Electronic Journal of Differential Equations, Vol. 2020 (2020), No. 124, pp. 1–24. ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu MAXIMAL REGULARITY FOR NON-AUTONOMOUS CAUCHY PROBLEMS IN WEIGHTED SPACES ACHACHE MAHDI, TEBBANI HOSSNI Abstract. We consider the regularity for the non-autonomous Cauchy prob- lem u′(t) +A(t)u(t) = f(t) (t ∈ [0, τ ]), u(0) = u0. The time dependent operatorA(t) is associated with (time dependent) sesquilin- ear forms on a Hilbert space H. We prove the maximal regularity result in temporally weighted L2-spaces and other regularity properties for the solution of the problem under minimal regularity assumptions on the forms and the initial value u0. Our results are motivated by boundary value problems. 1. Introduction The aim of this article is to study autonomous and non-autonomous evolution equation governed by time dependent sesquilinear forms. Let (H, (·, ·), ‖ · ‖) be a Hilbert space over R or C. We consider another Hilbert space V which is densely and continuously embedded in H. We denote by V ′ the (anti-) dual space of V, so that V ↪→d H ↪→d V ′. i.e. V is a dense subspace of H such that for some constant CH > 0, ‖u‖ ≤ CH‖u‖V (u ∈ V). We denote by 〈, 〉 the duality V ′ − V and note that 〈ψ, v〉 = (ψ, v) if ψ, v ∈ H. We consider a family of sesquilinear forms a : [0, τ ]× V × V → C such that (H1) D(a(t)) = V (constant form domain), (H2) |a(t, u, v)| ≤M‖u‖V‖v‖V (uniform boundedness), (H3) Re a(t, u, u) + ν‖u‖2 ≥ δ‖u‖2V for all u ∈ V, for some δ > 0 and some ν ∈ R (uniform quasi-coercivity). We denote by A(t),A(t) the usual associated operators with a(t) (as operators on H and V ′). 2010 Mathematics Subject Classification. 35A23. Key words and phrases. Maximal regularity; non-autonomous evolution equation; weighted space. c©2020 Texas State University. Submitted October 9, 2019. Published December 20, 2020. 1 2 A. MAHDI, T. HOSSNI EJDE-2020/124 In 1961 J. L. Lions proved that the non-autonomous Cauchy problem u̇(t) +A(t)u(t) = f(t) u(0) = u0 (1.1) has L2-maximal regularity in V ′. Theorem 1.1 (Lions’ theorem). Given f ∈ L2(0, τ ;V ′) and u0 ∈ H, there is a unique solution u ∈MR(V,V ′) := H1(0, τ ;V ′) ∩ L2(0, τ ;V) of problem (1.1). Note that MR(V,V ′) ↪→ C([0, τ ];H) so that the initial condition makes sense. In Theorem 1.1 only measurability of t→ a(t, ·, ·) with respect to the time variable is required to have a solution u ∈MR(V,V ′). However, considering boundary valued problems one is interested in strong solution, i.e. solution u ∈ H1(0, τ ;H) and not only in H1(0, τ ;V ′) (note that H ↪→ V ′ by the natural embedding). In the recent decades, the maximal regularity approach has become very useful in application to parabolic partial differential equations. The question of maximal regularity in H (autonomous or non-autonomous cases) is so important for several reasons. First of all, if Robin boundary conditions are considered, only the operator A(t) realizes these boundary conditions. The main reason for studying this problem is its importance for non-linear problems. They are mainly solved by applying Banach or Schauder fixed point theorems. Problem 1.2. Let f ∈ L2(0, τ ;H). Under which conditions on the forms a(·) the solution u ∈MR(V,V ′) of (1.1) satisfies u ∈ H1(0, τ ;H). Lions asked this question on maximal regularity for several conditions on the form and on the initial value. He also gave partial positive answers in [17, XVIII Chapter 3, p. 513]. More recently, this problem has been studied with a lot of progress. See the recent papers [3] or [4] for more details and references. The main focus of this work is the presence of the temporal weights. The choice of the weighted spaces has a big advantages. One of them is to reduce the necessary regularity for initial conditions of evolution equations. Time-weights can be used also to exploit parabolic regularization which is typical for quasilinear parabolic problems. This paper focuses on proving the maximal regularity in the non-autonomous case, i.e. we prove the existence and the uniqueness of solution to Problem (1.1). We shall allow considerably less restrictive assumptions on f and the initial data u0. Here, f belongs to the weighted Hilbert space L2(0, τ, tβdt;H), with β ∈ [0, 1[ and the initial data u0 takes its values in a certain interpolation space (H, D(A(0))) 1−β 2 ,2 between H and D(A(0)). The maximal regularity for the autonomous case in weighted spaces was the subject of treatment of many authors, see for instance [5]. In the non-autonomous case (Section 5) we prove that if f ∈ L2(0, τ, tβdt;H) and u0 ∈ (H, D(A(0))) 1−β 2 ,2 for arbitrary β ≥ 0 with the assumption that the operator A(·) belongs to the space W 1/2,2(0, τ ;L(V,V ′))∩Cε([0, τ ],L(V,V ′)) for some ε > 0, then problem (5.2) has a unique solution u such that u̇, A(·)u ∈ L2(0, τ, tβdt;H). Throughout this paper we assume that the Kato square root property (3.3) is satisfied. This property plays an important role in the questions of (non-autonomous) maximal regularity and optimal control. To prove our results we appeal to classical tools from