Non-autonomous Miyadera perturbation

Frank Räbiger^{1}, Abdelaziz Rhandi^{2,}^{∗}, Roland Schnaubelt^{1,}^{†}, and Jürgen Voigt^{3,}^{‡}

1Universität Tübingen, Mathematisches Institut, Auf der Morgenstelle 10, 72076 Tübingen, Germany 2Département de Mathématiques, Faculté des Sciences Semlalia, B.P.: S.15, 40000 Marrakech, Maroc

3Technische Universität Dresden, Fachrichtung Mathematik, 01062 Dresden, Germany

Abstract: We consider time dependent perturbations B(t) of a non–autonomous Cauchy problem v̇(t) = A(t)v(t) (CP) on a Banach space X. The existence of a mild solution u of the perturbed problem is proved under Miyadera type conditions on B(·). In the parabolic case and X = L^{d}(Ω), 1 < d < ∞, we show that u is differentiable a.e. and satisfies u̇(t) = (A(t)+B(t))u(t) for a.e. t. Our approach uses perturbation results due to one of the authors, [29], and S. Monniaux and J. Prüß, [14], which are applied to the evolution semigroup induced by the evolution family related to (CP). As an application we obtain solutions of a second order parabolic equation with singular lower order coefficients.

Mathematics Subject Classification (1991): 47D06, 34G10, 47A55, 35K10

1. Introduction

Consider the non–autonomous linear abstract Cauchy problems

(CP) \{ \begin{matrix}\frac{d}{dt}u(t) & = & A(t)u(t), \\ u(s) & = & x, t ≥ s, \end{matrix}  and (pCP) \{ \begin{matrix}\frac{d}{dt}u(t) & = & (A(t) + B(t)) u(t), \\ u(s) & = & x, t ≥ s, \end{matrix}

on a Banach space X. In order to solve (pCP) we assume that (CP) is “well–posed” and that the operators B(t) are “small with respect to A(t)”. In the autonomous case, i.e., A(t) ≡ A, B(t) ≡ B, and A generates a C_{0}–semigroup (T (t))_{t≥0}, there are various conditions on A and B ensuring that the operator A + B is also a generator, and hence (pCP) is well–posed. For instance, if B is A–bounded and satisfies the Miyadera condition

∃ α > 0, β ∈ [0, 1) such that  ∫_{0}^{α} ∥BT (t)x∥ dt ≤ β ∥x∥ , x ∈ D(A),  (1.1)

then (A + B, D(A)) generates a C_{0}–semigroup (T_{B}(t))_{t≥0} , [13], [29], where D(A) is the domain of A.

In the present paper we prove a corresponding perturbation theorem in the non– autonomous situation extending a previous result by part of the authors, [24]. Here, instead of a C_{0}–semigroup (T (t))_{t≥0} , we consider an evolution family U = (U(t, s))_{(t,s)∈D} in the space L(X) of bounded linear operators on X, that is, (E1) U(s, s) = Id, U(t, s) = U(t, r)U(r, s) for t ≥ r ≥ s in I and

∗Part of this work was done while the second author visited the Scuola Normale Superiore di Pisa. He wishes to express his gratitude to G. DaPrato and the Scuola Normale for their hospitality.

†Support by the Deutsche Forschungsgemeinschaft is gratefully acknowledged.

‡Partly supported by the Deutsche Forschungsgemeinschaft.

(E2) the mapping D ∋ (t, s) ↦ U(t, s) is strongly continuous,

where I is an interval in IR and D = D_{I} := {(t, s) ∈ I^{2} : t ≥ s}. Observe that for a C_{0}–semigroup (T (t))_{t≥0} the operators U(t, s) := T (t − s), t ≥ s, define an evolution family with index set D_{IR}. Further, for I ∈ {[a, b], [a, ∞), IR} the Cauchy problem (CP) is called well–posed (on spaces Y_{s}) if Y_{s}, s ∈ I, is dense in X and for x ∈ Y_{s} there is a unique solution u ∈ C^{1}(I ∩ [s, ∞), X) of (CP) depending continuously on initial data. In this case there exists an evolution family U such that u(t) = U(t, s)x, that is, U solves (CP), see [19].

In contrast to the autonomous case, it can happen that the mapping t ↦ U(t, s)x is differentiable only if x = 0 (e.g., take X = C and U(t, s) = p(t)/p(s), where p, 1/p ∈ C_{b}(IR) and p is nowhere differentiable). Moreover, regularity properties of U are not preserved even for bounded perturbations B(·), see [23, Ex. 6.4]. Thus, at first, we will not assume that the evolution family U is differentiable in any sense and look for an evolution family U_{B} such that the variation of constants formula

U_{B}(t, s)x = U(t, s)x + ∫_{s}^{t} U(t, τ)B(τ)U_{B}(τ, s)x dτ  (1.2)

holds. If U is related to (CP) then we say that t ↦ u(t) = U_{B}(t, s)x is a mild solution of (pCP). Note that if U solves (CP) on Y_{s} and τ ↦ B(τ)U_{B}(τ, s)x, x ∈ Y_{s}, is sufficiently smooth, then u satisfies (pCP), see e.g. [22, §5.5, 5.7].

In Theorem 3.4 we show the existence of U_{B} by assuming a non–autonomous version of the Miyadera condition (1.1). Our approach is based on the so–called evolution semigroup T = (T (t))_{t≥0} on E = L^{p}(I, X) given by (T (t)f) (s) := χ_{I}(s − t) U(s, s − t)f(s − t), f ∈ E, see Section 2. We perturb T by the multiplication operator B = B(·) on E. By applying a Miyadera type perturbation theorem for semigroups, [29], and an abstract characterization of evolution semigroups (see Theorem 2.4) we then obtain a perturbed evolution semigroup T_{B}. Finally, the associated evolution family U_{B} satisfies (1.2).

In Section 4 we consider evolution families U solving (CP) where the operators A(t) generate analytic semigroups and perturbations B(t) which are bounded operators from the domain of a fractional power of A(t) to X. For X = L^{d}(Ω), 1 < d < ∞, we can show that t ↦ U_{B}(t, s)x, x ∈ X, is differentiable for a.e. t > s and satisfies (pCP) for a.e. t > s. Our proof relies on a recent perturbation result of Dore–Venni type by S. Monniaux and J. Prüß, [14], which we apply to the evolution semigroup T . Finally, in the last section, we use the above results to solve a second order parabolic equation with singular lower order coefficients.

2. A characterization of evolution semigroups

In this section we consider an evolution family U = (U(t, s))_{(t,s)∈D} on a Banach space X defined for a right–closed interval I ⊆ IR. We call U exponentially bounded if there are constants M ≥ 1 and w ∈ IR such that ∥U(t, s)∥ ≤ Me^{w(t−s)} for (t, s) ∈ D. For compact intervals I this estimate always holds with w = 0 due to the principle of uniform boundedness. For exponentially bounded U,

(T (t)f) (s) := χ_{I}(s − t) U(s, s − t)f(s − t) = \{ \begin{matrix}U(s, s − t)f(s − t), & if s − t ∈ I, \\ 0, & if s − t ∉ I, \end{matrix}

defines a bounded operator T (t) on the Bochner–Lebesgue space E = L^{p}(I, X), 1 ≤ p < ∞, where χ_{I} is the characteristic function of I. It is easily seen that T = (T (t))_{t≥0} is a strongly continuous semigroup on E, see e.g. [27], [28]. We call T the evolution semigroup associated with U and denote its generator by (G, D(G)). For further information concerning this approach we refer to [5], [9], [11], [12], [16]–[19], [21], [24]–[28], and the references therein.

The resolvent R(λ, G) := (λ − G)^{−1}, λ ∈ ρ(G), has an important smoothing property which is stated below. For I = IR this result is contained in [25, Prop. 3.3]. The general case can be shown analogously and the proof is therefore omitted, see also [28, Prop. 2.2]. For a similar result in the case of bounded I we refer to [17, Cor. 4.7]. By C_{00}(I, X) we denote the space of continuous functions f : I → X vanishing at a := inf I and at infinity (if I is unbounded) endowed with the sup–norm ∥ · ∥_{∞}. (Notice that C_{00}(IR, X) is usually denoted by C_{0}(IR, X).)

Proposition 2.1. Let I be a right–closed interval in IR and (U(t, s))_{(t,s)∈D} an exponentially bounded evolution family on X. Let (G, D(G)) be the generator of the associated evolution semigroup on L^{p}(I, X), 1 ≤ p < ∞. Consider λ ∈ ρ(G). Then R(λ, G) : L^{p}(I, X) → C_{00}(I, X) is continuous with dense image. In particular, D(G) is dense in C_{00}(I, X).

Following an approach due to J. Howland, [9], we reduce the characterization of evolution semigroups to the characterization of multiplication operators on E. We consider the space C_{b}(I, L_{s}(X)) of bounded, strongly continuous functions M(·) : I → L(X), where I ⊆ IR is an interval. Clearly, M(·) ∈ C_{b}(I, L_{s}(X)) induces a bounded multiplication operator M on L^{p}(I, X) by setting Mf := M(·)f(·), and ∥M∥ = ∥M(·)∥_{∞}. We need the following notion (see also [17, Def. 4.4], [24, Def. 3.1]).

Definition 2.2. Let M ∈ L(L^{p}(I, X)), 1 ≤ p < ∞. A subspace F of L^{p}(I, X) is called M–determining if (D1) F and MF consist of continuous functions;

(D2) F^{s} := {f(s) : f ∈ F } is dense in X for all s ∈ I;

(D3) F is dense in L^{p}(I, X).

