DISCRETE AND CONTINUOUS Website: www.aimSciences.orgDYNAMICAL SYSTEMSSupplement 2013 pp. 259–272

REGULARIZATION FOR ILL-POSED INHOMOGENEOUS

EVOLUTION PROBLEMS IN A HILBERT SPACE

Matthew A. Fury

Division of Science and Engineering

Penn State Abington1600 Woodland Road

Abington, PA 19001, USA

Abstract. We prove regularization for ill-posed evolution problems that areboth inhomogeneous and nonautonomous in a Hilbert Space H. We consider

the ill-posed problem du/dt = A(t,D)u(t) + h(t), u(s) = χ, 0 ≤ s ≤ t < T

where A(t,D) =∑kj=1 aj(t)D

j with aj ∈ C([0, T ] : R+) for each 1 ≤ j ≤ k

and D a positive, self-adjoint operator in H. Assuming there exists a solution

u of the problem with certain stabilizing conditions, we approximate u by the

solution vβ of the approximate well-posed problem dv/dt = fβ(t,D)v(t)+h(t),v(s) = χ, 0 ≤ s ≤ t < T where 0 < β < 1. Our method implies the existence

of a family of regularizing operators for the given ill-posed problem with ap-

plications to a wide class of ill-posed partial differential equations includingthe inhomogeneous backward heat equation in L2(Rn) with a time-dependent

diffusion coefficient.

1. Introduction. Mathematical models formulated to describe physical processesoften do not possess desired properties such as existence and uniqueness of solutions,and/or continuous dependence of solutions on initial data. For instance, whileinitial value problems with such properties are called well-posed, the backward heatequation given by

∂

∂tu(t, x) = −∆u(t, x), (t, x) ∈ [0, T )× Rn

u(0, x) = ψ(x), x ∈ Rn

in L2(Rn) is called ill-posed since a small change in the initial data could yield avery large difference in the corresponding solutions. Letting A be the positive, self-adjoint operator A = −∆ in the Hilbert space H = L2(Rn), we have the ill-posedabstract Cauchy problem

du

dt= Au(t) 0 ≤ t < T (1)

u(0) = χ.

In general, for a positive, self-adjoint operator A on a Hilbert space H, (1) is ill-posed because A does not generate a C0 semigroup on H (cf. [8, Theorem II.1.2]).

Approximation techniques have been widely used to study ill-posed Cauchy prob-lems. One technique known as the regularization of ill-posed problems involves

2010 Mathematics Subject Classification. Primary: 47D06, 46C99; Secondary: 35K05.Key words and phrases. Regularizing families of operators, ill-posed problems, evolution equa-

tions, backward heat equation.

259

260 MATTHEW A. FURY

approximating a known solution of an ill-posed problem by the solution of an ap-proximate well-posed problem

dv

dt= fβ(A)v(t) 0 ≤ t < T (2)

v(0) = χ,

where β > 0 and the operator fβ(A) approaches A in some sense as β → 0. Forinstance, introduced by Lattes and Lions [12] and Miller [15], the quasi-reversibilitymethod yields a common approach in the regularization of problem (1) wherefβ(A) = A − βA2 (cf. [14, Chapter 3.1.1]). Specifically, regularization refers tothe following:

Definition 1.1. ([10, Definition 3.1]) A family {Rβ(t) | β > 0, t ∈ [0, T ]} ofbounded linear operators on H is called a family of regularizing operators for theproblem (1) if for each solution u(t) of (1) with initial data χ ∈ H, and for anyδ > 0, there exists β(δ) > 0 such that

(i) β(δ)→ 0 as δ → 0, and(ii) ‖u(t)−Rβ(δ)(t)χδ‖ → 0 as δ → 0 for 0 ≤ t ≤ T whenever ‖χ− χδ‖ ≤ δ.

Hence, in Definition 1.1, the solution u of (1) may be approximated given a smallperturbation in the operator (A to fβ(A)) and also in the initial data (χ to χδ).Here the regularizing operator Rβ(t) may be defined as the solution operator ofproblem (2); that is Rβ(t)x is the solution of (2) with initial condition v(0) = x.

Regularization has been established by authors such as Mel’nikova and Filinkov[14], Mel’nikova [13], Ames and Hughes [3], Huang and Zheng [9, 10], Trong andTuan [19, 20], and Fury and Hughes [7] for evolution problems in various settings.For example, Mel’nikova, Ames and Hughes, and Huang and Zheng each prove reg-ularization for problem (1) in the case that −A generates a holomorphic semigroupon a Banach space (cf. [13], [3], [9, 10]). Also, Fury and Hughes prove regularizationfor nonautonomous evolution problems where the operator A in (1) is replaced bythe nonconstant operator A(t) for t ∈ [0, T ] (cf. [7]).

This paper extends regularization to evolution problems that are both nonau-tonomous and inhomogeneous in a Hilbert space H, problems specifically of theform

du

dt= A(t,D)u(t) + h(t) 0 ≤ s ≤ t < T (3)

u(s) = χ

where D is a positive, self-adjoint operator in H, A(t,D) =∑kj=1 aj(t)D

j for

t ∈ [0, T ] with aj ∈ C([0, T ] : R+) for 1 ≤ j ≤ k, and h is a function from [s, T ]into H. Since D is positive, self-adjoint and aj(t) is nonnegative for 1 ≤ j ≤ k,(3) is generally ill-posed. Taking the simple example where A(t,D) = D = −∆for t ∈ [0, T ], H = L2(Rn), and h ≡ 0, (3) reduces to the ill-posed backward heatequation.

