Slowly Oscillating Solutions of a Parabolic Inverse Problem: Boundary Value Problems

Volume 2010, Article ID 471491,12pages doi:10.1155/2010/471491

Research Article

Slowly Oscillating Solutions of a Parabolic Inverse Problem: Boundary Value Problems

Fenglin Yang and Chuanyi Zhang

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

Correspondence should be addressed to Fenglin Yang,[email protected] Received 11 October 2010; Accepted 20 December 2010

Academic Editor: Colin Rogers

Copyrightq2010 F. Yang and C. 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.

The existence and uniqueness of a slowly oscillating solution to parabolic inverse problems for a type of boundary value problem are established. Stability of the solution is discussed.

1. Introduction

It is well known that the space APR of almost periodic functions and some of its generalizations have many applicationse.g.,1–13and references therein. However, little has been done forAPRto inverse problems except for our work in14–16. Sarason in17 studied the spaceSORof slowly oscillating functions. This is aC^∗-subalgebra ofCR, the space of bounded, continuous, complex-valued functionsf onR with the supremum norm fsup{|fx|:x∈R}. Compared withAPR,SORis a quite large spacesee17–20.

What we are interested inSORis based on the belief thatSORcertainly has a variety of applications in many mathematical areas too. In15, we studied slowly oscillating solutions of a parabolic inverse problem for Cauchy problems. In this paper, we devote such solutions for a type of boundary value problem.

SetJ ∈ {R,Rⁿ}. LetCJ resp.,CJ×Ω, whereΩ ⊂ R^mdenote theC^∗-algebra of bounded continuous complex-valued functions onJresp.,J×Ωwith the supremum norm.

Forf∈ CJ resp.,CJ×Ωands∈J, the translate offbysis the functionR_sft fts resp.,R_sft, Z fts, Z,t, Z∈J×Ω.

Definition 1.1. 1A functionf ∈ CJis called slowly oscillating if for everyτ∈J,Rτf−f∈ C₀J, the space of the functions vanishing at infinity. Denote bySOJthe set of all such functions.

2A functionf ∈ CJ×Ωis said to be slowly oscillating int ∈ J and uniform on compact subsets ofΩ if f·, Z ∈ SOJ for eachZ ∈ Ω and is uniformly continuous on

J×Kfor any compact subsetK ⊂Ω. Denote bySOJ×Ωthe set of all such functions. For convenience, such functions are also called uniformly slowly oscillating functions.

3 Let X be a Banach space, and let CJ, X be the space of bounded continuous functions fromJ to X. If we replace CJ in 1 by CJ, X, then we get the definition of SOJ, X.

As in17, we always assume thatf∈ SOJis uniformly continuous.

The following two propositions come from15, Section 1.

Proposition 1.2. Letf ∈ SOJ SOJ×Ωbe such that∂f/∂xi is uniformly continuous onJ.

Then∂f/∂x_i∈ SOJ SOJ×Ω.

ForH h1, h2, . . . , hn∈ CRⁿ, suppose thatHt∈Ωfor allt∈R. DefineH×ι → Ω×R by

H×ιt h1t, h2t, . . . , hnt, t t∈R. 1.1

The following proposition shows that the composite is also slowly oscillating.

Proposition 1.3. Letf ∈ SOR×Ω. IfH∈ SORⁿandHt∈Ωfor allt∈R, thenf◦H×ι∈ SOR.

In the sequel, we will use the notations:R^m_T R^m×0, T,F_T sup{|Fx, t| :x ∈ Rⁿ, 0 ≤t≤ T}.F ∈ SORⁿ×R^m_Tmeans thatFx¹, x², tis slowly oscillating inx¹ ∈Rⁿ and uniformly onx², t∈R^m_T;F ∈ SORⁿ×R^mmeans thatFx¹, x²is slowly oscillating inx¹∈Rⁿand uniformly onx²∈R^m.

Let

Zx, t;ξ, s 1

2

πt−s_nmexp

−

xi−ξi² 4t−s

x, ξ∈R^nm 1.2

be the fundamental solution of the heat equation21.