For I = IR the next result was shown in [24, Prop. 3.3]. The proof carries over to the case I ≠ IR with obvious changes. For related characterizations we refer to [5] and [9].

Proposition 2.3. Let I be an interval in IR and X a Banach space. Consider M ∈ L(L^{p}(I, X)), 1 ≤ p < ∞. Then M = M(·) ∈ C_{b}(I, L_{s}(X)) if and only if there is an M–determing subspace F of L^{p}(I, X) such that M(φf) = φ Mf for f ∈ F and φ ∈ L^{∞}(I).

The right translations R(t), t ∈ IR, defined by (R(t)f) (s) := χ_{I}(s − t) f(s − t) for s ∈ I and f ∈ E, yield a C_{0}–semigroup (R(t))_{t≥0} on E = L^{p}(I, X). For X = C we write R(t) instead of R(t). Let I_{t} := I ∩ (I + t) for t ∈ IR. We identify the space L^{p}(I_{t}, X) with a subspace of E = L^{p}(I, X), 1 ≤ p < ∞, by extending functions by 0. Recall that a core of a closed operator (A, D(A)) is a subspace of D(A) which is dense in the graph norm ∥x∥_{A} := ∥x∥ + ∥Ax∥. Further, C_{c}^{1}(I) is the space of continuously differentiable functions with compact support. Finally, we make use of the following condition, cf. Proposition 2.1:

(R) There is λ ∈ ρ(G) such that R(λ, G) : L^{p}(I, X) → C_{00}(I, X) is continuous with dense image D(G).

If I contains its left endpoint a and U is an exponentially bounded evolution family with index set D_{I}, then the restriction of U to the index set D_{I′}, I′ := I \ {a}, induces the same evolution semigroup as U on L^{p}(I, X). So, as in [17], in our characterization of evolution semigroups it suffices to consider a left half–open interval I, that is, I is left–open and right– closed. Observe that, in general, an evolution family defined for a left–open interval has no continuous extension to DĪ. For instance, let X = C, I = (0, ∞), and U(t, s) = p(t)/p(s) for p(t) = 2 + sin \frac{1}{t}.

We now formulate the main result of this section. It extends [24, Thm. 3.4] where the case I = IR was considered. A closely related result for bounded intervals I is due to H. Neidhardt, [17, Thm. 4.12]. Evolution semigroups on C_{00}(I, X) were characterized in [5], [12], [17], [21].

Theorem 2.4. Let I be a left half–open interval in IR. Consider a C_{0}–semigroup T = (T (t))_{t≥0} on E = L^{p}(I, X), 1 ≤ p < ∞, with generator (G, D(G)). Let D be a core of G. Then the following assertions are equivalent:

(a) T is an evolution semigroup induced by an exponentially bounded evolution family (U(t, s))_{(t,s)∈D}.

(b) Condition (R) holds, and we have T (t) (φf) = (R(t)φ) T (t)f for φ ∈ L^{∞}(I), f ∈ E, and t ≥ 0.

(c) Condition (R) holds, and for f ∈ D and φ ∈ C_{c}^{1}(I) we have φf ∈ D(G) and

G (φf) = −φ′f + φ Gf.  (2.1)

(d) Condition (R) holds, and there is μ ∈ ρ(G) such that R(μ, G) (φ′ R(μ, G)f) = φ R(μ, G)f − R(μ, G) (φf) for f ∈ E and φ ∈ C_{c}^{1}(I).

Proof. By Proposition 2.1, assertion (a) implies (R). The implications (a) ⇒ (c) ⇒ (b) and (c) ⇔ (d) can be shown as in [24, Thm. 3.4], cf. [28, Thm. 2.6].

(b) ⇒ (a): Observe that the assumption implies T (t)f ∈ L^{p}(I_{t}, X) for f ∈ E and t ≥ 0. Set M_{t} := T (t)R(−t) and F_{t} := R(t)D(G) for t ≥ 0. By (R) the space F_{t} consists of continuous functions vanishing on I \ I_{t}. We have

M_{t} (φf) = (R(t)R(−t)φ) T (t)R(−t)f = φ M_{t}f

for f ∈ E and φ ∈ L^{∞}(I) with support in I_{t}. In particular, M_{t} leaves L^{p}(I_{t}, X) invariant. From (R) we derive that F_{t} is dense in L^{p}(I_{t}, X) and C_{00}(I_{t}, X). Thus, F_{t} satisfies (D2) and (D3) for the interval I_{t}. Further, let f ∈ F_{t}. Then f = R(t)g for some g ∈ D(G), and

M_{t}f = T (t)R(−t)R(t)g = T (t) (χ_{I_{−t}} g) = χ_{I_{t}} T (t)g = T (t)g ∈ D(G) ⊆ C_{00}(I, X).

Hence, F_{t} is M_{t}–determining on L^{p}(I_{t}, X), t ≥ 0. By Proposition 2.3 there are bounded operators M(t, s), s ∈ I_{t}, such that M_{t} = M(t, ·) ∈ C_{b}(I_{t}, L_{s}(X)), t ≥ 0. Moreover, ∥M(t, s)∥ ≤ ∥M_{t}∥ ≤ Me^{wt} for some constants M ≥ 1 and w ∈ IR. Notice that T (t) = M(t, ·)R(t). We set U(t, s) := M(t − s, t) for (t, s) ∈ D_{I} . Then

T (t)f (s) = χ_{I}(s − t)U(s, s − t)f(s − t)

for f ∈ E, s ∈ I, and t ≥ 0. From the semigroup law for T one easily derives (E1) for U, see e.g. [17, p.291] or [24, p.524]. It remains to show the strong continuity of M(·, ·). Fix t ≥ 0 and s ∈ I_{t}. By (D2) and the exponential bound for M(t, s) it suffices to consider x ∈ F_{t}^{s}. So let f ∈ D(G) with f(s−t) = x. Set g = (λ−G)f. Then, T (t)f = R(λ, G)T (t)g ∈ C_{00}(I, X) by (R). Thus we obtain

∥M(t′, s′)x − M(t, s)x∥
= ∥M(t′, s′) (f(s − t) − f(s′ − t)) + M(t′, s′) (f(s′ − t) − f(s′ − t′))
+ M(t′, s′)f(s′ − t′) − M(t, s′)f(s′ − t) + M(t, s′)f(s′ − t) − M(t, s)f(s − t)∥
≤ Me^{wt′} (∥R(t)f (s) − R(t)f (s′)∥ + ∥R(t)f − R(t′)f∥_{∞})
+∥T (t′)f − T (t)f∥_{∞} + ∥T (t)f (s′) − T (t)f (s)∥

for t′ ≥ 0 and s′ ∈ I_{t} ∩ I_{t′}. The first, second, and fourth summand converge to 0 as (t′, s′) → (t, s) since f, T (t)f ∈ C_{00}(I, X). For the third summand we use

∥T (t′)f − T (t)f∥_{∞} ≤ ∥R(λ, G)∥_{L(L^{p},C_{00})} ∥T (t′)g − T (t)g∥_{p}

which yields the assertion. □

Remark 2.5.

(a) Implication “(b) ⇒ (a)” is false if we drop condition (R). Consider for instance X = C, E = L^{1}(IR), and U(t, s) = p(t)/p(s), where p, 1/p ∈ L^{∞}(IR) are discontinuous, but p(r) → p(t) as r ↗ t for a.e. t ∈ IR. See also [5, Thm. 6.4] and [9, Thm. 1].

(b) The results of this section remain valid if we replace L^{p}(I, X) by an X–valued Banach function space E(X) such that the norm on E is order continuous and the translations R(t), t ∈ IR, are uniformly bounded on E, see [24] and [28].

3. Miyadera perturbations

In this section we show a perturbation result for evolution families by applying Theorem 2.4 and the following Miyadera type perturbation theorem proved in [29, Thm. 1]. The proof of part (d) of the theorem uses an idea contained in the proof of [30, Prop. 1.3]. Let (A, D(A)) and (B, D(B)) be linear operators. Recall that B is A–bounded if D(A) ⊆ D(B) and ∥Bx∥ ≤ a∥x∥+b∥Ax∥ for x ∈ D(A) and constants a, b ≥ 0. Observe that if there exists λ ∈ ρ(A) then B is A–bounded if and only if D(A) ⊆ D(B) and BR(λ, A) is closed (and hence bounded).

Theorem 3.1. Let T = (T (t))_{t≥0} be a C_{0}–semigroup on a Banach space F with generator (G, D(G)). Consider a dense subspace D of F and a linear operator (B, D(B)) such that

(i) T (t)D ⊆ D ⊆ D(G) ∩ D(B) and [0, ∞) ∋ t ↦ BT (t)f ∈ F is continuous for f ∈ D;

(ii) there are constants α > 0 and γ ∈ [0, 1) such that ∫_{0}^{α} ∥BT (t)f∥ dt ≤ γ ∥f∥ for f ∈ D.

Then the following assertions hold:

(a) The operator (G+B, D) is closable. Its closure (G_{B}, D(G_{B})) generates a C_{0}–semigroup T_{B} = (T_{B}(t))_{t≥0} and satisfies D(G_{B}) = D(G). The operator (B, D) possesses a unique G–bounded extension B̂ to D(G) and we have G_{B} = G + B̂. Further, D is a core of G and G_{B}.

(b) For sufficiently large real λ ∈ ρ(G) ∩ ρ(G_{B}) there exists an invertible operator C_{λ} ∈ L(F) such that R(λ, G_{B}) = R(λ, G)C_{λ}.