Following techniques of Lattes and Lions [12], Miller [15], Ames [2], Showalter[18], and Ames and Hughes [3], we define the approximate well-posed problem

dv

dt= fβ(t,D)v(t) + h(t) 0 ≤ s ≤ t < T (4)

v(s) = χ

REGULARIZATION FOR ILL-POSED PROBLEMS 261

where 0 < β < 1. We will show that the assumptions made on the operatorsfβ(t,D) yield an evolution system Vβ(t, s), 0 ≤ s ≤ t ≤ T (cf. [16, Defini-tion 5.1.3]) associated with problem (4) such that the family of bounded operators{Vβ(t, s) | β > 0, 0 ≤ s ≤ t ≤ T} regularizes the ill-posed problem (3) in thesense of regularization for inhomogeneous problems. That is given a small changein initial data ‖χ− χδ‖ ≤ δ, there exists β(δ) > 0 such that β → 0 as δ → 0 and

‖u(t)− (Vβ(t, s)χδ +

∫ t

s

Vβ(t, r)h(r)dr)‖ → 0 as δ → 0

for each t ∈ [s, T ].The results of this paper apply to a wide class of ill-posed partial differential

equations that are both nonautonomous and inhomogeneous, with the simplestexample being the inhomogeneous backward heat equation in L2(Rn) with a time-dependent diffusion coefficient. In these applications, the regularizing family ofoperators stems from one of two main approximations fβ(t,D). The first, fβ(t,D) =

A(t,D)− βDk+1 =∑kj=1 aj(t)D

j − βDk+1 generalizes the approximation A− βA2

applied by Mel’nikova [13] and Ames and Hughes [3] in the autonomous case whereregularization is established for problem (1). The second approximation fβ(t,D) =A(t,D)(I + βDk)−1 is motivated by work of Showalter [18] and generalizes theapproximation A(I+βA)−1 applied by Huang and Zheng [10] and Ames and Hughes[3] also in the autonomous case.

The paper is organized as follows. In Section 2, we name conditions on fβ sothat (4) is well-posed, and we define an approximation condition, Condition (A, p)by which the operators fβ(t,D) approximate the operators A(t,D). In Section 3,we prove that given a solution u(t) of the ill-posed problem (3) adhering to certainstabilizing conditions,

‖u(t)− vβ(t)‖ ≤ CβT−tT−sM

t−sT−s , 0 ≤ s ≤ t ≤ T (5)

where vβ(t) is the solution of (4). Allowing β to approach 0, the inequality (5)establishes Holder-continuous dependence on modeling (cf. [12], [15], [2], [18], [3],[4], [6], [7]) meaning a small change from the models (3) to (4) produces only asmall change in the corresponding solutions. The calculations and results in thissection of the paper extend the work of Fury and Hughes who in [6] prove Holder-continuous dependence on modeling for the homogeneous version of (3) where h ≡ 0.In Section 4, we then use inequality (5) to prove the main result of the paper which isthe existence of a family of regularizing operators for problem (3). Finally, Section 5applies the theory to partial differential equations in L2(Rn).

Below, B(H) will denote the space of bounded linear operators on H. For asubspace Y of H, B(Y,H) denotes the space of bounded linear operators from Yinto H, and ‖·‖Y→H the corresponding operator norm. When referring to solutionsof (3) and (4), we will be concerned with classical solutions which (in the case of(3)) are functions u : [s, T ] → H such that u(t) ∈ Dom(A(t,D)) for s < t < T ,u ∈ C([s, T ] : H)∩C1((s, T ) : H), and u satisfies (3) in H. Also, a Y -valued solutionof (4) is a classical solution vβ of (4) such that vβ ∈ C([s, T ] : Y ).

2. The approximate well-posed problem. For 0 < β < 1, let fβ : [0, T ] ×[0,∞) → R be a function continuous in t ∈ [0, T ] and Borel in λ ∈ [0,∞). In thissection, we first determine conditions on the function fβ so that the approximateproblem (4) is well-posed, meaning a unique classical solution exists for each initial

262 MATTHEW A. FURY

data χ in a dense subspace Y of H, and solutions depend continuously on the initialdata (cf. [8, Chapter II.13]).

As fβ : [0, T ]×[0,∞)→ R is Borel in λ ∈ [0,∞), the operators fβ(t,D), 0 ≤ t ≤ Tare defined by means of the functional calculus for self-adjoint operators in theHilbert space H. In particular, since D is positive, self-adjoint, the spectrum σ(D)of D is contained in [0,∞) and for each t ∈ [0, T ],

fβ(t,D)x =

∫ ∞0

fβ(t, λ)dE(λ)x,

for x ∈ Dom(fβ(t,D)) = {x ∈ H |∫∞

0|fβ(t, λ)|2d(E(λ)x, x) < ∞}, where {E(·)}

denotes the resolution of the identity for the self-adjoint operator D.

Proposition 2.1. Let H be a Hilbert space and for 0 < β < 1, let fβ : [0, T ] ×[0,∞) → R be continuous in t ∈ [0, T ] and Borel in λ ∈ [0,∞). Assume thefollowing conditions.

(i) There exists ωβ ∈ R such that fβ(t, λ) ≤ ωβ for all (t, λ) ∈ [0, T ]× [0,∞).(ii) There exists a Borel function rβ : [0,∞) → [0,∞) such that |fβ(t, λ)| ≤ rβ(λ)for all (t, λ) ∈ [0, T ]× [0,∞), and Dom(fβ(t,D)) = Dom(rβ(D)) for all t ∈ [0, T ].(iii) The function t 7→ fβ(t,D) is continuous in the B(Y,H) norm ‖ · ‖Y→H whereY = Dom(rβ(D)) and ‖ · ‖Y denotes the graph norm associated with the operatorrβ(D).(iv) h ∈ C([s, T ] : Y ).

Then (4) is well-posed with unique classical solution (and Y -valued solution) vβ(t) =

e∫ tsfβ(τ,D)dτχ+

∫ tse∫ trfβ(τ,D)dτh(r)dr for every χ ∈ Y .

