We prove H¨older-continuous dependence results for the difference between certain ill-posed and well-posed evolution problems in a Hilbert space

Simulations. Electronic Journal of Differential Equations, Conference 19 (2010), pp. 99–

121.

ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu ftp ejde.math.txstate.edu

CONTINUOUS DEPENDENCE OF SOLUTIONS FOR ILL-POSED EVOLUTION PROBLEMS

MATTHEW FURY, RHONDA J. HUGHES

Abstract. We prove H¨older-continuous dependence results for the difference between certain ill-posed and well-posed evolution problems in a Hilbert space.

Specifically, given a positive self-adjoint operator D in a Hilbert space, we consider the ill-posed evolution problem

du(t)

dt =A(t, D)u(t) 0≤t < T u(0) =χ.

We determine functions f : [0, T]×[0,∞) → R for which solutions of the well-posed problem

dv(t)

dt =f(t, D)v(t) 0≤t < T v(0) =χ

approximate known solutions of the original ill-posed problem, thereby estab- lishing continuous dependence on modelling for the problems under consider- ation.

1. Introduction

LetDbe a positive self-adjoint operator in a Hilbert space H, and consider the evolution problem

du(t)

dt =A(t, D)u(t) 0≤t < T u(0) =χ

(1.1) whereχ is an arbitrary element ofH and

A(t, D) =

k

X

j=1

a_j(t)D^j,

2000Mathematics Subject Classification. 47A52, 42C40.

Key words and phrases. Continuous dependence on modelling; time-dependent problems;

Ill-posed problems.

c

2010 Texas State University - San Marcos.

Published September 25, 2010.

99

with aj : [0, T] →R continuous and nonnegative for each 1≤j ≤k. In general, (1.1) is ill-posed with formal solution given by

u(t) = expnX^k

j=1

Z t

0

aj(s)ds D^jo

χ.

Now let f : [0, T]×[0,∞)→Rbe a function continuous in t and Borel in λ, and consider the evolution problem

dv(t)

dt =f(t, D)v(t) 0≤t < T v(0) =χ ,

(1.2) where for each t ∈ [0, T], f(t, D) is defined by means of the functional calculus for self-adjoint operators inH. In particular, since D is positive, self-adjoint, the spectrumσ(D) ofD is contained in [0,∞) and for eacht∈[0, T],

f(t, D)x= Z ∞

0

f(t, λ)dE(λ)x, for x ∈ Dom(f(t, D)) = {x ∈ H : R∞

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

denotes the resolution of the identity for the self-adjoint operatorD.

We determine conditions onf so that (1.2) is well-posed and such that solutions of (1.2) approximate known solutions of (1.1). In this way, we illustrate how we might stabilize problems against errors that arise when formulating mathematical models such as (1.1) in attempts to describe some physical process. Ames and Hughes [3] established such structural stability results for the problem in the au- tonomous case, that is when A(t) = A is a positive self-adjoint operator in H, independent oft. This paper generalizes such work and yields a comparable result in the time-dependent case. Namely, ifu(t) andv(t) are solutions of (1.1) and (1.2) respectively, we prove the H¨older-continuous approximation

ku(t)−v(t)k ≤Cβ^1−TtM^t/T,

where 0< β <1, andCandM are constants independent ofβ. Our approximation establishes continuous dependence on modelling, meaning “small” changes to our model yield a “small” change in the corresponding solution.

In Section 2, we establish conditions under which (1.2) is well-posed using stable families of generators of semigroups and Kato’s stability conditions [8, 11]; our work also utilizes Tanaka’s results on evolution problems [14]. In Section 3, we present our approximation theorem which achieves H¨older-continuous dependence on modelling. Finally, Section 4 demonstrates our theorem with examples.

Below, for a closed operator A in a Banach space X, ρ(A) will denote the re- solvent set of A, and for λ ∈ ρ(A), R(λ;A) will denote the resolvent operator R(λ;A) = (λI−A)⁻¹ in X.

2. The Well-Posed Evolution Problem

We first clarify what we mean by solutions of evolution problems as well as the notions of ill-posed and well-posed evolution problems.

Definition 2.1 ([11]). LetX be a Banach space and for everyt∈[0, T], let A(t) be a linear operator inX. The initial value problem

du(t)

dt =A(t)u(t) 0≤s≤t < T u(s) =x

(2.1) is called an evolution problem. AnX-valued functionu: [s, T]→X is aclassical solution of (2.1) ifuis continuous on [s, T], u(t)∈Dom(A(t)) fors < t < T, uis continuously differentiable on (s, T), andusatisfies (2.1).

Theorem 2.2 ([11, Theorem 5.1.1]). Let X be a Banach space and for every t∈[0, T], let A(t) be a bounded linear operator on X. If the function t 7→A(t)is continuous in the uniform operator topology then for everyx∈X, the initial value problem (2.1)has a unique classical solution u.

The proof of Theorem 2.2 (cf. [11, Theorem 5.1.1]) shows that the mapping S:C([s, T] :X)→C([s, T] :X) defined by

(Su)(t) =x+ Z t

s

A(τ)u(τ)dτ

is a well-defined mapping with a unique fixed pointu. It is easily shown thatuis then a unique classical solution of (2.1). In this case we define thesolution operator of (2.1) by

U(t, s)x=u(t) for 0≤s≤t≤T.

Theorem 2.3([11, Theorem 5.1.2]). LetU(t, s)be the solution operator associated with (2.1)whereA(t)is a bounded linear operator on X for eacht∈[0, T]andt7→

A(t)is continuous in the uniform operator topology. Then for every0≤s≤t≤T, U(t, s)is a bounded linear operator such that

(i) kU(t, s)k ≤e^{RstkA(τ)kdτ}.

(ii) U(t, t) =I, U(t, s) =U(t, r)U(r, s)for0≤s≤r≤t≤T.

(iii) (t, s)7→U(t, s)is continuous in the uniform operator topology for 0≤s≤ t≤T.

(iv) ^∂U(t,s)_∂t =A(t)U(t, s)for0≤s≤t≤T. (v) ^∂U(t,s)_∂s =−U(t, s)A(s)for0≤s≤t≤T.

We now turn to the notions of well-posedness and ill-posedness for evolution problems.

Definition 2.4 ([6, Definition 7.1]). The evolution problem (2.1) is called well- posed in 0≤t < T if the following two assumptions are satisfied:

(i) (Existence of solutions for sufficiently many initial data) There exists a dense subspace Y of X such that for every s ∈ [0, T) and every x∈ Y, there exists a classical solutionu(t) of (2.1).

(ii) (Continuous dependence of solutions on their initial data) There exists a strongly continuousB(X)-valued functionU(t, s) defined in 0≤s≤t≤T such that ifu(t) is a classical solution of (2.1), then

u(t) =U(t, s)x.

Equation (2.1) is calledill-posed if it is not well-posed.

It is clear that under the hypotheses of Theorem 2.2, that Equation (2.1) is well- posed with unique classical solution given by u(t) = U(t, s)χ where U(t, s),0 ≤ s≤t≤T, is the solution operator given by Theorem 2.2. Otherwise, we use the construction of an evolution system and the theory of stable families of operators to obtain well-posedness of (2.1). We explore stability conditions for (2.1) first developed by Kato [8, 11], and later by Tanaka [14].

