note that∫_{𝐑𝑛+𝑚}𝑍(0,𝑡;𝜉,𝑠)𝑑𝜉=1, we get 𝑅_𝜏𝑢−𝑢≤𝑅_𝜏𝜑−𝜑_{𝐑𝑛+𝑚}𝑍(0,𝑡;𝜉,𝑠)𝑑𝜉<𝜖, (2.3) where 𝑡∈[𝑠,𝑇] and 𝑥⁽²⁾∈𝐵 with 𝐵 a bounded subset of 𝐑^𝑚. This shows that 𝑢∈𝒜𝒫(𝐑^𝑛×𝐑^𝑚×[𝑠,𝑇]).
To show that 𝑢∈𝒫𝒜𝒫(𝐑^𝑛×𝐑^𝑚×[𝑠,𝑇]) if 𝜑∈𝒫𝒜𝒫(𝐑^𝑛×𝐑^𝑚), we only need to show that 𝑢∈𝒫𝒜𝒫₀(𝐑^𝑛×𝐑^𝑚×[𝑠,𝑇]) if 𝜑∈𝒫𝒜𝒫₀(𝐑^𝑛×𝐑^𝑚). That is, lim_𝑟→∞1(2𝑟)^𝑛_{[−𝑟,𝑟]}^𝑛𝑢𝑥⁽¹⁾,𝑥⁽²⁾,𝑡;𝑠𝑑𝑠⁽¹⁾=0, (2.4) uniformly with respect to (𝑥⁽²⁾,𝑡)∈Ω, here Ω is any compact subset of 𝐑^𝑚×[𝑠,𝑇].
Since 𝜑∈𝒫𝒜𝒫₀(𝐑^𝑛×𝐑^𝑚), for 𝜖>0 there exist positive numbers 𝐴 and 𝑟₀ such that, when 𝑟≥𝑟₀ for all 𝜉⁽¹⁾∈[−𝐴,𝐴]^𝑛 and 𝜉⁽²⁾∈Ω∩𝐑^𝑚, one has 1(2𝑟)^𝑛_{[−𝑟,𝑟]}^𝑛𝜑𝑥⁽¹⁾+𝜉⁽¹⁾,𝜉⁽²⁾𝑑𝑥⁽¹⁾<𝜖3_{−𝐴−∞},+^∞_𝐴𝜑𝑥⁽¹⁾+𝜉⁽¹⁾,𝜉⁽²⁾𝑍0,𝑥⁽²⁾,𝑡;𝜉,𝑠𝑑𝜉<2𝜖3. (2.5) Therefore, 1(2𝑟)^𝑛_{[−𝑟,𝑟]}^𝑛𝑢𝑥⁽¹⁾,𝑥⁽²⁾,𝑡;𝑠𝑑𝑥⁽¹⁾≤1(2𝑟)^𝑛_{[−𝑟,𝑟]}^𝑛_𝐑^𝑛+𝑚𝜑𝑥⁽¹⁾+𝜉⁽¹⁾,𝜉⁽²⁾𝑍0,𝑥⁽²⁾,𝑡;𝜉,𝑠𝑑𝜉𝑑𝑥⁽¹⁾=^−𝐴_−∞+^𝐴_−𝐴+^∞_𝐴𝑍0,𝑥⁽²⁾1,𝑡;𝜉,𝑠𝑑𝜉(2𝑟)^𝑛_{[−𝑟,𝑟]}^𝑛𝜑𝑥⁽¹⁾+𝜉⁽¹⁾,𝜉⁽²⁾𝑑𝑥⁽¹⁾<𝜖 (2.6) uniformly with respect to (𝑥⁽²⁾,𝑡)∈Ω, where by ∫^𝑏_𝑎𝐹(𝜉)𝑑𝜉 we mean that ^𝑏_𝑎𝐹(𝜉)𝜉=_{[𝑎,𝑏]𝑛+𝑚}𝐹(𝜉)𝑑𝜉=^𝑏_𝑎⋯^𝑏_𝑎𝐹(𝜉)𝑑𝜉₁𝑑𝜉₂⋯𝑑𝜉_𝑛+𝑚. (2.7) This shows that 𝑢∈𝒫𝒜𝒫₀(𝐑^𝑛×𝐑^𝑚). The proof is complete.

Corollary 2.2. 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]).

Proof. Note that 𝜕𝑢(𝑥,𝑡;𝑠)𝜕𝑥_𝑖=_{𝐑𝑛+𝑚}𝜑(𝜉)𝜕𝑍(𝑥,𝑡;𝜉,𝑠)𝜕𝑥_𝑖𝑑𝜉=−_{𝐑𝑛+𝑚}𝜑(𝜉)𝜕𝑍(𝑥,𝑡;𝜉,𝑠)𝜕𝜉_𝑖=𝑑𝜉_{𝐑𝑛+𝑚}𝜕𝜑(𝜉)𝜕𝜉_𝑖𝑍(𝑥,𝑡;𝜉,𝑠)𝑑𝜉. (2.8)

By Lemma 2.1 we get the conclusion.