harmonic analysis such as square function estimate or functional calculus and from functional analysis such as interpolation theory or operator theory. EJDE-2020/124 MAXIMAL REGULARITY FOR NON-AUTONOMOUS PROBLEMS 3 This work is structured as follows. In Section 2 we present basic definitions and properties used throughout this paper, in particular those of weighted spaces. In Section 3, we prove some preparatory results. Section 4 uses this result to show the maximal regularity for the autonomous equations, while in Section 5 we prove our result on maximal regularity to the considered non-autonomous Cauchy problems in the weighted space L2(0, τ, tβdt;H) and other regularity properties for the solution. We illustrate our abstract results by two applications in the final section. One of them concerns the heat equation with Robin boundary conditions on a bounded Lipschitz domain Ω. Notation. We denote by L(E,F ) (or L(E)) the space of bounded linear operators from E to F (from E to E). The spaces Lp(a, b;E) and W 1,p(a, b;E) denote respectively the Lebesgue and Sobolev spaces of function on (a, b) with values in E. Cα(a, b;E) denote the space of Hölder continuous functions of order α. Recall that the norms of H and V are denoted by ‖ · ‖ and ‖ · ‖V . The scalar product of H is (·, ·). We denote by C, C ′ or c. all inessential positive constants. Their values may change from line to line. In some cases we will use the notation a . b to signify that there exists an inessential positive constant C such that a ≤ Cb. 2. Properties of weighted spaces In this section we briefly recall the definitions and we give the basic properties of vector-valued function spaces with temporal weights. Let (X, ‖ · ‖X) be a Banach space over R or C. For −1 < β < 1 we set L2 β(0, τ ;X) = L2(0, τ, tβdt;X), endowed with the norm ‖u‖2L2 β(0,τ,X) := ∫ τ 0 ‖u(t)‖2Xtβ dt. It known that L2 β(0, τ ;X) ↪→ L1 loc(0, τ ;X). Indeed, for u ∈ L2 β(0, τ ;X) we find by Hölder’s inequality∫ τ 0 ‖u(t)‖X dt ≤ (∫ τ 0 t−β dt )1/2 ‖u‖L2 β(0,τ ;X). It clearly holds that L2(0, τ ;X) ↪→ L2 β(0, τ ;X) for β > 0 and L2 β(0, τ ;X) ↪→ L2(0, τ ;X) for β < 0. We define the corresponding weighted Sobolev spaces W 1,2 β (0, τ ;X) := {u ∈W 1,1(0, τ ;X) s.t. u, u̇ ∈ L2 β(0, τ ;X)}, W 1,2 β,0(0, τ ;X) := {u ∈W 1,2 β (0, τ ;X), s.t. u(0) = 0}, which are Banach spaces for the norms, respectively, ‖u‖2 W 1,2 β (0,τ ;X) := ‖u‖2L2 β(0,τ ;X) + ‖u̇‖2L2 β(0,τ ;X), ‖u‖2 W 1,2 β,0(0,τ ;X) := ‖u̇‖2L2 β(0,τ ;X). We set also L∞β (0, τ ;X) := {u ∈ L1(0, τ ;X), s.t. s→ sβ/2u(s) ∈ L∞(0, τ ;X)}, 4 A. MAHDI, T. HOSSNI EJDE-2020/124 endowed with the norm ‖u‖L∞β (0,τ ;X) := ‖s 7→ sβ/2u(s)‖L∞(0,τ ;X). For s ∈ (0, 1) we define the fractional weighted Sobolev space W s,2 β (0, τ ;X) by W s,2 β (0, τ ;X) = (L2 β(0, τ ;X);W 1,2 β (0, τ ;X))s,2, endowed with the norm ‖u‖2 W s,2 β (0,τ ;X) := ‖u‖2L2 β(0,τ ;X) + ∫ τ 0 ∫ t 0 ‖u(t)− u(s)‖2X |t− s|1+2s sβ ds dt. Here, (·; ·)s,2 is the real interpolation space. For more details we refer the reader to [20, (2.6)]. Lemma 2.1 (Weighted Hardy inequality). For every f ∈ L2 β(0, τ,X), we have∫ τ 0 (1 t ∫ t 0 ‖f(s)‖X ds )2 tβ dt . ‖f‖L2 β(0,τ ;X). This lemma was proved in [23, Lemma 6]. Proposition 2.2. We have the following properties (1) (a) For p > 2 and β > 2 p − 1, Lp(0, τ ;X) ↪→ L2 β(0, τ,X), (b) For p < 2 and β < 2 p − 1, L2 β(0, τ,X) ↪→ Lp(0, τ ;X). (2) For all u ∈ L2 β(0, τ,X), we have t→ v(t) = 1 t ∫ t 0 u(s) ds ∈ L2 β(0, τ,X). (3) We define the operator Φ : L2 β(0, τ ;X) → L2(0, τ ;X), such that (Φf)(t) = tβ/2f(t) for f ∈ L2 β(0, τ ;X) and t ∈ [0, τ ]. Then Φ is an isometric iso- morphism. We note also that Φ ∈ L(L2(0, τ ;X), L2 −β(0, τ ;X)) and Φ ∈ L(W 1,2 β,0(0, τ ;X),W 1,2 0 (0, τ ;X)). (4) W 1,2 β,0(0, τ ;X) ↪→ L2 β−2(0, τ ;X) ∩ L∞β−1(0, τ ;X). (5) L2 −β(0, τ ;V ′) is the dual space of L2 β(0, τ ;H) by the duality defined for the space L2(0, τ ;H). (6) If u ∈W 1,2 β (0, τ ;X), we obtain that u has a continuous extension on X and W 1,2 β (0, τ ;X) ↪→ C([0, τ ];X). (7) C∞c ((0, τ);X) and C∞([0, τ ];X) are dense in L2 β(0, τ ;X) and W s,2 β (0, τ ;X) respectively, for all s ∈ [0, 1]. Proof. (1a) Let p > 2 and β > 2 p − 1, we set p′ = p 2 > 1, 1 p′ + 1 q = 1. This implies that q = p p−2 and by using Hölder’s inequality we obtain ‖u‖2L2 β(0,τ ;X) = ∫ τ 0 ‖u(t)‖2Xtβ dt ≤ (∫ τ 0 ‖u(t)‖pX dt )2/p(∫ τ 0 tβq dt )1/q = ( 1 βq + 1 τβq+1 )1/q ‖u‖2Lp(0,τ ;X). (1b) Similarly, for p < 2 and β < 2 p − 1 we have ‖u‖pLp(0,τ ;X) = ∫ τ 0 ‖u(t)‖pXt − βp2 t βp 2 dt ≤ (∫ τ 0 ‖u(t)‖2Xtβ dt )p/2(∫ τ 0 t βp p−2 dt ) 2−p 2 EJDE-2020/124 MAXIMAL REGULARITY FOR NON-AUTONOMOUS PROBLEMS 5 = C‖u‖p L2 β(0,τ ;X) . (2) Lemma 2.1 shows that ‖v‖2L2 β(0,τ ;X) = ∫ τ 0 ‖1 t ∫ t 0 u(t) ds‖2Xtβ dt . ‖u‖2L2 β(0,τ ;X). We obtain the result since u ∈ L2 β(0, τ ;X). (3) Note that ‖Φf‖L2(0,τ ;X) = ‖f‖L2 β(0,τ ;X) and Φ−1 : L2(0, τ ;X)→ L2 β(0, τ ;X) where (Φ−1g)(t) = t−β/2g(t) for all g ∈ L2(0, τ ;X). (4) Let u ∈W 1,2 β,0(0, τ ;X). We write u(t) = ∫ t 0 u̇(l) dl. Then ‖u(t)‖2Xtβ−2 = ‖ ∫ t 0 u̇(l) dl‖2Xtβ−2. This implies ‖u‖2L2 β−2(0,τ ;X) = ∫ τ 0 ‖u(t)‖2Xtβ−2 dt = ∫ τ 0 1 t2 ‖ ∫ t 0 