Definition 2.5([11, Definition 5.1.3]). A two parameter family of bounded linear operatorsU(t, s), 0≤s≤t≤T, on a Banach spaceX is called anevolution system if the following two conditions are satisfied:

(i) U(s, s) =I, U(t, r)U(r, s) =U(t, s) for 0≤s≤r≤t≤T.

(ii) (t, s)7→U(t, s) is strongly continuous for 0≤s≤t≤T.

Definition 2.6([11, Def. 5.2.1]). LetX be a Banach space. A family{A(t)}_t∈[0,T]

of infinitesimal generators ofC0 semigroups on X is calledstable if there are con- stantsM ≥1 andω (called the stability constants) such that

ρ(A(t))⊇(ω,∞) fort∈[0, T] and

k

k

Y

j=1

R(λ;A(tj))k ≤M(λ−ω)^−k forλ > ω and every finite sequence 0≤t₁≤t₂, . . . , t_k≤T,k= 1,2, . . ..

Remark. If for t ∈ [0, T], A(t) is the infinitesimal generator of a C0 semigroup {St(s)}_s≥0 satisfyingkSt(s)k ≤ e^ωs, then by the Hille-Yosida theorem (cf. [11]), the family{A(t)}_t∈[0,T] is stable with constantsM = 1 andω.

We now use the theory of stable families of operators to gain well-posedness of (2.1) in the following way. Let X and Y be Banach spaces with norms k · k and k · kY respectively. Assume thatY is densely and continuously imbedded inX, that isY is a dense subspace ofX and there is a constantC such that

kyk ≤CkykY fory∈Y.

For each t ∈ [0, T], let A(t) be the infinitesimal generator of a C₀ semigroup {S_t(s)}_s≥0 onX. Assume the following conditions (cf. [8, 11]):

(H1) {A(t)}_t∈[0,T] is a stable family with stability constantsM,ω.

(H2) For eacht∈[0, T],Y is an invariant subspace ofSt(s), s≥0, the restriction S˜t(s) ofSt(s) toY is aC0 semigroup inY, and the family {A(t)}˜ _t∈[0,T] of parts ˜A(t) ofA(t) inY, is a stable family inY.

(H3) Fort∈[0, T], Dom(A(t))⊇Y,A(t) is a bounded operator fromY intoX, andt7→A(t) is continuous in theB(Y, X) normk · kY→X.

Theorem 2.7([8, Theorem 4.1], [11, Theorem 5.3.1]). For eacht∈[0, T], letA(t) be the infinitesimal generator of a C0 semigroup {St(s)}_s≥0 on X. If the family {A(t)}_t∈[0,T] satisfies conditions (H1)–(H3), then there exists a unique evolution systemU(t, s),0≤s≤t≤T, inX satisfying

(E1) kU(t, s)k ≤M e^ω(t−s)for0≤s≤t≤T. (E2) ^∂_∂t⁺U(t, s)y

_t=s=A(s)y fory∈Y,0≤s≤T.

(E3) _∂s^∂U(t, s)y=−U(t, s)A(s)y fory∈Y,0≤s≤t≤T,

where the derivative from the right in (E2) and the derivative in (E3) are in the strong sense inX.

This theorem will help in obtaining a certain kind of classical solution of (2.1) in the case where the family{A(t)}_t∈[0,T] of infinitesimal generators ofC0 semigroups onX satisfies conditions (H1)–(H3).

Definition 2.8 ([11, Definition 5.4.1]). LetX andY be Banach spaces such that Y is densely and continuously imbedded in X and let{A(t)}_t∈[0,T] be a family of infinitesimal generators of C0 semigroups on X satisfying the assumptions (H1)–

(H3). A functionu∈C([s, T] :Y) is a Y-valued solution of (2.1) ifu∈C¹((s, T) : X) andusatisfies (2.1) inX.

Remark. A Y-valued solutionuof (2.1) is a classical solution of (2.1) such that u(t)∈Y ⊆Dom(A(t)) for t∈[s, T] andu(t) is continuous in the strongerY-norm rather than merely in theX-norm.

Theorem 2.9 ([11, Thm. 5.4.3]). Let {A(t)}_t∈[0,T] satisfy the conditions of The- orem 2.7 and let U(t, s),0≤s≤t≤T be the evolution system given in Theorem 2.7. If

(E4) U(t, s)Y ⊆Y for0≤s≤t≤T and

(E5) Forx∈Y,U(t, s)xis continuous in Y for0≤s≤t≤T, then for every x∈Y,U(t, s)xis the uniqueY-valued solution of (2.1).

We now use the above theory of stable families of generators to give criteria for well-posedness of the evolution problem (1.2). Let (2.2) denote the initial value problem (1.2) with 0 replaced bysfors∈[0, T); i.e.,

dv(t)

dt =f(t, D)v(t) 0≤s≤t < T v(s) =χ.

(2.2) We determine conditions on f so that the family of operators {f(t, D)}_t∈[0,T] is stable and such that (2.2) is well-posed.

Proposition 2.10. Let f : [0, T]×[0,∞)→Rbe continuous int and Borel in λ.

Assume there exist ω ∈Rsuch that f(t, λ)≤ω for all (t, λ)∈[0, T]×[0,∞)and a Borel function r: [0,∞)→[0,∞)such that|f(t, λ)| ≤r(λ)andDom(f(t, D)) = Dom(r(D))for all t∈[0, T]. Set Y = Dom(r(D))and let k · kY denote the graph norm associated with the operatorr(D). Further, assumet7→f(t, D)is continuous in theB(Y, H)normk · kY→H. Then (2.2)is well-posed and forχ∈Y,V(t, s)χ= e^{Rstf(τ,D)dτ}χis a unique Y-valued solution of (2.2).

Proof. By [5, Theorem XII.2.6],r(D) is a closed operator inH with dense domain.

SetY = Dom(r(D)) and endowY with the graph norm k · kY given by kykY =kyk+kr(D)yk

for all y ∈ Y. Since r(D) is a closed operator, it follows that (Y,k · kY) is a Banach space. It is also clear that Y is densely and continuously imbedded inH. Since f(t, λ)≤ ω for all (t, λ) ∈ [0, T]×[0,∞), we have that for eacht ∈ [0, T], f(t, D) is the infinitesimal generator of theC₀ semigroup{St(s)}s≥0 onH given by S_t(s) = e^sf(t,D). We show that the family {f(t, D)}_t∈[0,T] satisfies conditions (H1)–(H3).

Lett∈[0, T], x∈H. Then ke^sf(t,D)xk²=

Z ∞

0

|e^sf(t,λ)|²d(E(λ)x, x)≤(e^sω)² Z ∞

0

d(E(λ)x, x) = (e^sω)²kxk², showing thatkSt(s)k=ke^sf(t,D)k ≤ e^ωs. Thus, {f(t, D)}_t∈[0,T] is a stable family with stability constantsM = 1 andω, and so (H1) is satisfied.

Next, lett∈[0, T],y∈Y. For anys≥0, sincey∈Y = Dom(r(D)), we have Z ∞

0

|r(λ)e^sf(t,λ)|²d(E(λ)y, y)≤(e^sω)² Z ∞

0

|r(λ)|²d(E(λ)y, y)<∞.

Thus, St(s)y ∈Dom(r(D)) and so Y is an invariant subspace ofSt(s). Let ˜St(s) be the restriction ofSt(s) toY. For any positive constantc, for 0≤s≤c,

|r(λ)(e^sf(t,λ)−1)|²≤ |r(λ)|²(e^cω+ 1)²∈L¹(E(·)y, y).