(c) For f ∈ D(G) and t ≥ 0 we have

T_{B}(t)f = T (t)f + ∫_{0}^{t} T_{B}(t − τ)B̂T (τ)f dτ,  (3.1)
T_{B}(t)f = T (t)f + ∫_{0}^{t} T (t − τ)B̂T_{B}(τ)f dτ.  (3.2)

Moreover, T_{B} is the only C_{0}–semigroup satisfying (3.1) for f ∈ D.

(d) In addition, assume that B is closed. Then B̂f = Bf for f ∈ D(G). Moreover, for all f ∈ F we have T (t)f, T_{B}(t)f ∈ D(B) for a.e. t ≥ 0, the functions BT (·)f and BT_{B}(·)f are locally integrable on IR_{+} , and (3.1) and (3.2) hold for all t ≥ 0 and f ∈ F with B̂ replaced by B.

Proof. By [15, A-I.1.9] D is a core of G. Assertion (a) and (b) and uniqueness follow from [29], Thm. 1, Remark 2, and (1.7). Equation (3.1) is shown in [29, Thm. 1] for f ∈ D. Since D is a core of G and B̂ is G–bounded, (3.1) holds for f ∈ D(G) by the dominated convergence theorem. For 0 ≤ τ ≤ t and f ∈ D(G) = D(G_{B}) we have

\frac{d}{dτ} T (t − τ)T_{B}(τ)f = −T (t − τ)GT_{B}(τ)f + T (t − τ)G_{B}T_{B}(τ)f
= T (t − τ)B̂T_{B}(τ)f.

Integration from 0 to t yields (3.2).

Now assume that B is closed. Then B̂ is the restriction of B to D(G), [29, Cor. 4]. Further, (ii) holds for all f ∈ D(G), [30, Thm. 1.1], and [20, Lemma 1.1] yields ∫_{0}^{α} ∥BT_{B}(t)f∥ dt ≤ \frac{γ}{1−γ} ∥f∥ for every f ∈ D(G). One easily sees that this implies

∫_{0}^{t} ∥BT (τ)f∥ dτ ≤ c ∥f∥ and  ∫_{0}^{t} ∥BT_{B}(τ)f∥ dτ ≤ c̃∥f∥  (3.3)

for f ∈ D(G), t ≥ 0, and constants c, c̃ depending only on t. Next, fix f ∈ F. Choose f_{n} ∈ D(G) converging to f. By (3.3) the functions BT (·)f_{n} and BT_{B}(·)f_{n} converge in L^{1}([0, t], F), and thus (by passing to subsequences) they converge pointwise a.e.. Since B is closed, we derive T (t)f, T_{B}(t)f ∈ D(B) for a.e. t ≥ 0 and BT (·)f and BT_{B}(·)f are locally integrable on IR_{+}. Moreover, the estimates (3.3) hold for all f ∈ F and t ≥ 0. Observe that (3.1) and (3.2) are satisfied for f_{n} with B̂ replaced by B. Hence, due to (3.3) the last assertion in (d) follows by approximation. □

In the sequel we need the following measure theoretical lemma which is essentially shown in [17, Thm. 4.2], see also [26, Lemma 2.2]. Throughout “measurable” means “strongly measurable” and the integrals are Bochner integrals.

Lemma 3.2. Let X be a Banach space and let I be an interval in IR. Assume that u : [c, d] → L^{p}(I, X), 1 ≤ p < ∞, is integrable. Then there is a measurable function Φ : [c, d] × I → X such that for a.e. t ∈ [c, d] we have u(t) (s) = Φ(t, s) for a.e. s ∈ I. Moreover,

Φ(·, s) is integrable and (∫_{c}^{d} u(t) dt) (s) = ∫_{c}^{d} Φ(t, s) dt  (3.4)

for a.e. s ∈ I. If, in addition, there is a measurable function ũ : [c, d] × I → X such that for a.e. t ∈ [c, d] we have u(t) (s) = ũ(t, s) for a.e. s ∈ I, then (3.4) holds with Φ replaced by ũ.

We often use the fact that for a measurable function u : I × J → X and intervals I, J ⊆ IR we have u(t, s) = 0 for a.e. (t, s) ∈ I × J if and only if for a.e. t ∈ I we have u(t, s) = 0 for a.e. s ∈ J. This is an easy consequence of Fubini’s theorem.

Let I be a left half–open interval in IR. Consider an exponentially bounded evolution family U = (U(t, s))_{(t,s)∈D} on a Banach space X. Let T = (T (t))_{t≥0} be the associated evolution semigroup on E = L^{p}(I, X), 1 ≤ p < ∞, with generator (G, D(G)). In the next lemma we describe a certain class of functions contained in D(G). To simplify notation we set U(t, s) := 0 for t < s.

Lemma 3.3. Let x ∈ X, r ∈ I, and φ ∈ C_{c}^{1}(I) with φ(s) = 0 for s ≤ r. Set f(·) := φ(·)U(·, r)x. Then T (t)f = (R(t)φ) U(·, r)x and f ∈ D(G).

For the straightforward proof we refer to [28, Lemma 1.12], see also [25, Lemma 2.1]

We now introduce the assumptions on the operators (B(t), D(B(t))), t ∈ I, on X which are needed for our perturbation result. For q ≥ 1 we denote by q′ the conjugate exponent satisfying \frac{1}{q} + \frac{1}{q′} = 1.

(M) There are dense subspaces Y_{t} , t ∈ I, of X such that U(t, s)Y_{s} ⊆ Y_{t} for (t, s) ∈ D. Moreover, for all s ∈ I and y ∈ Y_{s} we have U(t, s)y ∈ D(B(t)) for a.e. t ∈ I ∩ [s, ∞), B(·)U(·, s)y is measurable, and

∫_{0}^{α} χ_{I}(s + t) ∥B(s + t)U(s + t, s)y∥^{p} dt ≤ β^{p} ∥y∥^{p}  (3.5)

for constants p ≥ 1, α > 0 and β ≥ 0 such that γ := α^{1/p′}β < 1.

It is straightforward to see that (M) implies

∫_{0}^{d} χ_{I}(s + t) ∥B(s + t)U(s + t, s)y∥^{p} dt ≤ c ∥y∥^{p}  (3.6)

for s ∈ I, d ≥ 0, and y ∈ Y_{s}, where the constant c is independent of s and y. Observe that if (M) is satisfied for p > 1, α > 0, and β ≥ 0 then, by Hölder’s inequality, (M) is also true for p = 1, α > 0, and γ ∈ [0, 1). Clearly, (M) holds if Y_{t} ≡ X and B(·) ⊆ L(X) such that ∥B(t)∥ ≤ C for a.e. t ∈ I and B(·)x is measurable for x ∈ X. Other examples will be studied in Section 4 and 5.

On E = L^{p}(I, X) we define the linear operator B by setting Bf := B(·)f(·) and D(B) := {f ∈ E : f(s) ∈ D(B(s)) for a.e. s ∈ I, B(·)f(·) ∈ E}. Further, we consider the space

D := lin {f ∈ E : f(·) = φ(·) U(·, r)y, r ∈ I, y ∈ Y_{r}, φ ∈ C_{c}^{1}(I) with φ(s) = 0, s ≤ r}.

We now state the main result of this section. It generalizes [24, Thm. 4.2] where the case U(t, s) = e^{(t−s)A} was considered; cf. [18, Thm 4.3] for the case U(t, s) = Id. Bounded perturbations B(·) were treated in [26, Thm. 2.3] with the same approach; see also the references therein.

Theorem 3.4. Let I be a left half–open interval in IR. Let U = (U(t, s))_{(t,s)∈D} be an exponentially bounded evolution family on X and (B(t), D(B(t))), t ∈ I, linear operators on X satisfying (M). Denote by (G, D(G)) the generator of the associated evolution semigroup T on E = L^{p}(I, X). Then the following assertions hold:

(a) There is a unique exponentially bounded evolution family U_{B} = (U_{B}(t, s))_{(t,s)∈D} satis-

fying

U_{B}(t, s)f(s) = U(t, s)f(s) + ∫_{s}^{t} U_{B}(t, τ)B(τ)U(τ, s)f(s) dτ  (3.7)

for f ∈ D and (t, s) ∈ D.

(b) Assume, in addition, that (B, D(B)) is G–bounded and denote by (G_{B}, D(G_{B})) the generator of the evolution semigroup T_{B} on E associated with U_{B}. Then D(G) = D(G_{B}) ⊆ D(B) and G_{B}f = Gf + Bf for f ∈ D(G). Moreover, (B, D(B)) is G_{B}– bounded.

(c) Suppose now that (B(t), D(B(t))) is closed for a.e. t ∈ I. Then B is G-bounded, hypothesis (M) is valid with Y_{t} = X for all t ∈ I, and

U_{B}(t, s)x = U(t, s)x + ∫_{s}^{t} U_{B}(t, τ)B(τ)U(τ, s)x dτ  (3.8)

holds for all x ∈ X and (t, s) ∈ D. Further, for x ∈ X and s ∈ I we have U_{B}(t, s)x ∈ D(B(t)) for a.e. t ∈ I ∩ [s, ∞), the function B(·)U_{B}(·, s)x is locally integrable and

U_{B}(t, s)x = U(t, s)x + ∫_{s}^{t} U(t, τ)B(τ)U_{B}(τ, s)x dτ  (3.9)

holds for all x ∈ X and (t, s) ∈ D.

Proof. (a) At first we verify conditions (i) and (ii) in Theorem 3.1 for T , G, B, and D as defined above. By Lemma 3.3 we have T (t)D ⊆ D ⊆ D(G). Using the arguments of the proof of [11, Prop. 2.9] it is easy to show that D is dense in E, cf. [28, Prop. 1.13].