Proof. As seen in [6, Proposition 2.10], condition (i) implies that for each t ∈ [0, T ],fβ(t,D) is the infinitesimal generator of the C0 semigroup esfβ(t,D), s ≥ 0, andconditions (i)–(iii) imply that {fβ(t,D)}t∈[0,T ] is a stable family satisfying Kato’sstability conditions (cf. [11, Theorem 4.1 (i)–(iii)], [16, Section 5.3]) with Y =Dom(rβ(D)) and with stability constants K = 1 and ωβ . The homogeneous version

of (4) with h ≡ 0 is then well-posed with unique Y -valued solution t 7→ e∫ tsfβ(τ,D)dτχ

for every χ ∈ Y where Vβ(t, s) := e∫ tsfβ(τ,D)dτ is an evolution system satisfying the

properties that Vβ(t, s)Y ⊆ Y and Vβ(t, s)y ∈ C([0, T ]2 : Y ) for y ∈ Y , for all0 ≤ s ≤ t ≤ T . The addition of (iv) then implies the desired result (cf. [11,Theorem 7.1], [16, Theorem 5.5.2]).

In order to establish Holder-continuous dependence on modeling and regulariza-tion, we will require additional conditions on fβ . The following definition is inspiredby results obtained by Ames and Hughes (cf. [3, Definition 1]) for continuous de-pendence on modeling in the autonomous or time-independent case.

Definition 2.2. For 0 < β < 1, let fβ : [0, T ] × [0,∞) → R be a function con-tinuous in t ∈ [0, T ] and Borel in λ ∈ [0,∞) and assume the conditions (i)–(iv)of Proposition 2.1. Then fβ is said to satisfy Condition (A, p) if there exists anonzero polynomial p(λ) independent of β such that for t ∈ [0, T ], Dom(p(D)) ⊆Dom(A(t,D)) ∩Dom(fβ(t,D)) and

‖(−A(t,D) + fβ(t,D))ψ‖ ≤ β‖p(D)ψ‖

for all ψ ∈ Dom(p(D)).

REGULARIZATION FOR ILL-POSED PROBLEMS 263

In light of Condition (A, p), for t ∈ [0, T ], we set gβ(t, λ) = −A(t, λ) + fβ(t, λ).For each n ≥ |ωβ |, set en = {λ ∈ [0,∞) | max{|gβ(t, λ)| : t ∈ [0, T ]} ≤ n}. Thenfor λ ∈ en, max{|A(t, λ)| : t ∈ [0, T ]} ≤ n + ωβ and max{|fβ(t, λ)| : t ∈ [0, T ]} ≤2n+ ωβ . Set En = E(en) and let ψn ∈ EnH. Consider the evolution problem

du

dt= A(t,D)Enu(t) + hn(t) 0 ≤ s ≤ t < T (6)

u(s) = ψn

in H where h ∈ C([s, T ] : Y ) and hn(t) = Enh(t) for all t ∈ [0, T ].

Lemma 2.3. The evolution problem (6) has a unique classical solution un(t) =

Un(t, s)ψn+∫ tsUn(t, r)hn(r)dr where Un(t, s), 0 ≤ s ≤ t ≤ T is an evolution system

on H such that Un(t, s) = e∫ tsA(τ,D)dτ when acting on EnH.

Proof. For each t ∈ [0, T ], A(t,D)En is a bounded operator on H by properties ofthe projection operators En and from the fact that |A(t, λ)| ≤ n + ωβ for (t, λ) ∈[0, T ]× en. Also, the function t 7→ A(t,D)En is continuous in the uniform operator

topology since A(t,D) =∑kj=1 aj(t)D

j and each aj is a continuous function. Hence,

it follows that the homogeneous version of (6) with hn ≡ 0 has a unique classicalsolution t 7→ Un(t, s)ψn where Un(t, s) is an evolution system such that Un(t, s) =

e∫ tsA(τ,D)dτ on EnH (cf. [6, Lemma 3.2]). Because h ∈ C([s, T ] : Y ) ⊆ C([s, T ] : H),

it may be shown that t 7→∫ tsUn(t, r)hn(r)dr is a classical solution of (6) with

the initial value ψn = 0. From this, it then follows that un(t) = Un(t, s)ψn +∫ tsUn(t, r)hn(r)dr is a unique classical solution of (6).

Note that if in (6), A(t,D) is replaced by either fβ(t,D) or gβ(t,D), then the

results of Lemma 2.3 hold with Un(t, s) = e∫ tsA(τ,D)dτ replaced by Vβ,n(t, s) =

e∫ tsfβ(τ,D)dτ or Wβ,n(t, s) = e

∫ tsgβ(τ,D)dτ respectively (all when acting on EnH).

Each of these evolution systems will aid in proving Holder-continuous dependenceon modeling in Section 3. We close the current section with an important relation-ship between these evolution systems which follows immediately from the relationgβ(t, λ) = −A(t, λ) + fβ(t, λ) and from properties of the functional calculus forself-adjoint operators (cf. [5, Corollary XII.2.7]).

Corollary 2.4. Let ψn ∈ EnH. Then

Un(t, s)Wβ,n(t, s)ψn = Vβ,n(t, s)ψn = Wβ,n(t, s)Un(t, s)ψn

for all 0 ≤ s ≤ t ≤ T .

