SOLVABILITY AND MAXIMAL REGULARITY OF PARABOLIC EVOLUTION EQUATIONS WITH COEFFICIENTS CONTINUOUS IN

TIME

JAN PRÜSS AND ROLAND SCHNAUBELT

Abstract. We establish maximal regularity of type L^{p} for a parabolic evolution equation u′(t) = A(t)u(t) + f(t) with A(·) ∈ C([0, T], L(D(A(0)), X)), and construct the corresponding evolution family on the underlying Banach space X. Our proofs are based on the operator sum method and the use of evolution semigroups. The results are applied to parabolic partial differential equations with continuous coefficients.

1. Introduction

We are concerned with the solvability and maximal regularity of the linear Cauchy problem

\begin{aligned}u′(t) & = & A(t)u(t) + f(t) for (a.e.) 0 ≤ t ≤ T, \\ u(0) & = & x, \end{aligned}  (1.1)

where A(t) generates an analytic semigroup on a Banach space X, f ∈ L^{p}([0, T], X) for 1 < p < ∞, x ∈ X, and T > 0 is fixed. One of our aims is to remove the usual Hölder condition on A(·) (or (λ − A(·))^{−1}). So we only assume that

D(A(t)) = D(A(0)) =: X_{1} and A(·) ∈ C([0, T], L(X_{1}, X)).  (1.2)

Since we allow for discontinuous inhomogeneities f, we can only expect to solve (1.1) for a.e. t ∈ [0, T] with solutions of class

u ∈ C([0, T], X) ∩ W_{loc}^{1,p}((0, T], X) ∩ L_{loc}^{p}((0, T], X_{1})

if x ∈ X and

u ∈ F := W^{1,p}([0, T], X) ∩ L^{p}([0, T], X_{1})

if x belongs to the real interpolation space (X, X_{1})_{1− \frac{1}{p} ,p}. In particular, in the latter case we show maximal regularity (of type L^{p}); that is, the existence of a unique solution u ∈ F of (1.1) satisfying

∥u∥_{L^{p}([0,T],X)} + ∥u∥_{L^{p}([0,T],X_{1})} + ∥u′∥_{L^{p}([0,T],X)} ≤ c_{p} (∥x∥_{1− \frac{1}{p} ,p} + ∥f∥_{L^{p}([0,T],X)}) (1.3)

for all x ∈ (X, X_{1})_{1− \frac{1}{p} ,p} and f ∈ L^{p}([0, T], X) and a constant c_{p}. To achieve this, we suppose that, besides (1.2),

the operator A(s) has maximal regularity in L^{p} for all fixed s ∈ [0, T].  (1.4)

Below we discuss sufficient conditions for this hypothesis.

1991 Mathematics Subject Classification. 35K22, 47D06.

Key words and phrases. Non-autonomous Cauchy problem, parabolic, maximal regularity, evolution family, evolution semigroup, Miyadera perturbation.

In order to solve (1.1) for initial values x ∈ X, it is necessary to construct an evolution family U(t, s), 0 ≤ s ≤ t ≤ T, of bounded linear operators on X. This means that

U(t, s) = U(t, r)U(r, s),  U(s, s) = Id,  (1.5)
(t, s) ↦ U(t, s) is strongly continuous  (1.6)

for 0 ≤ s ≤ r ≤ t ≤ T. Moreover, it will be shown that U(t, s) maps X into real interpolation spaces between X and X_{1}, satisfies corresponding norm estimates, and solves the homogeneous problem (1.1) with f = 0. See Theorem 2.5 for all this.

If A(t) is an elliptic operator on L^{q}(Ω), 1 < q < ∞, hypothesis (1.2) requires only continuity of the coefficients, while (1.4) may require additional regularity with respect to the space variables, see Section 4. So far, no complete characterization of (1.4) is known for elliptic operators on L^{q}(Ω). In this case, (1.3) leads to L^{p}–L^{q} a priori estimates which are very useful for the investigation of nonlinear equations, compare e.g. [3, 9, 19]. A perturbation result proved in Theorem 3.1 allows to reduce the regularity assumptions on the lower order coefficients to rather weak integrability conditions.

In the case of Hölder continuous A(·) : [0, T] → L(X_{1}, X) and f : [0, T] → X, it is known that there exists an evolution family U(t, s) on X and a unique solution u ∈ C^{1}([0, T], X) of (1.1) given by

u(t) = U(t, 0)x + ∫_{0}^{t} U(t, s)f(s) ds

for x ∈ X_{1}. In addition, U(t, s) maps X into X_{1} and is differentiable in L(X) with ∂_{t} U(t, s) = A(t)U(t, s) for t > s, see e.g. [3, 22, 24]. One also has maximal Hölder regularity which is extensively studied in the monographs [3] and [22]. These results remain essentially valid if the domains D(A(t)) depend on time t provided one imposes appropriate conditions on the resolvent of A(t). Among the variety of possible assumptions we only mention the estimate

∥A(t)(λ − A(t))^{−1}(A(t)^{−1} − A(s)^{−1})∥ ≤ L |t − s|^{α} |λ|^{−β}  (1.7)

for constants L ≥ 0 and α, β ∈ (0, 1] with α + β > 1, t, s ∈ [0, T], and λ in a sector with angle greater than π/2. This condition was introduced by P. Acquistapace and B. Terreni in [1, 2], see also [3, 23] and the references therein.

In the present work we concentrate on the case of time independent domains. Here it is possible to remove the Hölder continuity in time at the prize of increased space regularity. Namely, G. Da Prato and P. Grisvard assumed that, besides (1.2), the domain D_{A(t)}(θ + 1) of the part of A(t) in the interpolation space X_{θ} := D_{A(0)}(θ) is equal to a fixed space X_{θ+1} and A(·) ∈ C([0, T], L(X_{θ+1}, X_{θ})) for some θ ∈ (0, 1). They obtained solutions u ∈ C([0, T], X_{θ+1}) of (1.1) with x ∈ X_{θ+1} and f ∈ C([0, T], X_{θ}), see [11] and also [12]. Under (almost) the same assumptions, A. Buttu studied in detail maximal Hölder regularity and properties of the corresponding evolution family on interpolation spaces in [6, 7]. Very close to our work is the paper [18] by M. Giga, Y. Giga and H. Sohr, see also [3, III.4.10.10]. They showed maximal L^{p}–regularity of (1.1) imposing slight variants of (1.2) and (1.4). But we point out that initial values x ∈ X or the evolution operator on X were not considered in these papers. So the main aim of the present investigation is to construct the evolution family U(t, s) on X and to solve (1.1) for x ∈ X.

Maximal regularity of type L^{p} has been investigated thoroughly in the autonomous case A(·) = A_{0}. We refer to [3, 14, 21, 25] for a survey and further references. First we recall that by a result of A. Benedek, A.P. Calderón and R. Panzone, [5], hypothesis (1.4) is in fact independent of p ∈ (1, ∞); that is, if (1.4) is valid for some p_{0} ∈ (1, ∞), then it holds for all p ∈ (1, ∞), see [14] or [21]. It is known that a densely defined, closed operator A_{0} having maximal regularity already generates an analytic semigroup, [14, §2]. De Simon proved in [13] that the converse is true if X is a Hilbert space so that in this case (1.4) is a consequence of (1.2). There are examples of generators of analytic semigroups without maximal regularity if X is not a UMD space, cf. [14, §3]. Further, maximal regularity holds for generators of analytic semigroups on real interpolation spaces due to [10]. More recently, G. Dore and A. Venni showed in [15] maximal regularity in UMD-spaces X for operators having bounded imaginary powers with ∥(−A_{0})^{is}∥ ≤ K e^{θ|s|} for s ∈ ℝ and a constant 0 ≤ θ < π/2, see also [3, 25]. Notice that L^{q}(Ω) is an UMD-space if 1 < q < ∞. Finally, A_{0} possesses maximal regularity on L^{q}(Ω), 1 < q < ∞, if it generates an analytic semigroup on L^{2}(Ω) and satisfies certain heat kernel estimates, [21].

The results of [15] and [21] have been extended to the non-autonomous situation in [23] and [19, 20], respectively, where (1.7) was assumed. The latter condition ensures, in particular, the existence of an evolution family on X due to [1, 2].

Our approach combines the operator sum method as initiated by G. Da Prato and P. Grisvard in [10] and semigroup techniques. In Section 2 we first show that the operator

G = −\frac{d}{dt} + A(·) with D(G) = \{f ∈ W^{1,p}([a, b], X) ∩ L^{p}([a, b], X_{1}) : f(a) = 0\}

is invertible in L^{p}([a, b], X) if b − a is sufficiently small. Next, we approximate G by the operators G_{n} = −\frac{d}{dt} +A_{n}(·) for the Yosida approximations A_{n}(t) of A(t). The semigroups generated by G_{n} are uniformly bounded. The proof of this fact is relegated to Section 5. It uses another approximation argument and the Chernoff product formula. So the Trotter– Kato theorem shows that G is a generator. From a result in [26, 27] then follows that the semigroup T (·) generated by G is given by

(T (t)f) (s) = \begin{cases}U(s, s − t)f(s − t), & a ≤ s − t ≤ s ≤ b, \\ 0, & s − t < a ≤ s ≤ b, \end{cases}  (1.8)

for an evolution family U(t, s) on X and f ∈ L^{p}([a, b], X). We call semigroups of this form evolution semigroups, see [8, 27] and the references therein for more information. Using the representation of G and simple properties of T (·), one can now verify the asserted regularity of U(t, s) and solve (1.1). In Section 3 we establish a perturbation result for (1.1) which is a consequence of a non–autonomous Miyadera perturbation theorem from [26, 27]. Applications to parabolic partial differential equations are presented in Section 4.