Moreover, (3.6) yields D ⊆ D(B). Let f(·) = φ(·) U(·, r)y for r ∈ I, y ∈ Y_{r}, and φ ∈ C_{c}^{1}(I) with φ(s) = 0 for s ≤ r. Then for 0 ≤ t, t′ ≤ d we have

∥BT (t′)f − BT (t)f∥_{p}^{p} = ∫_{I} ∥(R(t′)φ (s) − R(t)φ (s)) B(s)U(s, r)y∥^{p} ds
≤ ∥R(t′)φ − R(t)φ∥_{∞} ∫_{r}^{c} ∥B(s)U(s, r)y∥^{p} ds,

where c only depends on d and φ. By (3.6) this yields (i). In order to check (ii) we consider a function f ∈ D given by

f(s) = ∑_{k=1}^{n} φ_{k}(s) U(s, s_{k})y_{k} .

Notice that due to (M) the mapping (t, s) ↦ χ_{I}(s+t) ∥B(s+t)U(s+t, s)f(s)∥^{p} is measurable. We compute

∫_{0}^{α} ∥BT (t)f∥_{p} dt = ∫_{0}^{α} (∫_{I} ∥∑_{k=1}^{n} χ_{I}(s − t)φ_{k}(s − t) B(s)U(s, s_{k})y_{k}∥^{p} ds)^{1/p} dt
= ∫_{0}^{α} (∫_{I_{−t}} ∥∑_{k=1}^{n} φ_{k}(s)B(s + t)U(s + t, s)U(s, s_{k})y_{k}∥^{p} ds)^{1/p} dt
= ∫_{0}^{α} (∫_{I} χ_{I}(s + t) ∥B(s + t)U(s + t, s)f(s)∥^{p} ds)^{1/p} dt
≤ α^{1/p′} (∫_{0}^{α} ∫_{I} χ_{I}(s + t) ∥B(s + t)U(s + t, s)f(s)∥^{p} ds dt)^{1/p}
= α^{1/p′} (∫_{I} ∫_{0}^{α} χ_{I}(s + t) ∥B(s + t)U(s + t, s)f(s)∥^{p} dt ds)^{1/p}
≤ α^{1/p′} β (∫_{I} ∥f(s)∥^{p} ds)^{1/p}
= α^{1/p′}β ∥f∥_{p} ,

where \frac{1}{p} + \frac{1}{p′} = 1. (Here we have used Hölder’s inequality, Fubini’s theorem, and (3.5).) By setting γ := α^{1/p′}β < 1, we obtain (ii).

Hence, by Theorem 3.1, the operator (B, D) can be extended to a G–bounded operator B̂ on D(G), and G_{B} := G + B̂ with D(G_{B}) := D(G) generates a C_{0}–semigroup T_{B} on E. Also, D is a core for G_{B}. Since G generates an evolution semigroup, G_{B} satisfies (2.1) in Theorem 2.4 for f ∈ D. Further, (R) holds for R(λ, G), λ ∈ ρ(G), due to Proposition 2.1. So, by Theorem 3.1(b), R(λ, G_{B}) satisfies (R) for λ ∈ IR sufficiently large. Thus Theorem 2.4 shows that T_{B} is an evolution semigroup induced by an exponentially bounded evolution family U_{B}.

Fix f ∈ D and t ≥ 0. Equation (3.1) implies

R(−t)T_{B}(t)f = R(−t)T (t)f + ∫_{0}^{t} u(τ)dτ,

where u(τ) := R(−t)T_{B}(t − τ)BT (τ)f, 0 ≤ τ ≤ t. The function u : [0, t] → E is continuous. For each τ ∈ [0, t] we have

(u(τ)) (s) = χ_{I}(s + t) U_{B}(s + t, s + τ)B(s + τ)U(s + τ, s)f(s) =: ũ(τ, s)

for a.e. s ∈ I. Since the mapping ũ : [0, t] × I → X is measurable, Lemma 3.2 implies

U_{B}(s + t, s)f(s) = U(s + t, s)f(s) + ∫_{s}^{s+t} U_{B}(s + t, τ)B(τ)U(τ, s)f(s) dτ (3.10) :::—zq��y���f ∈ D, t ≥ 0, and a.e. s ∈ I with s + t ∈ I. Next we show that in fact (3.10) holds for all s ∈ I such that s + t ∈ I.

For this it suffices to prove that the integral in (3.10) is continuous from the left with respect to s. First notice that f(s) ∈ Y_{s} for s ∈ I and f ∈ D so that due to (3.6) the integral in (3.10) is defined for all (s + t, s) ∈ D. Clearly, we only have to consider functions of the form f(·) = φ(·)U(·, r_{0})y for r_{0} ∈ I, y ∈ Y_{r_{0}} , and φ ∈ C_{c}^{1}(I) with φ(r) = 0 for r ≤ r_{0} . Further we may assume t > 0 and s > r_{0}. Let s′ ∈ [r_{0}, s] with s′ + t ≥ s. Then we obtain

∫_{s}^{s+t} U_{B}(s + t, τ)B(τ)U(τ, s)f(s) dτ − ∫_{s′}^{s′+t} U_{B}(s′ + t, τ)B(τ)U(τ, s′)f(s′) dτ
= ∫_{s}^{s+t} χ_{[s′+t,s+t]}(τ) U_{B}(s + t, τ)B(τ)U(τ, s)f(s) dτ
+ ∫_{s}^{s+t} χ_{[s,s′+t]}(τ) (U_{B}(s + t, τ) − U_{B}(s′ + t, τ)) B(τ)U(τ, s)f(s) dτ
+ ∫_{s}^{s′+t} U_{B}(s′ + t, τ)B(τ)U(τ, s) (f(s) − U(s, s′)f(s′)) dτ
−φ(s′) ∫_{r_{0}}^{s} χ_{[s′,s]}(τ) U_{B}(s′ + t, τ)B(τ)U(τ, r_{0})y dτ.

The integrands of the first, second, and fourth term tend to 0 for a.e. τ as s′ ↗ s, and they are bounded by C_{1} ∥B(τ)U(τ, s)f(s)∥ and C_{2} ∥B(τ)U(τ, r_{0})y∥, respectively, where C_{k} are constants independent of s′ and τ. So the first, second, and fourth integral converge to 0 as s′ ↗ s due to (3.6) and the dominated convergence theorem. Since f(s)−U(s, s′)f(s′) ∈ Y_{s}, by (3.6) the third summand can be estimated from above by C_{3} ∥f(s)−U(s, s′)f(s′)∥, where C_{3} does not depend on s′. Since f is continuous and U strongly continuous this term also tends to 0 as s′ ↗ s.

In order to show uniqueness of U_{B}, let V be an exponentially bounded evolution family satisfying (3.7) for f ∈ D. Let (S(t))_{t≥0} be the associated evolution semigroup on E. As above, Lemma 3.2 implies that (S(t))_{t≥0} satisfies (3.1) for f ∈ D and t ≥ 0. Consequently, by Theorem 3.1, S(t) = T_{B}(t), and thus V = U_{B}. This establishes (a).

(b) Since (B, D(B)) is G–bounded, we have B̂f = Bf for f ∈ D(G) by Theorem 3.1(a). Also, B is G_{B}–bounded by Theorem 3.1(b). The other assertions in (b) follow from Theorem 3.1(a).

(c) Observe that (B, D(B)) is closed due to our additional assumption. Hence, the first assertion in (c) holds by Theorem 3.1(d),(a). The remaining statements are shown for U and U_{B}, separately.

(i) Let s ∈ I, d ≥ 0, and c as in (3.6). Since Y_{s} is dense and B(t) is closed for a.e. t ∈ I one concludes that B(·)U(·, s)y ∈ L^{p}([s, s+d], X) and (3.6) holds for all y ∈ X, cf. the proof of Theorem 3.1(d). This shows (M) with Y_{s} = X. Let (t, s) ∈ D. Since {f(s) : f ∈ D} is dense in X and all terms in (3.7) are continuous with respect to x = f(s), this equation carries over to all x ∈ X.

10 (ii) Next, we treat the assertions concerning U_{B}. Fix f ∈ D(G) and t ≥ 0. We claim that

h̃ : [0, t] × I; (τ, s) ↦ χ_{I}(s − τ) B(s)U_{B}(s, s − τ)f(s − τ) is measurable.  (3.11)

We define g(τ) := T_{B}(τ)f for τ ∈ [0, t] and g̃(τ, s) := χ_{I}(s − τ) U_{B}(s, s − τ)f(s − τ) for (τ, s) ∈ [0, t] × I. Theorem 3.1(b) implies that τ ↦ g(τ) is continuous with respect to the graph norm of G. Fix n ∈ IN. There exist disjoint intervals J_{n,k} = [t_{n,k}, t_{n,k+1}) and functions h_{n,k} ∈ D, k = 1, ⋯, m(n), such that [0, t] = ⋃_{k} \overline{J_{n,k}} and

∥h_{n,k} − g(τ)∥_{p} + ∥G(h_{n,k} − g(τ))∥_{p} ≤ \frac{1}{n}  for τ ∈ J_{n,k}.  (3.12)

Let h_{n}(τ) := h_{n,k} and h̃_{n}(τ, s) := h_{n,k}(s) for τ ∈ J_{n,k}, s ∈ I, and n ∈ IN. From (3.12) we infer that h̃_{n} converges to g̃ in L^{p}([0, t] × I, X). Further, (M) implies that [0, t] × I ∋ (τ, s) ↦ B(s)h̃_{n}(τ, s) is measurable. Since B is G–bounded, inequality (3.12) yields

∥B h_{n}(τ) − B g(τ)∥_{p} ≤ c ∥h_{n,k} − g(τ)∥_{p} + c ∥G(h_{n,k} − g(τ))∥_{p} ≤ \frac{c}{n}

for each τ ∈ J_{n,k} and a constant c. Hence, the function (τ, s) ↦ B(s)h̃_{n}(τ, s) converges in L^{p}([0, t] × I, X) to a function Φ. By passing to subsequences, we can assume that h̃_{n}(τ, s) → g̃(τ, s) and B(s)h̃_{n}(τ, s) → Φ(τ, s) for a.e. (τ, s) as n → ∞. Since B(s) is closed for a.e. s ∈ I, we obtain that g̃(τ, s) ∈ D(B(s)) and B(s)h̃_{n}(τ, s) → B(s)g̃(τ, s) for a.e. (τ, s). This proves the claim.