3. Holder-continuous dependence on modeling. We prove inequality (5) whichestablishes Holder-continuous dependence on modeling for problems (3) and (4).Our strategy will be to extend into the complex plane, the solutions established

at the end of Section 2, un(t) = Un(t, s)χn +∫ tsUn(t, r)hn(r)dr and vβ,n(t) =

Vβ,n(t, s)χn +∫ tsVβ,n(t, r)hn(r)dr, where ψn in (6) has been replaced by χn =

Enχ. An application of Hadamard’s Three Lines Theorem (cf. [17, Theorem12.8]) to a related function will then yield inequality (5). For 1 ≤ j ≤ k, set

bj = maxt∈[0,T ] |aj(t)| and define b(λ) =∑kj=1 bjλ

j for λ ∈ [0,∞). We have

Theorem 3.1. Let D be a positive, self-adjoint operator in a Hilbert space Hand let A(t,D) be defined as above for all t ∈ [0, T ]. Let fβ satisfy Condition(A, p), and assume that there exists a constant γ, independent of β and ωβ such

264 MATTHEW A. FURY

that gβ(t, λ) ≤ γ for all (t, λ) ∈ [0, T ]× [0,∞). Then if u(t) and vβ(t) are classicalsolutions of (3) and (4) respectively with χ ∈ Y and h ∈ C([s, T ] : Y ), and if

there exist constants M ′,M ′′ ≥ 0 such that ‖p(D)b(D)e∫ TsA(τ,D)dτχ‖ ≤ M ′ and

‖p(D)b(D)e∫ TsA(τ,D)dτh(t)‖ ≤ M ′′ for all t ∈ [s, T ], then there exist constants C

and M independent of β such that for 0 ≤ s ≤ t < T ,

‖u(t)− vβ(t)‖ ≤ CβT−tT−sM

t−sT−s .

Proof. Let χn = Enχ for n ≥ |ωβ |, and define S to be the complex strip S ={t + iη | t ∈ [s, T ], η ∈ R}. Since D is self-adjoint, eiηD is a bounded operator onH for every η ∈ R. Hence, for α = t+ iη ∈ S, we may define φn : S → H by

φn(α) = eiηD(un(t)− vβ,n(t))

where, as above, un(t) = Un(t, s)χn+∫ tsUn(t, r)hn(r)dr and vβ,n(t) = Vβ,n(t, s)χn+∫ t

sVβ,n(t, r)hn(r)dr.The function φn(α) is not necessarily analytic, and so we may not apply the

Three Lines Theorem directly to it. To construct a function that is analytic on theinterior of S, following work of Agmon and Nirenberg [1, p. 148], we consider thefunction

Φn(α) = − 1

π

∫ ∫S

∂φn(z)

(1

z − α+

1

z + 1 + α

)dxdy,

where α = t + iη and z = x + iy are in S, and ∂ denotes the Cauchy-Riemann

operator ∂ = 12

(∂∂x + i ∂∂y

). Let z = x+ iy ∈ S. We first determine ∂φn(z). Since

eiyD is a bounded operator on H, and since un(x) and vβ,n(x) are both in EnH,we have by Lemma 2.3,

∂

∂xφn(z) = eiyD(

d

dxun(x)− d

dxvβ,n(x)) = eiyD(A(x,D)un(x)− fβ(x,D)vβ,n(x)).

Next, note un(x), vβ,n(x) ∈ Dom(D) since en is a bounded subset of [0,∞). Hence,

∂

∂yφn(z) =

∂

∂yeiyD(un(x)− vβ,n(x)) = eiyD(iD)(un(x)− vβ,n(x)),

and therefore,

2 ∂φn(z)

=∂

∂xφn(z) + i

∂

∂yφn(z)

=eiyD(A(x,D)un(x)− fβ(x,D)vβ,n(x))− eiyD(Dun(x)−Dvβ,n(x)).

(7)

Following the method of Agmon and Nirenberg [1, p. 148], we show that ∂φn(z) isbounded and continuous on S. Since ‖eiyD‖ = 1, we have from (7),

2‖∂φn(z)‖ ≤‖A(x,D)un(x)−A(x,D)vβ,n(x)‖+ ‖A(x,D)vβ,n(x)− fβ(x,D)vβ,n(x)‖+ ‖Dun(x)−Dvβ,n(x)‖.

(8)

REGULARIZATION FOR ILL-POSED PROBLEMS 265

Since χn ∈ EnH and hn(r) ∈ EnH for every s ≤ r ≤ x, we have by Corollary 2.4,

‖A(x,D)un(x)−A(x,D)vβ,n(x)‖≤‖A(x,D)Un(x, s)χn −A(x,D)Vβ,n(x, s)χn‖

+

∫ x

s

‖A(x,D)Un(x, r)hn(r)−A(x,D)Vβ,n(x, r)hn(r)‖dr

=‖A(x,D)Un(x, s)χn −A(x,D)Wβ,n(x, s)Un(x, s)χn‖

+

∫ x

s

‖A(x,D)Un(x, r)hn(r)−A(x,D)Wβ,n(x, r)Un(x, r)hn(r)‖dr

=‖(I −Wβ,n(x, s))A(x,D)Un(x, s)χn‖

+

∫ x

s

‖(I −Wβ,n(x, r))A(x,D)Un(x, r)hn(r)‖dr.

(9)

Now, A(x,D)Un(x, r)hn(r) ∈ Dom(p(D)) since en is a bounded subset of [0,∞). ByCondition (A, p), then A(x,D)Un(x, r)hn(r) ∈ Dom(A(t,D)) ∩ Dom(fβ(t,D)) ⊆Dom(gβ(t,D)) for each t ∈ [0, T ], and by standard properties of evolution systems(cf. [16, Theorem 5.1.2]),∫ x

s

‖(I −Wβ,n(x, r))A(x,D)Un(x, r)hn(r)‖dr

=

∫ x

s

‖(Wβ,n(x, x)−Wβ,n(x, r))A(x,D)Un(x, r)hn(r)‖dr

=

∫ x

s

∥∥∥∥∫ x

r

∂

∂qWβ,n(x, q)A(x,D)Un(x, r)hn(r)dq

∥∥∥∥ dr=

∫ x

s

∥∥∥∥∫ x

r

(−Wβ,n(x, q)gβ(q,D)En)A(x,D)Un(x, r)hn(r)dq

∥∥∥∥ dr≤∫ x

s

∫ x

r

‖Wβ,n(x, q)gβ(q,D)A(x,D)Un(x, r)hn(r)‖dq dr.