Lemma 2.3. If 𝑓(𝑥,𝑡)∈𝒜𝒫𝒯(𝐑^𝑛×𝐑^𝑚_𝑇) and 𝑢(𝑥,𝑡)=^𝑡₀𝑑𝑠_{𝐑𝑛+𝑚}𝑓(𝜉,𝑠)𝑍(𝑥,𝑡;𝜉,𝑠)𝑑𝜉, (2.9) then 𝑢 and 𝜕𝑢(𝑥,𝑡)/𝜕𝑥_𝑖(𝑖=1,2,…,𝑛+𝑚) are all in 𝒜𝒫𝒯(𝐑^𝑛×𝐑^𝑚_𝑇).

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

Theorem 2.4. Consider the heat problem 𝜕𝑢−𝜕𝑡^𝑛+𝑚_𝑖=1𝜕²𝑢𝜕𝑥²_𝑖+𝑏_𝑖(𝑥,𝑡)𝜕𝑢𝜕𝑥_𝑖−𝑐(𝑥,𝑡)𝑢=𝑓(𝑥,𝑡),(𝑥,𝑡)∈𝐑_𝑇^𝑛+𝑚,𝑢(𝑥,0)=𝜑(𝑥),𝑥∈𝐑^𝑛+𝑚. (2.10) If 𝑓(𝑥,𝑡),𝑏_𝑖(𝑥,𝑡),𝜕𝑏_𝑖/𝜕𝑥_𝑗(𝑖,𝑗=1,2,…,𝑛+𝑚),𝑐(𝑥,𝑡) are in 𝒜𝒫𝒯(𝐑^𝑛×𝐑^𝑚_𝑇) and 𝜑,𝜕𝜑/𝜕𝑥_𝑖(𝑖=1,2,…,𝑛+𝑚) are in 𝒜𝒫𝒯(𝐑^𝑛×𝐑^𝑚), then (2.10) has a unique solution 𝑢∈𝒜𝒫𝒯(𝐑^𝑛×𝐑^𝑚_𝑇).

Proof. Problem (2.10) has the standard solution (see [37]): 𝑢(𝑥,𝑡)=_{𝐑𝑛+𝑚}𝜑(𝜉)Γ(𝑥,𝑡;𝜉,0)𝑑𝜉+^𝑡₀𝑑𝑠_{𝐑𝑛+𝑚}𝑓(𝜉,𝑠)Γ(𝑥,𝑡;𝜉,𝑠)𝑑𝜉=𝐼₁+𝐼₂, (2.11) where Γ(𝑥,𝑡;𝜉,𝑠)=𝑍(𝑥,𝑡;𝜉,𝑠)+^𝑡_{𝑠𝐑𝑛+𝑚}𝑍Φ=𝑥,𝑡;𝑦,𝜂⋅Φ𝑦,𝜂;𝜉,𝑠𝑑𝑦𝑑𝜂,𝑦,𝜂;𝜉,𝑠^∞_𝑙=1(𝐿𝑍)_𝑙𝑦,𝜂;𝜉,𝑠,(𝐿𝑍)₁()=𝐿𝑍,𝐿𝑍_𝑙+1𝑦,𝜂;𝜉,𝑠=^𝑡_{𝑠𝐑𝑛+𝑚}𝐿𝑍𝑦,𝜂;𝑣,𝜎(𝐿𝑍)_𝑙(𝑣,𝜎;𝜉,𝑠)𝑑𝑣𝑑𝜎, (2.12) and 𝐿 is the parabolic operator 𝐿=^𝑛+𝑚_𝑖=1𝜕²𝜕𝑥²_𝑖+𝑏_𝑖()𝜕𝑥,𝑡𝜕𝑥_𝑖()−𝜕+𝑐𝑥,𝑡.𝜕𝑡 (2.13) Now, we show that 𝑢∈𝒜𝒫𝒯(𝐑^𝑛×𝐑^𝑚_𝑇)∶𝐼₁=_{𝐑𝑛+𝑚}𝜑(𝜉)𝑍()+𝑥,𝑡;𝜉,0𝑑𝜉_{𝐑𝑛+𝑚}𝜑(𝜉)𝑑𝜉^𝑡₀𝑑𝜂_{𝐑𝑛+𝑚}𝑍𝑥,𝑡;𝑦,𝜂^∞_𝑙=1(𝐿𝑍)_𝑙𝑦,𝜂;𝜉,0𝑑𝑦=𝑢₁(𝑥,𝑡)+^∞_𝑙=1^𝑡₀𝑑𝜂_{𝐑𝑛+𝑚}𝑍𝑥,𝑡;𝑦,𝜂𝑑𝑦_{𝐑𝑛+𝑚}𝜑(𝜉)(𝐿𝑍)_𝑙𝐼𝑦,𝜂;𝜉,0𝑑𝜉,₂=^𝑡₀𝑑𝑠_{𝐑𝑛+𝑚}𝑓()𝑍()+𝜉,𝑠𝑥,𝑡;𝜉,𝑠𝑑𝜉_𝑡0𝑑𝑠_{𝐑𝑛+𝑚}𝑓(𝜉,𝑠)𝑑𝜉^𝑡_𝑠𝑑𝜂_{𝐑𝑛+𝑚}𝑍𝑥,𝑡;𝑦,𝜂^∞_𝑙=1(𝐿𝑍)_𝑙𝑍𝑦,𝜂;𝜉,𝑠𝑑𝑦=𝑢₂(𝑥,𝑡)+^∞_𝑙=1^𝑡₀𝑑𝜂_{𝐑𝑛+𝑚}𝑍𝑥,𝑡;𝑦,𝜂𝑑𝑦^𝜂₀𝑑𝑠_{𝐑𝑛+𝑚}𝑓(𝜉,𝑠)(𝐿𝑍)_𝑙𝑦,𝜂;𝜉,𝑠𝑑𝜉. (2.14) By Lemmas 2.1 and 2.3, 𝑢_𝑖∈𝒜𝒫𝒯(𝐑^𝑛×𝐑^𝑚_𝑇)(𝑖=1,2,).
Obviously, 𝐿𝑍(𝑥,𝑡;𝜉,𝑠)=^𝑛+𝑚_𝑖=1𝑏_𝑖()()𝑥,𝑡𝜕𝑍𝑥,𝑡;𝜉,𝑠𝜕𝑥_𝑖+𝑐(𝑥,𝑡)𝑍(𝑥,𝑡;𝜉,𝑠). (2.15) By Lemmas 2.1 and 2.3 we only need to show that the functions 𝑤_𝑙(𝑥,𝑡)=_{𝐑𝑛+𝑚}𝜑(𝜉)(𝐿𝑍)_𝑙()𝑣𝑥,𝑡;𝜉,0𝑑𝜉,_𝑙(𝑥,𝑡)=^𝑡₀𝑑𝑠_{𝐑𝑛+𝑚}𝑓(𝜉,𝑠)(𝐿𝑍)_𝑙(𝑥,𝑡;𝜉,𝑠)𝑑𝜉, (2.16) are in 𝒜𝒫𝒯(𝐑^𝑛×𝐑^𝑚_𝑇). We do this by induction. By Lemmas 2.1 and 2.3 and Corollary 2.2, it is true for the case 𝑙=1. Suppose that 𝑤_𝑙(𝑥,𝑡),𝑣_𝑙(𝑥,𝑡)∈𝒜𝒫𝒯(𝐑^𝑛×𝐑^𝑚_𝑇). Then, 𝑤_𝑙+1(𝑥,𝑡)=_{𝐑𝑛+𝑚}𝜑(𝜉)𝑑𝜉^𝑡₀𝑑𝜂_{𝐑𝑛+𝑚}𝐿𝑍𝑥,𝑡;𝑦,𝜂(𝐿𝑍)_𝑙=𝑦,𝜂;𝜉,0𝑑𝑦_𝑡0𝑑𝜂_{𝐑𝑛+𝑚}𝐿𝑍𝑥,𝑡;𝑦,𝜂𝑑𝑦_{𝐑𝑛+𝑚}𝜑(𝜉)(𝐿𝑍)_𝑙=𝑦,𝜂;𝜉,0𝑑𝜉_𝑡0𝑑𝜂_{𝐑𝑛+𝑚}𝑤_𝑙𝑦,𝜂𝐿𝑍𝑥,𝑡;𝑦,𝜂𝑑𝑦. (2.17) By the induction assumption and Lemma 2.3, we have 𝑤_𝑙+1(𝑥,𝑡)∈𝒜𝒫𝒯(𝐑^𝑛×𝐑^𝑚_𝑇). Similarly one shows that 𝑣_𝑙+1(𝑥,𝑡)∈𝒜𝒫𝒯(𝐑^𝑛×𝐑^𝑚_𝑇). 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:𝑥=𝑥₁,𝑥₂,…,𝑥_𝑛−1,𝜉=𝜉₁,𝜉₂,…,𝜉_𝑛−1,𝑋=𝑥,𝑥_𝑛,𝜁=𝜉,𝜉_𝑛. (3.1)