Therefore, by Lebesgue’s Dominated Convergence Theorem, lim

s→0⁺kr(D)(St(s)−I)yk²= lim

s→0⁺

Z ∞

0

|r(λ)(e^sf(t,λ)−1)|²d(E(λ)y, y)

= Z ∞

0

lim

s→0⁺|r(λ)(e^sf(t,λ)−1)|²d(E(λ)y, y) = 0, and so

kS˜t(s)y−ykY =kS˜t(s)y−yk+kr(D)( ˜St(s)y−y)k

=kSt(s)y−yk+kr(D)(St(s)−I)yk

→0 ass→0⁺. Thus, ˜St(s) is a C0 semigroup onY.

Next, consider the family {f˜(t, D)}_t∈[0,T] of parts ˜f(t, D) of f(t, D) in Y. For eacht∈[0, T], ˜f(t, D) is defined by

Dom( ˜f(t, D)) ={x∈Dom(f(t, D))∩Y :f(t, D)x∈Y} and

f(t, D)x˜ =f(t, D)x forx∈Dom( ˜f(t, D)).

It is seen [11, Theorem 4.5.5] that ˜f(t, D) is the infinitesimal generator of the C₀ semigroup ˜S_t(s). Moreover, fory∈Y,

kS˜t(s)ykY =kS˜t(s)yk+kr(D) ˜St(s)yk

=kSt(s)yk+kr(D)St(s)yk

≤e^sωkyk+e^sωkr(D)yk

=e^sωkykY.

Thus, kS˜t(s)kY ≤e^ωs for all t∈[0, T] and so the family {f˜(t, D)}_t∈[0,T] is stable with stability constants ˜M = 1 andω. We have shown that (H2) is satisfied.

Finally, lett∈[0, T]. Since|f(t, λ)| ≤r(λ), we have fory∈Y, Z ∞

0

|f(t, λ)|²d(E(λ)y, y)≤ Z ∞

0

|r(λ)|²d(E(λ)y, y)<∞.

Thus Dom(f(t, D))⊇Y. Also, fory∈Y,

kf(t, D)yk ≤ kyk+kf(t, D)yk ≤ kyk+kr(D)yk=kykY,

showing that f(t, D) is a bounded operator from Y into H. By assumption, t 7→

f(t, D) is continuous in the B(Y, H) norm k · kY→H and so (H3) is satisfied. By Theorem 2.7, there exists a unique evolution system V(t, s), 0≤s≤t ≤T, inH satisfying conditions (E1)-(E3) with the operatorsf(t, D),t∈[0, T], andM = 1 in the condition (E1); that is we have

kV(t, s)k ≤e^ω(t−s) for 0≤s≤t≤T,

∂⁺

∂tV(t, s)y

_t=s=f(s, D)y fory∈Y, 0≤s≤T,

∂

∂sV(t, s)y=−V(t, s)f(s, D)y fory∈Y, 0≤s≤t≤T,

where the derivatives are in the strong sense in H. It can be shown using the Spectral Theorem thate^{Rstf(τ,D)dτ}is such an evolution system, and so by uniqueness we must haveV(t, s) =e^{Rstf(τ,D)dτ}. It is also readily seen thatV(t, s) =e^{Rstf(τ,D)dτ} satisfies (E4) and (E5). Therefore, by Theorem 2.9, for every χ ∈Y, V(t, s)χ = e^{Rstf(τ,D)dτ}χis the uniqueY-valued solution of (2.2).

Finally, supposev₁is a classical solution of (2.2). Thenv₁(q)∈Dom(f(q, D)) = Dom(r(D)) forq∈(s, T). AsV(t, s), 0≤s≤t≤T, satisfies condition (E3) with the operatorsf(t, D),t∈[0, T], the functionq7→V(t, q)v1(q) is then differentiable and

∂

∂qV(t, q)v1(q) =−V(t, q)f(q, D)v1(q) +V(t, q)d dqv1(q)

=−V(t, q)f(q, D)v₁(q) +V(t, q)f(q, D)v₁(q) = 0.

Thus V(t, q)v1(q) is constant for q ∈ (s, t). Since v1 is a classical solution, the functionV(t, q)v1(q) is also continuous for q∈[s, t]. Thus we have

v1(t) =V(t, t)v1(t) =V(t, s)v1(s) =V(t, s)χ.

Thus condition (ii) of Definition 2.4 is satisfied and we see that (2.2) is well-posed with unique classical solution given byv(t) =V(t, s)χ.

3. The Approximation Theorem

In order that solutions of (1.2) approximate known solutions of (1.1), we will require additional conditions on f. The following definition is inspired by results obtained by Ames and Hughes [3, Definition 1] for continuous dependence on mod- elling in the autonomous case, that is whenA(t) =A is independent oft.

Definition 3.1. Letf : [0, T]×[0,∞)→Rbe a function continuous intand Borel in λand assume the hypotheses of Proposition 2.10. Thenf is said to satisfy the approximation condition with polynomial por simplyCondition(A, p) if there exist a constant β, with 0 < β < 1, and a nonzero polynomial p(λ) independent of β such that for eacht∈[0, T], Dom(p(D))⊆Dom(A(t, D))∩Dom(f(t, D)), and

k(−A(t, D) +f(t, D))ψk ≤βkp(D)ψk, for allψ∈Dom(p(D)).

Now assumef satisfies Condition (A, p). For eacht∈[0, T], set g(t, λ) =−A(t, λ) +f(t, λ),

and for eachn≥ |ω|, set

en={λ∈[0,∞) : max

t∈[0,T]|g(t, λ)| ≤n}.

Then

λ∈en⇒ max

t∈[0,T]|g(t, λ)| ≤n

⇒ |g(t, λ)| ≤n ∀t∈[0, T]

⇒A(t, λ)≤n+f(t, λ) ∀t∈[0, T].

SinceA(t, λ)≥0 andf(t, λ)≤ω for all (t, λ)∈[0, T]×[0,∞), we have that onen, max

t∈[0,T]|A(t, λ)| ≤n+ω.

Sincef(t, λ) =A(t, λ) +g(t, λ), it then follows that one_n,

t∈[0,T]max |f(t, λ)| ≤2n+ω.

SetEn=E(en) and letψ∈H be arbitrary. Consider the following three evolution problems:

dun(t)

dt =A(t, D)Enun(t) 0≤s≤t < T un(s) =ψ,

(3.1) dvn(t)

dt =f(t, D)Envn(t) 0≤s≤t < T vn(s) =ψ,

(3.2) dwn(t)

dt =g(t, D)Enwn(t) 0≤s≤t < T wn(s) =ψ.

(3.3) Problems (3.1)–(3.3), as we will see, are well-posed due to the action of En and their solutions will aid in approximating known solutions of the ill-posed problem (1.1).