2. The main result

We first state our standing hypotheses on the operators A(t). (H1) A(t), 0 ≤ t ≤ T, generates an analytic semigroup of negative type on a Banach space

X, D(A(t)) = D(A(0)) =: X_{1}, and A(·) ∈ C([0, T], L(X_{1}, X)), where X_{1} is endowed with the graph norm of A(0).

(H2) For some p ∈ (1, ∞) and all f ∈ L^{p}([0, T], X) and s ∈ [0, T] there is a unique function u ∈ F_{0} := {f ∈ W^{1,p}([0, T], X) ∩ L^{p}([0, T], X_{1}) : f(0) = 0} such that

u′(t) = A(s)u(t) + f(t) for a.e. t ∈ [0, T].  (2.1)

Clearly, (H1) implies that, given s ∈ [0, T],

(λ − A(t))^{−1} = (λ − A(s))^{−1} [1 − (A(t) − A(s))A(0)^{−1}A(0)(λ − A(s))^{−1}]^{−1}

for t ∈ [0, T] with |t − s| ≤ δ_{s} and λ ∈ Σ_{ϕ_{s}} := {μ ∈ ℂ \ {0} : | arg μ| ≤ ϕ_{s}} ∪ {0} for some π/2 < ϕ_{s} ≤ π and δ_{s} > 0. So, due to the compactness of [0, T], there are constants M ≥ 0 and π/2 < ϕ ≤ π such that

∥(λ − A(t))^{−1}∥_{L(X,X_{k})} ≤ \frac{M}{1 + |λ|^{1−k}}  (2.2)

for λ ∈ Σ_{ϕ}, t ∈ [0, T], k = 0, 1, and X_{0} := X, cf. [11, Lemme 3.5]. In particular, the type of the semigroup (e^{τA(t)})_{τ≥0} generated by A(t) does not depend on t. Using (H1) and (2.2) one also sees that the graph norms of the operators A(t) are uniformly equivalent.

As mentioned in the introduction, condition (H2) does not depend on p ∈ (1, ∞), see e.g. [14, Thm. 4.2], [21, Prop. 2.4], and the references therein. Since A(s) generates a strongly continuous semigroup, the solution u ∈ F_{0} of (2.1) is given by

u(t) = ∫_{0}^{t} e^{(t−τ)A(s)}f(τ) dτ,

see [3, III.1.3.1] or [14, §1]. Due to this representation the mapping R_{s} : L^{p}([0, T], X) → F_{0}; f ↦ u is closed, and hence bounded. Therefore the solution of (2.1) satisfies

∥u∥_{F} := ∥u∥_{L^{p}([0,T],X)} + ∥u′∥_{L^{p}([0,T],X)} + ∥u∥_{L^{p}([0,T],X_{1})} ≤ c_{s,p} ∥f∥_{L^{p}([0,T],X)}  (2.3)

for a constant c_{s,p} depending on s ∈ [0, T] and p ∈ (1, ∞).

Let us fix some notation. For technical reasons, we set A(t) := A(0) for t ∈ [−1, 0), cf. [27, §2, 3]. Notice that (H1) and (H2) still hold for this extension. Given an interval I = [a, b] ⊆ [−1, T], we define the Banach spaces

F_{I} := W^{1,p}(I, X) ∩ L^{p}(I, X_{1}) and F_{0,I} := {f ∈ F_{I} : f(a) = 0}

endowed with the canonical norm ∥ · ∥_{F} and the operators

L_{I}f := −f′ + A(·)f, f ∈ F_{I}, and G_{I}f := −f′ + A(·)f, f ∈ F_{0,I},

on L^{p}(I, X) equipped with the usual p–norm ∥ · ∥_{p}. In the case I = [0, T] we omit the subscript I. Also, W_{0}^{1,p}(I, X) := {f ∈ W^{1,p}(I, X) : f(a) = 0}. We write (X, X_{1})_{θ,q} for the real interpolation space between X and X_{1} of exponent 0 < θ < 1 and parameter 1 ≤ q < ∞, see e.g. [28, §1.3]. Then

∥x∥_{θ,q} := ∥x∥ + (∫_{0}^{∞} ∥t^{1−θ} A(0)e^{tA(0)}x∥^{q} \frac{dt}{t})^{\frac{1}{q}}

is an equivalent norm on (X, X_{1})_{θ,q} by, for instance, [28, 1.14.5]. In particular,

e^{(·−a)A(0)}x ∈ F_{I} and ∥e^{(·−a)A(0)}x∥_{F} ≤ c ∥x∥_{1− \frac{1}{p} ,p} if x ∈ (X, X_{1})_{1− \frac{1}{p} ,p}.  (2.4)

Recall the continuous and dense embeddings

X_{1} ↪^{d} (X, X_{1})_{ξ,p} ↪^{d} (X, X_{1})_{ξ,q} ↪^{d} (X, X_{1})_{η,1} ↪^{d} X  (2.5)

for 0 < η < ξ < 1 and 1 ≤ p ≤ q < ∞, see e.g. [28, 1.3.3, 1.6.2]. We further need the continuous embedding

F_{I} ↪ C(I, (X, X_{1})_{1− \frac{1}{p} ,p})  (2.6)

where the corresponding norm estimate does not depend on I, [3, III.4.10.2].

Observe that in (H2) and (2.3) one can replace the interval [0, T] by I = [a, b] ⊆ [−1, T] with a possibly larger but I–independent constant, cf. [14, Thm. 2.5]. We start with a known perturbation argument, cf. [3, III.4.10.10] or [18].

Lemma 2.1. Assume that (H1) and (H2) hold. Let p ∈ (1, ∞) and s ∈ [−1, T]. Then there exists a constant δ (depending on p and s) such that the operator G_{I} defined on F_{0,I} is invertible in L^{p}(I, X) if I = [a, b] ⊆ [−1, T] with s ∈ I and 0 < b − a ≤ δ.

Proof. Let s ∈ I = [a, b] ⊆ [−1, T]. By (H2) the operator −\frac{d}{dt} + A(s) with domain F_{0,I} is invertible in L^{p}(I, X). Notice that we have

G_{I}f = [Id − (A(s) − A(·))(−\frac{d}{dt} + A(s))^{−1}] (−\frac{d}{dt} + A(s))f

for f ∈ F_{0,I}. Further, (H1) and (2.3) yield

∥(A(s) − A(·))(−\frac{d}{dt} + A(s))^{−1}f∥_{p} ≤ sup_{r,r′∈I} ∥A(r) − A(r′)∥_{L(X_{1},X)} c_{s,p} ∥f∥_{p}

for f ∈ L^{p}([0, T], X). This easily implies the assertion.

In order to show that G_{I} is a generator, we use the Yosida approximations A_{n}(t) = nA(t)(n − A(t))^{−1} of A(t) and the operator G_{n,I} = −\frac{d}{dt} + A_{n}(·) defined on W_{0}^{1,p}(I, X). Since A_{n}(·) ∈ C([−1, T], L(X)), there is an evolution family U_{n}(t, s), −1 ≤ s ≤ t ≤ T, such that

\frac{∂}{∂t} U_{n}(t, s) = A_{n}(t)U_{n}(t, s) and \frac{∂}{∂s} U_{n}(t, s) = −U_{n}(t, s)A_{n}(s)

in L(X) for t ≥ s. This easily implies that G_{n,I} generates the evolution semigroup T_{n,I}(·) on L^{p}(I, X) corresponding to U_{n}(t, s), cf. (1.8). It is further clear that

lim_{n→∞} ∥G_{n,I}f − G_{I}f∥_{p} = lim_{n→∞} ∥(A_{n}(·) − A(·))f∥_{p} = 0  (2.7)

for f ∈ F_{0,I}. The next result will be proved in Section 5.

Lemma 2.2. If (H1) holds, then ∥U_{n}(t, s)∥ ≤ C for a constant C ≥ 0, n ∈ ℕ, and −1 ≤ s ≤ t ≤ T. In particular, ∥T_{n,I}(t)∥ ≤ C for n ∈ ℕ, t ≥ 0, and intervals I ⊆ [−1, T].

Combining the above facts we now establish the existence of the evolution family U(t, s).