If f ∈ D(G) and t ≥ 0, then Theorem 3.1(d) yields

R(−t)T_{B}(t)f = R(−t)T (t)f + ∫_{0}^{t} v(τ)dτ, where
v(τ) := R(−t)T (t − τ)BT_{B}(τ)f

for τ ∈ [0, t]. Moreover, v : [0, t] → E is integrable and for each τ ∈ [0, t] we have

v(τ) (s) = χ_{I}(s + t) U(s + t, s + τ)B(s + τ)U_{B}(s + τ, s)f(s) =: ṽ(τ, s)

for a.e. s ∈ I. By (3.11) the function ṽ : [0, t] × I → X is measurable. Thus,

U_{B}(s + t, s)f(s) = U(s + t, s)f(s) + ∫_{s}^{s+t} U(s + t, τ)B(τ)U_{B}(τ, s)f(s) dτ  (3.13)

for a.e. s ∈ I with s + t ∈ I due to Lemma 3.2.

As mentioned above, (M) also holds for p = 1 and with β replaced by γ ∈ [0, 1). Therefore, for the remainder of the proof, we may and shall assume p = 1. For f ∈ D(G), the function h̃ defined in (3.11) is measurable, and from (3.3) we derive

∫_{0}^{t} ∫_{I} ∥h̃(τ, s)∥ ds dτ = ∫_{0}^{t} ∥BT_{B}(τ)f∥_{L^{1}(I,X)} dτ ≤ c̃∥f∥_{L^{1}(I,X)} .  (3.14)

Now, fix f ∈ L^{1}(I, X). Choose f_{n} ∈ D(G) such that f_{n} → f in L^{1}(I, X), and denote by h̃_{n} the measurable function corresponding to f_{n}. Then (3.14), applied to the difference f_{n}−f_{m}, shows that the functions h̃_{n} converge in L^{1}([0, t]×I, X); in particular, a subsequence converges for a.e. (τ, s). For a subsubsequence, U_{B}(s, s − τ)f_{n}(s − τ) tends to U_{B}(s, s − τ)f(s − τ) for a.e. (τ, s). Since B(s) is closed for a.e. s ∈ I we conclude that χ_{I}(s − τ) U_{B}(s, s − τ)f(s − τ) ∈ D(B(s)) for a.e. (τ, s) and that (3.11) and (3.14) hold for all f ∈ L^{1}(I, X). Note that, using Fubini’s theorem, equation (3.14) can be rewritten as

∫_{I} ∫_{0}^{t} χ_{I}(s + τ) ∥B(s + τ)U_{B}(s + τ, s)f(s)∥ dτ ds ≤ c̃∥f∥_{L^{1}(I,X)} .  (3.15)

Fix x ∈ X and s_{0} ∈ I. For 0 < r < \frac{t}{2} with s_{0}+r ∈ I we define f(s) = χ_{[s_{0},s_{0}+r]}(s) U_{B}(s, s_{0})x. With this f, we have B(s + τ)U_{B}(s + τ, s)f(s) = B(s + τ)U_{B}(s + τ, s_{0})x for s ∈ [s_{0}, s_{0} + r] and s + τ ∈ I. Hence, B(·)U_{B}(·, s_{0})x is measurable and estimate (3.15) implies

c̃ ∫_{s_{0}}^{s_{0}+r} ∥U_{B}(s, s_{0})x∥ ds ≥ ∫_{s_{0}}^{s_{0}+r} ∫_{0}^{t} χ_{I}(s + τ) ∥B(s + τ)U_{B}(s + τ, s_{0})x∥ dτ ds
≥ ∫_{s_{0}}^{s_{0}+r} ∫_{r}^{t−r} χ_{I}(s_{0} + τ) ∥B(s_{0} + τ)U_{B}(s_{0} + τ, s_{0})x∥ dτ ds
= r ∫_{r}^{t−r} χ_{I}(s_{0} + τ) ∥B(s_{0} + τ)U_{B}(s_{0} + τ, s_{0})x∥ dτ.

Dividing by r and letting r → 0 we obtain

∫_{0}^{t} χ_{I}(s_{0} + τ) ∥B(s_{0} + τ)U_{B}(s_{0} + τ, s_{0})x∥ dτ ≤ c̃∥x∥.  (3.16)

Applying (3.13) and (3.16) we can now conclude the proof in the same way as we showed that (3.8) holds for all t ≥ s and x ∈ X: First, instead of f ∈ D we take functions of the form g = φ(·)U_{B}(·, s)x, where x ∈ X, s ∈ I, and φ ∈ C_{c}^{1}(I) with φ(r) = 0 for r ≤ s. By Lemma 3.3 we have g ∈ D(G_{B}) = D(G). Further, we only used (3.10), strong continuity of U, and estimates as in (M) in the proof of (3.8). Since the corresponding hypotheses are satisfied this proof implies (3.9). □

The above result can be extended to closed intervals I ⊆ IR with a = min I ∈ IR. Assume that (M) holds for an exponentially bounded evolution family U = (U(t, s))_{(t,s)∈D_{I}} on X and linear operators (B(t), D(B(t))), t ∈ I. Set J := (a − 1, ∞) ∩ I. Define B̃(t) := B(t) and Ỹ_{t} := Y_{t} for t ∈ I and B̃(t) := 0 and Ỹ_{t} := Y_{a} for t ∈ (a − 1, a). Define an evolution family Ũ with index set D_{J} by setting

Ũ(t, s) := \{ \begin{matrix}Id, & a ≥ t ≥ s > a − 1, \\ U(t, max\{s, a\}), & otherwise. \end{matrix}

(See Section 4 for a different extension of U.) Then (M) also holds for Ũ, B̃(·), and Ỹ_{t} . Let (G̃, D(G̃)) be the generator of the evolution semigroup on L^{p}(J, X) associated with Ũ.

Corollary 3.5. Let U, Ũ, B(·), and B̃(·) be as considered above. Then there is an evolution family U_{B} with index set D_{J} satisfying the conclusions of Theorem 3.4 for Ũ, B̃(·), and G̃ on D_{J}. In particular, if B(t) is closed for a.e. t ∈ I, then U_{B} satisfies (3.8) and (3.9) for x ∈ X and (t, s) ∈ D_{I}.

Remark 3.6. Let I_{n} := I ∩ (−n, n], n ∈ IN. We weaken the hypotheses of Theorem 3.4 by only assuming that U is bounded on D_{I_{n}} and (M) holds on I_{n} for all n ∈ IN. In addition, let B(t) be closed for a.e. t ∈ I. Then there exists an evolution family U_{n,B} with index set D_{I_{n}} fulfilling Theorem 3.4(a)–(c) for I_{n}. By uniqueness, U_{B}(t, s) := U_{n,B}(t, s) for (t, s) ∈ D_{I_{n}} is a (well–defined) evolution family with index set D_{I} which satisfies (3.8) and (3.9) for x ∈ X and (t, s) ∈ D_{I}.

4. Differentiability in the parabolic case

In this section we suppose that the evolution family U solves the Cauchy problem related to operators (A(t), D(A(t))), t ≥ 0, which satisfy the following assumption. We set Σ = Σ_{ϕ} := {0} ∪ {λ ∈ C \ {0} : | arg λ| < ϕ} for 0 < ϕ < π.

(P1) D(A(t)) is dense, ρ(A(t)) ⊇ Σ_{ϕ} for some \frac{π}{2} < ϕ < π, ∥R(λ, A(t))∥ ≤ \frac{M}{1+|λ|} , and

∥A(t)R(λ, A(t)) (A(t)^{−1} − A(s)^{−1})∥ ≤ L \frac{|t − s|^{μ}}{1 + |λ|^{ν}}

for λ ∈ Σ, t, s ∈ I, and constants M, L ≥ 0 and μ, ν ∈ (0, 1] with μ + ν > 1.

In view of applications, cf. Section 5, this situation is referred to as the parabolic case. Observe that A(·)^{−1} is Hölder continuous and A(t) generates a bounded analytic semigroup. Condition (P1) was introduced (in a somewhat weaker form) by P. Acquistapace and B. Terreni, [2]. Assuming (P1), there is a unique evolution family U solving (CP) on the spaces D(A(t)), [1, Thm. 2.3], see also [3], [31], and the references therein. (To be precise, strong continuity of U at (s, s) follows from the proof of [1, Thm. 2.3(ii)].) Moreover, U is exponentially bounded, see (4.1) below.