Recall from the paragraph before Corollary 2.4 that Wβ,n(t, s) = e∫ tsgβ(τ,D)dτ on

EnH. By the assumption that gβ(t, λ) ≤ γ for all (t, λ) ∈ [0, T ] × [0,∞) and byCondition (A, p), we then have∫ x

s

∫ x

r

‖Wβ,n(x, q)gβ(q,D)A(x,D)Un(x, r)hn(r)‖dq dr

≤∫ x

s

∫ x

r

eγ(x−q)‖gβ(q,D)A(x,D)Un(x, r)hn(r)‖dq dr

≤∫ x

s

βT (1 + eγT )‖p(D)A(x,D)Un(x, r)hn(r)‖dr.

Similarly,

‖(I −Wβ,n(x, s))A(x,D)Un(x, s)χn‖ ≤ βT (1 + eγT )‖p(D)A(x,D)Un(x, s)χn‖,

and so by calculation (9), we have shown

‖A(x,D)un(x)−A(x,D)vβ,n(x)‖≤βT (1 + eγT )‖p(D)A(x,D)Un(x, s)χn‖

+

∫ x

s

βT (1 + eγT )‖p(D)A(x,D)Un(x, r)hn(r)‖dr.(10)

266 MATTHEW A. FURY

By similar calculations, it follows that

‖Dun(x)−Dvβ,n(x)‖ ≤βT (1 + eγT )‖p(D)DUn(x, s)χn‖

+

∫ x

s

βT (1 + eγT )‖p(D)DUn(x, r)hn(r)‖dr.(11)

Finally, by Corollary 2.4 and Condition (A, p),

‖A(x,D)vβ,n(x)− fβ(x,D)vβ,n(x)‖≤‖(−A(x,D) + fβ(x,D))Vβ,n(x, s)χn‖

+

∫ x

s

‖(−A(x,D) + fβ(x,D))Vβ,n(x, r)hn(r)‖dr

=‖Wβ,n(x, s)Un(x, s)(−A(x,D) + fβ(x,D))χn‖

+

∫ x

s

‖Wβ,n(x, r)Un(x, r)(−A(x,D) + fβ(x,D))hn(r)‖dr

≤(1 + eγT )(‖(−A(x,D) + fβ(x,D))Un(x, s)χn‖

+

∫ x

s

‖(−A(x,D) + fβ(x,D))Un(x, r)hn(r)‖dr)

≤β(1 + eγT )

(‖p(D)Un(x, s)χn‖+

∫ x

s

‖p(D)Un(x, r)hn(r)‖dr).

(12)

Combining calculations (8), (10), (11), and (12), together with the stabilizing as-

sumptions ‖p(D)b(D)e∫ TsA(τ,D)dτχ‖ ≤ M ′ and ‖p(D)b(D)e

∫ TsA(τ,D)dτh(t)‖ ≤ M ′′

for all t ∈ [s, T ], we have

‖∂φn(z)‖ ≤ βC ′ (13)

where C ′ is a constant independent of β, z and n (since γ was assumed to beindependent of β). Thus, ∂φn(z) is bounded on S.

It is easy to check that ∂φn(z) is also continuous on S. It follows then that

Φn(α) is absolutely convergent, ∂Φn(α) = ∂φn(α), and there exists a constant Ksuch that ∫ ∞

−∞

∣∣∣∣ 1

z − α+

1

z + 1 + α

∣∣∣∣ dy ≤ K (1 + log1

|x− t|

)(14)

for x 6= t (cf. [1, p. 148]). Hence, the function Ψn : S → H defined by

Ψn(α) = φn(α)− Φn(α)

is then analytic on the interior of S since ∂Ψn(α) = ∂φn(α)− ∂Φn(α) = 0 for α inthe interior of S (cf. [17, Theorem 11.2]).

In order to apply the Three Lines Theorem we show that Ψn is bounded andcontinuous on S. Let α = t+ iη ∈ S. Similar to calculations (10) and (11), we have

‖φn(α)‖ ≤ ‖un(t)− vβ,n(t)‖

≤ βT (1 + eγT )‖p(D)Un(t, s)χn‖+

∫ t

s

βT (1 + eγT )‖p(D)Un(t, r)hn(r)‖dr

≤ βK ′(15)

REGULARIZATION FOR ILL-POSED PROBLEMS 267

where K ′ is a constant independent of β, α, and n. Next, using (13) and (14),

‖Φn(α)‖ =

∥∥∥∥− 1

π

∫ ∫S

∂φn(z)

(1

z − α+

1

z + 1 + α

)dxdy

∥∥∥∥≤ 1

πβC ′

∫ T

s

(∫ ∞−∞

∣∣∣∣ 1

z − α+

1

z + 1 + α

∣∣∣∣ dy) dx≤ β K

πC ′∫ T

s

(1 + log

1

|x− t|

)dx

≤ βC ′

(16)

for a possibly different constant C ′ independent of β, α, and n. Therefore, from(15) and (16), we have for α = t+ iη ∈ S,

‖Ψn(α)‖ ≤ ‖φn(α)‖+ ‖Φn(α)‖ ≤ β(K ′ + C ′), (17)

proving that Ψn is bounded on S.It is easy to check that Ψn is also continuous on S, and so Ψn is bounded and

continuous on S, and analytic on the interior of S. It follows from the Cauchy-Schwarz inequality that for arbitrary ψ ∈ H, the mapping α 7→ (Ψn(α), ψ) fromS into C, where (·, ·) denotes the inner product in H, has the same properties.Therefore, by the Three Lines Theorem (cf. [17, Theorem 12.8]),

|(Ψn(t), ψ)| ≤Mn(s)T−tT−sMn(T )

t−sT−s (18)

for 0 ≤ s ≤ t ≤ T , where Mn(t) = maxη∈R|(Ψn(t+ iη), ψ)|.