Proposition 2.3. Assume that (H1) and (H2) hold. Let I = [a, b] ⊆ [0, T] be given as in Lemma 2.1. Then there is an evolution family U(t, s), 0 ≤ s ≤ t ≤ T, on X such that G_{I} generates the corresponding evolution semigroup

(T_{I}(t)f)(s) = \begin{cases}U(s, s − t)f(s − t), & a ≤ s − t ≤ s ≤ b, \\ 0, & s − t < a ≤ s ≤ b, \end{cases}

on L^{p}(I, X) and G_{I} has the inverse

(G_{I}^{−1}f)(t) = − ∫_{a}^{t} U(t, s)f(s) ds, t ∈ I, f ∈ L^{p}(I, X).

Moreover, T_{n}(t) converges strongly to T (t) as n → ∞, where T_{n}(·) and T (·) are the evolution semigroups on L^{p}([0, T], X) induced by U_{n}(t, s) and U(t, s), respectively.

Proof. Let [a, b] ⊆ [−1, T] be as in Lemma 2.1. Due to Lemma 2.1, Lemma 2.2, and (2.7), the Trotter–Kato theorem, [24, Thm. 3.4.5], shows that G_{I} generates a strongly continuous semigroup T_{I}(·) on L^{p}(I, X) and T_{n,I}(t) converges strongly to T_{I}(t) as n → ∞. Clearly, F_{0,I} is dense in C_{0}((a, b], X), and for φ ∈ C_{c}^{1}(a, b] and f ∈ F_{0,I} we have φf ∈ F_{0,I} and G_{I}(φf) = φ G_{I}f − φ′f. By virtue of [27, Thm. 2.4], this implies that T_{I}(·) is an evolution semigroup given by an evolution family U_{I}(t, s), a < s ≤ t ≤ b, on X. Using the convergence of T_{n,I}(·) to T_{I}(·), one easily sees that setting U(t, s) := U_{I}(t, s) for a < s ≤ t ≤ b defines an evolution family U(t, s), 0 ≤ s ≤ t ≤ T, on X. Since T_{I}(t) = 0 for large t, the inverse of G_{I} is given by

(G_{I}^{−1}f)(t) = − ∫_{0}^{∞}(T_{I}(τ)f)(t) dτ = − ∫_{a}^{t} U(t, s)f(s) ds

for t ∈ I and f ∈ L^{p}(I, X). (Notice that the first equality holds for a.e. t ∈ I due to e.g. [27, Lemma 3.2], but the functions on the left and right hand side are continuous in t.)

To show the last assertion, take 0 = t_{0} < t_{1} < ⋯ < t_{m} = T and 0 < t < t_{k} − t_{k−1} such that the intervals I_{k} = [t_{k−1}, t_{k}] and I_{k}^{t} = [t_{k} − t, t_{k} + t] satisfy the assumptions of Lemma 2.1. For f ∈ L^{p}([0, T], X), one obtains

∥T (t)f − T_{n}(t)f∥_{p}^{p} = ∑_{k=1}^{m−1} ∥(T_{I_{k}}(t) − T_{I_{k},n}(t)) f|I_{k}∥_{p}^{p} + ∥(T_{I_{k}^{t}}(t) − T_{I_{k}^{t},n}(t)) f|I_{k}^{t}∥_{p}^{p}
+ ∥(T_{I_{m}}(t)f − T_{I_{m},n}(t)) f|I_{m}∥_{p}^{p}

so that T_{n}(t)f → T (t)f as n → ∞ for small t ≥ 0. The semigroup property and the uniform boundedness of T_{n}(t) then imply convergence for all t ≥ 0.

The following lemma provides most of the required regularity properties of U(t, s).

Lemma 2.4. Assume that (H1) and (H2) hold. Then the operators U(t, s), 0 ≤ s ≤ t ≤ T, obtained in Proposition 2.3 also yield an evolution family on (X, X_{1})_{θ,q} for θ ∈ (0, 1) and q ∈ [1, ∞). Moreover, the function u = U(·, s)x belongs to F_{[s,T]}, L_{[s,T]}u = 0, and u satisfies ∥u∥_{F} ≤ c ∥x∥_{1− \frac{1}{p} ,p} for x ∈ (X, X_{1})_{1− \frac{1}{p} ,p}, p ∈ (1, ∞), s ∈ [0, T), and a constant c not depending on s and x.

Proof. (1) For s ∈ [0, T) and p ∈ (1, ∞), let δ = δ(s, p) be given by Lemma 2.1. Set I = [s, b] := [s, min{s + δ, T }]. We define

\begin{aligned}f_{n}(t) & = U_{n}(t, s)x − e^{(t−s)A_{n}(0)}x, & f(t) & = U(t, s)x − e^{(t−s)A(0)}x, \\ g_{n}(t) & = (A_{n}(0) − A_{n}(t))e^{(t−s)A_{n}(0)}x, & g(t) & = (A(0) − A(t))e^{(t−s)A(0)}x, \end{aligned}  (2.8)

for x ∈ X_{1}, t ∈ I, and n ∈ ℕ. It is straightforward to see that

f_{n}(t) = − ∫_{s}^{t} U_{n}(t, τ)g_{n}(τ) dτ = (G_{n,I}^{−1} g_{n})(t)

and that g_{n} converges to g in L^{p}(I, X). Then (2.7), Lemma 2.2, and Proposition 2.3 imply that f_{n} tends in L^{p}(I, X) to the function

I ∋ t ↦ (G_{I}^{−1}g)(t) = − ∫_{s}^{t} U(t, τ)g(τ) dτ.

On the other hand, we have e^{tA_{n}(0)}x → e^{tA(0)}x as n → ∞. Proposition 2.3, Lemma 2.2 and the dominated convergence theorem yield

∫_{0}^{T} ∫_{0}^{T −τ} ∥U_{n}(τ + t, τ)x − U(τ + t, τ)x∥^{p} dt dτ = ∫_{0}^{T} ∥T_{n}(t)h − T (t)h∥_{p}^{p} dt ⟶ 0

as n → ∞, where h ≡ x. Consequently, for all x ∈ X_{1} and a.e. s ∈ [0, T], there exists a subsequence such that f_{n_{k}}(t) converges to f(t) for a.e. t ∈ [s, b], and hence, f = G_{I}^{−1}, i.e.,

U(t, s)x = e^{(t−s)A(0)}x − ∫_{s}^{t} U(t, τ)g(τ) dτ

for a.e. t ∈ [s, b]. By continuity these equations hold for all s ∈ [0, T] and t ∈ [s, b].

Next, let x ∈ (X, X_{1})_{1− \frac{1}{p} ,p} in (2.8). Approximating x in (X, X_{1})_{1− \frac{1}{p} ,p} by x_{k} ∈ X_{1}, one easily deduces that f = G_{I}^{−1}g also holds for x ∈ (X, X_{1})_{1− \frac{1}{p} ,p} . As a result, u := U(·, s)x = f + e^{(·−s)A(0)}x belongs to F_{I} and

L_{I}u = g + (A(·) − A(0))e^{(·−s)A(0)}x = 0

for x ∈ (X, X_{1})_{1− \frac{1}{p} ,p}. Moreover, (2.6), Lemma 2.1, (2.4), and (H1) imply the estimate

∥U(t, s)x∥_{1− \frac{1}{p} ,p} ≤ c_{1} ∥U(·, s)x∥_{F} ≤ c_{1} (∥G_{I}^{−1}g∥_{F} + ∥e^{(·−s)A(0)}x∥_{F})
≤ c_{2} (∥g∥_{p} + ∥x∥_{1− \frac{1}{p} ,p}) ≤ c_{3} ∥x∥_{1− \frac{1}{p} ,p}

for x ∈ (X, X_{1})_{1− \frac{1}{p} ,p}, t ∈ I, and constants c_{k} depending on I and p but not on t and x.

(2) Now cover [0, T] by finitely many intervals of the above type. Using (2.6) and part (1), we see that U(·, s)x belongs to the kernel of L_{[s,T]} and satisfies

∥U(t, s)x∥_{1− \frac{1}{p} ,p} ≤ c ∥U(·, s)x∥_{F} ≤ c̃∥x∥_{1− \frac{1}{p} ,p}  (2.9)

for x ∈ (X, X_{1})_{1− \frac{1}{p} ,p}, 0 ≤ s ≤ t ≤ T, and constants independent of t, s, and x. Finally, from (2.5), (2.9), and the strong continuity of U(t, s) on X, we deduce the strong continuity of U(t, s) on the spaces (X, X_{1})_{θ,q} , θ ∈ (0, 1), q ∈ [1, ∞), using the reiteration theorem [28, 1.10.2].

We can now prove our main result.