Lemma 3.2. For eacht∈[0, T],A(t, D)E_n is a bounded operator onH such that kA(t, D)Enk ≤n+ω,

and (3.1)has a unique classical solution un(t) =Un(t, s)ψ. The solution operator Un(t, s)is a bounded operator onH with

kUn(t, s)k ≤e^T(n+ω)

for alls, t such that 0≤s≤t≤T. Furthermore, if ψis replaced by ψ_n=E_nψ in (3.1), then

Un(t, s)ψn =e^{RstA(τ,D)dτ}ψn. Proof. Fixt∈[0, T]. For allx∈H, by [5, Theorem XII.2.6],

kA(t, D)Enxk²= Z ∞

0

|A(t, λ)|²d(E(λ)Enx, Enx)

= Z

en

|A(t, λ)|²d(E(λ)x, x)

≤(n+ω)² Z

e_n

d(E(λ)x, x)

≤(n+ω)² Z ∞

0

d(E(λ)x, x)

= (n+ω)²kxk²,

showing thatA(t, D)En is a bounded operator onH withkA(t, D)Enk ≤n+ω.

Next, let t0 ∈ [0, T]. Since en is a bounded subset of [0,∞), we have that D^jE_n∈B(H) for each 1≤j≤k. Then by continuity ofa_j for each 1≤j≤k, we have

kA(t, D)En−A(t0, D)Enk=k

k

X

j=1

(aj(t)−aj(t0))D^jEnk

≤

k

X

j=1

|a_j(t)−a_j(t₀)| kD^jE_nk →0 as t→t₀, showing thatt7→A(t, D)Enis continuous in the uniform operator topology. It fol- lows from Theorem 2.2 that (3.1) has a unique classical solutionu_n(t) =U_n(t, s)ψ.

That

kUn(t, s)k ≤e^T(n+ω)

follows directly from Theorem 2.3 (i) and the fact thatkA(t, D)Enk ≤n+ωfor all t∈[0, T].

Next, set ψ_n = E_nψ and let (3.4) denote the evolution problem (3.1) with ψ replaced byψ_n; i.e.,

du_n(t)

dt =A(t, D)Enun(t) 0≤s≤t < T, un(s) =ψn.

(3.4) Using the Spectral Theorem it can be shown that e^{RstA(τ,D)dτ}ψn is a classical so- lution of (3.4). In particular, using properties of the projection operator En, we have

d

dte^{RstA(τ,D)dτ}ψn=A(t, D)e^{RstA(τ,D)dτ}ψn

=A(t, D)Ene^{RstA(τ,D)dτ}ψn, and

e^{RssA(τ,D)dτ}ψ_n=ψ_n.

Therefore, by uniqueness guaranteed by Theorem 2.2, we have Un(t, s)ψn =e^{RstA(τ,D)dτ}ψn.

Lemma 3.3. For each t∈[0, T],f(t, D)En is a bounded operator onH such that

kf(t, D)E_nk ≤2n+ω,

and (3.2) has a unique classical solutionvn(t) =Vn(t, s)ψ. The solution operator Vn(t, s)is a bounded operator onH with

kV_n(t, s)k ≤e^T(2n+ω)

for alls, t such that 0≤s≤t≤T. Furthermore, if ψis replaced by ψn=Enψ in (3.2), then

V_n(t, s)ψ_n=e^{Rstf(τ,D)dτ}ψ_n.

Proof. Using the fact that onen, max_t∈[0,T]|f(t, λ)| ≤2n+ω, it is easily shown that for eacht∈[0, T],f(t, D)En is a bounded operator onH such thatkf(t, D)Enk ≤ 2n+ω. Next, let t0 ∈ [0, T]. Since EnH ⊆ Dom(f(t, D)) = Dom(r(D)) for all t∈[0, T], we haver(D)En∈B(H), and so

kf(t, D)En−f(t0, D)Enk

= sup

x∈H,kxk≤1

k(f(t, D)−f(t₀, D))E_nxk

≤ sup

x∈H,kxk≤1

kf(t, D)−f(t0, D)kY→HkEnxkY

= sup

x∈H,kxk≤1

kf(t, D)−f(t₀, D)k_Y→H(kE_nxk+kr(D)E_nxk)

≤ kf(t, D)−f(t0, D)kY→H(kEnk+kr(D)Enk)→0 ast→t0

by the assumption that t7→f(t, D) is continuous in theB(Y, H) norm k · k_Y→H. Therefore,t7→f(t, D)En is continuous in the uniform operator topology. It follows from Theorem 2.2 that (3.2) has a unique classical solutionvn(t) =Vn(t, s)ψ. That

kVn(t, s)k ≤e^T(2n+ω)

follows directly from Theorem 2.3 (i) and the fact thatkf(t, D)E_nk ≤2n+ω for allt∈[0, T]. The rest of the proof is similar to that of Lemma 3.2.

Lemma 3.4. For each t∈[0, T],g(t, D)E_n is a bounded operator onH such that kg(t, D)Enk ≤n,

and (3.3)has a unique classical solution w_n(t) =W_n(t, s)ψ. The solution operator Wn(t, s)is a bounded operator onH with

kWn(t, s)k ≤e^{T n}

for alls, t such that 0≤s≤t≤T. Furthermore, if ψis replaced by ψn=Enψ in (3.3), then

W_n(t, s)ψ_n=e^{Rstg(τ,D)dτ}ψ_n.

Proof. Using the fact that one_n, max_t∈[0,T]|g(t, λ)| ≤n, it is easily shown that for eacht∈[0, T], g(t, D)E_n is a bounded operator onH such that kg(t, D)Enk ≤n.

Also, by the relation g(t, D)E_n = −A(t, D)En +f(t, D)E_n, it follows that t 7→

g(t, D)E_n is continuous in the uniform operator topology. Therefore, by Theorem 2.2, (3.3) has a unique classical solutionw_n(t) =W_n(t, s)ψ. That

kW_n(t, s)k ≤e^{T n}

follows directly from Theorem 2.3 (i) and the fact that kg(t, D)Enk ≤ n for all t∈[0, T]. The rest of the proof is similar to that of Lemma 3.2.

Corollary 3.5. Let ψ∈H andψn=Enψ. Then

U_n(t, s)W_n(t, s)ψ_n =V_n(t, s)ψ_n=W_n(t, s)U_n(t, s)ψ_n for all0≤s≤t≤T.

The corollary above follows immediately from Lemmas 3.2, 3.3, and 3.4, and from properties of the functional calculus for unbounded self-adjoint operators [5, Corollary XII.2.7].

We now have all the necessary machinery to prove our approximation theorem.

Our strategy will be to extend the solutionsu_n(t) of (3.1) withψ=χ_n, and v_n(t)

of (3.2) withψ=χn, into the complex strip S ={t+iη:t∈[0, T], η∈R}, and eventually employ Hadamard’s Three Lines Theorem (cf. [12]). To make use of such extensions we will need the following results. Our approach is motivated by work of Agmon and Nirenberg [1].

Definition 3.6 ([12, Definition 11.1]). Letφ(α) be a complex function defined in a plane open set Ω. Assume all partial derivatives of φexist and are continuous.

Define theCauchy-Riemann operator ∂¯as

∂¯= 1 2

∂

∂t+i ∂

∂η ,

whereα=t+iη.

Theorem 3.7([12, Theorem 11.2]). Supposeφ(α)is a complex function inΩsuch that all partial derivatives ofφ exist and are continuous. Thenφ is analytic in Ω if and only if the Cauchy-Riemann equation

∂φ(α) = 0¯ holds for every α∈Ω.

Lemma 3.8 ([1]). Letφ(z)be a complex function withz=x+iy. Assumeφ(z)is continuous and bounded onS={z=x+iy:x∈[0, T], y∈R}. Forα=t+iη∈S, define

Φ(α) =−1 π

Z Z

S

φ(z) 1

z−α+ 1

¯

z+ 1 +α

dx dy.