We aim to prove (5) and so we bound Mn(s) in terms of β. For η ∈ R, from (16),|(Ψn(s+ iη), ψ)| ≤ ‖Ψn(s+ iη)‖‖ψ‖ =

∥∥eiηD(un(s)− vβ,n(s))− Φn(s+ iη)∥∥ ‖ψ‖ =∥∥eiηD(χn − χn)− Φn(s+ iη)

∥∥ ‖ψ‖ = ‖Φn(s+ iη)‖‖ψ‖ ≤ βC ′‖ψ‖, and so

Mn(s) = maxη∈R|(Ψn(s+ iη), ψ)| ≤ βC ′‖ψ‖. (19)

Next, for η ∈ R, using (17) and the fact that 0 < β < 1, we have |(Ψn(T + iη), ψ)| ≤‖Ψn(T + iη)‖‖ψ‖ ≤ (K ′ + C ′)‖ψ‖, and so

Mn(T ) = maxη∈R|(Ψn(T + iη), ψ)| ≤M‖ψ‖. (20)

where M is a constant independent of β and n.It follows from (18), (19), and (20) that there exist constants C ′ and M , each

independent of β and n, such that |(Ψn(t), ψ)| ≤ (βC ′‖ψ‖)T−tT−s (M‖ψ‖)

t−sT−s =

(βC ′)T−tT−sM

t−sT−s ‖ψ‖ for 0 ≤ s ≤ t < T . Taking the supremum over all ψ ∈ H

with ‖ψ‖ ≤ 1, we have found constants C and M , independent of β and n, such

that ‖Ψn(t)‖ ≤ CβT−tT−sM

t−sT−s for 0 ≤ s ≤ t < T . Hence, using (16),

‖un(t)− vβ,n(t)‖ =‖φn(t)‖ ≤ ‖Ψn(t)‖+ ‖Φn(t)‖ ≤ CβT−tT−sM

t−sT−s + βC ′

= (C + βt−sT−sM

s−tT−sC ′)β

T−tT−sM

t−sT−s ≤ Cβ

T−tT−sM

t−sT−s

for a possibly different constant C independent of β and n. Letting n → ∞, wehave found constants C and M , independent of β, such that

‖u(t)− vβ(t)‖ ≤ CβT−tT−sM

t−sT−s

for 0 ≤ s ≤ t < T as desired.

268 MATTHEW A. FURY

4. Regularization for problem (3). In this section, we utilize the results fromTheorem 3.1 to prove the main result of the paper, that is regularization for problem(3). Following [10, Definition 3.1], we have

Definition 4.1. A family {Vβ(t, s) | β > 0, 0 ≤ s ≤ t ≤ T} ⊆ B(H) is called afamily of regularizing operators for the problem (3) if for each classical solution u(t)of (3) with initial data χ ∈ Y and h ∈ C([s, T ] : Y ), and for any δ > 0, there existsβ(δ) > 0 such that

(i) β(δ)→ 0 as δ → 0,

(ii) ‖u(t) − (Vβ(δ)(t, s)χδ +∫ tsVβ(δ)(t, r)h(r)dr)‖ → 0 as δ → 0 for s ≤ t ≤ T

whenever ‖χ− χδ‖ ≤ δ.

We show that the evolution system Vβ(t, s) = e∫ tsfβ(τ,D)dτ given in Proposi-

tion 2.1 provides a family of regularizing operators for the problem (3) where thefunction fβ(t, λ) is given by two different approximations each satisfying Condition(A, p). The first approximation fβ : [0, T ] × [0,∞) → R, motivated by work ofMiller [15], Lattes and Lions [12], Ames [2], and Ames and Hughes [3] will be givenby

fβ(t, λ) =

k∑j=1

aj(t)λj − βλk+1. (21)

The second approximation, motivated by work of Showalter [18] is defined by

fβ(t, λ) =

∑kj=1 aj(t)λ

j

1 + βλk. (22)

Theorem 4.2. Let D be a positive, self-adjoint operator in a Hilbert space H andlet A(t,D) be defined as above for all t ∈ [0, T ]. Let fβ : [0, T ] × [0,∞) → R bedefined by either (21) or (22). Then the family

{Vβ(t, s) = e∫ tsfβ(τ,D)dτ | β > 0, 0 ≤ s ≤ t ≤ T}

is a family of regularizing operators for the ill-posed problem (3).

Proof. The content of the proof is to first show that each approximation satisfiesthe hypotheses of Theorem 3.1, and then to use inequality (5) to prove regular-ization. Set bj = maxt∈[0,T ] |aj(t)| for 1 ≤ j ≤ k, and for λ ∈ [0,∞), define

b(λ) =∑kj=1 bjλ

j .

Approximation (21): For the first approximation, we have fβ(t, λ) ≤ φβ(λ) :=

b(λ)− βλk+1. Note, for 0 ≤ λ ≤ 1, φβ(λ) ≤ B := b(1) =∑kj=1 bj . Next, fix λ ≥ 1.

Then λj ≤ λk for 1 ≤ j ≤ k and φβ(λ) ≤ Bλk − βλk+1. Now, it is readily shown

that on [0,∞), Bλk − βλk+1 has a maximum value at λ = Bkβ(k+1) , and so

φβ(λ) ≤ B(

Bk

β(k + 1)

)k− β

(Bk

β(k + 1)

)k+1

=Bk+1kk

βk(k + 1)k+1.