Theorem 2.5. Assume that (H1) and (H2) hold. Let p ∈ (1, ∞). Then, for x ∈ (X, X_{1})_{1− \frac{1}{p} ,p} and f ∈ L^{p}([0, T], X), there exists a unique function u ∈ W^{1,p}([0, T], X) ∩ L^{p}([0, T], X_{1}) satisfying

\begin{aligned}u′(t) & = & A(t)u(t) + f(t)  for a.e. 0 ≤ t ≤ T, \\ u(0) & = & x, \end{aligned}  (2.10)

and we have the estimate

∥u∥_{L^{p}([0,T],X)} + ∥u∥_{L^{p}([0,T],X_{1})} + ∥u′∥_{L^{p}([0,T],X)} ≤ c_{p} (∥x∥_{1− \frac{1}{p} ,p} + ∥f∥_{L^{p}([0,T],X)}) (2.11)

for a constant c_{p} independent of x and f. The solution u is given by

u(t) = U(t, 0)x + ∫_{0}^{t} U(t, s)f(s) ds,  0 ≤ t ≤ T,  (2.12)

for an evolution family U(t, s), 0 ≤ s ≤ t ≤ T, on X. If x ∈ X and f ∈ L^{p}([0, T], X), then (2.12) gives the unique solution of (2.10) in the space F_{loc} := C([0, T], X) ∩ W_{loc}^{1,p}((0, T], X) ∩ L_{loc}^{p}((0, T], X_{1}). The operators U(t, s) also yield an evolution family on the real interpolation space (X, X_{1})_{θ,q} for θ ∈ (0, 1) and q ∈ [1, ∞). Moreover, U(t, s) maps X into (X, X_{1})_{1− \frac{1}{p} ,p} and there is a constant c̃_{p} such that

∥U(t, s)x∥_{1− \frac{1}{p} ,p} ≤ c̃_{p} (t − s)^{\frac{1}{p} −1} ∥x∥  (2.13)

for 1 < p < ∞ and 0 ≤ s < t ≤ T. Finally, the evolution semigroup on L^{p}([0, T], X) corresponding to U(t, s) is generated by the operator Gf = −f′ + A(·)f with D(G) = W_{0}^{1,p}([0, T], X) ∩ L^{p}([0, T], X).

Proof. Let U(t, s), 0 ≤ s ≤ t ≤ T, be the evolution family obtained in Proposition 2.3. Fix f ∈ L^{p}([0, T], X), x ∈ (X, X_{1})_{1− \frac{1}{p} ,p}, and p ∈ (1, ∞). Cover [0, T] by finitely many intervals I_{k} = [t_{k−1}, t_{k}] with t_{0} = 0 and t_{n} = T such that the conclusion of Proposition 2.3 holds for I_{k}. Let G_{k} = G_{I_{k}} , L_{k} = L_{I_{k}} , and g_{k} = g|I_{k} for g ∈ L^{p}([0, T], X). We define on L^{p}(I_{k}, X), k = 1, ⋯, n, the functions

v_{1} = −G_{1}^{−1}f_{1} , v_{2} = −G_{2}^{−1}f_{2} + U(·, t_{1})v_{1}(t_{1}), ⋯, v_{n} = −G_{n}^{−1}f_{n} + U(·, t_{n−1})v_{n−1}(t_{n−1}) .

These functions belong to F_{I_{k}} and L_{k}v_{k} = −f_{k} by virtue of Proposition 2.3, Lemma 2.4, and (2.6). Therefore, the function u given by

u(t) := U(t, 0)x + v_{k}(t) = U(t, 0)x + ∫_{0}^{t} U(t, s)f(s) ds, t ∈ I_{k},

belongs to F and solves (2.10). To show uniqueness, let w ∈ F_{0} satisfy w′(t) = A(t)w(t). Then w_{1} ∈ D(G_{1}) and G_{1}w_{1} = 0 so that w_{1} = 0 since G_{1} is invertible. Proceeding to the right, we derive w = 0.

As a result, the operator G = G_{[0,T]} with domain D(G) = F_{0} in L^{p}([0, T], X) is bijective and has the bounded inverse given by Rf(t) = − ∫_{0}^{t} U(t, s)f(s) ds. In particular, G is closed and its graph norm is equivalent to ∥ · ∥_{F} because of the inverse mapping theorem. Together with Lemma 2.4 this implies the estimate (2.11). As in the proof of Proposition 2.3 one sees that R is also the inverse of the generator of the evolution semigroup T (·) on L^{p}([0, T], X) induced by U(t, s). So the last assertion follows.

Given 0 ≤ s < t ≤ T, take the absolutely continuous function φ = φ_{t,s} which is equal to 0 on [0, s] and to 1 on [t, T] and is linear on (s, t). Notice that ∥φ′∥_{p} = (t − s)^{\frac{1}{p} −1} . Define u = φ(·) U(·, s)x for x ∈ X, where we let U(r, s) := 0 for 0 ≤ r < s. Then, (T (h)u) (r) = φ(r − h) U(r, s)x which implies

u ∈ D(G) = F_{0} and Gu = −φ′(·) U(·, s)x.  (2.14)

Using (2.6) and (2.11), we now obtain

∥U(t, s)x∥_{1− \frac{1}{p} ,p} ≤ c_{1} ∥u∥_{F} ≤ c_{2} ∥φ′(·) U(·, s)x∥_{p} ≤ c_{3} (t − s)^{\frac{1}{p} −1} ∥x∥

for constants c_{k} depending on p. From (2.14) we also derive that U(·, 0)x belongs to the kernel of L_{[ε,T]} for all ε > 0. So (2.12) gives a solution of (2.10) in F_{loc} if x ∈ X. It remains to show uniqueness of such solutions. Let w ∈ F_{loc} satisfy (2.10) with x = 0 and f = 0. Then the restriction w|[ε, T] ∈ F_{[ε,T]} solves (2.10) with f = 0 and initial value w(ε) at initial time ε. The first part of the proof implies that w(t) = U(t, ε)w(ε) for all t ≥ ε > 0. Hence, w = 0 by the continuity of w.

3. A perturbation result

In this section we consider the perturbed problem

\begin{aligned}u′(t) & = & A(t)u(t) + B(t)u(t) + f(t) for a.e. 0 ≤ t ≤ T, \\ u(0) & = & x, \end{aligned}  (3.1)

where we suppose that the operators A(t) satisfy the conditions (H1) and (H2). For the perturbations B(t) we require that (B) B(t) ∈ L((X, X_{1})_{1− \frac{1}{q} ,q} , X) for some q ∈ [1, ∞), B(·)x is measurable for x ∈

(X, X_{1})_{1− \frac{1}{q} ,q} , and ∥B(t)∥_{L((X,X_{1})_{1− \frac{1}{q} ,q} ,X)} ≤ β(t) for a.e. t ∈ [0, T] and a function β ∈ L^{r}[0, T] with r ∈ (q, ∞].

Here we let (X, X_{1})_{0,1} := X. Condition (B) can be verified for lower order terms of parabolic partial differential equations with coefficients belonging to suitable Lebesgue spaces, cf. Section 4. We now show that under these assumptions the problem (3.1) has essentially the same properties as (2.10).

Theorem 3.1. Assume that (H1), (H2), and (B) hold. Let p ∈ (1, \frac{r}{q}). Then, for x ∈ (X, X_{1})_{1− \frac{1}{p} ,p} and f ∈ L^{p}([0, T], X), there exists a unique function u ∈ W^{1,p}([0, T], X) ∩ L^{p}([0, T], X_{1}) satisfying (3.1) and the estimate (2.11). Moreover, u is given by

u(t) = U_{B}(t, 0)x + ∫_{0}^{t} U_{B}(t, s)f(s) ds,  0 ≤ t ≤ T,  (3.2)

for an evolution family U_{B}(t, s), 0 ≤ s ≤ t ≤ T, on X. If x ∈ X and f ∈ L^{p}([0, T], X), then (3.2) gives the unique solution of (3.1) in the space F_{loc} = C([0, T], X) ∩ W_{loc}^{1,p}((0, T], X) ∩ L_{loc}^{p}((0, T], X_{1}). The operators U_{B}(t, s) also yield an evolution family on the real interpolation space (X, X_{1})_{θ,τ} for θ ∈ (0, 1 − \frac{q}{r}) and τ ∈ [1, ∞). Moreover, U_{B}(t, s) maps X into (X, X_{1})_{1− \frac{1}{p} ,p} and there is a constant c̃_{p} such that

∥U_{B}(t, s)x∥_{1− \frac{1}{p} ,p} ≤ c̃_{p} (t − s)^{\frac{1}{p} −1} ∥x∥

for x ∈ X, 1 < p < \frac{r}{q}, and 0 ≤ s < t ≤ T. Finally, the evolution semigroup on L^{p}([0, T], X) corresponding to U_{B}(t, s) is generated by the operator G_{B}f = −f′ + A(·)f + B(·)f with D(G_{B}) = W_{0}^{1,p}([0, T], X) ∩ L^{p}([0, T], X_{1}).

Proof. (1) Let p ∈ (1, \frac{r}{q}) and f ∈ F_{0} = W_{0}^{1,p}([0, T], X) ∩ L^{p}([0, T], X_{1}). Set \frac{1}{q′} = 1 − \frac{1}{q}. Since \frac{1}{q′} < 1 − \frac{p}{r}, we can estimate

∥B(·)f∥_{p} ≤ (∫_{0}^{T} β(t)^{p} ∥f(t)∥_{\frac{1}{q′} ,q}^{p} dt)^{\frac{1}{p}}

≤ ∥β∥_{r} (∫_{0}^{T} ∥f(t)∥_{\frac{1}{q′} ,q}^{p(1− \frac{p}{r})^{−1}} dt)^{\frac{1}{p} (1− \frac{p}{r})}

≤ ∥β∥_{r} (∫_{0}^{T} ∥f(t)∥_{X}^{\frac{p}{q} (1− \frac{p}{r})^{−1}}∥f(t)∥_{X_{1}}^{\frac{p}{q′} (1− \frac{p}{r})^{−1}} dt)^{\frac{1}{p} (1− \frac{p}{r})}

≤ ∥β∥_{r} ∥f∥\frac{p}{q} ( \frac{1}{q} − \frac{p}{r})^{−1}^{\frac{1}{q}} ∥f∥L^{p}([0,T],X_{1})^{\frac{1}{q′}}

≤ ∥β∥_{r} ∥f∥∞^{\frac{p}{r}} ∥f∥_{p}^{\frac{1}{q} − \frac{p}{r}} ∥f∥L^{p}([0,T],X_{1})^{\frac{1}{q′}} .