We extend A(·) to the interval I := (−1, ∞) by setting A(t) := A(0) for t ∈ (−1, 0). Clearly, the extension still satisfies (P1) with the same constants. Further, we define

U(t, s) := \{ \begin{matrix}e^{(t−s)A(0)}, & 0 ≥ t ≥ s > −1, \\ U(t, 0)e^{−sA(0)}, & t > 0 > s > −1. \end{matrix}

Then U = (U(t, s))_{t≥s>−1} is an exponentially bounded evolution family solving (CP) on I. Let (G, D(G)) be the generator of the evolution semigroup T on E = L^{p}(I, X) associated with U (where p ∈ [1, ∞) is specified below in condition (B)).

For θ ∈ [0, 1] and t > −1, let Y_{t} be the domain of the fractional power (−A(t))^{θ} endowed with the norm ∥x∥_{Y_{t}} := ∥(−A(t))^{θ}x∥. (See, e.g., [22, §2.6] for the basic theory of fractional powers.) By (P1) we obtain sup_{t∈I} ∥(−A(t))^{−θ}∥ < ∞ and sup_{t∈I} ∥(−A(t))^{θ}A(t)^{−1}∥ < ∞ (see [22], proof of Lemma 2.6.3). It is shown in [6, Thm. 2.3] that U(t, s)X ⊆ Y_{t} for t > s > −1 and

∥(−A(t))^{θ}U(t, s)∥ ≤ C e^{w(t−s)} (t − s)^{−θ}  (4.1)

for (t, s) ∈ D′ = D′_{I} := {(t, s) ∈ D_{I} : t > s} and constants C ≥ 1 and w ∈ IR.

We denote by L_{loc,u}^{q}(I) the space of locally uniformly q-integrable functions on I endowed with the norm

∥φ∥_{L_{loc,u}^{q}} := sup_{s∈I} (∫_{s}^{s+1} |φ(τ)|^{q} dτ )^{\frac{1}{q}} .

On the perturbations B(t), t ≥ 0, we impose the following condition.

(B) Let p ≥ 1 and 0 < θ < 1 with pθ < 1. Let q = (\frac{1}{p} − \frac{θ}{ρ})^{−1} for some ρ ∈ (pθ, 1). Let B(t) ∈ L(Y_{t}, X) and ∥B(t)∥_{L(Y_{t},X)} ≤ ψ(t) for a.e. t ≥ 0, where ψ ∈ L_{loc,u}^{q}(IR_{+}). Finally, let B(·)U(·, s)y be measurable for s ≥ 0 and y ∈ Y_{s}.

Notice that q > p(1−pθ)^{−1} and B(t)A(t)^{−1} ∈ L(X). We remark that in our application B(t) is a differential operator and the assumptions on q, θ, p, ρ lead to an integrability condition for the coefficients of this operator. Let B(t) := 0 and ψ(t) := 0 for t ∈ (−1, 0). On E = L^{p}(I, X) we define a linear operator by setting Bf := B(·)f(·) for f ∈ D(B) := {f ∈ E : f(t) ∈ Y_{t} for a.e. t ∈ I, B(·)f(·) ∈ E}. In the next lemma we check the assumptions of Theorem 3.4(a),(b) for U and B(·).

Lemma 4.1. Assume (P1) and (B). Then (M) holds for U, B(·), and Y_{t} on I. Assume, in addition, that B(·)A(·)^{−1}x is measurable for all x ∈ X. Then (B, D(B)) is G–bounded.

Proof. (i) For the first assertion we only have to verify (3.5). Consider x ∈ X, s > −1, t, r > 0, and μ ∈ IR. Notice that by (B) and (4.1) we obtain

∥B(s + t)U(s + t, s)x∥ ≤ Ct^{−θ}e^{wt} ψ(s + t) ∥x∥  (4.2)

for a.e. t > 0 and all x ∈ X. Set ε := (μ − w)p. Then (4.2) and Hölder’s inequality imply

∫_{0}^{r} ∥e^{−μt}B(s + t)U(s + t, s)x∥^{p} dt
≤ C^{p} ∥x∥^{p} ∫_{0}^{r} e^{(w−μ)pt} t^{−pθ} ψ(s + t)^{p} dt
≤ C^{p} ∥x∥^{p} (∫_{0}^{r} (e^{− \frac{εt}{2}} t^{−pθ})^{\frac{ρ}{pθ}} dt)^{\frac{pθ}{ρ}} (∫_{0}^{r} (e^{− \frac{εt}{2}} ψ(s + t)^{p})^{(1− \frac{pθ}{ρ})^{−1}} dt)^{1− \frac{pθ}{ρ}}
= C^{p} ∥x∥^{p} (∫_{0}^{r} e^{−ε_{1}t} t^{−ρ} dt)^{\frac{pθ}{ρ}} (∫_{0}^{r} e^{−ε_{2}t} ψ(s + t)^{q} dt)^{\frac{p}{q}}  (4.3)

for some constants ε_{k} ∈ IR. For r ≤ r_{0} and μ = 0, estimate (4.3) yields

∫_{0}^{r} ∥B(s + t)U(s + t, s)x∥^{p} dt ≤ C_{1} ∥x∥^{p} (∫_{0}^{r} t^{−ρ} dt)^{\frac{pθ}{ρ}} (∫_{s}^{s+r} ψ(t)^{q} dt)^{\frac{p}{q}}
≤ C_{2} ∥ψ∥_{L_{loc,u}^{q}}^{p} r^{κ} ∥x∥^{p}

for some constants C_{k}, κ > 0. Letting r → 0 establishes (3.5). Further, let μ > w. Hence, ε_{k} > 0. We derive from (4.3) that

∫_{0}^{∞} ∥e^{−μt}B(s + t)U(s + t, s)x∥^{p} dt
≤ C^{p} ∥x∥^{p} (∫_{0}^{∞} \frac{e^{−ε_{1}t}}{t^{ρ}} dt)^{\frac{pθ}{ρ}} (∑_{k=0}^{∞} e^{−ε_{2}k} ∫_{s+k}^{s+k+1} ψ(t)^{q} dt)^{\frac{p}{q}} ≤ C_{3} ∥ψ∥_{L_{loc,u}^{q}}^{p} ∥x∥^{p} (4.4)

for a constant C_{3} > 0.

(ii) Next assume that B(·)A(·)^{−1}x, x ∈ X, is measurable. Consider the space

F := {f ∈ E : f(t) ∈ Y_{t} for a.e. t ∈ I, (−A(·))^{θ}f(·) ∈ E}

endowed with the norm ∥f∥_{F} := ∥(−A(·))^{θ}f(·)∥_{E}. Observe that F is a Banach space which is continuously embedded in E. Further we consider B as an operator on D(B_{F}) := {f ∈ F : Bf ∈ E}. It can easily be verified that B : D(B_{F}) ⊆ F → E is a closed operator. Let f ∈ E and t > 0. By (4.1) we obtain

∥(−A(s))^{θ} (T (t)f)(s)∥ ≤ ∥(−A(s))^{θ}U(s, s − t)∥ ∥R(t)f (s)∥ ≤ C t^{−θ} e^{wt} ∥R(t)f (s)∥ .

Thus, T (t) ∈ L(E, F) for t > 0. For t, t′ ≥ t_{0} > 0 and f ∈ E we have T (t)f − T (t′)f = T (t_{0})(T (t − t_{0})f − T (t′ − t_{0})f), and hence the function u : (0, ∞) → F : t ↦ e^{−λt}T (t)f is continuous, where λ > w and f ∈ E. Moreover,

∥e^{−λt}T (t)f∥_{F} ≤ C e^{(w−λ)t} t^{−θ} (∫_{I} ∥R(t)f(s)∥^{p} ds)^{\frac{1}{p}} ≤ C e^{(w−λ)t} t^{−θ} ∥f∥_{E} .

As a consequence, u : IR_{+} → F is integrable. Since λ dominates the growth bound of T , we infer that R(λ, G) : E → F is bounded for λ > w.

It follows from [1, Thm. 2.3] that D′ ∋ (t, s) ↦ A(t)U(t, s)x is continuous for x ∈ X. Hence, by the measurability of B(·)A(·)^{−1}x, x ∈ X, the mapping

D′ ∋ (t, s) ↦ B(t)U(t, s)f(s) = B(t)A(t)^{−1}A(t)U(t, s)f(s)

is measurable for f ∈ E. Now let f ∈ C_{c}(I, X) with support J. Then

∥B(s)(T (t)f)(s)∥^{p} ≤ C^{p} e^{wpt} t^{−pθ} ∥f∥_{∞}^{p} ψ(s)^{p} χ_{J}(s − t),

for t > 0. Because of q > p we have ψ ∈ L_{loc}^{p}(I), and thus T (t)f ∈ D(B_{F}) ⊆ D(B), t > 0. Moreover,

∥BT (t)f − BT (t′)f∥_{E}^{p} ≤ ∫_{I} ψ(s)^{p} ∥(−A(s))^{θ}U(s, s − t_{0})∥^{p}
∥U(s − t_{0}, s − t) (R(t)f)(s) − U(s − t_{0}, s − t′) (R(t′)f)(s)∥^{p} ds

for t′, t ≥ t_{0} > 0. Clearly, the integrand converges to 0 for a.e. s ∈ I as t′ → t. Also, the integrand is bounded by C_{4} ψ(·)^{p}χ_{K}(·) for a constant C_{4} and a compact interval K ⊆ I. Therefore the mapping (0, ∞) ∋ t ↦ BT (t)f ∈ E is continuous for f ∈ C_{c}(I, X) due to the dominated convergence theorem. Further, we derive for λ > μ > w and u(t) = e^{−λt}T (t)f, f ∈ C_{c}(I, X), that