2. A Type of Boundary Value Problem

We will keep the notation in Section 1 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, Dⁿ{X ∈Rⁿ:x_n>0}. 2.1

In this section, we always assume the following:f,f_xnxn ∈ SORⁿ⁻¹×D_T0,hx, t ≥ const>0,h,Δh−ht∈ SORⁿ⁻¹_T0 ,ϕ,ϕ_xnxn ∈ SORⁿ⁻¹×D,ϕ∈C³Rⁿ⁻¹×D, andg,Δg−gt∈ SORⁿ⁻¹_T0 .

Let

GX, t;ζ, τ ZX, t;ξ, ξn, τ ZX, t;ξ,−ξn, τ 2.2

be Green’s function for the boundary value problems22,23.

The following estimates are easily obtained:

t 0

ds

Dⁿ

GX, t;ζ, sdζ

≤m1T,

t 0

ds

Rⁿ⁻¹ZX, t;ξ,0, sdξ

≤m₂T,

t 0

ds

Rⁿ

∂ZX, t;ζ, s

∂x_n dζ

≤m3T,

2.3

wherem_iT i1,2,3are positive and increasing forT ≥0 andm_iT → 0 asT → 0.

To show the main results of this section, the following lemmas are needed. The first lemma is Lemma 3.1 on page 15 in24.

Lemma 2.1. Letϕ,φ, andχbe real, continuous functions on0, Twithχ≥0. If

ϕt≤φt ^t

0

χsϕsds t∈0, T, 2.4

then

ϕt≤φt ^t

0

χsφsexp

t s

χ ρ

dρ

ds t∈0, T. 2.5

Lemma 2.2. Let ϕ be a continuous function on 0, T. If φ, χ1, and χ2 are nondecreasing and nonnegative on0, Tand

ϕt≤φt χ1t ^t

0

ϕsdsχ2t ^t

0

√ϕs

t−sds t∈0, T, 2.6 then

ϕt≤φt

1tχ1t 2√ tχ2t

e^tχt, 2.7

where

χt tχ²₁t 4√

tχ1tχ2t πχ²₂t. 2.8

Proof. Replacing ϕs in the two integrals of 2.6 by the expression on the right hand side in2.6, changing the integral order of the resulting inequality and making use of the monotonicity ofφ,χ₁andχ₂, one gets

ϕt≤φt

1tχ1t 2√ tχ2t

tχ²₁t 4√

tχ1tχ2t πχ²₂t ^t

0

ϕsds. 2.9

ApplyLemma 2.1to get the conclusion.

Lemma 2.3. Let FX, t ∈ SOD_Tⁿ,φx, t, qx, t ∈ SORⁿ⁻¹_T , andϕ ∈ SODⁿ. Then the problem

u_t−ΔuquFX, t, X, t∈Dⁿ_T, uX,0 ϕX, X∈Dⁿ, u_xnx,0, t φx, t, x, t∈Rⁿ⁻¹_T

2.10

has a unique solutionu, anduis inSODⁿ_Tand satisfies

u_T≤KT

TF_Tϕ

√T 2 φ

T

, 2.11

whereKT 21Tq_Te^TqT.

One sees thatKTdepends onq_Tonly and is bounded near zero.

Proof. The existence and uniqueness of the solution comes from Theorem 5.3 on page 320 in 25.

As in22,23, the solutionucan be written as

uX, t

Dⁿ

ϕζGX, t;ζ,0dζ ^t

0

ds

Dⁿ

Fζ, sGX, t;ζ, sdζ

− ^t

0

ds

Dⁿ

qξ, suζ, sGX, t;ζ, sdζ−2

t 0

ds

Rⁿ⁻¹φξ, sZX, t;ξ,0, sdξ vx, t− ^t

0

ds

Dⁿ

qξ, suζ, sGX, t;ζ, sdζ.

2.12

So,

u_t≤2ϕ2

t 0

F_sds2

t 0

φ

√ s

t−sds2

t 0

q

su_sds. 2.13

ByLemma 2.1, one gets the desired inequality.