Here we have used Hölder’s inequality in the second and fourth line, interpolation for the third estimate, and the Riesz-Thorin convexity theorem for the last one. Due to F ↪ L^{∞}([0, T], X) and Young’s inequality, we further obtain

∥B(·)f∥_{p} ≤ c_{1} ∥β∥_{r} (ε^{−1} ∥f∥_{p}^{\frac{1}{q} − \frac{p}{r}} ε ∥f∥_{F}^{1−( \frac{1}{q} − \frac{p}{r})})
≤ c_{1} ∥β∥_{r} (α ε^{− \frac{1}{α}} ∥f∥_{p} + (1 − α) ε^{\frac{1}{1−α}} ∥f∥_{F} )
≤ c_{2}ε^{− \frac{1}{α}} ∥f∥_{p} + c_{3} ε^{\frac{1}{1−α}} ∥f∥_{F}  (3.3)

for constants c_{k}, α := \frac{1}{q} − \frac{p}{r} ∈ (0, 1), and all ε > 0. Let G = −\frac{d}{dt} + A(·) with domain F_{0} in L^{p}([0, T], X), cf. Theorem 2.5. Then (3.3) and (2.11) yield

∥B(·)(λ − G)^{−1}f∥_{p} ≤ c_{2}ε^{− \frac{1}{α}} ∥(λ − G)^{−1}f∥_{p} + c_{3} ε^{\frac{1}{1−α}} ∥(λ − G)^{−1}f∥_{F}

≤ c_{4} (ε^{− \frac{1}{α}} λ^{−1} + ε^{\frac{1}{1−α}}) ∥f∥_{p} ≤ \frac{1}{2} ∥f∥_{p}

for f ∈ L^{p}([0, T], X), where we have chosen first a sufficiently small ε > 0 and then a sufficiently large λ > 0 to derive the last inequality. As a result, the operator λ−(G+B(·)) with domain F_{0} is invertible in L^{p}([0, T], X) for 1 < p < \frac{r}{q} and large λ.

(2) Now let 1 − \frac{1}{q} + \frac{1}{r} < \frac{1}{p} < 1. (Notice that \frac{q}{r} ≤ 1 − \frac{1}{q} + \frac{1}{r}.) As in the preceding section we set A(t) = A(0), B(t) = 0, and β(t) = 0 for t ∈ [−1, 0). Using (B) and (2.13) we estimate

∫_{s}^{s+t} ∥B(τ)U(τ, s)x∥^{p} dτ ≤ c_{1} ∫_{s}^{s+t} β(τ)^{p} (τ − s)^{− \frac{p}{q′}} ∥x∥^{p} dτ
≤ c_{1} ∥β∥_{r}^{p} (∫_{0}^{t} τ^{− \frac{p}{q′} (1− \frac{p}{r})^{−1}} dτ)^{1− \frac{p}{r}} ∥x∥^{p}
= c_{2} t^{1− \frac{p}{r} − \frac{p}{q′}} ∥x∥^{p} =: c(t)^{p} ∥x∥^{p}

for s, s+t ∈ [−1, T] and x ∈ X. Hence, c(t) < 1 for sufficiently small t > 0. Moreover, B(·) is G-bounded by (3.3). Therefore the non–autonomous Miyadera perturbation theorem [27, Thm. 3.4] shows the existence of an evolution family U_{B}(t, s), 0 ≤ s ≤ t ≤ T, on X such that the corresponding evolution semigroup on L^{p}([0, T], X) is generated by G+B(·) with domain F_{0}. In addition, [27, Thm. 3.4] gives

U_{B}(t, s)x = U(t, s)x + ∫_{s}^{t} U_{B}(t, τ)B(τ)U(τ, s)x dτ  (3.4)

for 0 ≤ s ≤ t ≤ T and x ∈ ⋃_{−1<r<s} U(s, r)X.

(3) The evolution family U_{B}(t, s) induces an evolution semigroup on all spaces L^{p}([0, T], X), 1 ≤ p < ∞. Denote the corresponding generator by G_{B}^{p}. Due to parts (1) and (2) we have G_{B}^{p_{0}} = −\frac{d}{dt} + A(·) + B(·) with D(G_{B}^{p_{0}}) = F_{0}^{p_{0}} if 1 − \frac{1}{q} + \frac{1}{r} < \frac{1}{p_{0}} < 1. (For a moment we express the dependence on p explicitely.) Notice that G_{B}^{p} is the part of G_{B}^{p_{0}} in L^{p}([0, T], X) for p ≥ p_{0}. Take p ∈ (1, \frac{r}{q}). Part (1) implies that G_{B}^{p} extends (−\frac{d}{dt} + A(·) + B(·), F_{0}^{p}) and that there exists λ ∈ ρ(G_{B}^{p}) ∩ ρ(−\frac{d}{dt} + A(·) + B(·)). Hence, G_{B}^{p} = −\frac{d}{dt} + A(·) + B(·) on F_{0}^{p} for 1 < p < \frac{r}{q}. Also, the graph norm of G_{B}^{p} is equivalent to ∥ · ∥_{F} by virtue of (3.3) and the inverse mapping theorem. As in the proof of Lemma 2.4 one sees that G_{B} has the inverse

(G_{B}^{−1}f) (t) = − ∫_{0}^{t} U_{B}(t, s)f(s) ds.

Moreover, u := U_{B}(·, s)x belongs to F_{[s+ε,T]}^{p}, −u′ + A(·)u + B(·)u = 0, and

∥U_{B}(t, s)x∥_{1− \frac{1}{p} ,p} ≤ c_{p} (t − s)^{\frac{1}{p} −1} ∥x∥

for x ∈ X, 0 ≤ s < t ≤ T, a constant c_{p}, and 1 < p < \frac{r}{q}, compare the proof of Theorem 2.5.

It remains to study the behaviour of U_{B}(t, s) on real interpolation spaces. Let g(t) = (A(0)−A(t))e^{(t−s)A(0)}x for 0 ≤ s < t ≤ T and x ∈ (X, X_{1})_{1− \frac{1}{p} ,p}. Then the function u(t) = e^{(t−s)A(0)}x − (G_{[s,T]}^{−1}g)(t) belongs to the kernel of L_{[s,T]} by (2.4). Hence, u(t) = U(t, s)x due to Theorem 2.5. From (3.3), (H1), and (2.4) we deduce

∥B(·)G_{[s,T]}^{−1}g∥_{p} ≤ c_{1} ∥g∥_{p} ≤ c_{2} ∥x∥_{1− \frac{1}{p} ,p}.

One further obtains

∥e^{tA(0)}x∥_{1− \frac{1}{q} ,q} ≤ c_{3} t^{\frac{1}{q} − \frac{1}{p}} ∥x∥_{1− \frac{1}{p} ,q} ≤ c_{4} t^{\frac{1}{q} − \frac{1}{p}} ∥x∥_{1− \frac{1}{p} ,p}

for 1 < p < q < ∞, x ∈ (X, X_{1})_{1− \frac{1}{p} ,p}, and 0 < t ≤ T by [22, Prop. 2.2.9] and (2.5). Thus,

∥B(·)e^{(·−s)A(0)}x∥_{p} ≤ c_{4} (∫_{s}^{T} β(t)^{p} (t − s)^{p( \frac{1}{q} − \frac{1}{p})} ∥x∥_{1− \frac{1}{p} ,p}^{p} dt)^{\frac{1}{p}}

≤ c_{4} ∥β∥_{r} ∥x∥_{1− \frac{1}{p} ,p} (∫_{0}^{T} t^{( \frac{p}{q} −1)(1− \frac{p}{r})^{−1}} dt)^{\frac{1}{p} − \frac{1}{r}}
= c_{5} ∥x∥_{1− \frac{1}{p} ,p}

for p < q < r. The case 1 ≤ q ≤ p can be handled similarly. Consequently, ∥B(·)U(·, s)x∥_{p} ≤ c_{6} ∥x∥_{1− \frac{1}{p} ,p}. A straightforward approximation argument then shows that (3.4) holds for every x ∈ (X, X_{1})_{1− \frac{1}{p} ,p}. We can rewrite this identity as

U_{B}(t, s)x = U(t, s)x − (G_{B,[s,T]}^{−1}B(·)U(·, s)x) (t).