The following estimates are easily obtained:^𝑡₀𝑑𝑠_𝐑𝑛𝑍(𝑋,𝑡;𝜁,𝑠)𝑑𝜁_𝑇≤𝑚(𝑇), (3.2) 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 of [38].

Lemma 3.1. Let 𝜑, 𝜙, and 𝜒 be real, continuous functions on [0,𝑇] with 𝜒≥0. If 𝜑(𝑡)≤𝜙(𝑡)+^𝑡₀𝜒(𝑠)𝜑(𝑠)𝑑𝑠(𝑡∈[0,𝑇]), (3.3) then 𝜑(𝑡)≤𝜙(𝑡)+^𝑡₀𝜒(𝑠)𝜙(𝑠)exp^𝑡_𝑠𝜒𝜌𝑑𝜌𝑑𝑠(𝑡∈[0,𝑇]). (3.4)

Lemma 3.2. Let 𝜑 be a continuous function on [0,𝑇]. If 𝜙, 𝜒₁, and 𝜒₂ are nondecreasing and nonnegative on [0,𝑇] and 𝜑(𝑡)≤𝜙(𝑡)+𝜒₁(𝑡)^𝑡₀𝜑(𝑠)𝑑𝑠+𝜒₂(𝑡)^𝑡₀𝜑(𝑠)√([𝑡−𝑠𝑑𝑠𝑡∈0,𝑇]), (∗) then 𝜑(𝑡)≤𝜙(𝑡)1+𝑡𝜒₁(𝑡)+2√𝑡𝜒₂(𝑡)𝑒^{𝑡𝜒(𝑡)}, (3.5) where 𝜒(𝑡)=𝑡𝜒²₁(𝑡)+4√𝑡𝜒₁(𝑡)𝜒₂(𝑡)+𝜋𝜒²₂(𝑡). (3.6)

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, 𝜒₁, and 𝜒₂, one gets 𝜑(𝑡)≤𝜙(𝑡)1+𝑡𝜒₁(𝑡)+2√𝑡𝜒₂(𝑡)+𝑡𝜒²₁(𝑡)+4√𝑡𝜒₁(𝑡)𝜒₂(𝑡)+𝜋𝜒²₂(𝑡)^𝑡₀𝜑(𝑠)𝐷𝑠. (3.7) Using Lemma 3.1 leads to the conclusion.