u̇(s) ds‖2Xtβ dt ≤ ∫ τ 0 (1 t ∫ t 0 ‖u̇(s)‖X ds )2 tβ dt . ‖u̇‖L2 β(0,τ ;X) ≤ ‖u‖W 1,2 β (0,τ ;X), where we used Lemma 2.1. For t ∈ [0, τ ], by Hölder’s inequality we have ‖u(t)‖Xt β−1 2 ≤ ∫ t 0 ‖u̇(s)‖X ds t β−1 2 ≤ ‖u‖W 1,2 β,0(0,τ ;X). It follows that W 1,2 β,0(0, τ ;X) ↪→ L2 β−2(0, τ ;X) ∩ L∞β−1(0, τ ;X). (5) For this proof we use the simple functions in L2 −β(0, τ ;V ′) and the Cauchy- Schwartz inequality (the proof is analogous to the non-weighted case, for more details see [11, p.98]. (6) For u ∈W 1,2 β (0, τ ;X) and (t, s) ∈ [0, τ ]2, we obtain ‖u(t)− u(s)‖X = ‖ ∫ t s u̇(l) dl‖X ≤ (∫ t s l−β dl )1/2 ‖u̇‖L2 β(0,τ ;X) = 1√ 1− β ( t−β+1 − s−β+1 )1/2‖u̇‖L2 β(0,τ ;X). Letting s→ t we obtain u(s)→ u(t) in X. Therefore u has a continuous extension on X. Thus we can always identify a function in W 1,2 β (0, τ ;X) by its continuous representative. (7) First we note that C∞c ((0, τ);X) is dense L2(0, τ ;X). Then for all f ∈ L2 β(0, τ ;X) and for any given ε > 0 there exists a function ψ ∈ C∞c ((0, τ);X) such that ‖(Φf)− ψ‖2L2(0,τ ;X) ≤ ε. 6 A. MAHDI, T. HOSSNI EJDE-2020/124 It follows that ‖f − (Φ−1ψ)‖2L2 β(0,τ ;X) ≤ ‖Φ‖L(L2 β(0,τ ;X);L2(0,τ ;X))‖(Φf)− ψ‖2L2(0,τ ;X) ≤ ε. Thus C∞c ((0, τ);X) is dense in L2 β(0, τ ;X). As in [24, Theorem 2.9.1] for the scalar-valued case, one sees that the space of all function f in C∞([0, τ ];X) such that f(0) = 0 is dense in W 1,2 0 (0, τ ;X). Then for all g ∈ W 1,2 β,0(0, τ ;X) and ε > 0 there exists φ ∈ C∞([0, τ ];X) with φ(0) = 0 such that ‖φ− Φg‖2W 1,2(0,τ ;X) ≤ ε. Then ‖Φ−1φ − g‖2 W 1,2 β (0,τ ;X) ≤ ε. This shows that the space of all function f in C∞([0, τ ];X) such that f(0) = 0, is dense in W 1,2 β,0(0, τ ;X). Let f ∈ W 1,2 β (0, τ ;X) and φ ∈ C∞([0, τ ];X) such that φ(0) = f(0). Then f − φ ∈ W 1,2 β,0(0, τ ;X) and there is ξ ∈ C∞([0, τ ];X) with ξ(0) = 0, such that ‖f−ξ−φ‖2 W 1,2 β (0,τ ;X) ≤ ε. Since ξ + φ ∈ C∞([0, τ ];X), then C∞([0, τ ];X) is dense in W 1,2 β (0, τ ;X). Since C∞([0, τ ];X) is dense in W 1,2 β (0, τ ;X) and W s,2 β (0, τ ;X) = (L2 β(0, τ ;X);W 1,2 β (0, τ ;X))s,2, we obtain that C∞([0, τ ];X) is also dense in W s,2 β (0, τ ;X) by [24, p.39]. � 3. Preliminaries In this section we prove several estimates which will play an important role in the proof of our results. From now we assume without loss of generality that the forms are coercive, that is (H3) holds with ν = 0. The reason is that by replacing A(t) by A(t) + ν, the solution v of (1.1) is v(t) = e−νtu(t) and it is clear that u ∈W 1,2 β (0, τ ;H) ∩ L2 β(0, τ ;V) if and only if v ∈W 1,2 β (0, τ ;H) ∩ L2 β(0, τ ;V). Proposition 3.1. The solution of problem (1.1) is unique. Proof. We suppose that there are two solutions u1, u2 to Problem (1.1). Obviously, v = u1 − u2 satisfies v̇(t) +A(t)v(t) = 0 v(0) = 0. (3.1) Then for all t ∈ [0, τ ] we have 2 Re ∫ t 0 (v̇(s), v(s))sβ ds+ 2 Re ∫ t 0 (A(s)v(s), v(s))sβ ds = 0. Integration by parts gives tβ‖v(t)‖2 − β ∫ t 0 ‖v(s)‖2sβ−1 ds+ 2δ ∫ t 0 ‖v(s)‖2Vsβ ds ≤ 0. It is clear that for the case β ≤ 0 we obtain v(t) = 0 for all t ∈ [0, τ ]. Therefore u1 = u2 and then the solution of Problem (1.1) is unique. For the case β ≥ 0 we have tβ‖v(t)‖2 + ∫ t 0 ‖v(s)‖2(2δC2 Hs β − βsβ−1) ds ≤ 0. So for the case t ≤ 2δC2 H β we have v(t) = 0 for all t ∈ [0, δC2 H β ]. Now we proceed inductively to obtain v = 0 on [0, τ ]. � EJDE-2020/124 MAXIMAL REGULARITY FOR NON-AUTONOMOUS PROBLEMS 7 We denote by Sθ the open sector Sθ = {z ∈ C∗ : |arg(z)| < θ} with vertex 0. It is known that −A(t) is sectorial operator and generates a bounded holomorphic semigroup on H. The same is true for −A(t) on V ′. From [14] (Proposition 2.1), we have the following lemma which point out that the constants involved in the estimates are uniform with respect to t. Lemma 3.2. For any t ∈ [0, τ ], the operators −A(t) and −A(t) generate strongly continuous analytic semigroups of angle γ = π 2 − arctan(Mδ ) on H and V ′, respec- tively. In addition, there exist real constants C > 0, Cθ > 0 independent of t, such that (1) ‖e−zA(t)‖L(H) ≤ 1 and ‖e−zA(t)‖L(V′) ≤ C for all z ∈ Sγ . (2) ‖A(t)e−sA(t)‖L(H) ≤ C s and ‖A(t)e−sA(t)‖L(V′) ≤ C s for all s ∈ (0,∞). (3) ‖e−sA(t)‖L(H,V) ≤ C√ s for all s ∈ (0,∞). (4) ‖(z−A(t))−1‖L(H,V) ≤ Cθ√ |z| and ‖(z−A(t))−1‖L(V′,H) ≤ Cθ√ |z| for all z /∈ Sθ with fixed θ > γ. The following lemma is proved in [19, Corollary 4.3.12] Lemma 3.3. Let H1,H2 be two Hilbert spaces, with H2 ⊂ H1, and H2 dense in H1. Then for every θ ∈ (0, 1), [H1,H2]θ = (H1,H2)θ,2, with ‖u‖[H1,H2]θ = C‖u‖(H1,H2)θ,2 , where C is a positive constant independent of H1 and H2. As a consequence from the previous lemma and [19, Theorem 4.2.6] we have that for all γ ∈ (0, 1), t ∈ [0, τ ], (H, D(A(t)))γ,2 = [H, D(A(t))]γ = D(A(t)γ). Lemma 3.4. For all x ∈ (H, D(A(t)))1/2,2 one has∫ ∞ 0 ‖A(t)e−sA(t)x‖2 ds ≤ C‖x‖2(H,D(A(t))) 1 2 ,2 , where C > 0 is independent of t. Proof. Note that ‖e−sA(t)‖L(H) ≤ 1 and ‖sA(t)e−sA(t)‖L(H) ≤ M1, where M1 is independent of t. Let x ∈ (H, D(A(t))) 1 2 ,2 . We write x = a + b, where a ∈ H and b ∈ D(A(t)) to obtain s1/2‖A(t)e−sA(t)x‖ ≤ inf x=a+b; a∈H, b∈D(A(t)) M1s −1/2‖a‖+ s1/2‖b‖D(A(t)) ≤ max{M1, 1} inf x=a+b; a∈H, b∈D(A(t)) s−1/2{‖a‖+ s‖b‖D(A(t))} ≤ max{M1, 1} inf x=a+b; a∈H, b∈D(A(t)) s−1/2K(s, x;H, D(A(t))). So ‖A(t)e−sA(t)x‖ ≤ max{M1, 1}s−1K(s, x;H, D(A(t))), where K(s, x;H, D(A(t))) = inf x=a+b; a∈H, b∈D(A(t)) ( ‖a‖+ s‖b‖D(A(t)) ) . Since ‖x‖2(H,D(A(t))) 1 2 ,2 = ∫∞ 0 |K(s, x;H, D(A(t)))|2 ds s2 [19, Definition 1.1.1],∫ ∞ 0 ‖A(t)e−sA(t)x‖2 ds ≤ max{M1, 1}‖x‖2(H,D(A(t))) 1 2 ,2 . 