Differentiability of convolutions,integrated semigroups of

bounded semi-variation, and theinhomogeneous Cauchy problem

Horst R. Thiemea

Department of Mathematics and Statistics

Arizona State University

J. Evolution Equations 8 (2008), 283-305

apartially supported by NSF grant 0314529

– p. 1/31

Inhomogeneous Cauchy Problems

u′(t) = Au(t) + f(t), t ∈ [0, b], u(0) = 0, (1)

where A a closed linear operator in a Banach space X ,

b ∈ (0,∞) and f : [0, b] → X.

A function u : [0, b] → X is called a classical solution of (1) if

u is continuously differentiable on [0, b],

u(t) ∈ D(A) for all t ∈ [0, b],

and u satisfies equation (1).

– p. 2/31

C0-Semigroups

Travis (1981)

A is the generator of a C0-semigroup T :

(1) has a classical solution for every continuousf : [0, b] → X if and only if

the semigroup T is of bounded semi-variation.

Quite restrictive (Travis): implies

semigroup is analytic

A is bounded if X is reflexive or an abstract L space.

– p. 3/31

Where to go from here

smaller classes of inhomogeneities f :

Crandall, Pazy (1969) Hölder continuous.

Webb (1977) continuous and of bounded variation.

Sell, You (2002) sufficient conditions.

Alternative: weaker notion of solution

– p. 4/31

Integral solutions

If f ∈ L1(0, b,X), a function u is an integral solution of (1):

u is continuous on [0, b],∫ t

0u(s)ds ∈ D(A) for all t ∈ [0, b],

u(t) = A

∫ t

0u(s)ds+

∫ t

0f(s)ds, t ∈ [0, b].

Every classical solution is an integral solution.

– p. 5/31

Hille-Yosida Operators

There exist M ≥ 1, ω ∈ R such that (ω,∞) is contained inthe resolvent set of A and

‖(λ− A)−n‖ ≤M(λ− ω)−n, λ > ω, n = 1, 2, . . . (2)

Theorem [Da Prato, Sinestrari, 1987]. Let A be aHille-Yosida operator, 0 < b <∞.

For all f ∈ L1(0, b,X), there exist a unique integral solutionu of (1).

Estimate

‖u(t)‖ ≤M

∫ t

0eω(t−s)‖f(s)‖ds, 0 ≤ t ≤ b.

– p. 6/31

Characterization

Proposition. The following are equivalent for a closed linearoperator A in a Banach space X.

1. A is Hille-Yosida operator.

2. A is the generator of an integrated semigroup T that is

Lipschitz continuous on some interval [0, b], b > 0.

Arendt (1987), Kellermann&Hieber (1989)

– p. 7/31

Integrated Semigroups

Focus on integrated semigroups rather than C0-semigroups.

Arendt, Batty, Hieber, Neubrander (2000)

T is a strongly continuous operator family T = {T (t); t ≥ 0},

T (t)T (r) =

∫ t+r

0T (s)ds−

∫ t

0T (s)ds−

∫ r

0T (s)ds, t, r ≥ 0,

T (0) = 0.

Example: T (t) =

∫ t

0S(r)dr, with C0-semigroup S.

Non-degenerate: T (t)x = 0 for all t > 0 occurs only for x = 0.

– p. 8/31

Generator

If T is exponentially bounded, definition via Laplacetransform,

(λ− A)−1 = λ

∫ ∞

0e−λtT (t)dt

for sufficiently large λ > 0.

Otherwise (Th 1990): if x, y ∈ X,

x ∈ D(A), y = Ax ⇐⇒ T (t)x−tx =

∫ t

0T (s)y ds ∀t ≥ 0.

– p. 9/31

Integrated solutions

Variation of constants formula, convolution

(T ∗ f)(t) =

∫ t

0T (t− s)f(s)ds.

Proposition. Let A be the generator of an integratedsemigroup T , f ∈ L1(0, b,X).

Then v = T ∗ f is the unique solution of

v(t) = A

∫ t

0v(s)ds+

∫ t

0(t− s)f(s)ds, 0 ≤ t ≤ b.

Can we differentiate?

– p. 10/31

Semivariation

Hönig (1975). The semi-variation of T on an interval [a, b] is

V∞(T ; a, b) = sup∥

∥

∥

k∑

j=1

[

T (tj) − T (tj−1)]

xj

∥

∥

∥,

where the supremum is taken over

all partitions P = {t0, . . . , tk} of [a, b], a = t0 < · · · < tk = b,

and all elements x1, . . . , xk ∈ X, ‖xj‖ ≤ 1, k ∈ N.

If V∞(T ; a, b) <∞, we will say that T is of boundedsemi-variation on [a, b].

– p. 11/31

Semi-p-variation

A larger net (Magal Ruan, 2007):

The semi-p-variation Vp(T ; a, b) is defined by the sameformula as before ,

and the supremum is taken over all partitions P (as before)

but x1, . . . , xk ∈ X with

k∑

j=1

(tj − tj−1)‖xj‖p ≤ 1.