Lemma 3.3. Let 𝐹(𝑋,𝑡)∈𝒞(𝐑^𝑛_𝑇) and 𝜑∈𝒞(𝐑^𝑛). If 𝑢(𝑋,𝑡) is a solution of the problem 𝑢_𝑡−Δ𝑢+𝑞𝑢=𝐹(𝑋,𝑡),(𝑋,𝑡)∈𝐑^𝑛_𝑇,𝑢(𝑋,0)=𝜑(𝑋),𝑋∈𝐑^𝑛, (3.8) then ‖𝑢‖_𝑇≤𝐾(𝑇)𝑇‖𝐹‖_𝑇+𝜑, (3.9) where 𝐾(𝑇)=1+𝑇‖𝑞‖_𝑇𝑒^{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)=_𝐑𝑛𝜑(𝜁)𝑍(𝑋,𝑡;𝜁,0)𝑑𝜁+^𝑡₀𝑑𝑠_𝐑𝑛𝐹()𝑍()−𝜁,𝑠𝑋,𝑡;𝜁,𝑠𝑑𝜁_𝑡0𝑑𝑠_𝐑𝑛𝑞(𝜉,𝑠)𝑢(𝜁,𝑠)𝑍(𝑋,𝑡;𝜁,𝑠)𝑑𝜁, (3.10) so, ‖𝑢‖_𝑡≤𝜑+^𝑡₀‖𝐹‖_𝑠𝑑𝑠+^𝑡₀𝑞_𝑠‖𝑢‖_𝑠𝑑𝑠. (3.11) By Lemma 3.1 one gets the desired result. The proof is complete.

Problem 1. Find functions 𝑢(𝑋,𝑡)∈𝒜𝒫𝒯(𝐑^𝑛−1×𝐑_𝑇) and 𝑞(𝑥,𝑡)∈𝒜𝒫𝒯(𝐑_𝑇^𝑛−1) such that 𝑢_𝑡−Δ𝑢+𝑞𝑢=𝐹(𝑋,𝑡),(𝑋,𝑡)∈𝐑^𝑛_𝑇,𝑢(𝑋,0)=𝜑(𝑋),𝑋∈𝐑^𝑛𝑢()(),()∈,𝑥,0,𝑡=ℎ𝑥,𝑡𝑥,𝑡𝐑_𝑇^𝑛−1, (3.12)(3.13)(3.14) where 𝜑(𝑥,𝑥_𝑛),𝜑_{𝑥𝑛𝑥𝑛}(𝑥,𝑥_𝑛)∈𝒜𝒫𝒯(𝐑^𝑛−1×𝐑),ℎ(𝑥,𝑡)≥const>0,ℎ(𝑥,𝑡),(Δℎ−ℎ_𝑡)∈𝒜𝒫𝒯(𝐑_𝑇^𝑛−1₀), and 𝐹(𝑥,𝑥_𝑛,𝑡),𝐹_{𝑥𝑛𝑥𝑛}(𝑥,𝑥_𝑛,𝑡)∈𝒜𝒫𝒯(𝐑^𝑛−1×𝐑_𝑇0). 𝑇₀>𝑇>0 are constants.

By (3.13) and (3.14), one sees that ℎ(𝑥,0)=𝜑(𝑥,0).

We have the following additional problem.