∫_{0}^{∞} ∥Bu(t)∥_{E} dt = ∫_{0}^{∞} e^{( \frac{μ}{p} −λ)t} (∫_{I} e^{−μt} ∥B(s)U(s, s − t)χ_{I}(s − t)f(s − t)∥^{p} ds)^{\frac{1}{p}} dt
≤ C_{5} (∫_{0}^{∞} ∫_{I} e^{−μt} ∥B(s)U(s, s − t)χ_{I}(s − t)f(s − t)∥^{p} ds dt)^{\frac{1}{p}}
= C_{5} (∫_{I} ∫_{0}^{∞} e^{−μt} ∥B(s + t)U(s + t, s)f(s)∥^{p} dt ds)^{\frac{1}{p}}
≤ C_{6} (∫_{I} ∥f(s)∥^{p} ds)^{\frac{1}{p}} = C_{6} ∥f∥_{E}  (4.5)

for constants C_{k} > 0. (Here we have used Hölder’s inequality, Fubini’s theorem and estimate (4.4).) Thus, Bu : IR_{+} → E is integrable. Now Hille’s theorem, [8, Thm. 3.7.12], yields R(λ, G)f = ∫_{0}^{∞} u(t) dt ∈ D(B_{F}) ⊆ D(B) and BR(λ, G)f = ∫_{0}^{∞} Bu(t) dt. In particular, by (4.5) we have ∥BR(λ, G)f∥_{E} ≤ C_{6}∥f∥_{E} for f ∈ C_{c}(I, X).

Finally, we show that (B, D(B)) is G–bounded. Let g ∈ D(G) and f = (λ − G)g ∈ E. Choose a sequence (f_{n}) ⊆ C_{c}(I, X), converging to f. Then g_{n} := R(λ, G)f_{n} ∈ D(B_{F}) tends to g in the norm of F as n → ∞. Moreover, there exists E − lim_{n→∞} BR(λ, G)f_{n} =: h. Since B is closed in F × E, we infer that g ∈ D(B_{F}) ⊆ D(B) and Bg = h. Hence, ∥Bg∥_{E} ≤ C_{6} ∥f∥_{E} = C_{6} ∥(λ − G)g∥_{E} for g ∈ D(G). □

So we can apply the results of the preceding section. In fact, we can even improve them considerably and show that the function t ↦ U_{B}(t, s)x, s ∈ I, x ∈ X, solves (pCP) for a.e. t ∈ (s, ∞). For this we have to introduce some new concepts.

Let 1 < p, d < ∞ and let X = L^{d}(Ω) for a σ-finite measure space (Ω, μ). Then E = L^{p}(I, X) is a UMD-space, cf. e.g. [14]. We assume (P1) and that the operators (A(t), D(A(t))), t ≥ 0, satisfy the following condition, where ϕ is determined by (P1).

(P2) The operators (A(t), D(A(t))) admit bounded imaginary powers (−A(t))^{ir} and ∥(−A(t))^{ir}∥ ≤ c e^{(π−ϕ)|r|} for t ≥ 0, r ∈ IR, and a constant c,

see [14] and the references cited therein. Clearly, the extension of A(·) to I = (−1, ∞) given by A(t) = A(0), −1 < t < 0, also satisfies (P2). In Section 5 we will see that (P1) and (P2) can be verified for a large class of elliptic operators A(t). We define a linear operator A on E by setting Af := A(·)f(·) for f ∈ D(A) := {f ∈ E : f(s) ∈ D(A(s)) for a.e. s > −1, A(·)f(·) ∈ E}. Notice that (A, D(A)) is closed. Consider the first derivative G_{0}f = −f′ on E with domain D(G_{0}) = {f ∈ W^{1,p}(I, X) : f(−1) = 0}. Then it was shown in [14], Cor. 2, Thm. 2, that the operator (G_{0} +A, D(G_{0})∩D(A)) is closed and (ω, ∞) ⊆ ρ(G_{0} +A) for some ω ∈ IR. Further, we set

D_{1} := lin {f ∈ E : f(·) = φ(·)U(·, r)y; r ∈ I, y ∈ D(A(r)), φ ∈ C_{c}^{1}(I) with φ(s) = 0, s ≤ r}.

Since the Cauchy problem (CP) related to A(·) is well–posed, it follows from [28, Prop. 1.13], cf. [11, Prop. 2.9], that D_{1} ⊆ D(A) ∩ D(G_{0}) and the generator (G, D(G)) of the evolution semigroup is the closure of (G_{0} + A, D_{1}). Hence, G_{0} + A extends G. Since there exists λ ∈ ρ(G) ∩ ρ(G_{0} + A), this implies

(G, D(G)) = (G_{0} + A, D(G_{0}) ∩ D(A)).  (4.6)

This fact plays a central role in the proof of the next theorem.

Theorem 4.2. Assume that the operators (A(t), D(A(t))) and (B(t), Y_{t}), t ≥ 0, on X = L^{d}(Ω) satisfy (P1), (P2), and (B) for 1 < p, d < ∞. Moreover, assume that B(·)A(·)^{−1}x is measurable for x ∈ X. Then there is a perturbed evolution family U_{B} = (U_{B}(t, s))_{t≥s>−1} satisfying the conclusions of Theorem 3.4(a) and (b). Set u(t) = U_{B}(t, s)x for t ≥ s ≥ 0 and x ∈ X. Then u ∈ C([s, ∞), X) and u(s) = x. Further, we have u(t) ∈ D(A(t)) for a.e. t ∈ (s, ∞), u ∈ W_{loc}^{1,1}((s, ∞), X), (A(·) + B(·))u(·) ∈ L_{loc}^{1}((s, ∞), X), and

\frac{d}{dt} u(t) = (A(t) + B(t))u(t) for a.e. t ∈ (s, ∞).

Proof. (1) The existence of U_{B} = (U_{B}(t, s))_{t≥s>−1} follows from Lemma 4.1 and Theorem 3.4. Let T_{B} be the associated evolution semigroup on E = L^{p}(I, X) with generator G_{B} = G + B, where D(G_{B}) = D(G) ⊆ D(B) and B = B(·). Observe that the generator (G̃_{0}, D(G̃_{0})) of the left translation semigroup (R(−t))_{t≥0} is given by G̃_{0}g = g′ for g ∈ D(G̃_{0}) = W^{1,p}(I, X). Let f ∈ D(G). We have T_{B}(t)f ∈ D(G) = D(A) ∩ D(G_{0}) ⊆ D(G̃_{0}) and G_{B} = G_{0} + A + B due to (4.6). Hence,

\frac{d}{dτ} R(−τ)T_{B}(τ)f = R(−τ)G_{B}T_{B}(τ)f + R(−τ)G̃_{0}T_{B}(τ)f
= R(−τ)(A + B)T_{B}(τ)f =: v(τ),

and v : [0, t] → E is continuous. Integration from 0 to t yields

R(−t)T_{B}(t)f = f + ∫_{0}^{t} R(−τ)(A + B)T_{B}(τ)f dτ.  (4.7)

Moreover, for τ ∈ [0, t] we have

v(τ) (s) = (A(s + τ) + B(s + τ))U_{B}(s + τ, s)f(s) =: ṽ(τ, s)

for a.e. s > −1. Since (A, D(A)) is closed and AT_{B}(τ)f = AR(λ, G_{B})T_{B}(τ)(λ − G_{B})f, we derive from (4.6) the continuity of τ ↦ AT_{B}(τ)f. By Lemma 3.2 there is a measurable function Φ : [0, t]×I → X such that for a.e. τ ∈ [0, t] we have (R(−τ)AT_{B}(τ)f)(s) = Φ(τ, s) for a.e. s > −1. That is, for a.e. τ ∈ [0, t]

A(s + τ)U_{B}(s + τ, s)f(s) = Φ(τ, s),  and hence  (4.8)
U_{B}(s + τ, s)f(s) = A(s + τ)^{−1}Φ(τ, s)  (4.9)

for a.e. s > −1. Observe that both sides of (4.9) are measurable with respect to (τ, s). Thus (4.9) and, hence, (4.8) hold for a.e. (τ, s), and so the left–hand side of (4.8) is measurable on [0, t] × I. Then the function

[0, t] × I ∋ (τ, s) ↦ B(s + τ)U_{B}(s + τ, s)f(s) = B(s + τ)A(s + τ)^{−1} A(s + τ)U_{B}(s + τ, s)f(s)

is measurable by the assumption. As a consequence, ṽ : [0, t] × I → X is measurable. From Lemma 3.2 we deduce that ṽ(·, s) is integrable for a.e. s > −1 and (4.7) implies after a change of variables

U_{B}(t, s)f(s) = f(s) + ∫_{s}^{t}(A(τ) + B(τ))U_{B}(τ, s)f(s)dτ  (4.10)

for s ∈ (−1, ∞) \ N(f), t ∈ [s, ∞), and a null set N(f). (Notice that both sides of (4.10) are continuous with respect to t.)