8 A. MAHDI, T. HOSSNI EJDE-2020/124 This completes the proof. � In the next lemma we prove the quadratic estimate that was proved in [3] under assumption (3.3). Here we prove it without the assumption. Lemma 3.5. Let x ∈ H and t ∈ [0, τ ]. We have∫ τ 0 ‖A(t)1/2e−sA(t)x‖2 ds ≤ c‖x‖2, (3.2) where c is a positive constant independent of t. Proof. Note that by [16, (A1) p. 269], A(t)−β = 1 π ∫ ∞ 0 µ−β(µ+A(t))−1 dµ. Then Lemma 3.2 gives ‖A(t)−1/2‖L(H) ≤ C ′, where C ′ is a positive constant inde- pendent of t. Let x ∈ H and t ∈ [0, τ ]. By Lemma 3.4 we have∫ 1 0 ‖A(t)1/2e−sA(t)x‖2 ds = ∫ 1 0 ‖A(t)e−sA(t)A(t)−1/2x‖2 ds ≤ ‖A(t)−1/2x‖2(H;D(A(t))) 1 2 ,2 = ‖x‖2 + ‖A(t)−1/2x‖2 ≤ (C ′2 + 1)‖x‖2. This completes the proof. � In the sequel, we assume that D(A(t)1/2) = V for all t ∈ [0, τ ] and there exist c1, c 1 > 0 such that for all v ∈ V c1‖v‖V ≤ ‖A(t)1/2v‖ ≤ c1‖v‖V , (3.3) this also holds for adjoint-operators and we find c1‖v‖V ≤ ‖A∗(t)1/2v‖ ≤ c1‖v‖V . Note that this assumption is always true for symmetric forms such that c1 = √ δ and c1 = √ M . Lemma 3.6. For all t ∈ [0, τ ] we have D(A(t)1/2) = H and D(A(t)∗ 1 2 ) = V. Proof. We write A(t)1/2u = A(t)A(t)−1/2u. Therefore α c1 ‖u‖ ≤ ‖A(t)1/2u‖V′ ≤ M c1 ‖u‖. So A(t)1/2 ∈ L(H,V ′) and by duality we find A(t)∗ 1 2 ∈ L(V,H). � Let t ∈ [0, τ ]. For f ∈ L2(0, t;H), we define the operator (R(t)f) := ∫ t 0 e−(t−s)A(t)f(s) ds. The next lemma shows that R(t) is bounded in L(L2(0, t;H),V), and it was proved in [3, Lemma 4.1]. Lemma 3.7. We have R(t) ∈ L(L2(0, t;H),V) for all t ∈ [0, τ ]. EJDE-2020/124 MAXIMAL REGULARITY FOR NON-AUTONOMOUS PROBLEMS 9 Lemma 3.8. Assume that A(·) ∈ Cε([0, τ ];L(V,V ′)), ε > 0. Then for all λ ∈ (0,∞), we obtain (λ+A(·))−1 ∈ Cε([0, τ ];L(H)) and ‖(λ+A(·))−1‖Cε([0,τ ];L(H)) ≤ C λ . Proof. Let λ ∈ (0,∞), t, s ∈ [0, τ ]. We obtain (λ+A(t))−1 − (λ+A(s))−1 = (λ+A(t))−1(A(t)−A(s))(λ+A(s))−1. Therefore by Lemma 3.2 we have ‖(λ+A(t))−1 − (λ+A(s))−1‖L(H) ≤ ‖(λ+A(t))−1‖L(V′,H)‖A(t)−A(s)‖L(V′,V)‖(λ+A(t))−1‖L(H,V) ≤ C |t− s| ε |λ| . � We denote by L2 β(0, τ ;D(A(·))) the space of all measurable functions f : [0, τ ]→ H for which f(t) ∈ D(A(t)) for almost all t ∈ [0, τ ] and A(·)f ∈ L2 β(0, τ ;H). Then the following density result holds. Lemma 3.9. Suppose that A(·) ∈ Cε([0, τ ];L(V,V ′)). Then L2 β(0, τ ;D(A(·))) is dense in L2 β(0, τ ;H). Proof. Let f ∈ L2 β(0, τ ;H) and set fn(t) = n(n + A(t))−1f(t) for n ∈ N. Since the map t 7→ (n + A(t))−1 ∈ Cε([0, τ ];L(H)), then for all n ∈ N the function fn : [0, τ ] → H is measurable and satisfies fn(t) ∈ D(A(t)) almost everywhere as well as ‖A(t)fn(t)‖ ≤ Cn‖f(t)‖. Moreover ‖fn(t)− f(t)‖ = ‖(n(n+A(t))−1 − I)f(t)‖. Hence, the convergence fn → f in L2 β(0, τ ;H) holds by the dominated convergence theorem. � Proposition 3.10. Assume that A(·) ∈ Cε([0, τ ];L(V,V ′)), for some ε > 0. Then for all f ∈ L2 β(0, τ ;H), with β < 1 the operator L defined by (Lf)(t) := A(t) ∫ t 0 e−(t−s)A(t)f(s) ds is bounded on L2 β(0, τ ;H). Proof. Let f ∈ L2 β(0, τ ;D(A(·))). We split the integral into two parts to obtain (Lf)(t) = A(t) ∫ t/2 0 e−(t−s)A(t)f(s) ds+A(t) ∫ t t/2 e−(t−s)A(t)f(s) ds := I1(t) + I2(t). We begin by estimating the first integral ‖I1(t)‖ = ‖A(t) ∫ t/2 0 e−(t−s)A(t)f(s) ds‖ . ∫ t/2 0 1 t− s ‖f(s)‖ ds . 2 t ∫ t/2 0 ‖f(s)‖ ds. 10 A. MAHDI, T. HOSSNI EJDE-2020/124 Lemma 2.1 gives ∫ τ 0 ‖A(t) ∫ t/2 0 e−(t−s)A(t)f(s) ds‖2tβ dt . ∫ τ 0 ( 2 t ∫ t/2 0 ‖f(s)‖ ds)2tβ dt . ‖f‖2L2 β(0,τ ;H). Similarly, we estimate the second integral. For x ∈ H we obtain |(I2(t), x)| = | ∫ t t/2 (A(t)1/2e− 1 2 (t−s)A(t)f(s), A(t) 1 2∗e− 1 2 (t−s)A(t)∗x) ds| ≤ (∫ t t/2 ‖A(t)1/2e− 1 2 (t−s)A(t)f(s)‖2 ds )1/2(∫ t t/2 ‖A(t)∗ 1 2 e− 1 2 (t−s)A(t)∗x‖2 ds )1/2 . (∫ t t/2 ‖A(t)1/2e− 1 2 (t−s)A(t)f(s)‖2 ds )1/2 ‖x‖. In the above inequality we used the quadratic estimate (3.5). Taking the supremum over all x ∈ H, we obtain∫ τ 0 tβ‖I2(t)‖ dt = ∫ τ 0 tβ‖A(t) ∫ t t/2 e−(t−s)A(t)f(s) ds‖2 dt . ∫ τ 0 tβ ∫ t t/2 ‖A(t)1/2e− 1 2 (t−s)A(t)f(s)‖2 ds dt . ∫ τ 0 ∫ t t/2 ‖A(t)1/2e− 1 2 (t−s)A(t) ( sβ/2f(s) ) ‖2 ds dt. Let g be the function defined by g = (Φf). Using Fubini’s theorem and the basic inequality (x+ y)2 ≤ 2x2 + 2y2, we obtain∫ τ 0 ∫ t t/2 ‖A(t)1/2e− 1 2 (t−s)A(t)[sβ/2f(s)]‖2 ds dt ≤ 2 ∫ τ 0 ∫ t t/2 ‖A(s)1/2e− 1 2 (t−s)A(s)g(s)‖2 ds dt + 2 ∫ τ 0 ∫ t t/2 ‖(A(s)1/2e− 1 2 (t−s)A(s) −A(t)1/2e− 1 2 (t−s)A(t))g(s)‖2 ds dt ≤ 2 ∫ τ 0 ∫ 2s s ‖A(s)1/2e− 1 2 (t−s)A(s)g(s)‖2 dt ds + 2 ∫ τ 0 ∫ t t/2 ‖(A(s)1/2e− 1 2 (t−s)A(s) −A(t)1/2e− 1 2 (t−s)A(t))g(s)‖2 ds dt . ‖g‖2L2(0,τ ;H) + ∫ τ 0 ∫ t t/2 ‖(A(s)1/2e− 1 2 (t−s)A(s) −A(t)1/2e− 1 2 (t−s)A(t))g(s)‖2 ds dt. The functional calculus for the sectorial operators A(t), A(s) gives A(s)1/2e− 1 2 (t−s)A(s) −A(t)1/2e− 1 2 (t−s)A(t) EJDE-2020/124 MAXIMAL REGULARITY FOR NON-AUTONOMOUS PROBLEMS 11 = ∫ Γ λ1/2e− 1 2 (t−s)λ(λ−A(t))−1(A(t)−A(s))(λ−A(s))−1 dλ. Hence, ‖A(s)1/2e− 1 2 (t−s)A(s) −A(t)1/2e− 1 2 (t−s)A(t)‖L(H) ≤ ∫ Γ |λ|1/2e− 1 2 (t−s) Reλ‖(λ−A(t))−1‖L(V′,H) × ‖(A(t)−A(s))‖L(V,V′)‖(λ−A(s))−1‖L(H,V) |dλ|. Thus ‖A(s)1/2e− 1 2 (t−s)A(s) −A(t)1/2e− 1 2 (t−s)A(t)‖L(H) ≤ ∫ ∞ 0 |λ|−1/2e− 1 2 (t−s)cos(γ)|λ| d|λ|‖(A(t)−A(s))‖L(V,V′). where γ is the angle mentioned in Lemma 3.2. Then ‖A(s)1/2e− 1 2 (t−s)A(s) −A(t)1/2e− 1 2 (t−s)A(t)‖L(H) . ‖A(t)−A(s)‖L(V,V′) (t− s)1/2 . Therefore ∫ τ 0 ∫ t t/2 ‖(A(s)1/2e−(t−s)A(s) −A(t)1/2e−(t−s)A(t))g(s)‖2 ds dt . ∫ τ 0 ∫ t t/2 ‖A(t)−A(s)‖2L(V,V′) t− s ‖g(s)‖2 ds dt . sup s∈[0,τ ] ∫ τ s ‖A(t)−A(s)‖2L(V,V′) t− s dt‖g‖2L2(0,τ ;H) . τ2ε‖A‖2Cε([0,τ ];L(V,V′))‖f‖ 2 L2 β(0,τ ;H). This completes the proof. � Proposition 3.11. For β ≥ 1 the operator L is not bounded on L2 β(0, τ ;H) in general. Proof. Let u ∈ H and g ∈ L2 −β(0, τ ;H). Noting that (L∗g)(t) = ∫ τ t A(s)∗e−(s−t)A(s)∗g(s) ds, t ∈ (0, τ) and L ∈ L(L2 β(0, τ ;H)) if and only if L∗ ∈ L(L2 −β(0, τ ;H)). If A(s)∗ = A(0)∗ for all s ∈ [0, τ ], then (L∗g)(t) = ∫ τ t A(0)∗e−(s−t)A(0)∗g(s) ds. Assume now that t < 1 < τ and take g(s) = 1[1,τ ](s)u, so (L∗g)(t) = e−(1−t)A(0)∗u− e−(τ−t)A(0)∗u, which converges to e−A(0)∗u− e−τA(0)∗u as t→ 0. We claim that e−A(0)∗u− e−τA(0)∗u 6= 0, then ‖L∗g‖2L2 −β(0,τ ;H) ≥ ‖L ∗g‖2L2 −β(0,1;H) = ∫ 1 0 ‖e−(1−t)A(0)∗u− e−(τ−t)A(0)∗u‖2 dt tβ =∞. 12 A. MAHDI, T. HOSSNI EJDE-2020/124 Now, suppose that e−A(0)∗u− e−τA(0)∗u = 0, thus e−A(0)∗u = e−(2τ−1)A(0)∗u. Using induction, for all n ∈ N we obtain e−A(0)∗u− e−(n(τ−1)+1)A(0)∗u = 0. Since ‖A(0)∗e−(n(τ−1)+1)A(0)∗A(0)∗−1u‖ . 1 (n(τ−1)+1)‖A(0)∗−1u‖, by letting n → ∞ it follows that e−A(0)∗u = 0. Hence e−tA(0)∗u = 0 for all t ≥ 1, and we deduce that u = 0 by an application of the isolated point theorem and the analyticity of the semigroup. � Lemma 3.12. For all f ∈ L2 β(0, τ ;H), β < 1 we have (L1f)(t) ∈ V, where (L1f)(t) = tβ/2 ∫ t 0 e−(t−s)A(t)f(s) ds, t ∈ [0, τ ]. Proof. We write (L1f)(t) = tβ/2 ∫ t/2 0 e−(t−s)A(t)f(s) ds+ tβ/2 ∫ t t/2 e−(t−s)A(t)f(s) ds. A straightforward computation gives ‖tβ/2 ∫ t/2 0 e−(t−s)A(t)f(s) ds‖V . tβ/2 ∫ t/2 0 ‖e−(t−s)A(t)‖L(H,V)‖f(s)‖ ds . tβ/2( ∫ t/2 0 s−β−1 ds)1/2‖f‖L2 β(0,τ ;H) . ‖f‖L2 β(0,τ ;H). Using Lemma 3.5 we deduce ‖tβ/2 ∫ t t/2 e−(t−s)A(t)f(s) ds‖V . ‖ ∫ t t/2 e−(t−s)A(t)(sβ/2f(s)) ds‖V . ‖f‖L2 β(0,τ ;H). This completes the proof. � Lemma 3.13. For all u0 ∈ (H;D(A(0))) 1−β 2 ,2 and β ∈ [0, 1), we have∫ τ 0 ‖tβ/2A(0)e−tA(0)u0‖2 dt ' ‖u0‖2(H;D(A(0))) 1−β 2 ,2 . Proof. Note that (H;D(A(0))) 1−β 2 ,2 = D(A(0) 1−β 2 ). Let β ∈ [0, 1). In light of the quadratic estimate we obtain∫ τ 0 ‖tβ/2A(0)e−tA(0)u0‖2 dt = ∫ τ 0 ‖tβ/2A(0) 1+β 2 e−tA(0)A(0) 1−β 2 u0‖2 dt . ∫ τ 0 ‖A(0)1/2e− t 2A(0)A(0) 1−β 2 u0‖2 dt . ‖A(0) 1−β 2 u0‖2 = ‖u0‖2[H;D(A(0))] 1−β 2 . ‖u0‖2(H;D(A(0))) 1−β 2 ,2 . EJDE-2020/124 MAXIMAL REGULARITY FOR NON-AUTONOMOUS PROBLEMS 13 Conversely, we know that [19, Definition 1.1.1] ‖u0‖2(H;D(A(0))) 1−β 2 ,2 = ∫ 1 0 tβ−2‖K(t, u0)‖2 dt, where K(t, u0) = inf u0=a+b;a∈H,b∈D(A(0)) ( ‖a‖+ t‖b‖D(A(0)) ) . This allows us to write, for t ∈ [0, τ ], u0 = (u0 − e−tA(0)u0) + e−tA(0)u0 = − ∫ t 0 A(0)e−lA(0)u0 dl + e−tA(0)u0. Since e−tA(0)u0 ∈ D(A(0)) a.e. t ∈ [0, τ ] and (u0 − e−tA(0)u0) ∈ H, it follows that ‖K(t, u0)‖ ≤ ∫ t 0 ‖A(0)e−lA(0)u0‖ dl + t‖A(0)e−tA(0)u0‖. Roughly speaking, by Lemma 2.1 we find ‖u0‖2(H;D(A(0))) 1−β 2 ,2 . ∫ τ 0 ‖tβ/2A(0)e−tA(0)u0‖2 dt. This completes the proof. � Remark 3.14. From the previous lemma, the orbit the map t 7→ e−tA(0)u0 belongs to the space W 1,2 β (0, τ ;H)∩L2 β(0, τ ;D(A(0))) if and only if u0 ∈ (H;D(A(0))) 1−β 2 ,2. We define the space Wβ(D(A(·)),H) := {u ∈W 1,1(0, τ ;H), s.t. A(·)u ∈ L2 β(0, τ ;H), u̇ ∈ L2 β(0, τ ;H)}, with norm ‖u‖Wβ(D(A(·),H) = ‖A(·)u‖L2 β(0,τ ;H) + ‖u̇‖L2 β(0,τ ;H). It is easy to see that Wβ(D(A(·),H) ↪→W 1,2 β (0, τ ;H). Lemma 3.15. For all γ ≤ 1/2, we have (H, D(A(0)))γ,2 = [H,V]2γ and for γ > 1/2 we have (H, D(A(0)))γ,2 ↪→ V. Proof. As a consequence of the interpolation method [19, Remark 1.3.6], for γ ≤ 1/2 we have (H, D(A(0)))γ,2 = (H, D(A(0)1/2))2γ,2 = (H,V)2γ,2. Since H and V are Hilbert spaces we obtain by Lemma 3.3 (H, D(A(0)))γ,2 = (H,V)2γ,2 = [H,V]2γ . Let v ∈ D(A(0)) and γ > 1 2 . We obtain δ‖v‖2V ≤ Re(A(0)v, v) . ‖A(0)γv‖‖A(0)∗(1−γ)v‖ . ‖A(0)γv‖‖v‖[H,V]2(1−γ) . ‖A(0)γv‖‖v‖V . Therefore we have that for all γ > 1 2 and v ∈ D(A(0)), ‖v‖V . ‖v‖D(A(0)γ). Finally, by the density of D(A(0)) in D(A(0)γ) we obtain the desired result. � 14 A. MAHDI, T. HOSSNI EJDE-2020/124 4. Maximal regularity for autonomous problems In this section we are interested in the regularity of the problem u̇(t) +A(0)u(t) = f(t) u(0) = u0. (4.1) The following is our main result in this section. Theorem 4.1. Let f ∈ L2 β(0, τ,H) and u0 ∈ (H;D(A(0))) 1−β 2 ,2 for β ≥ 0 and u0 = 0 if β < 0. There exists a unique u ∈ Wβ(D(A(0)),H) ∩ L∞β (0, τ ;V) be the solution to Problem (4.1). Moreover, we have the following embeddings Wβ(D(A(0)),H) ↪→ C([0, τ ]; (H;D(A(0))) 1−β 2 ,2) Wβ(D(A(0)),H) ↪→W 1 2 ,2 β (0, τ ;V), β ∈ [0, 1[. Proof. Since