Problem 2. Find functions 𝑊(𝑋,𝑡)∈𝒜𝒫𝒯(𝐑^𝑛−1×𝐑_𝑇) and 𝑞(𝑥,𝑡)∈𝒜𝒫𝒯(𝐑_𝑇^𝑛−1) such that 𝑊_𝑡−Δ𝑊+𝑞𝑊=𝐹_{𝑥𝑛𝑥𝑛}(𝑋,𝑡),(𝑋,𝑡)∈𝐑^𝑛_𝑇,𝑊(𝑋,0)=𝜑_{𝑥𝑛𝑥𝑛}(𝑋),𝑋∈𝐑^𝑛𝑊(),𝑥,0,𝑡=ℎ_𝑡−Δℎ+𝑞ℎ−𝐹(𝑥,0,𝑡),(𝑥,𝑡)∈𝐑_𝑇^𝑛−1. (3.15)(3.16)(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_𝑥𝑛(𝑋,𝑡). Then, 𝑉(𝑋,𝑡) satisfies 𝑉_𝑡−Δ𝑉+𝑞𝑉=𝐹_𝑥𝑛(𝑋,𝑡),(𝑋,𝑡)∈𝐑^𝑛_𝑇,𝑉(𝑋,0)=𝜑_𝑥𝑛(𝑋),𝑋∈𝐑^𝑛𝑉_𝑥𝑛,(𝑥,0,𝑡)=ℎ_𝑡−Δℎ+𝑞ℎ−𝐹(𝑥,0,𝑡),𝜑(𝑥,0)=ℎ(𝑥,0),(𝑥,𝑡)∈𝐑_𝑇^𝑛−1. (3.18)(3.19)(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_𝑥𝑛(𝑋,𝑡). Obviously 𝑉(𝑋,𝑡)∈𝒜𝒫𝒯(𝐑^𝑛−1×𝐑_𝑇) if 𝑢(𝑋,𝑡)∈𝒜𝒫𝒯(𝐑^𝑛−1×𝐑_𝑇).
On the other hand, if 𝑉(𝑋,𝑡)∈𝒜𝒫𝒯(𝐑^𝑛−1×𝐑_𝑇) and 𝑞(𝑥,𝑡)∈𝒜𝒫𝒯(𝐑_𝑇^𝑛−1) 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_𝑥𝑛. Thus, we define 𝑢(𝑋,𝑡)=^𝑥𝑛₀𝑉𝑥,𝑦,𝑡𝑑𝑦+Φ(𝑥,𝑡). (3.21) It follows from (3.14) that Φ=ℎ. Now, 𝑢 satisfies (3.13) because 𝑢(𝑋,0)=^𝑥𝑛₀𝑉𝑥,𝑦,0𝑑𝑦+ℎ(𝑥,0)=𝜑(𝑋)−𝜑(𝑥,0)+ℎ(𝑥,0)=𝜑(𝑋). (3.22) It follows from (3.18), (3.20), and (3.21) that 𝑢_𝑡−Δ𝑢+𝑞𝑢=^𝑥𝑛₀𝑉_𝑡−Δ𝑉+𝑞𝑉𝑑𝑦+^𝑥𝑛₀𝜕²𝑉𝑥,𝑦,𝑡𝜕𝑦²−𝜕𝑑𝑦₂𝜕𝑥²_𝑛^𝑥𝑛₀𝑉𝑥,𝑦,𝑡𝑑𝑦+ℎ_𝑡−∇ℎ+𝑞ℎ=𝐹𝑥,𝑥_𝑛,𝑡−𝐹(𝑥,0,𝑡)+𝑉_𝑥𝑛𝑥,𝑥_𝑛,𝑡−𝑉_𝑥𝑛(𝑥,0,𝑡)−𝑉_𝑥𝑛𝑥,𝑥_𝑛,𝑡+ℎ_𝑡−Δℎ+𝑞ℎ=𝐹(𝑋,𝑡). (3.23) Thus, 𝑢 satisfies (3.12) and (𝑢,𝑞) is a unique solution of Problem 1.
It follows from 𝑉(𝑋,𝑡)∈𝒜𝒫𝒯(𝐑^𝑛−1×𝐑_𝑇,ℎ(𝑥,𝑡)∈𝒜𝒫𝒯(𝐑_𝑇^𝑛−1), and (3.21) that 𝑢(𝑋,𝑡) in (3.21) is in 𝒜𝒫𝒯(𝐑^𝑛−1×𝐑_𝑇).
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_𝑥𝑛(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)=^𝑥𝑛₀𝑊𝑥,𝑦,𝑡𝑑𝑦+Φ(𝑥,𝑡), (3.24) where Φ is the solution of the Cauchy problem Φ_𝑡−ΔΦ+𝑞Φ=𝑊_𝑥𝑛(𝑥,0,𝑡)+𝐹_𝑥𝑛(𝑥,0,𝑡),(𝑥,𝑡)∈𝐑_𝑇^𝑛−1Φ(𝑥,0)=𝜑_𝑥𝑛(𝑥,0),𝑥∈𝐑^𝑛−1. (3.25) Since 𝑊(𝑋,𝑡)∈𝒜𝒫𝒯(𝐑^𝑛−1×𝐑_𝑇), 𝑊_𝑥𝑛(𝑥,0,𝑡)∈𝒜𝒫𝒯(𝐑_𝑇^𝑛−1). By Theorem 2.4, Φ(𝑥,𝑡)∈𝒜𝒫𝒯(𝐑_𝑇^𝑛−1) and so 𝑉(𝑋,𝑡)∈𝒜𝒫𝒯(𝐑^𝑛−1×𝐑_𝑇).