ThenΦ(α) is absolutely convergent, ∂Φ(α) =¯ φ(α), and there exists a constantK such that

Z ∞

−∞

1

z−α+ 1

¯

z+ 1 +α

dy≤K

1 + log 1

|x−t|

if x6=t.

We now state and prove our approximation theorem.

Theorem 3.9. LetDbe a positive self-adjoint operator acting onH and letA(t, D) be defined as above for allt∈[0, T]. Letf satisfy Condition(A, p), and assume that there exists a constant γ, independent ofβ, ω, andt such that g(t, λ)≤γ, for all (t, λ)∈[0, T]×[0,∞). Then ifu(t)andv(t)are classical solutions of(1.1)and(1.2) respectively, and if there exist constantsM⁰, M⁰⁰, M⁰⁰⁰≥0 such that ku(T)k ≤M⁰, kp(D)χk ≤M⁰⁰, and kp(D)A(t, D)u(T)k ≤M⁰⁰⁰ for all t∈[0, T], then there exist constants C andM independent of β such that for 0≤t < T,

ku(t)−v(t)k ≤Cβ^1−TtM^t/T.

Proof. Letχn=Enχ and setS ={t+iη:t∈[0, T], η∈R}. Lettingψ=χn and s= 0 in (3.1) and (3.2), we extend the solutionsun(t) andvn(t) into the complex strip S in the following way. Since A(0, D) andf(0, D) are self-adjoint, e^iηA(0,D) ande^iηf(0,D)are bounded operators onH for allη∈R, and so we define

u_n(α) =e^iηA(0,D)U_n(t,0)χ_n, vn(α) =e^iηf(0,D)Vn(t,0)χn, forα=t+iη∈S. Finally defineφn:S→H by

φ_n(α) =u_n(α)−v_n(α).

We first determine ¯∂φn(α). Sincee^iηA(0,D)ande^iηf(0,D)are bounded operators on H, andA(t, D) andf(t, D) are bounded when acting onEnH, we have

∂

∂tφn(α) = ∂

∂te^iηA(0,D)Un(t,0)χn− ∂

∂te^iηf(0,D)Vn(t,0)χn

=e^iηA(0,D)d

dtU_n(t,0)χ_n−e^iηf(0,D)d

dtV_n(t,0)χ_n

=e^iηA(0,D)A(t, D)Un(t,0)χn−e^iηf(0,D)f(t, D)Vn(t,0)χn

=A(t, D)un(α)−f(t, D)vn(α).

Next, by standard properties of semigroups of linear operators (cf. [11, Theorem 1.2.4 (c)]), since Un(t,0)χn ∈ Dom(A(0, D)), and Vn(t,0)χn ∈Dom(f(0, D)), we have

∂

∂ηφ_n(α) = ∂

∂ηe^iηA(0,D)U_n(t,0)χ_n− ∂

∂ηe^iηf(0,D)V_n(t,0)χ_n

=iA(0, D)e^iηA(0,D)U_n(t,0)χ_n−if(0, D)e^iηf(0,D)V_n(t,0)χ_n

=i(A(0, D)un(α)−f(0, D)vn(α)).

Therefore,

∂φ¯ _n(α) =1 2

∂

∂tφ_n(α) +i ∂

∂ηφ_n(α)

=1

2[(A(t, D)u_n(α)−f(t, D)v_n(α))−(A(0, D)u_n(α)−f(0, D)v_n(α))]

=1

2[(A(t, D)−A(0, D))u_n(α)−(f(t, D)−f(0, D))v_n(α)].

(3.5) Since, in general, this quantity is not identically zero, φ_n is not analytic and so we cannot apply the Three Lines Theorem toφn. To amend this, we introduce a related function. Define

Φ_n(α) =−1 π

Z Z

S

e^z2∂φ¯ _n(z) 1

z−α+ 1

¯

z+ 1 +α

dx dy,

where z = x+iy and α = t+iη are in S. To apply Lemma 3.8, we show that e^z2∂φ¯ _n(z) is bounded and continuous onS. Letz =x+iy ∈S be arbitrary. We have from (3.5),

ke^z2∂φ¯ _n(z)k= 1

2|e^z2| k(A(x, D)−A(0, D))u_n(z)−(f(x, D)−f(0, D))v_n(z)k

≤ 1

2e^T2(kA(x, D)un(z)k+kA(0, D)un(z)k +kf(x, D)vn(z)k+kf(0, D)vn(z)k).

Sinceke^iyA(0,D)k= 1, we have by Lemma 3.2 and properties of En, kA(x, D)un(z)k=kA(x, D)e^iyA(0,D)Un(x,0)χnk

=kA(x, D)Ene^iyA(0,D)Un(x,0)χnk

≤(n+ω)ke^iyA(0,D)Un(x,0)χnk

≤(n+ω)kU_n(x,0)χ_nk

≤(n+ω)e^T(n+ω)kχ_nk.

Note that sincex∈[0, T] is arbitrary, the same bound holds for kA(0, D)un(z)k.

Similarly,ke^iyf(0,D)k= 1, and using Lemma 3.3,

kf(x, D)vn(z)k=kf(x, D)e^iyf(0,D)Vn(x,0)χnk

=kf(x, D)Ene^iyf(0,D)Vn(x,0)χnk

≤(2n+ω)ke^iyf(0,D)V_n(x,0)χ_nk

≤(2n+ω)kVn(x,0)χnk

≤(2n+ω)e^T(2n+ω)kχnk.

Again, sincex∈[0, T] is arbitrary, the same bound holds forkf(0, D)v_n(z)k. Set- ting

Cn= (n+ω)e^T(n+ω)+ (2n+ω)e^T(2n+ω), we have

ke^z2∂φ¯ n(z)k ≤e^T2Cnkχnk, (3.6) showing thate^z2∂φ¯ n(z) is indeed bounded onS. It can also easily be shown using continuity of A(t, D)En and f(t, D)En in the B(H) norm, continuity of Un(t, s) and Vn(t, s) in the B(H) norm (Theorem 2.3 (iii)), and strong continuity of the groups{e^iyA(0,D)}_y∈Rand{e^iyf(0,D)}_y∈R, thate^z2∂φ¯ n(z) is continuous onS. Hav- ing satisfied the hypotheses of Lemma 3.8, it follows that

Φ_n(α) =−1 π

Z Z

S

e^z2∂φ¯ _n(z) 1

z−α+ 1

¯

z+ 1 +α

dx dy is absolutely convergent,

∂Φ¯ n(α) =e^α2∂φ¯ n(α), and there exists a constantK such that

Z ∞

−∞

1

z−α+ 1

¯

z+ 1 +α

dy≤K

1 + log 1

|x−t|

forx6=t.

We now construct a candidate for the Three Lines Theorem. Define Ψn :S→H by

Ψn(α) =e^α2φn(α)−Φn(α).

Forαin the interior ofS, using the product rule and results from Lemma 3.8,

∂Ψ¯ n(α) = ¯∂[e^α2φn(α)]−∂Φ¯ n(α)

= [( ¯∂e^α2)φ_n(α) +e^α2∂φ¯ _n(α)]−∂Φ¯ _n(α)

= [(0)φn(α) + ¯∂Φn(α)]−∂Φ¯ n(α) = 0.