– p. 12/31

Differentiability of convolutions

Theorem. The following hold for a strongly continuousfamily T = {T (t); 0 ≤ t ≤ b} of bounded linear operatorsbetween Banach spaces X, Y :

(a) T ∗ f is continuously differentiable on [0, b] for allf ∈ C(0, b,X) if and only if T is of boundedsemi-variation on [0, b].

(b) If 1 ≤ p <∞, T ∗ f is continuously differentiable on[0, b] for all f ∈ Lp(0, b,X) if and only if T is of boundedsemi-p-variation on [0, b] and T (0) = 0.

In (a), C(0, b,X) can be replaced by R(0, b,X), the space ofregulated functions.

– p. 13/31

Tools

=⇒: Closed graph theorem

⇐=: Stieltjes convolution.

If T is strongly continuous and if T (0) = 0 or g is continuous,then T ∗ g is continuously differentiable on [0, b], and

(T ∗ g)′(t) = (T ⋆ g)(t) + T (0)g(t).

where

(T ⋆ g)(t) =

∫ t

0T (ds)g(t− s).

Estimatesup

t∈[0,b]‖(T ⋆ g)(t)‖ ≤ Vp(T ; 0, b) ‖g‖p.

– p. 14/31

Back to the Cauchy problem

Theorem. Let A be the generator of an integratedsemigroup T , 0 < b <∞.

(a) If T is of bounded semi-variation on [0, b], then forevery f ∈ R(0, b,X) there exists a unique integralsolution u of (1) on [0, b], u = (T ∗ f)′ = T ⋆ f .

(b) T is of bounded semi-variation on [0, b]if for every f ∈ C(0, b,X) there exists an integralsolution u of (1) on [0, b].

Estimate leading to semilinear theory

supt∈[0,b]

‖(T ⋆ f)(t)‖ ≤ V∞(T ; 0, b) supt∈[0,b]

‖f(t)‖.

– p. 15/31

Cauchy problem in Lp

Theorem. Let A be the generator of an integratedsemigroup T , 1 ≤ p <∞.

Then T is of bounded semi-p-variation on [0, b]

if and only if,

for every f ∈ Lp(0, b,X), there exists an integral solution u of(1) on [0, b].

The solution u is uniquely determined by f ,u = (T ∗ f)′ = T ⋆ f .

Duality characterization in Magal&Ruan (2007).

– p. 16/31

(q-) variation

The variation of a function g : [a, b] → X∗ is

v(g; a, b) = supk∑

j=1

∥

∥g(tj) − g(tj−1)∥

∥,

where the supremum is taken over all partitionsP = {t0, . . . , tk} with a = t0 < · · · < tk = b,

and the q-variation is

vq(g; a, b) = sup

(

k∑

j=1

∥

∥g(tj) − g(tj−1)∥

∥

q

(tj − tj−1)q−1

)1/q

.

– p. 17/31

The duality connection

T ∗ is called to be of bounded strong (q-)variation if T ∗(·)y∗ isof bounded (q-)variation for each y∗ ∈ Y ∗.

Operator families of bounded strong (p-) variation arestudied by Th&Voßeler (2002) and Voßeler (2000).

Proposition. T is of bounded semi-variation if and only ifT ∗ is of bounded strong variation.

If 1 < p, q <∞ and1

p+

1

q= 1,

T is of bounded semi-p-variation

if and only if

T ∗ is of bounded strong q-variation.

– p. 18/31

More on semi-p-variation

Lemma. (a) If T is of bounded semi-p-variation, then it isHölder continuous with exponent 1/p and

‖T (t) − T (s)‖ ≤ Vp(T ; a, b)|t− s|1/p.

(b) T is of bounded semi-1-variation if and only if it isLipschitz continuous, and

V1(T ; a, b) = supa≤s<t≤b

‖T (t) − T (s)‖

t− s.

– p. 19/31

Characterization of Hille-Yosida Ops.

Theorem. The following are equivalent for a closed linearoperator A in a Banach space X.

1. A is a Hille-Yosida operator.

2. (i) For each x ∈ X, there exists a unique integralsolution of

u′ = Au+ x on R+, u(0) = 0,

which is exponentially bounded.

(ii) There exists some b ∈ (0,∞) such that, for eachf ∈ L1(0, b,X), there exists a unique integralsolution u of (1) on [0, b].

– p. 20/31

Example

Let p > 1, 0 < α < p−1p , X = Lp[0, 1],

(Af)(x) = −f ′(x) +α

xf(x).

D(A): absolutely continuous functions f on [0, 1]with f(0) = 0 and f ′ ∈ Lp[0, 1],

Arendt (1987): A generates the integrated semigroup

[T (t)f ](x) =

∫ t

0xα(x− s)−αf(x− s)H(x− s)ds, x ∈ [0, 1].