As a result, U_{B}(·, s)x belongs to the kernel of (L + B(·))_{[s,T]}^{p} if 1 < p < \frac{r}{q} and x ∈ (X, X_{1})_{1− \frac{1}{p} ,p} and U_{B}(t, s) is uniformly bounded on (X, X_{1})_{1− \frac{1}{p} ,p}. Now one can complete the proof as in the case of Theorem 2.5.

Remark 3.2. Notice that Theorem 3.1 yields maximal regularity of type L^{p} not for all values of p ∈ (1, ∞) if B(·) is unbounded in time, i.e., r < ∞. The following simple example shows that our upper bound for p is almost optimal if q = 1. Take X = ℂ, A(t) = −1, B(t) = (1 − t)^{− \frac{1}{2}} for t ∈ [0, 1] so that (B) holds for q = 1 und each r ∈ (1, 2). Let u be the solution of (3.1) for x = 0 and f ≡ 1. Then, u′ ∈ L^{p}[0, T] if and only if 1 ≤ p < 2. It is known that maximal L^{p}–regularity does not depend on p in the autonomous case, see e.g. [14, 21], and in the non-autonomous case for more regular A(t), [20].

4. Applications to parabolic initial boundary value problems

To obtain an idea of the strength of our main results, consider the parabolic system

\begin{aligned}∂_{t} u(t, x) & = & −A(t, x)u(t, x) + B(t, x)u(t, x) + f(t, x),  0 ≤ t ≤ T, x ∈ ℝ^{n}, \\ u(0, x) & = & u_{0}(x),  x ∈ ℝ^{n}, \end{aligned} (4.1)

in X = L^{q}(ℝ^{n}, ℂ^{N}), 1 < q < ∞, where

A(t, x) = (−1)^{m} ∑_{|α|=2m} a_{α}(t, x) D^{α} and B(t, x) = ∑_{|β|<2m} b_{β}(t, x) D^{β}

for some m, n, N ∈ ℕ. Here D^{α} = ∂_{1}^{α_{1}} ⋯ ∂_{n}^{α_{n}} denotes a spatial derivative in multi-index notation. The coefficients of the principal part A(t, x) are assumed to be of class

a_{α} ∈ BUC([0, T] × ℝ^{n}, ℂ^{N×N})

and to be uniformly (K, θ)–elliptic in the sense of [4, 16]. This means that there are constants θ ∈ [0, \frac{π}{2}) and K ≥ 1 such that

∑_{|α|=2m} ∥a_{α}∥_{∞} ≤ K, σ(Ã(t, x, ξ)) ⊂ Σ_{θ} \ {0}, and |Ã(t, x, ξ)^{−1}| ≤ K

for t ∈ [0, T], x, ξ ∈ ℝ^{n} with |ξ| = 1, where

Ã(t, x, ξ) := ∑_{|α|=2m} a_{α}(t, x) ξ^{α}

denotes the principal symbol of A. We define the L^{q}–realization A(t) of A(t, x) by

(A(t)u)(x) := −A(t, x)u(x), x ∈ ℝ^{n}, t ∈ [0, T], D(A(t)) := W^{2m,q}(ℝ^{n}, ℂ^{N}).

Let θ < θ′ < \frac{π}{2}. Due to [16, Thm. 6.1], there is a constant w > 0 such that A(t) − w generates an analytic semigroup of negative type and w −A(t) admits bounded imaginary powers with ∥(w − A(t))^{is}∥ ≤ Ne^{θ′|s|} for s ∈ ℝ and a constant N ≥ 1, see also [4, §9]. By the Dore–Venni Theorem [15, Thm. 3.2], we then obtain maximal regularity of type L^{p} for the operator A(t), i.e., (H2) holds. Hypothesis (H1) is easily verified. Observe that no extra regularity of the coefficients a_{α}(t, x) besides BUC is needed here.

Thus we may apply Theorem 2.5 to conclude that for each fixed p, q ∈ (1, ∞) the problem (4.1) with B = 0 has maximal regularity of type L^{p} − L^{q} as well. More precisely, for each f ∈ L^{p}([0, T], L^{q}(ℝ^{m}, ℂ^{N})) and each u_{0} ∈ (X, X_{1})_{1− \frac{1}{p} ,p} ≅ B_{q,p}^{2m(1− \frac{1}{p})}(ℝ^{n}, ℂ^{N}), a Besov space (see Remark 4 in [28, 2.4.2]), there is a unique solution

u ∈ W^{1,p}([0, T], L^{q}(ℝ^{n}, ℂ^{N})) ∩ L^{p}([0, T], W^{2m,q}(ℝ^{n}, ℂ^{N}))

of (4.1) and estimate (2.11) is valid. Moreover, there is an evolution family U(t, s) on X = L^{q}(ℝ^{n}, ℂ^{N}) for (4.1) which satisfies the interpolation estimate (2.13) and for each u_{0} ∈ X and f ∈ L^{p}([0, T], X) there is unique solution

u ∈ C([0, T], X) ∩ W_{loc}^{1,p}((0, T], X) ∩ L_{loc}^{p}((0, T], W^{2m,q}(ℝ^{n}, ℂ^{N}))

of (4.1).

To include lower order terms, i.e. B ≠ 0, we may use Theorem 3.1. On L^{q}(ℝ^{n}, ℂ^{N}) we define

(B(t)u)(x) := B(t, x)u(x), x ∈ ℝ^{n}, t ∈ [0, T], D(B(t)) := W^{2m,q}(ℝ^{n}, ℂ^{N}),

assuming that, for given p, q ∈ (1, ∞), the coefficients belong to class

b_{β} ∈ L^{r_{k}}([0, T], L^{s_{k}}(ℝ^{n}, ℂ^{N}))

for |β| = k < 2m, where r_{k}, s_{k} ∈ (1, ∞] satisfy

\frac{p}{r_{k}} + \frac{n}{2ms_{k}} + \frac{k}{2m} < 1, r_{k} > p, s_{k} > q.

Set σ_{k} := s_{k}/q and \frac{1}{σ′_{k}} := 1 − \frac{1}{σ_{k}} and choose ρ_{k} > 1 such that

1 − \frac{n}{2mqσ_{k}} − \frac{k}{2m} > \frac{1}{ρ_{k}} > \frac{p}{r_{k}} .

By 2.3.2(4a), 2.3.3, and 2.8.1(12) in [28] we have the continuous embedding

B_{q,ρ_{k}}^{2m(1− \frac{1}{ρ_{k}})}(ℝ^{n}, ℂ^{N}) ↪ W^{k,qσ′_{k}}(ℝ^{n}, ℂ^{N})

which yields, for |β| = k and u ∈ (X, X_{1})_{1− \frac{1}{ρ_{k}} ,ρ_{k}} ,

∥b_{β}(t, ·) D^{β}u∥_{L^{q}} ≤ ∥b_{β}(t, ·)∥_{L^{qσ_{k}}} ∥u∥_{W^{k,qσ′_{k}}} ≤ c ∥b_{β}(t, ·)∥_{L^{s_{k}}} ∥u∥_{1− \frac{1}{ρ_{k}} ,ρ_{k}}.

As a result, the k–order part of B(t) satisfies (B) with q replaced by ρ_{k} and r by r_{k}. Since 1 < p < p_{0} := min\{\frac{r_{k}}{ρ_{k}} : k = 0, ⋯, 2m−1\} by assumption, a slight variant of Theorem 3.1 applies, and we obtain again maximal regularity of type L^{p} − L^{q} for 1 < p < p_{0} and the existence of an evolution family U(t, s) on X = L^{q}(ℝ^{n}, ℂ^{N}).

As another application of our main results, consider a bounded domain Ω ⊂ ℝ^{n} with smooth boundary ∂Ω, a differential operator

A(t, x) = − ∑_{k,l=1}^{n} a_{kl}(t, x) ∂_{k}∂_{l} + ∑_{k=1}^{n} a_{k}(t, x) ∂_{k} + a_{0}(t, x),

and a boundary operator

B(x) = ∑_{k=1}^{n} b_{k}(x) ∂_{k} + b_{0}(x).

Here we assume that

a_{kl}, a_{k}, a_{0} ∈ C([0, T] × \overline{Ω}, ℝ), a_{kl}(t, ·), a_{k}(t, ·), a_{0}(·) ∈ C^{ρ}(\overline{Ω}), b_{k}, b_{0} ∈ C^{1+ρ}(∂Ω, ℝ),

that A(t, x) is uniformly elliptic, i.e.,

∑_{k,l=1}^{n} a_{kl}(t, x) ξ_{k}ξ_{l} ≥ c |ξ|^{2}

for all x ∈ Ω, 0 ≤ t ≤ T, ξ ∈ ℝ^{n}, and a constant c > 0, and that

∑_{k=1}^{n} b_{k}(x) ν_{k}(x) ≥ c_{0} > 0, x ∈ ∂Ω,

where ν(x) denotes the outer normal at x ∈ ∂Ω. Then the L^{q}–realization A(t) of A with boundary conditions B = 0 is given by

(A(t)u)(x) := −A(t, x)u(x), x ∈ Ω, t ∈ [0, T],
D(A(t)) := W_{B}^{2m,q} = {u ∈ W^{2,q}(Ω) : Bu = 0}.