Therefore, by Theorem 3.7, Ψ_n is analytic on the interior of S. Next, for α = t+iη∈S, from (3.6) and the results from Lemma 3.8,

kΦn(α)k=k − 1 π

Z Z

S

e^z2∂φ¯ n(z) 1

z−α+ 1

¯

z+ 1 +α

dx dyk

≤ 1 π

Z ∞

−∞

Z T

0

e^T2Cnkχnk

1

z−α+ 1

¯

z+ 1 +α dx dy

= 1

πe^T2Cnkχnk Z T

0

Z ∞

−∞

1

z−α+ 1

¯

z+ 1 +α dy

dx

≤ K

πe^T2C_nkχnk Z T

0

1 + log 1

|x−t|

dx.

Also, using Lemmas 3.2 and 3.3,

kφn(α)k=ke^iηA(0,D)Un(t,0)χn−e^iηf(0,D)Vn(t,0)χnk

≤(kUn(t,0)k+kVn(t,0)k)kχnk

≤(e^T(n+ω)+e^T(2n+ω))kχnk.

Therefore,

kΨn(α)k=ke^α2φ_n(α)−Φ_n(α)k

≤ |e^α2| kφn(α)k+kΦn(α)k

≤e^T2(e^T(n+ω)+e^T(2n+ω))kχnk +K

πe^T2C_nkχnk max

t∈[0,T]

Z T

0

1 + log 1

|x−t|

dx ,

proving that Ψn is bounded on S. From (3.6) and the results from Lemma 3.8, it follows via a dominated convergence argument that Φn is continuous onS. It is also easily shown thatφn is continuous onS, and therefore Ψn is continuous onS.

We have shown that Ψn is bounded and continuous onS, and analytic on the interior of S. It follows from the Cauchy-Schwarz Inequality that for arbitrary h∈H, the mapping

α7→(Ψn(α), h)

fromSintoC, where (·,·) denotes the inner product inH, has the same properties.

Therefore, by the Three Lines Theorem,

|(Ψn(α), h)| ≤M(0)^1−TtM(T)^t/T, fort∈[0, T], whereα=t+iηand

M(t) = max

η∈R|(Ψn(t+iη), h)|.

We aim to find bounds onM(0) andM(T). First, forη∈R,

|(Ψn(iη), h)| ≤ kΨn(iη)kkhk

=ke^−η2

e^iηA(0,D)Un(0,0)χn−e^iηf(0,D)Vn(0,0)χn

−Φn(iη)kkhk

=ke^−η2

e^iηA(0,D)χn−e^iηf(0,D)χn

−Φn(iη)kkhk

≤

e^−η2ke^iηA(0,D)χ_n−e^iηf(0,D)χ_nk+kΦn(iη)k khk.

Now, sinceke^iηA(0,D)k= 1,

ke^iηA(0,D)χn−e^iηf(0,D)χnk=ke^iηA(0,D)χn−e^iηA(0,D)e^iηg(0,D)χnk

=ke^iηA(0,D)(I−e^iηg(0,D))χ_nk

≤ k(I−e^iηg(0,D))χ_nk.

For ψ∈ Dom(g(0, D)) and η ∈ R, we have by standard properties of semigroups (cf. [11, Theorem 1.2.4]) that

k(I−e^iηg(0,D))ψk=k −i Z η

0

e^isg(0,D)g(0, D)ψdsk ≤ |η|kg(0, D)ψk.

Noteχ_n∈Dom(A(0, D))∩Dom(f(0, D))⊆Dom(g(0, D)) and sincee_nis a bounded subset of [0,∞),χ_n∈Dom(p(D)). Thus we have by Condition (A, p) and the above inequality that

ke^iηA(0,D)χn−e^iηf(0,D)χnk ≤β|η|kp(D)χnk.

Next we would like a bound onkΦn(iη)kin terms ofβ. Letz=x+iy ∈S. Then from (3.5),

2k∂φ¯ n(z)k=k(A(x, D)−A(0, D))un(z)−(f(x, D)−f(0, D))vn(z)k

≤ kA(x, D)un(z)−f(x, D)vn(z)k+kA(0, D)un(z)−f(0, D)vn(z)k.

Now,

kA(x, D)un(z)−f(x, D)vn(z)k

=kA(x, D)e^iyA(0,D)Un(x,0)χn−f(x, D)e^iyf(0,D)Vn(x,0)χnk

≤ kA(x, D)e^iyA(0,D)U_n(x,0)χ_n−A(x, D)e^iyf(0,D)U_n(x,0)χ_nk (3.7) +kA(x, D)e^iyf(0,D)U_n(x,0)χ_n−A(x, D)e^iyf(0,D)V_n(x,0)χ_nk (3.8) +kA(x, D)e^iyf(0,D)V_n(x,0)χ_n−f(x, D)e^iyf(0,D)V_n(x,0)χ_nk. (3.9) Setψ_n=A(x, D)U_n(x,0)χ_n. We note that ψ_n∈Dom(A(t, D))∩Dom(f(t, D))⊆ Dom(g(t, D)) for all t ∈ [0, T], and ψ_n ∈ Dom(p(D)). Then expression (3.7) is equal to

ke^iyA(0,D)ψ_n−e^iyf(0,D)ψ_nk ≤β|y|kp(D)ψnk, as above.

Next, using Theorem 2.3, Lemma 3.4, and the assumption that g(t, λ)≤γ for all (t, λ)∈[0, T]×[0,∞), we have

k(I−Wn(x,0))ψnk=k(Wn(x, x)−Wn(x,0))ψnk

=k Z x

0

∂

∂sW_n(x, s)ψ_ndsk

=k Z x

0

(−Wn(x, s)g(s, D)En)ψndsk

≤ Z x

0

kWn(x, s)g(s, D)ψnkds

≤ Z x

0

(1 +e^γT)kg(s, D)ψnkds.

It then follows from Corollary 3.5 and Condition (A, p) that expression (3.8) is equal to

ke^iyf(0,D)(U_n(x,0)−V_n(x,0))A(x, D)χ_nk

≤ k(Un(x,0)−Vn(x,0))A(x, D)χnk

=k(Un(x,0)−Wn(x,0)Un(x,0))A(x, D)χnk

=k(I−W_n(x,0))ψ_nk

≤ Z x

0

(1 +e^γT)kg(s, D)ψnkds

≤ Z x

0

(1 +e^γT)βkp(D)ψnkds

≤βT(1 +e^γT)kp(D)ψnk.

Finally, since Un(x,0)χn ∈Dom(p(D)), Corollary 3.5 and Condition (A, p) imply that expression (3.9) is equal to

ke^iyf(0,D)Vn(x,0)(−A(x, D) +f(x, D))χnk

≤ kVn(x,0)(−A(x, D) +f(x, D))χnk

=kWn(x,0)U_n(x,0)(−A(x, D) +f(x, D))χ_nk

≤(1 +e^γT)k(−A(x, D) +f(x, D))Un(x,0)χnk

≤β(1 +e^γT)kp(D)U_n(x,0)χ_nk.