H is the Heaviside function. T is not a C0-semigroup.Since D(A) is dense, A is not a Hille-Yosida operator.

– p. 21/31

Example cont.

The dual family on Lq[0, 1], 1q + 1

p = 1, is given by

[T ∗(t)g](x) =

∫ t

0(x+ s)αx−αg(x+ s)H(1 − x− s)ds,

T ∗ is the integrated semigroup generated by A∗.

T ∗ is of locally bounded strong q-variation.

T is of locally bounded semi-p-variation.

Notice: X is an ordered Banach space,T and T ∗ are increasing,A and A∗ have positive resolvents.

– p. 22/31

Age-structure

Find u(t, a) ∈ X, t time, a age,

X Banach space, additional structure (space e.g.)

ut + ua = B(a)u+ g(t, a), t > 0, a > 0,

u(t, 0) = h(t), t > 0,

u(0, a) = u0(a), a > 0.

u(t, ·) ∈ Lp(R+, X), 1 ≤ p <∞.

B(a) closed linear operators associated with anevolutionary system {U(a, s); a ≥ s}.

– p. 23/31

A larger space

X = X × Lp(R+, X), X0 = {0} × Lp(R+, X).

B(0, φ) =(

−φ(0), −φ′ +B(·)φ)

.

Set v(t) = (0, u(t, ·)), Cauchy problem

v′(t) = Bv(t) + (h(t), g(t, ·)).

B is a Hille-Yosida operator if p = 1 (Th 1989),

and the generator of an integrated semigroup of locallybounded semi-p-variation if p > 1 (Magal&Ruan 2007).

– p. 24/31

Evolution semigroups

Let U = {U(t, s); 0 ≤ s ≤ t <∞} be the forwardevolutionary system associated with B(t).

We define the associated (Howland) evolution semigroupon Y = Lp(R+, X), 1 ≤ p <∞, by

[S(t)φ](a) =

{

U(a, a− t)φ(a− t); 0 ≤ t ≤ a

0; 0 ≤ a < t

}

, φ ∈ Y.

Integrated semigroup T (t)(x, φ) = (0, ψ(t)),

ψ(t)(a) = H(t− a)U(a, 0)x+

∫ t

0[S(r)φ](a)dt.

– p. 25/31

Perturbation Theory

Generalization of a result by Da Prato and Grisvard (1975).

Theorem. Let S = {S(t); t ≥ 0} be a C0-SG, generator A,

T = {T (t)} an ISG, generator B.

Assume that T is of locally bounded semi-variation andthat T and S commute, i.e. T (t)S(r) = S(r)T (t) for allt, r ≥ 0. Then

T̃ (t)x =

∫ t

0T (dr)S(r)x

defines an ISG of locally bounded semi-variation whosegenerator extends A+B (with domain D(A) ∩D(B)).T̃ commutes with both T and S.

– p. 26/31

commuting families

If 0 < b <∞ and T is of bounded semi-p-variation on [0, b],so is T̃ and

Vp(T̃ ; 0, b) ≤ mVp(T ; 0, b)

m = sup0≤t≤b

‖S(t)‖.

Da Prato and Grisvard (1975): p = 1

– p. 27/31

Bounded perturbations

Theorem. Let A be the generator of an ISG T which is oflocally bounded semi-(p-)variation and the linear operatorB : D(A) → X satisfy

‖B‖ = sup{‖Bx‖;x ∈ D(A), ‖x‖ ≤ 1} <∞.

If T is of locally bounded semi-variation, assume inaddition that ‖B‖V∞(T, 0, b) < 1 for some b > 0.Then A+ B generates an ISG V of locally bounded semi-(p-)variation which solves the equations

V (t)x− T (t)x =

∫ t

0T (ds)B̄V (t− s)x =

∫ t

0V (ds)B̄T (t− s)x,

where B is the extension of B to D(A).

– p. 28/31

Summary

(CP) u′ = Au+ f(t) on [0, b], u(0) = 0.

(CP) has integral solutions for all continuous(regulated) f if and only if A generates an ISG ofbounded semi-variation.

(CP) has integral solutions for all f ∈ Lp if and only ifA generates an ISG of bounded semi-p-variation.

the generator A of an ISG is a Hille-Yosida operator ifand only if there is some b > 0 such that (CP) hasintegral solutions for all f ∈ L1.

– p. 29/31

Perturbations

ISGs of locally bounded semi-(p-)variation arepreserved under bounded additive perturbations oftheir generators.

The commutative sum A+ B of generators of aC0-semigroup and of an ISG of locally boundedsemi-(p-)variation generates an ISG of the same type.

– p. 30/31

Examples

ISGs of bounded semi-p-variation occur naturally inage-structured population models (cf. Magal Ruan,2007)

where, at any time, the age-distribution of thepopulation is a vector-valued function in Lp(R+, X).

Natural examples of resolvent positive operatorsgenerate ISGs of bounded semi-(p-)variation.

– p. 31/31