Now we show thatu∈ SODⁿ_T. As in the proofs of Lemmas 2.1 and 2.3 in15, one getsv∈ SODⁿ_T. Forx,τ ∈Rⁿ⁻¹with|x| ≥A >0,

uxτ, xn, t−ux, xn, t vxτ, x_n, t−vx, xn, t− ^t

0

ds

Dⁿ

qξ, suζ, sGxτ, x_n, t;ζ, s−Gx, xn, t;ζ, sdζ vxτ, xn, t−vx, xn, t

− ^t

0

ds

Dⁿ

qxτξ, suxτξ, x_nξ_n, s−qxξ, suxξ, x_nξ_n, s

Gθ, t;ζ, sdζ vxτ, x_n, t−vx, xn, t

− ^t

0

ds

Dⁿ

qxτξ, s−qxξ, s

uxτξ, xnξn, sGθ, t;ζ, sdζ

− ^t

0

ds

Dⁿ

uxτξ, xnξn, s−uxξ, xnξn, sqxξ, sGθ, t;ζ, sdζ.

2.14

Note that

t 0

ds

Dⁿ

qxτξ, s−qxξ, s

uxτξ, x_nξ_n, sGθ, t;ζ, sdζ

≤B·dist_A

R_τq−q

t

Dⁿ

qξ, sGθ, t;ζ, sdζ

≤Bq

s,

2.15

whereBis a constant and distA

Rτq, q

t sup

s∈0,t,|x|≥A

qxτ, s−qx, s. 2.16

So,

dist_ARτu, u_t≤dist_ARτv, v_tB·dist_A R_τq, q

tB

t 0

dist_ARτu, u_sq

sds. 2.17

ByLemma 2.1, one has

dist_ARτu, u_t≤m

dist_ARτv, v_tB·dist_A R_τq, q

t

, 2.18

where m is a constant. Since v and q are slowly oscillating, the right-hand sides of the inequality above approaches zero as A → ∞. This means that u ∈ SODⁿ_T. The proof is complete.

Consider the following problem.

Problem 1. Find functionsu∈ SORⁿ⁻¹×DTandq∈ SORⁿ⁻¹_T such that

ut−Δuqx, tufX, t, X, t∈D_Tⁿ, 2.19

uX,0 ϕX, X ∈Dⁿ, 2.20

uxnx,0, t gx, t, x, t∈Rⁿ⁻¹_T , 2.21 ux, a, t hx, t, x, t∈Rⁿ⁻¹_T , a∈0,∞. 2.22

One sees that

hx,0 ϕx, a, ϕxnx,0 gx,0, x∈Rⁿ⁻¹, 2.23 h_tx,0 u_t|_xna,t0

Δu−qufX, t

xna,t0 ΔϕX

xna−qx,0ϕx, a fx, a,0, gtx,0 utxn|_xn0,t0 ΔϕxnX

xn0−qx,0ϕxnx,0 fxnx,0,0.

2.24

It follows from2.24that ϕ_xnx,0ΔϕX

xnafx, a,0ϕxnx,0−h_tx,0ϕxnx,0 ϕx, aΔϕxnX

xn0fxnx,0,0ϕx, a−gtx,0ϕx, a. 2.25 Let VX, t uxnX, t, and let WX, t VxnX, t. We have the following two additional problems forV andW, respectively.

Problem 2. Find functionsV ∈ SORⁿ⁻¹×D_Tandq∈ SORⁿ⁻¹_T such that

V_t−ΔVqx, tV f_xnX, t, X, t∈Dⁿ_T, 2.26

VX,0 ϕ_xnX, X∈Dⁿ, 2.27

Vx,0, t gx, t, x, t∈Rⁿ⁻¹_T , 2.28

V_xnx, a, t h_t−Δhqh−fx, a, t, x, t∈Rⁿ⁻¹_T . 2.29

Problem 3. Find functionsW∈ SORⁿ⁻¹×D_Tandq∈ SORⁿ⁻¹_T such that

W_t−ΔWqx, tWf_xnxnX, t, X, t∈Dⁿ_T, 2.30

WX,0 ϕxnxnX, X∈Dⁿ, 2.31

W_xnx,0, t g_t−Δgqg−f_xnx,0, t, x, t∈Rⁿ⁻¹_T , 2.32 Wx, a, t ht−Δhhq−fx, a, t, x, t∈Rⁿ⁻¹_T . 2.33

Lemma 2.4. Problems1,2, and3are equivalent to each other.