Therefore, we have shown

kA(x, D)un(z)−f(x, D)v_n(z)k

≤β(1 +e^γT) ((|y|+T)kp(D)A(x, D)Un(x,0)χnk+kp(D)Un(x,0)χnk). Sincexis arbitrary, it follows similarly that

kA(0, D)un(z)−f(0, D)vn(z)k

≤β(1 +e^γT) ((|y|+T)kp(D)A(0, D)Un(x,0)χnk+kp(D)Un(x,0)χnk). From the assumptions ku(T)k ≤ M⁰ and kp(D)A(t, D)u(T)k ≤ M⁰⁰⁰ for all t ∈ [0, T], it follows thatkp(D)u(T)k ≤N⁰ for some constantN⁰ ≥0. It then follows from these estimates that forz=x+iy∈S,

k∂φ¯ _n(z)k ≤β(1 +e^γT) ((|y|+T)M⁰⁰⁰+N⁰), so that by Lemma 3.8,

kΦn(iη)k

=k − 1 π

Z Z

S

e^z2∂φ¯ n(z) 1

z−iη+ 1

¯

z+ 1 +iη

dx dyk

≤ 1 π

Z ∞

−∞

Z T

0

|e^z2|k∂φ¯ _n(z)k

1

z−iη + 1

¯

z+ 1 +iη dx dy

≤ 1 π

Z ∞

−∞

Z T

0

e^x2−y2β(1 +e^γT) ((|y|+T)M⁰⁰⁰+N⁰)

1

z−iη+ 1

¯

z+ 1 +iη

dx dy

= 1 π

Z ∞

−∞

Z T

0

e^x2β(1 +e^γT)

(|y|e^−y2+T e^−y2)M⁰⁰⁰+e^−y2N⁰

×

1

z−iη+ 1

¯

z+ 1 +iη dx dy

≤βh1

πe^T2(1 +e^γT) ((1 +T)M⁰⁰⁰+N⁰) Z T

0

Z ∞

−∞

1

z−iη+ 1

¯

z+ 1 +iη dy

dxi

≤βhK

πe^T2(1 +e^γT) ((1 +T)M⁰⁰⁰+N⁰) Z T

0

1 + log1 x

dxi .

Therefore, M(0)

= max

η∈R

|(Ψn(iη), h)|

≤max

η∈R

e^−η2ke^iηA(0,D)χ_n−e^iηf(0,D)χ_nk+kΦn(iη)k khk

≤max

η∈R

β|η|e^−η2kp(D)χnk+kΦn(iη)k khk

≤β

M⁰⁰+hK

πe^T2(1 +e^γT) ((1 +T)M⁰⁰⁰+N⁰) Z T

0

1 + log1 x

dxi khk.

(3.10)

Next, forη∈R,

|(Ψn(T+iη), h)|

≤ kΨ_n(T+iη)kkhk

=ke^(T+iη)2

e^iηA(0,D)Un(T,0)χn−e^iηf(0,D)Vn(T,0)χn

−Φn(T+iη)kkhk

≤

e^T2−η2ke^iηA(0,D)U_n(T,0)χ_n−e^iηf(0,D)V_n(T,0)χ_nk+kΦn(T+iη)k khk.

Using the assumption thatku(T)k ≤M⁰,

ke^iηA(0,D)Un(T,0)χn−e^iηf(0,D)Vn(T,0)χnk

≤ kU_n(T,0)χ_nk+kV_n(T,0)χ_nk

=kUn(T,0)χnk+kWn(T,0)Un(T,0)χnk

≤(1 + (1 +e^γT))kUn(T,0)χnk

≤(2 +e^γT)M⁰.

Next, the assumptionsku(T)k ≤M⁰andkp(D)A(t, D)u(T)k ≤M⁰⁰⁰for allt∈[0, T] imply thatkA(t, D)u(T)k ≤N⁰⁰for allt∈[0, T], for some constantN⁰⁰≥0. Thus, forz=x+iy∈S, we have

k(A(x, D)−A(0, D))un(z)k

≤ ke^iyA(0,D)A(x, D)Un(x,0)χnk+ke^iyA(0,D)A(0, D)Un(x,0)χnk

≤ kA(x, D)Un(x,0)χ_nk+kA(0, D)Un(x,0)χ_nk

≤2N⁰⁰.

Meanwhile, using Condition (A, p) and the fact that 0< β <1, k(f(x, D)−f(0, D))vn(z)k

≤ ke^iyf(0,D)f(x, D)V_n(x,0)χ_nk+ke^iyf(0,D)f(0, D)V_n(x,0)χ_nk

≤ kf(x, D)Vn(x,0)χnk+kf(0, D)Vn(x,0)χnk

≤ kA(x, D)Vn(x,0)χnk+k(−A(x, D) +f(x, D))Vn(x,0)χnk +kA(0, D)Vn(x,0)χnk+k(−A(0, D) +f(0, D))Vn(x,0)χnk

≤ kVn(x,0)A(x, D)χ_nk+kVn(x,0)A(0, D)χ_nk+ 2βkp(D)Vn(x,0)χ_nk

≤(1 +e^γT)(kA(x, D)Un(x,0)χnk+kA(0, D)Un(x,0)χnk+ 2kp(D)Un(x,0)χnk)

≤2(1 +e^γT)(N⁰⁰+N⁰).

Therefore, forz=x+iy∈S, from (3.5), k∂φ¯ n(z)k ≤ 1

2(k(A(x, D)−A(0, D))un(z)k+k(f(x, D)−f(0, D))vn(z)k)

≤N⁰⁰+ (1 +e^γT)(N⁰⁰+N⁰) so that by Lemma 3.8,

kΦ_n(T+iη)k

=k − 1 π

Z Z

S

e^z2∂φ¯ n(z)

1

z−(T+iη)+ 1

¯

z+ 1 + (T +iη)

dx dyk

≤ 1 π

Z ∞

−∞

Z T

0

|e^z2|k∂φ¯ n(z)k

1

z−(T+iη)+ 1

¯

z+ 1 + (T+iη) dx dy

≤ 1 π

Z ∞

−∞

Z T

0

e^T2(N⁰⁰+ (1 +e^γT)(N⁰⁰+N⁰))

×

1

z−(T+iη)+ 1

¯

z+ 1 + (T+iη) dx dy

= 1

πe^T2(N⁰⁰+ (1 +e^γT)(N⁰⁰+N⁰))

× Z T

0

Z ∞

−∞

1

z−(T+iη)+ 1

¯

z+ 1 + (T+iη) dy

dx

≤ K

πe^T2(N⁰⁰+ (1 +e^γT)(N⁰⁰+N⁰)) Z T

0

1 + log 1

|x−T|

dx.

Thus, M(T)

= max

η∈R

|(Ψn(T+iη), h)|

≤max

η∈R

e^T2−η2(2 +e^γT)M⁰+K

πe^T2(N⁰⁰+ (1 +e^γT)(N⁰⁰+N⁰))

× Z T

0

1 + log 1

|x−T| dx

khk

≤

e^T2(2 +e^γT)M⁰+K

πe^T2(N⁰⁰+ (1 +e^γT)(N⁰⁰+N⁰))

× Z T

0

1 + log 1

|x−T| dx

khk.

(3.11)

It follows from (3.10) and (3.11) that there exist constantsC⁰ andM, independent ofβ, such that for 0≤t < T,

|(Ψn(t), h)| ≤(C⁰βkhk)^1−Tt(Mkhk)^t/T = (C⁰β)^1−TtM^t/Tkhk.

Taking the supremum over allh∈H withkhk ≤1, we have constantsC and M, independent ofβ, such that for 0≤t < T,

kΨn(t)k ≤Cβ^1−TtM^t/T. Consequently,

kun(t)−vn(t)k=kφn(t)k

=e^−t2kΨ_n(t) + Φ_n(t)k

≤ kΨn(t)k+kΦn(t)k

≤Cβ^1−TtM^t/T+kΦn(t)k.