Obviously, 𝑉_𝑥𝑛(𝑥,0,𝑡)=𝑊(𝑥,0,𝑡)=ℎ_𝑡−Δℎ+𝑞ℎ−𝐹(𝑥,0,𝑡), and this shows that 𝑉 satisfies (3.20). 𝑉 satisfies (3.19) because 𝑉(𝑋,0)=^𝑥𝑛₀𝑊𝑥,𝑦,0𝑑𝑦+Φ(𝑥,0)=^𝑥𝑛₀𝜑_{𝑥𝑛𝑥𝑛}𝑥,𝑦𝑑𝑦+Φ(𝑥,0)=𝜑_𝑥𝑛𝑥,𝑥_𝑛−𝜑_𝑥𝑛(𝑥,0)+Φ(𝑥,0)=𝜑_𝑥𝑛(𝑋). (3.26) Finally, 𝑉_𝑡−Δ𝑉+𝑞𝑉=^𝑥𝑛₀𝑊_𝑡−Δ𝑊+𝑞𝑊𝑑𝑦+^𝑥𝑛₀𝜕²𝑊𝑥,𝑦,𝑡𝜕𝑦²−𝜕𝑑𝑦₂𝜕𝑥²_𝑛^𝑥𝑛₀𝑊𝑥,𝜂,𝑡𝑑𝜂+Φ_𝑡−ΔΦ+𝑞Φ=𝐹_𝑥𝑛𝑥,𝑥_𝑛,𝑡−𝐹_𝑥𝑛(𝑥,0,𝑡)+𝑊_𝑥𝑛𝑥,𝑥_𝑛,𝑡−𝑊_𝑥𝑛(𝑥,0,𝑡)−𝑊_𝑥𝑛𝑥,𝑥_𝑛,𝑡+Φ_𝑡−ΔΦ+𝑞Φ=𝐹_𝑥𝑛(𝑋,𝑡). (3.27) 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):𝑊(𝑋,𝑡)=_𝐑𝑛𝜑_{𝑥𝑛𝑥𝑛}(𝜁)𝑍(𝑋,𝑡;𝜁,0)𝑑𝜁+^𝑡₀𝑑𝑠_𝐑𝑛𝐹_{𝑥𝑛𝑥𝑛}()𝑍()−𝜁,𝑠𝑋,𝑡;𝜁,𝑠𝑑𝜁_𝑡0𝑑𝑠_𝐑𝑛𝑞()𝑊()𝑍()=𝜉,𝑠𝜁,𝑠𝑋,𝑡;𝜁,𝑠𝑑𝜁,𝑞=𝐿𝑞Δℎ−ℎ_𝑡+𝐹(𝑥,0,𝑡)ℎ⁻¹+ℎ⁻¹_𝐑𝑛𝜑_{𝑥𝑛𝑥𝑛}(𝜁)𝑍(𝑥,0,𝑡;𝜁,0)𝑑𝜁+ℎ^−1𝑡₀𝑑𝑠_𝐑𝑛𝐹_{𝑥𝑛𝑥𝑛}(𝜁,𝑠)𝑍(𝑥,0,𝑡;𝜁,𝑠)𝑑𝜁−ℎ^−1𝑡₀𝑑𝑠_𝐑𝑛𝑞(𝜉,𝑠)𝑊(𝜁,𝑠)𝑍(𝑥,0,𝑡;𝜁,𝑠)𝑑𝜁(𝑥,𝑡)∈𝐑_𝑇^𝑛−1, (3.28)(3.29) 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_𝑇^𝑛−1), 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)∈𝒜𝒫𝒯(𝐑_𝑇^𝑛−1). That is, 𝐿 has a fixed point in 𝒜𝒫𝒯(𝐑_𝑇^𝑛−1). To this end, letΔℎ−ℎ_𝑡+𝐹(𝑥,0,𝑡)ℎ⁻¹+ℎ⁻¹_𝐑𝑛𝜑_{𝑥𝑛𝑥𝑛}(𝜁)𝑍(𝑥,0,𝑡;𝜁,0)𝑑𝜁+ℎ^−1𝑇₀𝑑𝑠_𝐑𝑛𝐹_{𝑥𝑛𝑥𝑛}(𝜁,𝑠)𝑍(𝑥,0,𝑡;𝜁,𝑠)𝑑𝜁_𝑇0=𝑀2. (3.30) Set 𝐵(𝑀,𝑇)={𝑞(𝑥,𝑡)∈𝒜𝒫𝒯(𝐑_𝑇^𝑛−1)∶‖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 𝜑_{𝑥𝑛𝑥𝑛}∈𝒜𝒫𝒯(𝐑^𝑛−1×𝐑), (Δℎ−ℎ_𝑡)∈𝒜𝒫𝒯(𝐑_𝑇^𝑛−1), and 𝐹_{𝑥𝑛𝑥𝑛}∈𝒜𝒫𝒯(𝐑^𝑛−1×𝐑_𝑇). It follows from Lemma 2.3, Theorem 2.4, and (3.29) that 𝐿𝑞∈𝒜𝒫𝒯(𝐑_𝑇^𝑛−1) and𝐿𝑞_𝑇≤Δℎ−𝑡_𝑡+𝐹(𝑥,0,𝑡)ℎ⁻¹+ℎ⁻¹_𝐑𝑛𝜑_{𝑥𝑛𝑥𝑛}(𝜁)𝑍(𝑥,0,𝑡;𝜁,0)𝑑𝜁+ℎ^−1𝑡₀𝑑𝑠_𝐑𝑛𝐹_{𝑥𝑛𝑥𝑛}(𝜁,𝑠)𝑍(𝑥,0,𝑡;𝜁,𝑠)𝑑𝜁_𝑇+ℎ⁻¹_𝑇𝑞_𝑇‖𝑊‖_𝑇𝑚(𝑇)≤12𝑀+ℎ⁻¹_𝑇𝑀𝐾₀(𝑇)𝜑_{𝑥𝑛𝑥𝑛}+𝑇𝐹_{𝑥𝑛𝑥𝑛𝑇}𝑚(𝑇), (3.31) where 𝐾₀ comes from Lemma 3.3. Noting that 𝑚(𝑇)→0 ad 𝑇→0, we choose 𝑇₁≤𝑇₀ such that when 𝑇<𝑇₁ one hasℎ⁻¹_𝑇𝐾₀(𝑇)𝜑_{𝑥𝑛𝑥𝑛}+𝑇𝐹_{𝑥𝑛𝑥𝑛𝑇}𝑚(𝑇)≤12. (3.32) So, 𝐿𝑞∈𝐵(𝑀,𝑇). For 𝑞₁,𝑞₂∈𝐵(𝑀,𝑇), by (3.29)𝐿𝑞₁−𝐿𝑞₂≤ℎ⁻¹_𝑇𝑊₁𝑞₁−𝑊₂𝑞_2𝑇𝑚(𝑇)≤𝑚(𝑇)ℎ⁻¹_𝑇𝑊_1𝑇𝑞₁−𝑞_2𝑇+𝑞_2𝑇𝑊₁−𝑊_2𝑇. (3.33) The function 𝑉=𝑊₁−𝑊₂ is a solution of the Cauchy problem𝑉_𝑡−Δ𝑉+𝑞₁𝑉=𝑊₂𝑞₂−𝑞₁,(𝑋,𝑡)∈𝐑^𝑛_𝑇,𝑉(𝑋,0)=0,𝑋∈𝐑^𝑛. (3.34) Thus, by Lemma 3.3‖𝑉‖_𝑇≤𝐾₁(𝑇)𝑊_2𝑇𝑞₂−𝑞₁. (3.35) Applying Lemma 3.3 to (𝑊₁,𝑞₁) and (𝑊₂,𝑞₂), respectively, one gets𝑊₁≤𝐾₁(𝑇)𝜑_{𝑥𝑛𝑥𝑛}+𝑇𝐹_{𝑥𝑛𝑥𝑛},𝑊₂≤𝐾₂(𝑇)𝜑_{𝑥𝑛𝑥𝑛}+𝑇𝐹_{𝑥𝑛𝑥𝑛}. (3.36) If we choose 𝑇₂≤𝑇₁ so that when 𝑇≤𝑇₂2𝑚(𝑇)ℎ⁻¹_𝑇𝜑_{𝑥𝑛𝑥𝑛}+𝑇𝐹_{𝑥𝑛𝑥𝑛}𝐾₁(𝑇)1+𝑀𝐾₂(𝑇)≤1, (3.37) then