Proof. The existence and uniqueness of the solutionV, qof Problem2can be easily obtained from that of the solutionu, qof Problem1. Conversely, letV, qbe the solution of Problem 2. We show that Problem 1 has a unique solution u, q. The uniqueness comes from the uniqueness of2.19–2.21. For the existence, let

uX, t ^xn

a

V x, y, t

dyhx, t. 2.34

Obviously,uX, t∈ SORⁿ⁻¹×DTand satisfies2.22. Alsousatisfies2.21because uxnx,0, t Vx,0, t gx, t. By2.23and2.27, one sees that2.20is true. Finally, we show thatusatisfies2.19and therefore, along withq, constitutes a solution of Problem1. In fact,

u_t−Δuquh_t−Δhqh ^xn

a

V_t x, y, t

−ΔV x, y, t

qV x, y, t

dy

^xn

a

∂²

∂y²V x, y, t

dy− ∂²

∂x_n²

xn

a

V x, y, t

dy

ht−ΔhqhfX, t−fx, a, t VxnX, t−Vxnx, a, t−VxnX, t fX, t.

by2.29

2.35

Thus, we have shown the equivalence of Problems1and2. Replacing2.34by the function

VX, t ^xn

a

W x, y, t

dygx, t, 2.36

the equivalence of Problems2and3can be proved similarly. The proof is complete.

ByLemma 2.4, to solve Problem1, we only need to solve Problem3. By2.30–2.32, we have the integral equation aboutW:

WX, t

Dⁿ

ϕξnξnζGX, t;ζ,0dζ ^t

0

ds

Dⁿ

fξnξnζ, sGX, t;ζ, sdζ

− ^t

0

ds

Dⁿ

qξ, sWζ, sGX, t;ζ, sdζ

−2

t 0

ds

Rⁿ⁻¹

g_s−Δgqg−f_ξnξ,0, s

ZX, t;ξ,0, sdξ.

2.37

Rewrite2.33as

qLqh⁻¹x, t

Δh−htfx, a, t Wx, a, t

, 2.38

whereWis determined by2.37.

One can directly test that Problem3is equivalent to2.37-2.38.

Note that for a given qx, t ∈ SORⁿ⁻¹_T , Lemma 2.3 shows that 2.30–2.32 or equivalently,2.37have a unique solutionW ∈ SORⁿ⁻¹×D_T. Thus,2.38does define an operatorL. Therefore, we only need to show that the integral2.38has a unique solution qandq∈ SORⁿ⁻¹_T . That is,Lhas a fixed point inSORⁿ⁻¹_T . Let

Δh−htfx, a, t

T02ϕξnξn

t 0

ds

Dⁿ

fξnξnζ, sGx, a, t;ζ,sdζ

T0

2

t 0

ds

Rⁿ⁻¹

Δg−gsfξnξ,0, s

Zx, a, t;ξ,0, sdξ

T0

h⁻¹

T0

M 2 .

2.39

SetBM, T {q∈ SORⁿ⁻¹_T :q_T ≤M}, whereT ≤T₀. Ifq∈BM, t, then, byLemma 2.3, WX, tis inSORⁿ⁻¹×D_T, and so, by2.38,Lqis inSORⁿ⁻¹_T with

Lq

T ≤ M

2 h⁻¹

T0

2m2Tg

T0m1TW_T

M. 2.40

Equation2.37gives the estimate

W_T ≤2ϕξ_nξ_n2m₂T0g_t−Δg−f_xnx,0, t

T02Mm₂T0g

T0

m₁T0f_xnxn

T0Mm₁TW_T.

2.41

Chooset0< T0such that whenT ≤t0, one has 1<21−Mm1T. It follows that

W_T ≤2

2ϕ_xnxn2m₂T0g_t−Δg−f_xnx,0, t

T02Mm₂T0g

T0m₁T0f_xnxn

T0

. 2.42

ChooseT₁≤t₀such that whenT ≤T₁, one has

2h⁻¹

T0

m2Tg

T0m1T

×

2ϕxnxn2m2T0gt−Δg−fxnx,0, t

T02Mm2T0g

T0m1T0fxnxn< 1 2, 2.43

and therefore,Lq_T ≤M.

Let q1,q2 ∈ BM, T. By 2.38, Lq1−Lq2_T ≤ h⁻¹_TW1−W2_T. Note that the functionW W₁−W₂is the solution of the problem

W_t−ΔWqW W₂ q₂−q₁

, X, t∈Dⁿ_T, WX,0 0, X ∈Dⁿ,

W_xnx,0, t q₂−q₁

gx, t, x, t∈Rⁿ⁻¹_T .