A(0) is a generator of an analytic semigroup in H, it is well known that by the variation of constants formula the solution of Problem (4.1) is u(t) = e−tA(0)u0 + ∫ t 0 e−(t−s)A(0)f(s) ds. Thus, A(0)u(t) = A(0)e−tA(0)u0 +A(0) ∫ t 0 e−(t−s)A(0)f(s) ds := (Fu0)(t) + (Lf)(t). Lemmas 3.12, 3.13 and Proposition 3.10 gives ‖A(0)u‖L2 β(0,τ ;H) ≤ ‖Fu0‖L2 β(0,τ ;H) + ‖Lf‖L2 β(0,τ ;H) ≤ C ( ‖u0‖(H;D(A(0))) 1−β 2 ,2 + ‖f‖L2 β(0,τ ;H) ) . Since u̇ = f −A(0)u ∈ L2 β(0, τ ;H), we obtain finally ‖u‖Wβ(D(A(0)),H) ≤ C ′ ( ‖u0‖(H;D(A(0))) 1−β 2 ,2 + ‖f‖L2 β(0,τ ;H) ) . (4.2) Using Proposition 5.1 and (4.2), for all t ∈ [0, τ ] we obtain ‖u(t)‖(H;D(A(0))) 1−β 2 ,2 . ‖u‖Wβ(D(A(0)),H)∩L∞β (0,τ ;V) . ‖u0‖(H;D(A(0))) 1−β 2 ,2 + ‖f‖L2 β(0,τ ;H). (4.3) For 0 ≤ s ≤ l ≤ t ≤ τ , we set v(l) = e−(t−l)A(0)u(l). This yields u(t)− u(s) = v(s)− u(s) + ∫ t s v̇(l) dl = (e−(t−s)A(0) − I)u(s) + ∫ t s e−(t−l)A(0)f(l) dl. (4.4) Observe that e−(t−s)A(0) is strongly continuous on (H;D(A(0))) 1−β 2 ,2. In particular, this ensures that ‖(e−(t−s)A(0) − I)u(s)‖(H;D(A(0))) 1−β 2 ,2 → 0 as t→ s. EJDE-2020/124 MAXIMAL REGULARITY FOR NON-AUTONOMOUS PROBLEMS 15 The estimate (4.3) for the case u0 = 0 gives that ‖ ∫ t s e−(t−l)A(0)f(l) dl‖(H;D(A(0))) 1−β 2 ,2 . ‖f‖L2 β(s,t;H). It follows that u(t) is right continuous on (H;D(A(0))) 1−β 2 ,2. Now, set v(l) = e−(l−s)A(0)u(l), for 0 ≤ s ≤ l ≤ t. Then u(s)− u(t) = v(t)− u(t)− ∫ t s v̇(l) dl = (e−(t−s)A(0) − I)u(t)− ∫ t s e−(l−s)A(0)(f(l)− 2A(0)u(l)) dl. The same argument shows that u is left continuous in (H;D(A(0))) 1−β 2 ,2. Thus, u ∈ C([0, τ ]; (H;D(A(0))) 1−β 2 ,2). Now, we prove that Wβ(D(A(0)),H) ↪→W 1 2 ,2 β (0, τ ;V). Indeed, let β ∈ [0, 1[ and u ∈ C∞([0, τ ];D(A(0))). We recall that ‖u‖2 W 1 2 ,2 β (0,τ ;V) = ‖u‖2L2 β(0,τ ;V) + ∫ τ 0 ∫ t 0 ‖u(t)− u(s)‖2V |t− s|2 sβ ds dt. By (4.4) it holds that for all 0 ≤ s ≤ t ≤ τ u(t)− u(s) = (e−(t−s)A(0)u(s)− u(s)) + ∫ t s e−(t−l)A(0)f(l) dl := L1(t, s) + L2(t, s), where f(l) = A(0)u(l) + u̇(l). So ‖u‖2 W 1 2 ,2 β (0,τ ;V) ≤ ‖u‖2L2 β(0,τ ;V) + 2 ∫ τ 0 ∫ t 0 ‖L1(t, s)‖2V |t− s|2 sβ ds dt + 2 ∫ τ 0 ∫ t 0 ‖L2(t, s)‖2V |t− s|2 sβ ds dt. We write L1(t, s) = e−(t−s)A(0)u(s)− u(s) = ∫ t−s 0 e−lA(0)A(0)u(s) dl. Lemma 2.1 and the quadratic estimate gives∫ τ 0 ∫ t 0 ‖L1(t, s)‖2V |t− s|2 sβ ds dt ≤ ∫ τ 0 ∫ τ s (∫ t−s 0 ‖e−lA(0)A(0)u(s)‖Vdl |t− s| )2 dtsβ ds ≤ C ∫ τ 0 ∫ τ s ‖e−tA(0)A(0)u(s)‖2V dtsβ ds ≤ C ′ ∫ τ 0 ‖A(0)u(s)‖2sβ ds = C ′‖A(0)u‖2L2 β(0,τ ;H). Similarly, we obtain∫ τ 0 ∫ t 0 ‖L2(t, s)‖2V |t− s|2 sβ ds dt ≤ ∫ τ 0 ∫ t 0 (∫ t s ‖e(t−l)A(0)(Φf)(l)‖V dl |t− s| )2 ds dt 16 A. MAHDI, T. HOSSNI EJDE-2020/124 ≤ C ∫ τ 0 ∫ t 0 ‖e(t−s)A(0)(Φf)(s)‖2V ds dt = C ∫ τ 0 ∫ τ s ‖e(t−s)A(0)(Φf)(s)‖2V dt ds ≤ C‖Φf‖2L2(0,τ ;H) = C‖f‖2L2 β(0,τ ;H). Therefore, ‖u‖ W 1 2 ,2 β (0,τ ;V) . ‖A(0)u‖L2 β(0,τ ;H) + ‖f‖L2 β(0,τ ;H) . ‖u‖Wβ(D(A(0)),H). We note that C∞([0, τ ];D(A(0))) is dense in Wβ(D(A(0)),H). This shows that Wβ(D(A(0)),H) ↪→W 1 2 ,2 β (0, τ ;V). which completes the proof � Remark 4.2. The following embeddings hold (1) Wβ(D(A(0)),H) ↪→ C([0, τ ]; [H,V]1−β), for 0 ≤ β < 1. (2) Wβ(D(A(0)),H) ↪→ C([0, τ ];V), for β ≤ 0. Theorem 4.3. For all f ∈W 1,2 β,0(0, τ,H), there exists a unique u ∈ C1([0, τ ]; (H;D(A(0))) 1−β 2 ,2) ∩ C([0, τ ];D(A(0))), which satisfies the equation u̇(t) +A(0)u(t) = f(t) u(0) = 0. (4.5) In addition, ‖u‖C1([0,τ ];(H;D(A(0))) 1−β 2 ,2 )∩C([0,τ ];D(A(0))) ≤ C‖f‖W 1,2 β (0,τ ;H). Assume now that τ = +∞ and f is a periodic function with period p. Then u satisfies u(t+ p) = e−tA(0)u(p) + u(t), t ∈ [0,∞), and it is periodic with the same period p if and only if u(p) = 0. Proof. According to Theorem 4.1, there exists a unique solution u to Problem (4.5) and for all f ∈ L2 β(0, τ ;H) u(t) = ∫ t 0 e−(t−s)A(0)f(s) ds, t ∈ [0, τ ]. (4.6) Moreover u ∈Wβ(D(A(0)),H) and ‖u‖Wβ(D(A(0)),H) ≤ C‖f‖L2 β(0,τ ;H). (4.7) Integrating by parts, we obtain for t ∈ [0, τ ] and f ∈W 1,2 β,0(0, τ,H) A(0)u(t) = A(0) ∫ t 0 e−(t−s)A(0)f(s) ds = f(t)− ∫ t 0 e−(t−s)A(0)ḟ(s) ds = u̇(t) +A(0)u(t)− ∫ t 0 e−(t−s)A(0)ḟ(s) ds. EJDE-2020/124 MAXIMAL REGULARITY FOR NON-AUTONOMOUS PROBLEMS 17 Hence, u̇(t) = ∫ t 0 e−(t−s)A(0)ḟ(s) ds = (Lḟ)(t). Theorem 4.1 shows that u ∈ C1([0, τ ]; (H;D(A(0))) 1−β 2 ,2). Since A(0)u = f − u̇ we deduce that A(0)u ∈ C([0, τ ];H). As a consequence, we obtain the final estimate ‖u‖C1([0,τ ];(H;D(A(0))) 1−β 2 ,2 )∩C([0,τ ];D(A(0))) ≤ C‖f‖W 1,2 β (0,τ ;H). Consider now the case where τ = +∞ and f is a periodic function with some period p > 0, i.e. f(t + p) = f(t) for all t ∈ [0,+∞). It is clear that if u is periodic with period p, then u(p) = u(0) = 0. Formula (4.6) yields u(t+ p) = ∫ t+p 0 e−(t+p−s)A(0)f(s) ds. Hence, u(t+ p) = ∫ p 0 e−(t+p−s)A(0)f(s) ds+ ∫ p+t p e−(t+p−s)A(0)f(s) ds = e−tA(0) ∫ p 0 e−(p−s)A(0)f(s) ds+ ∫ t 0 e−(t−l)A(0)f(l + p) dl = e−tA(0)u(p) + u(t). In the previous equality, we made a change of variables, and in the last equality we used the periodicity of f . Then u is periodic with period p if and only if e−tA(0)u(p) = 0 for all t ∈ [0,∞). Therefore, the analyticity of the semigroup shows that u(p) = 0 is a necessary condition for u to be periodic. � 