Setting ωβ = Bk+1kk

βk(k+1)k+1 , we have that fβ(t, λ) ≤ ωβ for small β and so Proposi-

tion 2.1 (i) is satisfied. Also,

|fβ(t, λ)| ≤ rβ(λ) := b(λ) + βλk+1 (23)

REGULARIZATION FOR ILL-POSED PROBLEMS 269

for all (t, λ) ∈ [0, T ]× [0,∞) and Dom(fβ(t,D)) = Dom(rβ(D)) = Dom(Dk+1) forall t ∈ [0, T ]. It is easily shown that Proposition 2.1 (iii) holds (cf. [6, Example 4.1]).Hence if h(t) is continuous from [s, T ] to Y = Dom(rβ(D)) in the graph norm ‖ ·‖Yassociated with the operator rβ(D), then the hypotheses (i)–(iv) of Proposition 2.1hold so that (4) is well-posed.

It is easy to check that fβ satisfies Condition (A, p) with p(λ) = λk+1. Also,gβ(t, λ) = −βλk+1 ≤ 0 for all (t, λ) ∈ [0, T ] × [0,∞), so we may choose γ = 0.

Theorem 3.1 then yields the result ‖u(t)−vβ(t)‖ ≤ CβT−tT−sM

t−sT−s for 0 ≤ s ≤ t < T ,

where u(t) and vβ(t) are classical solutions of (3) and (4) respectively.

Approximation (22): For the second approximation, we have for small β,

fβ(t, λ) ≤ rβ(λ) :=b(λ)

1 + βλk≤ B

β(24)

where again B =∑kj=1 bj . Setting ωβ = B

β , we have that fβ(t, λ) ≤ ωβ for small β

and so Proposition 2.1 (i) is satisfied. Since rβ(λ) is a bounded Borel function on[0,∞), the Spectral Theorem implies that rβ(D) = b(D)(I + βDk)−1 is a boundedeverywhere-defined operator on H and so we may choose Y = Dom(rβ(D)) = H.It is easy to check then that hypotheses (ii) and (iii) of Proposition 2.1 hold wherefβ(t,D) = A(t,D)(I + βDk)−1 ∈ B(H) for each t ∈ [0, T ]. Therefore, if h ∈C([s, T ] : H), then Proposition 2.1 (i)–(iv) hold so that (4) is well-posed.

It is not hard to show that fβ satisfies Condition (A, p) with p(λ) =∑kj=1 bjλ

k+j

(cf. [6, Example 4.2]). Further, gβ(t, λ) = −A(t, λ) + A(t, λ)(1 + βλk)−1 ≤ 0 forall (t, λ) ∈ [0, T ]× [0,∞), so we may choose γ = 0. Again, Theorem 3.1 yields the

result ‖u(t)− vβ(t)‖ ≤ CβT−tT−sM

t−sT−s for 0 ≤ s ≤ t ≤ T .

Regularization: Now, for regularization, we assume u(t) is a classical solution of (3)where χ ∈ Y and h ∈ C([s, T ] : Y ). Let δ > 0 be given and assume ‖χ − χδ‖ ≤ δ.Also, let vβ(t) be a classical solution of (4) with fβ defined by either (21) or (22).

By Proposition 2.1, then vβ(t) = Vβ(t, s)χ+∫ tsVβ(t, r)h(r)dr. Note, for x ∈ H and

0 ≤ s ≤ t ≤ T ,

‖Vβ(t, s)x‖2 =

∫ ∞0

|e∫ tsfβ(τ,λ)dτ |2d(E(λ)x, x)

≤ (eωβT )2

∫ ∞0

d(E(λ)x, x)

= (eωβT )2‖x‖2

so that ‖Vβ(t, s)‖ ≤ eωβT . Hence, for 0 ≤ s ≤ t ≤ T , we have

‖u(t)− (Vβ(t, s)χδ +

∫ t

s

Vβ(t, r)h(r)dr)‖

≤‖u(t)− vβ(t)‖+ ‖vβ(t)− (Vβ(t, s)χδ +

∫ t

s

Vβ(t, r)h(r)dr)‖

=‖u(t)− vβ(t)‖+ ‖Vβ(t, s)χ− Vβ(t, s)χδ‖≤‖u(t)− vβ(t)‖+ δ eωβT .

(25)

First, consider s ≤ t < T . For the first approximation (21), choose

β =

(−2Bk+1kkT

(k + 1)k+1ln δ

)1/k

.

270 MATTHEW A. FURY

Then β → 0 as δ → 0 and by (25),

‖u(t)− (Vβ(t, s)χδ +

∫ t

s

Vβ(t, r)h(r)dr)‖

≤CβT−tT−sM

t−sT−s + δ e

Bk+1kkT

βk(k+1)k+1

=C

(−2Bk+1kkT

(k + 1)k+1ln δ

) 1kT−tT−s

Mt−sT−s +

√δ

→0 as δ → 0.

(26)

In the case of the second approximation (22), choose β = −2BT/ln δ. Then β → 0as δ → 0 and

‖u(t)− (Vβ(t, s)χδ +

∫ t

s

Vβ(t, r)h(r)dr)‖

≤CβT−tT−sM

t−sT−s + δ e

BTβ

=C(−2BT/ln δ)T−tT−sM

t−sT−s +

√δ

→0 as δ → 0.

(27)

Finally, for the case that t = T , from inequalities (17) and (18), it is easily shownby an argument similar to that in the last paragraph of the proof of Theorem 3.1with t = T that ‖u(T )−vβ(T )‖ ≤ βN for some constant N independent of β. Thenby (25), in the case of either approximation (21) or (22), we have that β → 0 asδ → 0 and

‖u(T )− (Vβ(T, s)χδ +

∫ T

s

Vβ(T, r)h(r)dr)‖

≤‖u(T )− vβ(T )‖+ δ eωβT

≤βN +√δ

→0 as δ → 0.