It is wellknown that A(t) generates an analytic semigroup on X = L^{q}(Ω), cf. [22, Thm. 3.1.3], which satisfies Gaussian bounds, see [17, Thm. VI.3.1]. Therefore hypothesis (H2) holds due to [21, Thm. 3.1]. Condition (H1) is easy to check, but notice that we have to restrict the boundary operator to the time independent case since otherwise D(A(t)) would depend on t. As a consequence of Theorem 2.5, the parabolic second order initial boundary value problem

∂_{t} u(t, x) = −A(t, x)u(t, x) + f(t, x),  0 ≤ t ≤ T, x ∈ Ω,
B(x)u(t, x) = 0,  0 ≤ t ≤ T, x ∈ ∂Ω,
u(0, x) = u_{0}(x),  x ∈ Ω,

has maximal regularity of type L^{p} − L^{q}, and it exists the corresponding evolution family U(t, s) on L^{q}(Ω). Observe that by means of the perturbation result Theorem 3.1 the regularity of the lower order terms a_{k}, a_{0} can be relaxed considerably, but we are not going into details here.

5. Proof of Lemma 2.2

Throughout this section we assume that condition (H1) holds. Recall that (H1) implies the uniform resolvent estimate (2.2) and that the graph norms of A(t) are uniformly equivalent. In this section we want to show uniform boundedness of the evolution families U_{n}(t, s) solving the Cauchy problem for the Yosida approximations A_{n}(t) = nA(t)(n − A(t))^{−1}. To that purpose, we use the integrals ∫_{s}^{t} A(τ)x dτ defined for x ∈ X_{1} and 0 ≤ s < t ≤ T. Set ω(t) := sup_{|s−r|≤t} ∥A(s) − A(r)∥_{L(X_{1},X)} .

Lemma 5.1. Assume that (H1) holds. Let 0 < t ≤ δ with ω(δ) ≤ \frac{1}{2M} and 0 ≤ s − t < s ≤ T. Then the operator \frac{1}{t} ∫_{s−t}^{s} A(τ) dτ defined on X_{1} generates an analytic semigroup on X satisfying (2.2) with constants ϕ and 2M. Moreover,

∥(λ − \frac{1}{t} ∫_{s−t}^{s} A(τ) dτ)^{−1} − (λ − \frac{1}{t} ∫_{r−t}^{r} A(τ) dτ)^{−1}∥ ≤ \frac{4M^{2}}{1 + |λ|} ω(|s − r|)

for λ ∈ Σ_{ϕ} and s − t, r − t, s, r ∈ [0, T].

Proof. Notice that (H1) and (2.2) yield, for x ∈ X_{1} and λ ∈ Σ_{ϕ},

λ − \frac{1}{t} ∫_{s−t}^{s} A(τ)x dτ = (Id − \frac{1}{t} ∫_{s−t}^{s}(A(τ) − A(s))(λ − A(s))^{−1} dτ) (λ − A(s))x,

∥\frac{1}{t} ∫_{s−t}^{s}(A(τ) − A(s))(λ − A(s))^{−1}x dτ∥ ≤ Mω(t) ∥x∥ ≤ \frac{1}{2} ∥x∥.

This implies the first assertion. The proof of the second one is straightforward.

In the sequel, let 0 ≤ s − t < s ≤ T and 0 < t ≤ δ with ω(δ) ≤ \frac{1}{2M}. We further introduce the Yosida approximations

B_{n}(s, t) = \frac{n}{t} ∫_{s−t}^{s} A(τ) dτ (n − \frac{1}{t} ∫_{s−t}^{s} A(τ) dτ)^{−1} = n^{2} (n − \frac{1}{t} ∫_{s−t}^{s} A(τ) dτ)^{−1} − nId,

n ∈ ℕ, of the above integrals. Lemma 5.1 implies that

∥e^{τB_{n}(s,t)}∥ ≤ K  and ∥B_{n}(s, t) − B_{n}(r, t)∥ ≤ 4nM^{2} ω(|s − r|)  (5.1)

for n ∈ ℕ and a constant K. We also need the following properties.

Lemma 5.2. Let 0 < t ≤ δ, n ∈ ℕ, s, r, s − t, r − t ∈ [0, T], and τ ≥ 0. Then 1. ∥ exp(τtB_{n}(s, t)) − exp(τtB_{n}(r, t))∥ ≤ 4nM^{2}K^{2} τt ω(|s − r|) and

2. [\frac{d}{dt} exp(tB_{n}(s, t))]_{|t=0} = A_{n}(s) in L(X) and uniformly in s.

Proof. The first assertion follows from (5.1) and

e^{τtB_{n}(s,t)} − e^{τtB_{n}(r,t)} = ∫_{0}^{τt} \frac{∂}{∂σ} [e^{σB_{n}(s,t)} e^{(τt−σ)B_{n}(r,t)}] dσ
= ∫_{0}^{τt} e^{σB_{n}(s,t)} [B_{n}(s, t) − B_{n}(r, t)] e^{(τt−σ)B_{n}(r,t)} dσ.

To show the second assertion, we observe that

∥B_{n}(s, t)∥ ≤ n2M sup_{0≤t≤T} ∥A(t)∥_{L(X_{1},X)} =: cn

Together with (5.1), this yields

∥e^{αB_{n}(s,t)} − Id∥ ≤ ∫_{0}^{1} ∥α B_{n}(s, t) e^{ταB_{n}(s,t)}∥ dτ ≤ ncKα  (5.2)

for α > 0. Using the identity

\frac{1}{t} (e^{tB_{n}(s,t)} − Id) − A_{n}(s) = ∫_{0}^{1}(B_{n}(s, t) e^{τtB_{n}(s,t)} − A_{n}(s)) dτ,

we infer from (5.1) and (5.2) that

∥\frac{1}{t} (e^{tB_{n}(s,t)} − Id) − A_{n}(s)∥
≤ ∫_{0}^{1} ∥e^{τtB_{n}(s,t)}∥ ∥B_{n}(s, t) − A_{n}(s)∥ dτ + ∫_{0}^{1} ∥e^{τtB_{n}(s,t)} − Id∥ ∥A_{n}(s)∥ dτ
≤ K ∥B_{n}(s, t) − A_{n}(s)∥ + n^{2}c^{2}K t.

So it remains to show that ∥B_{n}(s, t) − A_{n}(s)∥ → 0 as t → 0 uniformly in s. In fact,

B_{n}(s, t) − A_{n}(s) =\frac{n}{t} ∫_{s−t}^{s}(A(τ) − A(s)) dτ (n − \frac{1}{t} ∫_{s−t}^{s} A(τ) dτ)^{−1}

+ nA(s) (n − \frac{1}{t} ∫_{s−t}^{s} A(τ) dτ)^{−1} \frac{1}{t} ∫_{s−t}^{s}(A(τ) − A(s)) dτ (n − A(s))^{−1},

and Lemma 5.1 gives

∥B_{n}(s, t) − A_{n}(s)∥ ≤ 2nM ω(t) + ncM ω(t).  □

For intervals I = [a, a+δ] ⊆ [0, T] we now establish a uniform bound for the semigroups T_{n,I}(·) on L^{p}(I, X) generated by G_{n} = −\frac{d}{dt} +A_{n}(·) with domain W_{0}^{1,p}(I, X), see Section 2. Lemma 2.2 then follows from this estimate. To that purpose, we define bounded operators V_{n,I}(t), n ∈ ℕ, on L^{p}(I, X) by setting

(V_{n,I}(t)f)(s) = \begin{cases}e^{tB_{n}(s,t)} f(s − t), & a ≤ s − t ≤ s ≤ a + δ, \\ 0, & s − t < a ≤ s ≤ s + δ, \end{cases}

and V_{n,I}(0) = Id. Notice that V_{n,I}(t) = 0 for t > δ. We will use the operators V_{n,I}(t) in order to approximate T_{n,I}(t) in the sense of Chernoff’s product formula. One of the conditions to apply this formula is a straightforward consequence of Lemma 5.2.

Lemma 5.3. We have \frac{1}{t}(V_{n,I}(t)f − f) → G_{n,I}f as t → 0 for n ∈ ℕ and f ∈ W_{0}^{1,p}(I, X).

Further, we have to estimate powers of V_{n,I}(t). So let us compute

(V_{n,I}(t)^{m}f) (s) = e^{tB_{n}(s,t)} (V_{n,I}(t)^{m−1}f) (s − t) = ⋯ =
= e^{tB_{n}(s,t)} ⋯ e^{tB_{n}(s−(m−1)t,t)} f(s − mt)
=: Π_{n,I}(s, t, m) f(s − mt)

for m ∈ ℕ and s − mt ≥ a. If s − mt < a, we set Π_{n,I}(s, t, m) := 0. To handle these products we renorm the space X by

∥x∥_{s,t,n} := sup_{τ≥0} ∥e^{−ετ} e^{τtB_{n}(s,t)}x∥,

where ε > 0 is fixed. From (5.1) and Lemma 5.2(1) one easily derives the following properties.