‖𝐿𝑞₁−𝐿𝑞₂‖_𝑇≤𝑚(𝑇)‖ℎ⁻¹‖_𝑇[‖𝑊₁‖+𝑀𝐾₁(𝑇)‖𝑊₂‖]‖𝑞₂−𝑞₁‖_𝑇≤𝑚(𝑇)‖ℎ⁻¹‖_𝑇𝐾₁(𝑇)(1+𝑀𝐾₂(𝑇))(‖𝜑_{𝑥𝑛𝑥𝑛}‖+𝑇‖𝐹_{𝑥𝑛𝑥𝑛}‖_𝑇)‖𝑞₂−𝑞₁‖_𝑇≤12‖𝑞₂−𝑞₁‖_𝑇.

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.

Theorem 3.5. 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_𝑇^𝑛−1).

Furthermore, we have the following.

Theorem 3.6. Let 𝐹,𝜑,andℎ be as in Problem 1. Then, there exists an almost periodic type solution for Problem 1 in 𝐑^𝑛_𝑇0.

Proof. We show that the conclusion of Theorem 3.5 can be extended to 𝐑^𝑛_𝑇0. Let 𝑇=sup{𝑠∶ Problem 1 has solution in 𝐑^𝑛_𝑠}. By Theorem 3.5, 𝑇>0. Suppose that 𝑇<𝑇₀. Consider the problem 𝑢_𝑡−Δ𝑢+𝑞𝑢=𝐹(𝑋,𝑡),𝑋∈𝐑^𝑛,𝑇≤𝑡≤𝑇₀,𝑢(𝑋,𝑇)=𝑓(𝑋),𝑋∈𝐑^𝑛,𝑢(𝑥,0,𝑡)=ℎ(𝑥,𝑡),(𝑥,𝑡)∈𝐑^𝑛−1×𝑇,𝑇₀. (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,𝑇₀]. As the proof above, define the ball 𝐵₁(𝑀,𝑆) in 𝒜𝒫𝒯(𝐑^𝑛−1×[𝑇,𝑇₀]); then there exists a 𝑡₀>0 such that the operator 𝐿_𝑡0 is a contraction from 𝐵₁(𝑀,𝑆) into itself. So, (3.29) has a solution for the domain 𝑥∈𝐑^𝑛−1,𝑡∈[𝑇,𝑇+𝑡₀]. This contradicts the definition of 𝑇. We must have 𝑇=𝑇₀.

For the stability, we have the following.

Theorem 3.7. Let functions ℎ_𝑖,𝜑_𝑖, and 𝐹_𝑖(𝑖=1,2) be as in Problem 1. If 𝑊_𝑖(𝑋,𝑡)∈𝒜𝒫𝒯(𝐑^𝑛−1×𝐑_𝑇) and 𝑞_𝑖∈𝒜𝒫𝒯(𝐑_𝑇^𝑛−1)(𝑖=1,2) are solutions to (3.15)–(3.17), then 𝑞₂−𝑞_1𝑡≤𝑐₁ℎ₂−ℎ_1𝑡+𝑐₂𝜕ℎ𝜕𝑡−Δ₂−ℎ_1𝑡+𝑐₃𝜕²𝜕𝑥²_𝑛𝜑₂−𝜑₁+𝑐₄𝐹₂−𝐹_1𝑡+𝑐₅𝜕²𝜕𝑥²_𝑛𝐹₂−𝐹_1𝑡, (3.39) where 𝑐_𝑗(1≤𝑗≤5) depends on 𝑇,‖ℎ₁‖_𝑡,‖(𝜕²/𝜕𝑥²_𝑛)𝜑₁‖, ‖𝐹₁‖_𝑡,‖(𝜕²/𝜕𝑥²_𝑛)𝐹₁‖_𝑡, and ‖𝑞_𝑖‖_𝑡(𝑖=1,2) only.