2.44

So, byLemma 2.3, one has

W_T≤KT √

T

2 q1−q2

Tg

TTq1−q2

TW2_T

. 2.45

ChooseT2 < t0such that for T ≤ T2,h⁻¹_T0W1−W2_T ≤ 1/2q1−q2_T. Now, set T ≤min{T1, T₂}. ThenLis a contraction fromBM, Tinto itself, and therefore, has a unique fixed point. Thus, we have shown.

Theorem 2.5. Let functionsf,g,h, andϕbe as above. Then, for smallT, Problem3has a unique solution (W, q) inRⁿ_TwithW∈ SORⁿ⁻¹×DTandq∈ SORⁿ⁻¹_T .

LetWⁱ, q_i i1.2be the solutions of Problem3inDⁿ_Tfor the functionsfⁱ,gⁱ,hⁱ, and ϕⁱ. Seth⁰h¹−h²,f⁰f¹−f²,ϕ⁰ ϕ¹−ϕ², andg⁰g¹−g². For the stability of the solution, we have the following.

Theorem 2.6. For 0≤t≤T, one has q1−q2

t≤c1h⁰

tc2g⁰

tc3f_x⁰_nxn

tc4ϕ⁰_xnxnc5h⁰_t−Δh⁰−f⁰x, a, t

t

c₆g_t⁰−Δg⁰−f_x⁰_nx,0, t

t,

2.46

where ci1 ≤ i ≤ 6 depends on t, h⁻¹_{1 t}, g¹_t, f_x¹_nxnt, ϕ¹_xnxn, q1_t, q2_t, and g_t¹−Δg¹−f_x¹_nx,0, t_t.

Proof. By2.33,

q₁−q₂ h¹₋₁

Δh⁰−h⁰_t f⁰x, a, t−q₂h⁰W₁−W₂

. 2.47

So,

q1−q2

t≤

h¹₋₁ t

Δh⁰−h⁰_tf⁰x, a, t

tq2

th⁰

tW1−W2_t

. 2.48

Note that the functionWW1−W2is the solution of the problem W_t−ΔWq₂Wf_x⁰_nxn −W₁

q₁−q₂

, X, t∈D_Tⁿ, WX,0 ϕ⁰_xnxnX, X∈Dⁿ,

W_xnx,0, t g_t⁰−Δg⁰q₂g⁰−f_x⁰_nx,0, t q₁−q₂

g¹, x, t∈Rⁿ⁻¹_T .

2.49

Using a formula similar to2.37andLemma 2.2for the functionW, one gets

W_t≤

tf_x⁰_nxn

tϕ⁰_xnxn2 t

πq2

tg⁰

t2 t

π

g_t⁰−Δg⁰−f_x⁰_nx,0, t

t

W1_t ^t

0

q1−q2

sdsg¹

√ t

π

t 0

q₁−q₂ s

t−s

exp

t 0

q2

ρdρ ds

.

2.50

ApplyingLemma 2.2and2.48, one gets the desired conclusion with c₁φt

h¹₋₁

t

q₂

t,

c₂2φt t

π

h¹₋₁ t

q₂

texp

t 0

q₂

sds

,

c3tφt

h¹₋₁ t

exp

t 0

q2

sds

,

c₄φt

h¹₋₁ t

exp

t 0

q₂

sds

,

c₅φt

h¹₋₁ t

,

c₆2φt t

π

h¹₋₁ t

exp

t 0

q₂

sds

,

2.51

where

φt

1tχ₁t 2√ tχ₂t

e^tχt, χt tχ²₁t 4√

tχ1tχ2t πχ²₂t, χ1t

h¹₋₁

tΦtexp

t 0

q2

sds

,

χ2t π^−1/2

h¹₋₁ t

g¹

texp

t 0

q2

sds

2.52

andΦtis majorant ofW1_t. One can specially assume that

Φt

ϕ¹_xnxntf_x¹_nxn

t ^t

0

g_s¹−Δg¹−f_x¹_nx,0, s πt−s ds

exp

t s

q2

sds

. 2.53

The proof is complete.

Corollary 2.7. Under the conditions inTheorem 2.6, the solution of Problem3is unique.

Acknowledgment

The research is supported by the NSF of Chinano. 11071048.

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