(2) Now fix x ∈ X and s ≥ 0. Choose r_{n} ↘ s and φ_{n} ∈ C_{c}^{1}(I) with φ_{n} ≡ 0 on (−1, s] and φ_{n} ≡ 1 on [r_{n}, r_{n} + 1]. Set f_{n}(·) := φ_{n}(·)U_{B}(·, s)x. By Lemma 3.3 we have f_{n} ∈ D(G_{B}) = D(G). There exist s_{n} ∈ [r_{n}, r_{n} + 1] such that s_{n} ∉ N(f_{n}) and s_{n} ↘ s. From the first part of the proof follows that U_{B}(t, s_{n})f_{n}(s_{n}) ∈ D(A(t)) for a.e. t ∈ [s_{n}, ∞), (A(·) + B(·))U_{B}(·, s_{n})f_{n}(s_{n}) ∈ L_{loc}^{1}([s_{n}, ∞), X) and (4.10) holds for t ∈ [s_{n}, ∞). As a consequence, U_{B}(·, s_{n})f_{n}(s_{n}) ∈ W_{loc}^{1,1}([s_{n}, ∞), X) and

\frac{d}{dt} U_{B}(t, s_{n})f_{n}(s_{n}) = (A(t) + B(t))U_{B}(t, s_{n})f_{n}(s_{n})

for a.e. t ∈ [s_{n}, ∞). Set u(t) := U_{B}(t, s)x for t ≥ s. Since u(t) = U_{B}(t, s_{n})f_{n}(s_{n}) for t ≥ s_{n} the theorem is established. □

Other perturbation results in the parabolic case can be found, for instance, in [10, Thm. 6.1] or [7, Thm. 7.1.3]. There B(·) satisfies certain Hölder conditions which lead to stronger regularity properties of U_{B}.

5. An application

In this section we investigate the following intial value problem with mixed boundary conditions:

D_{t}u(t, ξ) = ∑_{k,l=1}^{n} D_{k} (a_{kl}(t, ξ)D_{l}u(t, ξ)) + ∑_{k=1}^{n} b_{k}(t, ξ) D_{k}u(t, ξ) + c(t, ξ)u(t, ξ), t > s, ξ ∈ Ω,

u(t, ξ) = 0, ξ ∈ Γ_{0} , t > s, (IV P)

∑_{k,l=1}^{n} n_{k}(ξ) a_{kl}(t, ξ) D_{l}u(t, ξ) = 0, ξ ∈ Γ_{1} , t > s,

u(s, ξ) = x(ξ), ξ ∈ Ω.

Here D_{t} = \frac{∂}{∂t} and D_{k} = \frac{∂}{∂ξ_{k}} denote derivatives in the sense of distributions and the boundary conditions are understood in the sense of traces. Moreover, we consider the following hypotheses:

(A1) Ω ⊆ IR^{n} is a bounded domain with compact C^{2}–boundary ∂Ω, Γ_{i} are open and closed in ∂Ω such that ∂Ω = Γ_{0} ∪ Γ_{1}, Γ_{0} ∩ Γ_{1} = ∅, Γ_{0} ≠ ∅, and n(ξ) is the outer normal of ∂Ω at ξ ∈ ∂Ω.

(A2) a_{kl} = a_{lk} : IR_{+} × \overline{Ω} → IR is continuous with bounded partial derivatives with respect to ξ ∈ Ω, a_{kl}, D_{j}a_{kl} ∈ C^{μ}(IR_{+}, C(\overline{Ω})), 1 ≤ k, l, j ≤ n, with μ > \frac{1}{2}, and

\frac{1}{δ} |η|^{2} ≥ ∑_{k,l=1}^{n} a_{kl}(t, ξ) η_{k}η_{l} ≥ δ |η|^{2}

for a constant δ > 0, η ∈ IR^{n}, t ≥ 0, and ξ ∈ Ω.

(A3) b_{k}, c ∈ L_{loc,u}^{q}(IR_{+}, L^{r}(Ω)), where r > n and q > (\frac{1}{2} − \frac{n}{2r} )^{−1}.

(A3’) b_{k} ≡ 0 and c ∈ L_{loc,u}^{q}(IR_{+}, L^{r}(Ω)), where 2r > n, r > 1, and q > (1 − \frac{n}{2r} )^{−1}.

Assumptions (A1) and (A2) were already considered in [14, §6].

On the space X = L^{d}(Ω), 1 < d < ∞, d ≤ r, we define the second order operator

A(t)x = ∑_{k,l=1}^{n} D_{k} (a_{kl}(t, ·)D_{l}x(·)), t ≥ 0,
D(A(t)) = {x ∈ W^{2,d}(Ω) : x(ξ) = 0, ξ ∈ Γ_{0}, ∑_{k,l=1}^{n} n_{k}(ξ)a_{kl}(t, ξ)D_{l}x(ξ) = 0, ξ ∈ Γ_{1}}.

Then it is shown in [14, §6] that the operators (A(t), D(A(t))), t ≥ 0, satisfy (P1) (with constants μ and ν = \frac{1}{2}) and (P2). (See also [1] and [31] for the verification of (P1) and the references in [14] concerning (P2).) Denote by U the evolution family solving the Cauchy problem (CP) corresponding to A(·).

Next, we introduce the (formal) first order operator

B(t)x = ∑_{k=1}^{n} b_{k}(t, ·) D_{k}x(·) + c(t, ·)x(·), t ≥ 0.

Assuming (A3), resp. (A3’), one can see that there are numbers θ > \frac{1}{2} + \frac{n}{2r} , resp. θ > \frac{n}{2r} , p > 1, and ρ < 1 such that 1 > ρ > pθ and q = (\frac{1}{p} − \frac{θ}{ρ})^{−1}. As in the preceding section Y_{t} is defined as the domain of (−A(t))^{θ} endowed with the norm ∥ · ∥_{Y_{t}}. Let \frac{1}{ϑ} = \frac{1}{d} − \frac{1}{r} ≥ 0. Then ϑ ≥ d and \frac{j}{2} + \frac{n}{2r} = \frac{j}{2} + \frac{n}{2d} − \frac{n}{2ϑ} for j ∈ \{0, 1\}. Since θ > \frac{j}{2} + \frac{n}{2r} , we obtain by [7, Thm. 1.6.1] that Y_{t} ⊆ W^{j,ϑ}(Ω) and ∥x∥_{W^{j,ϑ}} ≤ K∥x∥_{Y_{t}}. The proof of [7, Thm. 1.6.1] and the estimate

∥A(t)^{−1}x∥_{W^{2,d}} ≤ K′ ∥x∥_{d} ,  x ∈ X,  (5.1)

(see for instance [14, (6.6)] or [3, (IV.2.6.19)]) imply that the constant K does not depend on t. Moreover, we have \frac{1}{r} + \frac{1}{d′} + \frac{1}{ϑ} = 1, where \frac{1}{d} + \frac{1}{d′} = 1. Hence, by Hölder’s inequality we derive for x ∈ Y_{t} and y ∈ L^{d′}(Ω)

|⟨B(t)x, y⟩| ≤ ∫_{Ω} (|c(t, ξ)x(ξ)y(ξ)| + ∑_{k=1}^{n}(|b_{k}(t, ξ)D_{k}x(ξ)y(ξ)|) dξ
≤ ∑_{k=1}^{n} (∥D_{k}x∥_{ϑ} ∥b_{k}(t)∥_{r} ∥y∥_{d′}) + ∥x∥_{ϑ} ∥c(t)∥_{r} ∥y∥_{d′}
≤ ψ̃(t) ∥x∥_{W^{1,ϑ}} ∥y∥_{d′}
≤ Kψ̃(t) ∥x∥_{Y_{t}} ∥y∥_{d′},

where ψ̃(t) := max{∥c(t)∥_{r}, ∥b_{k}(t)∥_{r}, k = 1, ⋯, n}. Thus B(t) ∈ L(Y_{t}, X), ∥B(t)∥_{L(Y_{t},X)} ≤ ψ(t) := Kψ̃(t), and ψ ∈ L_{loc,u}^{q}(IR_{+}).

Now we show that B(·)A(·)^{−1}x, x ∈ X, is measurable. The Gagliardo–Nirenberg inequality (see e.g. [22, Lemma 8.4.1]), (5.1), and (P1) imply

∥D_{k} (A(t)^{−1} − A(s)^{−1})x∥_{d} ≤ C_{1} ∥(A(t)^{−1} − A(s)^{−1})x∥d^{\frac{1}{2}} ∥(A(t)^{−1} − A(s)^{−1})x∥W^{2,d}^{\frac{1}{2}}
≤ C_{2} ∥x∥_{d} |t − s|^{\frac{μ}{2}}

for constants C_{l} . Hence, B(·)A(·)^{−1}x is measurable if b_{k} , c ∈ L^{∞}(IR_{+} × Ω), cf. [26, Lemma 2.4]. The general case follows by truncation of the coefficients b_{k} and c and an application of the dominated convergence theorem. Finally, from B(t)U(t, s)x = B(t)A(t)^{−1} A(t)U(t, s)x we derive the measurability of B(·)U(·, s)x for x ∈ X.

As a consequence, Theorem 4.2 yields the following result which extends [24, Ex. 4.6], where the case b_{k} ≡ 0 and A(t) ≡ A was considered.

Proposition 5.1. Assume (A1), (A2), and (A3), resp. (A3’). Let X = L^{d}(Ω), 1 < d < ∞, d ≤ r, x ∈ X, and s ≥ 0. Then there is a function u ∈ W_{loc}^{1,1}((s, ∞), X) ∩ C([s, ∞), X) with u(s) = x such that u(t) ∈ D(A(t)) for a.e. t ∈ (s, ∞), (A(·) + B(·))u(·) ∈ L_{loc}^{1}((s, ∞), X), and u satisfies (IVP) for a.e. t ∈ (s, ∞) and ξ ∈ Ω.

In [4] D.G. Aronson has studied a more general version of (IVP) (however with Dirichlet boundary conditions) and obtained weak solutions of (IVP) by PDE methods. We point out that in [4] the conditions r > n and q > (\frac{1}{2} − \frac{n}{2r} )^{−1}, cf. (A3), also play a crucial role.

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