Proof. By (3.29), ℎ₁𝑞₂−𝑞₁=−𝑞₂ℎ₂−ℎ₁−𝜕ℎ𝜕𝑡−Δ₂−ℎ₁+𝐹₂(𝑥,0,𝑡)−𝐹₁(𝑥,0,𝑡)+_𝐑𝑛𝜕²𝜕𝜉²_𝑛𝜑₂(𝜁)−𝜑₁(𝜁)𝑍()+𝑥,0,𝑡;𝜁,0𝑑𝜁_𝑡0𝑑𝑠_𝐑𝑛𝜕²𝜕𝜉²_𝑛𝐹₂−𝐹₁𝑍()−𝑥,0,𝑡;𝜁,𝑠𝑑𝜁_𝑡0𝑑𝑠_𝐑𝑛𝑊₂−𝑊₁𝑞₂+𝑊₁𝑞₂−𝑞₁𝑍(𝑥,0,𝑡;𝜁,𝑠)𝑑𝜁. (3.40) By Lemma 3.3, 𝑊_1𝑡≤𝐾₁(𝑡𝜕)²𝜑₁𝜕𝑥²_𝑛𝜕+𝑡²𝐹₁𝜕𝑥²_𝑛𝑡. (3.41) Since the function 𝑉=𝑊₂−𝑊₁ is the solution of the Cauchy problem 𝑉_𝑡−Δ𝑉+𝑞₂𝜕𝑉=²𝜕𝑥²_𝑛𝐹₂−𝐹₁−𝑊₁𝑞₂−𝑞₁,(𝑋,𝑡)∈𝐑^𝑛_𝑇𝑉()=𝜕𝑋,0²𝜕𝑥²_𝑛𝜑₂(𝑋)−𝜑₁(𝑋),𝑋∈𝐑^𝑛, (3.42) one has ‖𝑉‖_𝑡≤𝐾₂(𝑇𝜕)²𝜕𝑥²_𝑛𝜑₂−𝜑₁𝜕+𝑡²𝜕𝑥²_𝑛𝐹₂−𝐹_1𝑡+𝐾₁(𝑡)𝜕²𝜕𝑥²_𝑛𝜑₁𝜕+𝑡²𝜕𝑥²_𝑛𝐹₁𝑞₂−𝑞_1𝑡. (3.43) Therefore, 𝑞₂−𝑞_1𝑡≤ℎ₁⁻¹_𝑡𝑞_2𝑡ℎ₂−ℎ_1𝑡+𝜕ℎ𝜕𝑡−Δ₂−ℎ_1𝑡+𝐹₂(𝑥,0,𝑡)−𝐹₁(𝑥,0,𝑡)_𝑡+𝜕²𝜕𝑥²_𝑛𝜑₂−𝜑₁𝜕+𝑇²𝜕𝑥²_𝑛𝐹₂−𝐹_1𝑡+^𝑡₀𝐾₂(𝑠𝜕)²𝜕𝑥²_𝑛𝜑₂−𝜑₁𝜕+𝑠²𝜕𝑥²_𝑛𝐹₂−𝐹_1𝑠𝑞_2𝑠+𝑑𝑠_𝑡0𝐾₁(𝑠𝜕)²𝜕𝑥²_𝑛𝜑₁𝜕+𝑠²𝜕𝑥²_𝑛𝐹_1𝑠𝑞₂−𝑞_1𝑠𝑑𝑠+^𝑡₀𝐾₁(𝑠)𝐾₂(𝑠𝜕)²𝜕𝑥²_𝑛𝜑₁𝜕+𝑠²𝜕𝑥²_𝑛𝐹_1𝑠𝑞₂−𝑞_1𝑠𝑞_2𝑠𝑑𝑠. (3.44) Using Lemma 3.1, we get the estimates desired if we let 𝑐₁=ℎ₁⁻¹_𝑡𝑞_2𝑡1+ℎ₁⁻¹_𝑡^𝑡₀𝜒(𝑠)expℎ₁⁻¹_𝑡^𝑡_𝑠𝜒𝜌,𝑐𝑑𝜌𝑑𝑠₂=ℎ₁⁻¹_𝑡1+ℎ₁⁻¹_𝑡^𝑡₀𝜒(𝑠)expℎ₁⁻¹_𝑡^𝑡_𝑠𝜒𝜌,𝑐𝑑𝜌𝑑𝑠₃=ℎ₁⁻¹_𝑡1+^𝑡₀𝑞_2𝑠𝐾₂(𝑠)𝑑𝑠1+ℎ₁⁻¹_𝑡^𝑡₀𝜒(𝑠)expℎ₁⁻¹_𝑡^𝑡_𝑠𝜒𝜌,𝑐𝑑𝜌𝑑𝑠₄=𝑐₂𝑐₅=𝑐₃, (3.45) where 𝜒(𝑡)=𝜕²𝜕𝑥²_𝑛𝜑₁𝜕+𝑡²𝜕𝑥²_𝑛𝐹_1𝑡𝐾₁(𝑡)1+𝐾₂(𝑡)𝑞_2𝑡. (3.46) The proof is complete.

Corollary 3.8. If Problem 1 has a solution in 𝐑^𝑛_𝑇0, then it has a unique one.

Acknowledgment

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