It follows from an earlier estimate onkΦn(iη)k, that kΦ_n(t)k ≤βhK

πe^T2(1 +e^γT) ((1 +T)M⁰⁰⁰+N⁰) Z T

0

1 + log 1

|x−t|

dxi .

Setting K⁰= K

πe^T2(1 +e^γT) ((1 +T)M⁰⁰⁰+N⁰)n max

t∈[0,T]

Z T

0

1 + log 1

|x−t|

dxo ,

we have

ku_n(t)−v_n(t)k ≤Cβ^1−TtM^t/T +kΦ_n(t)k

≤Cβ^1−TtM^t/T +βK⁰

=

C+β^t/TK⁰M^−Tt

β^1−TtM^t/T

≤Cβ^1−TtM^t/T,

for a possibly different constant C. Letting n → ∞, we have found constants C andM, independent ofβ, such that for 0≤t < T,

ku(t)−v(t)k ≤Cβ^1−TtM^t/T,

as desired.

4. Examples

Below, we give examples illustrating our approximation theorem. Each is a general case of the following universal example. Let H = L²(Rⁿ) and D = −∆

where ∆ denotes the Laplacian defined by

∆h=

n

X

i=1

∂²h

∂x²_i.

The operator−∆ is a positive self-adjoint operator onL²(Rⁿ) and so we compare the ill-posed evolution problem

∂

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

(4.1) inL²(Rⁿ) to the well-posed approximate problem

∂

∂tv(t, x) =f(t,−∆)v(t, x), (t, x)∈[0, T)×Rⁿ v(0, x) =h(x), x∈Rⁿ.

(4.2)

Example 4.1. As initially defined, let A(t, D) =

k

X

j=1

a_j(t)D^j.

Let 0 < < 1 and set Bj = max_t∈[0,T]|aj(t)| for each 1 ≤j ≤ k. Consider the problem

∂

∂tv(t, x) =A(t,−∆)v(t, x)−(−∆)^k+1v(t, x), (t, x)∈[0, T)×Rⁿ v(0, x) =h(x), x∈Rⁿ.

Motivated by approximations used by Lattes and Lions, Miller, and Ames and Hughes [2, 3, 9, 10], we definef : [0, T]×[0,∞)→Rby

f(t, λ) =

k

X

j=1

aj(t)λ^j−λ^k+1. Then for each (t, λ)∈[0, T]×[0,∞),

f(t, λ)≤h(λ) :=

k

X

j=1

Bjλ^j−λ^k+1.

The polynomial h(λ) has at most k+ 1 real roots. Ifh(λ) has no real roots on [0,∞), then h(λ)<0 for allλ≥0. Otherwise, let R be the maximum of all such roots of h(λ) on [0,∞). Then h(λ) is bounded above on [0, R] and is negative on (R,∞). Therefore, in any case, there existsω∈Rsuch thath(λ)≤ωfor allλ≥0.

Consequently,

f(t, λ)≤ω for all (t, λ)∈[0, T]×[0,∞). Also

|f(t, λ)| ≤r(λ) :=

k

X

j=1

B_jλ^j+λ^k+1

for all (t, λ) ∈ [0, T]×[0,∞). We setY = Dom(r(D)) and let k · kY denote the graph norm associated with the operator r(D). We note thatY = Dom(r(D)) = Dom(f(t, D)) for all t ∈ [0, T], and that Y is the Sobolev space W^2(k+1),2(Rⁿ), consisting of functionsh∈L²(Rⁿ) whose derivatives, in the sense of distributions, of orderj≤2(k+ 1) are inL²(Rⁿ) (cf. [11, Chapter 7.1]).

Now, let t0 ∈ [0, T]. It follows from the definition of r(D) thatD^j ∈B(Y, H) for each 1≤j≤k. Then since aj is continuous for each 1≤j ≤k,

kf(t, D)−f(t₀, D)k_Y→H =k(A(t, D)−D^k+1)−(A(t₀, D)−D^k+1)k_Y→H

=kA(t, D)−A(t0, D)kY→H

=k

k

X

j=1

(a_j(t)−a_j(t₀))D^jk_Y→H

≤

k

X

j=1

|aj(t)−aj(t0)| kD^jkY→H→0 ast→t0, showing thatt7→f(t, D) is continuous in theB(Y, H) normk · kY→H.

Next, set

p(λ) =λ^k+1.

Then Dom(p(D))⊆Dom(D^j) for all 1≤j≤k+ 1 so that

Dom(p(D))⊆Dom(A(t, D))∩Dom(f(t, D)) = Dom(f(t, D))

for eacht∈[0, T]. Furthermore, forψ∈Dom(p(D)) andt∈[0, T], k(−A(t, D) +f(t, D))ψk=k(−D^k+1)ψk=kD^k+1ψk.

Thus f satisfies Condition (A, p) with r(λ) = Pk

j=1Bjλ^j +λ^k+1, β = , and p(λ) =λ^k+1. Moreover,g(t, λ) =−λ^k+1 ≤0 for all (t, λ)∈[0, T]×[0,∞), so we may chooseγ= 0. Theorem 3.9 then yields the result

ku(t)−v(t)k ≤Cβ^1−TtM^t/T

for 0≤t < T, where u(t) andv(t) are solutions of (4.1) and (4.2) respectively.

Example 4.2. As in Example 4.1, let A(t, D) =

k

X

j=1

aj(t)D^j.

Let 0 < < 1 and set Bj = max_t∈[0,T]|aj(t)| for each 1 ≤j ≤ k. Consider the problem

∂

∂tv(t, x)−A(t,−∆)v(t, x) +(−∆)^k ∂

∂tv(t, x) = 0, (t, x)∈[0, T)×Rⁿ v(0, x) =h(x), x∈Rⁿ.

Motivated by work of Showalter [13], we definef : [0, T]×[0,∞)→Rby f(t, λ) =

Pk

j=1aj(t)λ^j 1 +λ^k . Then for each (t, λ)∈[0, T]×[0,∞),

f(t, λ)≤ Pk

j=1Bjλ^j 1 +λ^k . The rational function r(λ) =

Pk j=1Bjλ^j

1+λ^k is continuous on [0,∞) and tends to ^Bk as λ → ∞. Therefore, there exists ω ∈ R such that r(λ) ≤ ω for all λ ≥ 0.

Consequently,

f(t, λ)≤ω

for all (t, λ)∈ [0, T]×[0,∞). Asr(λ) is a bounded Borel function on [0,∞), the Spectral Theorem yields that r(D) is a bounded everywhere-defined operator on H. Thus, we may chooseY = Dom(r(D)) =H.

Now, lett0∈[0, T]. It follows from the definition ofr(D) thatD^j(I+D^k)⁻¹∈ B(H) for each 1≤j≤k. Then since aj is continuous for each 1≤j≤k,

kf(t, D)−f(t0, D)k=kA(t, D)(I+D^k)⁻¹−A(t0, D)(I+D^k)⁻¹k

=k

k

X

j=1

(a_j(t)−a_j(t₀))D^j(I+D^k)⁻¹k

≤

k

X

j=1

|aj(t)−aj(t0)| kD^j(I+D^k)⁻¹k →0 as t→t0, showing thatt7→f(t, D) is continuous in theB(H) norm.