(28)

Combining (26), (27), and (28), we have in the case of either approximation, that{Vβ(t, s) | β > 0, 0 ≤ s ≤ t ≤ T} is a family of regularizing operators for problem(3).

5. Examples. The main application of our theory is to partial differential equa-tions in the Hilbert space (H, ‖ · ‖) = (L2(Rn), ‖ · ‖2) with D = −∆ where ∆

denotes the Laplacian defined by ∆ψ =∑ni=1

∂2ψ∂x2i

. The operator −∆ is a positive,

self-adjoint operator in L2(Rn) and hence the partial differential equation

∂

∂tu(t, x) = A(t,−∆)u(t, x) + h(t, x), (t, x) ∈ [s, T )× Rn (29)

u(s, x) = ψ(x), x ∈ Rn

in L2(Rn), where A(t,−∆) =∑kj=1 aj(t)(−∆)j and aj ∈ C([0, T ] : R+) for 1 ≤ j ≤

k, is generally ill-posed. By the results in Section 4, regularization for (29) stemsfrom the approximate well-posed problem (4) where fβ(t,D) = fβ(t,−∆) is definedby two different functional calculi fβ :

REGULARIZATION FOR ILL-POSED PROBLEMS 271

Example 5.1. By the first approximation fβ(t, λ) = A(t, λ)− βλk+1, the approx-imate well-posed problem (4) becomes

∂

∂tv(t, x) = A(t,−∆)v(t, x)− β(−∆)k+1v(t, x) + h(t, x), (t, x) ∈ [s, T )× Rn

(30)

v(s, x) = ψ(x), x ∈ Rn.

Here, as in the proof of Theorem 4.2, we note that Y = Dom(rβ(−∆)) = Dom(∆k+1)

where rβ(λ) = b(λ) + βλk+1 as in (23), and Y is the Sobolev space W 2(k+1),2(Rn)consisting of functions ψ ∈ L2(Rn) whose derivatives, in the sense of distributionsof order j ≤ 2(k + 1) are in L2(Rn) (cf. [16, Section 7.1]). Hence, in accordancewith Proposition 2.1 (iv), we require t→ h(t, ·) to be continuous in the graph norm‖ · ‖Y associated with the operator rβ(−∆) so that (30) is indeed well-posed.

For a simple example, consider problem (29) in H = L2(R), where A(t,−∆) =

a(t)(−∆) with a ∈ C([0, T ] : R+), and h(t, x) is defined by h(t, x) = e−(x−t)2 . Thus,the ill-posed problem (29) becomes the inhomogeneous backward heat equation

∂

∂tu(t, x) = −a(t)

∂2

∂x2u(t, x) + e−(x−t)2 , (t, x) ∈ [s, T )× R

u(s, x) = ψ(x), x ∈ R

with a time-dependent diffusion coefficient a(t). In this case, fβ(t, λ) = a(t)λ −βλ2, rβ(λ) = Bλ + βλ2 where B = maxt∈[0,T ] |a(t)|, and Y = Dom(rβ(−∆)) =

Dom(∆2) = W 4,2(R). The approximate well-posed problem in accordance withfβ(t, λ) = a(t)λ− βλ2 becomes

∂

∂tv(t, x) = −a(t)

∂2

∂x2v(t, x)− β ∂4

∂x4v(t, x) + e−(x−t)2 , (t, x) ∈ [s, T )× R (31)

v(s, x) = ψ(x), x ∈ R.

It is easy to check that e−(x−t)2 , and its derivatives with respect to x up to the

fourth order are each in L2(R) since q(x)e−(x−t)2 ∈ L2(R) for any polynomial q(x).

Using a dominated convergence argument, it may also be shown that t→ e−(x−t)2

is continuous in the graph norm ‖ · ‖Y associated with the operator rβ(−∆) =−B∆ + β∆2. Hence, (31) is well-posed.

Returning to the general case, as seen in Section 4, Theorem 4.2 implies that theapproximation fβ(t, λ) = A(t, λ)−βλk+1 of Example 5.1 yields an evolution system

Vβ(t, s) = e∫ tsfβ(τ,−∆)dτ , 0 ≤ s ≤ t ≤ T which serves as a family of regularizing

operators for the ill-posed problem (29).

Example 5.2. Using the second approximation fβ(t, λ) = A(t,λ)1+βλk

, the approximate

well-posed problem (4) becomes

∂

∂tv(t, x)−A(t,−∆)v(t, x) + β(−∆)k

∂

∂tv(t, x) = h(t, x), (t, x) ∈ [s, T )× Rn

(32)

v(s, x) = ψ(x), x ∈ Rn.

In this case, rβ(λ) = b(λ)1+βλk

as in (24), and since rβ(−∆) is a bounded everywhere-

defined operator, we have Y = Dom(rβ(−∆)) = L2(Rn). Thus, we require t →h(t, ·) to be continuous in the ordinary L2-norm ‖ · ‖2 so that (32) is indeed well-posed.

272 MATTHEW A. FURY

Again, as seen in Section 4, by Theorem 4.2, the approximation fβ(t, λ) = A(t,λ)1+βλk

of Example 5.2 yields an evolution system Vβ(t, s) = e∫ tsfβ(τ,−∆)dτ , 0 ≤ s ≤ t ≤ T

which serves as a family of regularizing operators for the ill-posed problem (29).

Acknowledgments. The author would like to thank Atousa Jahanshahi and YasirMughal for their contribution to the calculations in Section 4, and also Rhonda J.Hughes for her invaluable guidance.

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Received for publication July 2012.

E-mail address: maf44@psu.edu