Lemma 5.4. Let 0 < t ≤ δ as in Lemma 5.1, s, s − t ∈ [0, T], and n ∈ ℕ. Then ∥ · ∥_{s,t,n} is a norm on X satisfying

1. ∥x∥ ≤ ∥x∥_{s,t,n} ≤ K∥x∥,

2. ∥e^{−ε} e^{tB_{n}(s,t)}x∥_{s,t,n} ≤ ∥x∥_{s,t,n},

3. | ∥x∥_{s,t,n} − ∥x∥_{s−t,t,n} | ≤ nC_{ε} tω(t)∥x∥_{s−t,t,n} if s − 2t ∈ [0, T], where C_{ε} is a constant only depending on ε > 0.

Lemma 5.5. Under the above assumptions we have 1. ∥e^{−εm} Π_{n,I}(s, t, m)∥ ≤ K exp(nC_{ε} ω(δ) mt) and

2. lim sup_{m→∞} sup_{s∈I} ∥e^{−εm} Π_{n,I}(s, \frac{t}{m}, m)∥ ≤ K

Proof. (1) We only have to consider the case s − mt ≥ a. From Lemma 5.4 we obtain

∥e^{−εm} Π_{n,I}(s, t, m)x∥ ≤ ∥e^{−ε} e^{tB_{n}(s−t,t)} ⋯ e^{−ε} e^{tB_{n}(s−(m−1)t,t)}x∥_{s,t,n}

≤ (1 + nC_{ε}ω(t)t) ∥e^{−ε} e^{tB_{n}(s−t,t)} ⋯ e^{−ε} e^{tB_{n}(s−(m−1)t,t)}x∥_{s−t,t,n}

≤ e^{nC_{ε}ω(t)t} ∥e^{−ε} e^{tB_{n}(s−t,t)} ⋯ e^{−ε} e^{tB_{n}(s−(m−1)t,t)}x∥_{s−t,t,n}
≤ ⋯ ≤
≤ e^{nC_{ε}ω(t)t(m−1)} ∥e^{−ε} e^{tB_{n}(s−(m−1)t,t)}x∥_{s−(m−1)t,t,n}
≤ K e^{nC_{ε}ω(t)tm} ∥x∥ ≤ K e^{nC_{ε}ω(δ)tm} ∥x∥.

(2) After replacing t by \frac{t}{m} , step (1) of the proof yields

∥e^{−εm} Π_{n,I}(s, \frac{t}{m}, m)x∥ ≤ K exp(nC_{ε} ω(\frac{t}{m})t) → K as m → ∞.  □

Proof of Lemma 2.2. Given I = [a, a + δ] ⊆ [0, T] with ω(δ) ≤ \frac{1}{2M}, let V_{n,I}(t), G_{n,I}, and T_{n,I}(t) be defined as above on L^{p}(I, X). Lemma 5.5 yields

∥(e^{−ε}V_{n,I}(t))^{m}∥ ≤ K e^{ωtm}

for t ≥ 0 and m ∈ ℕ and a constant ω ≥ 0 independent of m and t. Moreover, by Lemma 5.3 we have

\frac{d}{dt} (e^{−ε}V_{n,I}(t)f)_{|t=0} = e^{−ε} G_{n,I}f

for f ∈ W_{0}^{1,p}(I, X). Since G_{n,I} generates T_{n,I}(·), Chernoff’s product formula, see e.g. [24, Cor. 3.5.4], implies

T_{n,I}(e^{−ε}t)f = lim_{m→∞} e^{−εm}V_{n,I}(\frac{t}{m})^{m}f

for f ∈ L^{p}(I, X) and t ≥ 0. Consequently,

∥T_{n,I}(e^{−ε}t)∥ ≤ lim sup_{m→∞} sup_{s∈I} ∥e^{−εm} Π_{n}(s, \frac{t}{m}, m)∥ ≤ K.

Letting ε → 0, we see that the corresponding evolution family U_{n}(t, s) satisfies ∥U_{n}(s, s − t)∥ ≤ K for 0 ≤ s − t ≤ s ≤ T and 0 ≤ t ≤ δ. So we may take C = K^{l} for a natural number l ≥ T/δ.

References

[1] P. Acquistapace, Evolution operators and strong solutions of abstract linear parabolic equations, Differential Integral Equations 1 (1988), 433–457.

[2] P. Acquistapace, B. Terreni, A unified approach to abstract linear nonautonomous parabolic equations, Rend. Semin. Mat. Univ. Padova 78 (1987), 47–107.

[3] H. Amann, “Linear and Quasilinear Parabolic Problems. Volume 1: Abstract Linear Theory,” Birkhäuser, Basel, 1995.

[4] H. Amann, M. Hieber, G. Simonett, Bounded H_{∞}–calculus for elliptic operators, Differential Integral Equations 7 (1994), 613–653.

[5] A. Benedek, A.P. Calderón, R. Panzone, Convolution operators on Banach space valued functions, Proc. Nat. Acad. Sci. USA 48 (1962), 356–365.

[6] A. Buttu, On the evolution operator for a class of non–autonomous abstract parabolic equations, J. Math. Anal. Appl. 170 (1992), 115–137.

[7] A. Buttu, A construction of the evolution operator for a class of abstract parabolic equations, Dyn. Syst. Appl. 3 (1994), 221–234.

[8] C. Chicone, Y. Latushkin, “Evolution Semigroups in Dynamical Systems and Differential Equations,” book manuscript.

[9] Ph. Clément, S. Li, Abstract parabolic quasilinear equations and applications to a groundwater flow problem, Adv. Math. Sci. Appl. 3 (1994), 17-32.

[10] G. Da Prato, P. Grisvard, Sommes d’opérateurs linéaires et équations différentielles opérationelles, J. Math. Pures Appl. 54 (1975), 305–387.

[11] G. Da Prato, P. Grisvard, Equations d’évolution abstraites non linéaires de type parabolique, Ann. Mat. Pura Appl. 120 (1979), 329–396.

[12] G. Da Prato, P. Grisvard, Maximal regularity for evolution equations by interpolation and extrapolation, J. Funct. Anal. 58 (1984), 107–124.

[13] L. De Simon, Un’applicazione della teoria degli integrali singolari allo studio delle equazioni differenziali lineari astratte del primo ordine, Rend. Sem. Mat. Univ. Padova 34 (1964), 205–232.

[14] G. Dore, L^{p} regularity for abstract differential equations, in: H. Komatsu (ed.), “Functional Analysis and Related Topics (Proceedings Kyoto 1991)”, Lecture Notes in Mathematics 1540, Springer, Berlin, 1993, 25–38.

[15] G. Dore, A. Venni, On the closedness of the sum of two operators, Math. Z. 196 (1987), 189–201.

[16] X.T. Duong, G. Simonett, H_{∞}–calculus for elliptic operators with nonsmooth coefficients, Differential Integral Equations 10 (1997), 201–217.

[17] M.G. Garroni, J.L. Menaldi, “Green Functions for Second Order Parabolic Integro–Differential Problems,” Longman, Harlow, 1992.

[18] M. Giga, Y. Giga, H. Sohr, L^{p} estimate for abstract linear parabolic equations, Proc. Japan Acad. 67 Ser. A (1991), 197–202.

[19] M. Hieber, S. Monniaux, Heat–kernels and maximal L^{p}–L^{q}–estimates: the non–autonomous case, preprint.

[20] M. Hieber, S. Monniaux, Pseudo-differential operators and maximal regularity results for nonautonomous parabolic equations, preprint.

[21] M. Hieber, J. Prüss, Heat kernels and maximal L^{p}–L^{q} estimates for parabolic evolution equations, Commun. Partial Differential Equations 22 (1997), 1647–1669.

[22] A. Lunardi, “Analytic Semigroups and Optimal Regularity in Parabolic Problems,” Birkhäuser, Basel, 1995.

[23] S. Monniaux, J. Prüss, A theorem of Dore–Venni type for non–commuting operators, Trans. Amer. Math. Soc. 349 (1997), 4787–4814.

[24] A. Pazy, “Semigroups of Linear Operators and Applications to Partial Differential Equations,” Springer, Berlin, 1983.

[25] J. Prüss, “Evolutionary Integral Equations and Applications,” Birkhäuser, Basel, 1993.

[26] F. Räbiger, A. Rhandi, R. Schnaubelt, Perturbation and an abstract characterization of evolution semigroups, J. Math. Anal. Appl. 198 (1996), 516–533.

[27] F. Räbiger, A. Rhandi, R. Schnaubelt, J. Voigt, Non-autonomous Miyadera perturbations, to appear in Differential Integral Equations.

[28] H. Triebel, “Interpolation Theory, Function Spaces, and Differential Operators,” North Holland, Amsterdam, 1978.

J.P., FB Mathematik und Informatik, Martin–Luther–Universität, Theodor–Lieser– Str. 5, 60120 Halle, Germany.

  E-mail address: anokd@volterra.mathematik.uni-halle.de

R.S., Mathematisches Institut, Universität Tübingen, Auf der Morgenstelle 10, 72076 Tübingen, Germany.

  E-mail address: rosc@michelangelo.mathematik.uni-tuebingen.de