5. Maximal regularity for non-autonomous problems In this section we focus on the maximal regularity for the non-autonomous prob- lem (which is our main aim), i.e. we prove the existence and the uniqueness of the solution to Problem (1.1) in the weighted space W 1,2 β (0, τ ;H). We start by stating and proving some estimates which we will need in the proof of the main result. Proposition 5.1. (1) Assume that∫ τ 0 ‖A(t)−A(0)‖2L(V,V′) t dt <∞. Then for all s ∈ [0, τ ], TRs : Wβ(D(A(·),H) ∩ L∞β (0, τ ;V) −→ (H;D(A(s))) 1−β 2 ,2 u 7−→ u(s) is a bounded operator. (2) For u0 ∈ (H;D(A(0))) 1−β 2 ,2, we have t→ (Fu0)(t) = tβ/2A(t)e−tA(t)u0 ∈ L2(0, τ ;H). Proof. (1) First we consider the case s = 0. We have ‖u(0)‖2(H;D(A(0))) 1−β 2 ,2 = ∫ 1 0 ‖tβ/2A(0)e−tA(0)u(0)‖2 dt+ ‖u(0)‖2 18 A. MAHDI, T. HOSSNI EJDE-2020/124 ≤ 2 ∫ 1 0 ‖tβ/2A(0)e−tA(0)(u(0)− u(t))‖2 dt+ ‖u(0)‖2 + 2 ∫ 1 0 ‖tβ/2A(0)e−tA(0)u(t)‖2 dt . ∫ 1 0 tβ (1 t ∫ t 0 ‖u̇(l)‖ ds )2 dl + ∫ τ 0 tβ‖A(t)u(t)‖2 dt + ∫ τ 0 ‖tβ/2(A(0)e−tA(0) −A(t)e−tA(t))u(t)‖2 dt+ ‖u(0)‖2 . ‖u̇‖2L2 β(0,τ ;H) + ‖A(·)u‖2L2 β(0,τ ;H) + ∫ τ 0 ‖A(t)−A(0)‖2L(V,V′) t dt‖u‖L∞β (0,τ ;V) + ‖u(0)‖2 . ‖u‖2Wβ(D(A(·),H) + ‖u‖2L∞β (0,τ ;V) + ‖u(0)‖2, where we have used the quadratic estimate, Hardy inequality and the estimate ‖A(0)e−tA(0) −A(t)e−tA(t)‖L(V,H) . ‖A(t)−A(0)‖L(V,V′) t1/2 . Now, we prove the result for all s ∈]0, τ ]. Indeed, let l ∈]0, τ [ and set v(t) := { u(t+ s), t ∈ [0, τ − s]. u( τs (τ − t)), t ∈ [τ − s, τ ]. Similarly, B(t) := { A(t+ s), t ∈ [0, τ − s]. A( τs (τ − t)), t ∈ [τ − s, τ ]. Since v(t) ∈Wβ(D(B(·),H), therefore v(0) = u(s) ∈ (H;D(B(0))) 1−β 2 ,2 = (H;D(A(s))) 1−β 2 ,2. For the case s = τ , we take v(t) = u(τ − t) and B(t) = A(τ − t). (2) Note that (Fu0)(t) = tβ/2A(t)e−tA(t)u0 = tβ/2(A(t)e−tA(t)u0 −A(0)e−tA(0)u0) + tβ/2A(0)e−tA(0)u0. For β > 0 we have by interpolation ‖(λ−A(0))−1‖L((H;D(A(0))) 1−β 2 ,2 ,V) . 1 |λ|1− β2 . Therefore ‖(Fu0)(t)‖ . ‖A(0)−A(t)‖L(V,V′) t ‖u0‖(H;D(A(0))) 1−β 2 + ‖tβ/2A(0)e−tA(0)u0‖. Hence, ‖(Fu0)‖2L2(0,τ ;H) . ∫ τ 0 ‖A(0)−A(t)‖2L(V,V′) t dt‖u0‖(H;D(A(0))) 1−β 2 ,2 + ∫ τ 0 ‖tβ/2A(0)e−tA(0)u0‖2 dt EJDE-2020/124 MAXIMAL REGULARITY FOR NON-AUTONOMOUS PROBLEMS 19 . ‖u0‖2(H;D(A(0))) 1−β 2 ,2 . This shows the second assertion. � In the sequel we consider only the case β ∈ [0, 1[. Proposition 5.2. Suppose A ∈ Cε([0, τ ];L(V,V ′)). Then for each f ∈ L2 β(0, τ ;H), u0 ∈ (H;D(A(0))) 1−β 2 ,2 and for τ small enough, there exists a unique solution u in L∞β (0, τ ;V) for (1.1). Proof. Let f ∈ L2 β(0, τ ;H). We set v(s) = e−(t−s)A(t)u(s). Since u(t) = e−tA(t)u0 +∫ t 0 v̇(s) ds, therefore u(t) = e−tA(t)u0 + ∫ t 0 e−(t−s)A(t)(A(t)−A(s))u(s) ds + ∫ t 0 e−(t−s)A(t)f(s) ds := (Mu0)(t) + (M1u)(t) + (L1f)(t). (5.1) For β > 0 and u0 ∈ (H, D(A(0))) 1−β 2 ,2 we have by interpolation ‖e−tA(t)u0‖V . t−β/2‖u0‖(H,D(A(0))) 1−β 2 ,2 . (5.2) In view of Lemma 3.12 and (5.2), tβ/2(Mu0)(t), tβ/2(L1f)(t) are bounded in V for all t ∈ [0, τ ]. Now, we show that M1u ∈ L∞β (0, τ ;V) for all u ∈ L∞β (0, τ ;V). We write (M1u)(t) = ∫ t/2 0 e−(t−s)A(t)(A(t)−A(s))u(s) ds + ∫ t t/2 e−(t−s)A(t)(A(t)−A(s))u(s) ds := (M11u)(t) + (M12u)(t). By taking x ∈ V ′ we obtain |((M12u)(t), x)V′×V | = ∣∣ ∫ t t/2 (e− (t−s) 2 A(t)(A(t)−A(s))u(s), A(t)∗ 1 2 e− (t−s) 2 A(t)∗A(t)∗− 1 2x) ds ∣∣ ≤ (∫ t t/2 ‖e− (t−s) 2 A(t)‖2L(V′,H)‖A(t)−A(s)u(s)‖2V′ ds )1/2 × (∫ t t/2 ‖A(t)∗ 1 2 e− (t−s) 2 A(t)∗A(t)∗− 1 2x‖2 ds )1/2 . Now, we estimate the norm of (M11v)(t) in V as follows tβ/2‖(M11v)(t)‖V . tβ/2 ∫ t/2 0 ‖e− (t−s) 2 A(t)‖L(V′,V)‖A(t)−A(s)‖L(V,V′)s −β/2 ds‖s → sβ/2u(s)‖L∞(0, t2 ;V) 20 A. MAHDI, T. HOSSNI EJDE-2020/124 . tβ/2 ∫ t/2 0 s−β/2 (t− s)1−ε ds sup s∈[0,t/2] ‖A(t)−A(s)‖L(V,V′) (t− s)ε ‖s → sβ/2u(s)‖L∞(0, t2 ;V). Note that tβ/2 ∫ t/2 0 s−β/2 (t− s)1−ε ds = tε ∫ 1/2 0 l−β/2 (1− l)1−ε dl. Therefore, tβ/2‖(M1v)(t)‖V . tε‖A‖Cε([0,τ ];L(V,V′))‖s → sβ/2u(s)‖L∞(0, t2 ;V) + (∫ t t/2 ‖A(t)−A(s)‖2L(V,V′) t− s ds)1/2‖u‖L∞β ( t2 ,t;V) . tε‖A(·)‖Cε([0,τ ];L(V,V′))‖u‖L∞β (0,t;V). Choosing τ small enough, M1 ∈ L(L∞β (0, τ ;V)), with norm ‖M1‖L(L∞β (0,τ ;V)) < 1. Therefore (I −M1) is invertible in L∞β (0, τ ;V). Hence, u = (I −M1)−1(Mu0 + L1f) ∈ L∞β (0, τ ;V). This completes the proof. � Our main result reads as follows. Theorem 5.3. Suppose that A ∈ W 1 2 ,2(0, τ ;L(V,V ′)) ∩ Cε([0, τ ],L(V,V ′)) with ε > 0, then for all f ∈ L2 β(0, τ ;H) and u0 ∈ (H;D(A(0))) 1−β 2 , there exists a unique u ∈Wβ(D(A(·),H) be the solution of (1.1). Proof. Let τ be small enough and f ∈ L2 β(0, τ ;H), u0 ∈ (H;D(A(0))) 1−β 2 ,2. By Proposition 5.2, u belongs to L∞β (0, τ ;V), where u is the unique solution to the Cauchy problem (1.1). Using (5.1), for 0 ≤ t ≤ τ , we have A(t)u(t) = A(t)e−tA(t)u0 +A(t) ∫ t 0 e−(t−s)A(t)(A(t)−A(s)u(s) ds +A(t) ∫ t 0 e−(t−s)A(t)f(s) ds := (Fu0)(t) + (Su)(t) + (Lf)(t). Thanks to Propositions 3.10, 5.1, Fu0 and Lf are bounded in L2 β(0, τ ;H). Then to prove that A(·)u ∈ L2 β(0, τ ;H) it is sufficient to show that Su belongs to L2 β(0, τ ;H). Taking g ∈ L2(0, τ ;H) we find that |(·β/2Su, g)L2(0,τ ;H)| = ∣∣ ∫ τ 0 tβ/2 ∫ t 0 〈(A(t)−A(s))u(s), A(t)∗e−(t−s)A(t)∗g(t)〉V′×V ds dt ∣∣ ≤ | ∫ τ 0 tβ/2 ∫ t/2 0 〈(A(t)−A(s))u(s), A(t)∗e−(t−s)A(t)∗g(t)〉V′×V ds dt| + | ∫ τ 0 tβ/2 ∫ t t/2 〈(A(t)−A(s))u(s), A(t)∗e−(t−s)A(t)∗g(t)〉V′×V ds dt| := I1 + I2. EJDE-2020/124 MAXIMAL REGULARITY FOR NON-AUTONOMOUS PROBLEMS 21 For I2 we find, I2 . ∫ τ 0 tβ/2 ∫ t t/2 ‖A(t)−A(s)‖L(V,V′)‖e− (t−s) 2 A(t)∗‖L(H,V) × ‖A(t)∗ 1 2 e− (t−s) 4 A(t)∗‖L(H)‖A(t)∗ 1 2 e− (t−s) 4 A(t)∗g(t)‖s− β 2 ds dt‖ ·β/2 u‖L∞(0,τ ;V) . ∫ τ 0 ∫ t t/2 ‖A(t)−A(s)‖L(V,V′) t− s ‖A(t)∗ 1 2 e− (t−s) 4 A(t)∗g(t)‖ ds dt‖ ·β/2 u‖L∞(0,τ ;V) . ‖A‖ W 1 2 ,2(0,τ ;L(V,V′)) (∫ τ 0 ∫ t t/2 ‖A(t)∗ 1 2 e− (t−s) 4 A(t)∗g(t)‖2 ds dt )1/2 ‖u‖L∞β (0,τ ;V) . ‖A‖ W 1 2 ,2(0,τ ;L(V,V′)) ‖g‖L2(0,τ,H)‖u‖L∞β (0,τ ;V). Similarly, I1 . ∫ τ 0 tβ/2 ∫ t/2 0 s −β 2 (t− s) 3 2−ε ‖g(t)‖ ds dt × ‖A‖Cε([0,τ ];L(V,V′))‖ ·β/2 u‖L∞(0,τ ;V) . ‖A‖Cε([0,τ ];L(V,V′))‖g‖L2(0,τ,H)‖u‖L∞β (0,τ ;V). Now, we obtain the final estimate ‖A(·)u‖L2 β(0,τ ;H) . ‖Fu0‖L2 β(0,τ ;H) + ‖Su‖L2 β(0,τ ;H) + ‖Lf‖L2 β(0,τ ;H) . ‖u0‖(H;D(A(0))) 1−β 2 ,2 + ‖u‖L∞β (0,τ ;V) + ‖f‖L2 β(0,τ ;H) . ‖u0‖(H;D(A(0))) 1−β 2 ,2 + ‖f‖L2 β(0,τ ;H). Therefore A(·)u ∈ L2 β(0, τ ;H) and since u̇ = f − Au, one has u̇ ∈ L2 β(0, τ ;H). So u belongs to Wβ(D(A(·),H). Moreover, by Proposition 5.1 we have u(t) ∈ (H;D(A(t))) 1−β 2 ,2 for all t ∈ [0, τ ]. For arbitrary τ we split the interval [0, τ ] into union of small intervals and argue exactly as before to each subinterval. Finally we stick the solutions and we obtain the desired result. � Proposition 5.4. For all g ∈ L2(0, τ ;H) and 0 ≤ β < 1 there exists a unique v ∈W0(D(A(·),H) be the solution of the singular equation v̇(t) +A(t)v(t) + β 2 v(t) t = g(t) v(0) = 0. (5.3) Proof. We set f(t) = (Φg)(t) = tβ/2g(t) with t ∈ [0, τ ], so that f ∈ L2 β(0, τ ;H). Let u ∈Wβ(D(A(·),H) be the solution to the problem u̇(t) +A(t)u(t) = f(t) u(0) = 0. (5.4) Now, set v = (Φ−1u). Then v ∈ W0(D(A(·),H) and v is the unique solution to Problem (5.3). � 22 A. MAHDI, T. HOSSNI EJDE-2020/124 6. Applications This section is devoted to some applications of the results given in the previous sections. We give examples illustrating the theory without seeking for generality. 6.1. Elliptic operators in the divergence form. Let Ω be a bounded Lipschitz domain of Rn. We set H := L2(Ω) and V := H1(Ω) and we define the sesquilinear forms a(t, u, v) := ∫ Ω C(t, x)∇u∇v dx where here u, v ∈ V and C : [0, τ ] × Ω → Cn×n is a bounded and measurable function for which there exists α,M > 0 such that α|ξ|2 ≤ Re(C(t, x)ξ.ξ̄) and |C(t, x)ξ.ν| ≤M |ξ||ν| for all t ∈ [0, τ ] and a.e x ∈ Ω, and all ξ, ν ∈ Cn. We define the gradient operator ∇ : V → H and ∇∗ : H → V ′. The non-autonomous form a(t) induces the operators A(t) := −∇∗C(t, x)∇ ∈ L(V,V ′). The form a(t) is H1(Ω)-bounded and coercive. The part of A(t) inH is the operator A(t) := −div C(t, x)∇ under Neumann boundary conditions. We note that ‖A(t)‖L(V,V′) ' ‖C(t, ·)‖L∞(Ω;Cn×n) = M. Next, we suppose that C ∈ W 1 2 ,2(0, τ ;L∞(Ω;Cn×n)) ∩ Cε([0, τ ];L∞(Ω;Cn×n)), with ε > 0, which is equivalent to∫ τ 0 ∫ τ 0 sup x∈Ω ‖C(t, x)− C(s, x)‖2Cn×n |t− s|2 ds dt <∞, ‖C(t, x)− C(s, x)‖Cn×n < C|t− s|ε a.e. for x ∈ Ω and t, s ∈ [0, τ ]. Note that ‖A(t)−A(s)‖L(V,V′) . ‖C(t, .)− C(s, .)‖L∞(Ω;Cn×n). Hence A ∈W 1 2 ,2(0, τ ;L(V,V ′)) ∩ Cε([0, τ ];L(V,V ′)). Remark 6.1. D(A(t)1/2) = V = H1(Ω) for all t ∈ [0, τ ] and c1‖u‖H1(Ω) ≤ ‖u‖D(A(t)1/2) ≤ c1‖u‖H1(Ω) where c1, c 1 are two positive constants independents of t [6, Theorem 1]. In the next proposition we assume that β ∈ [0, 1[. Proposition 6.2. For all f ∈ L2 β(0, τ ;L2(Ω)), u0 ∈ H1−β(Ω) there is a unique u ∈Wβ(D(A(·), L2(Ω)), be the solution of the problem u̇(t)− divC(t, x)∇u(t) = f(t) ∂u(t, σ) ∂n = 0 (σ ∈ ∂Ω) u(0) = u0. (6.1) The above proposition follows by Theorem 5.3. EJDE-2020/124 MAXIMAL REGULARITY FOR NON-AUTONOMOUS PROBLEMS 23 6.2. Robin boundary conditions. Let Ω be a bounded domain of Rd with Lipschitz boundary ∂Ω. We denote by Tr the classical trace operator. Let β : [0, τ ]× ∂Ω→ [0,∞) be a bounded function and H := L2(Ω). We define the form a(u, v) := ∫ Ω ∇u.∇v dx+ ∫ ∂Ω β(·)Tr(u)Tr(v) dσ, for all u, v ∈ V := H1(Ω). The form a is H1(Ω)-bounded, symmetric and quasi-coercive. The first state- ment follows readily from the continuity of the trace operator and the boundedness of β. The second one is a consequence of the inequality∫ ∂Ω |u|2dσ ≤ δ‖u‖2H1(Ω) + Cδ‖u‖2L2(Ω) which is valid for all δ > 0 (Cδ is a constant depending on δ). Note that this is a con- sequence of compactness of the trace as an operator from H1(Ω) into L2(∂Ω, dσ). Formally, the associated operator A is (minus) the Laplacian with the time depen- dent Robin boundary condition ∂u ∂n + β(·)u = 0 on ∂Ω. Here, ∂u∂n denotes the normal derivative in the weak sense. For more general bound- ary conditions with an indefinite weight we refer the reader to the recent paper [10]. Theorems 4.1 combined with Theorem 4.3 yields the following result. Proposition 6.3. Let β ∈]− 1, 1[ and f ∈ L2 β(0, τ ;L2(Ω)). There exists a unique u ∈Wβ(D(A), L2(Ω))∩C([0, τ ], (L2(Ω);D(A)) 1−β 2 ,2) be the solution to the problem u̇(t)−∆u(t) = f(t) ∂u ∂n + β(·)u = 0 on ∂Ω u(0) = 0. (6.2) If we assume moreover that f ∈ W 1,2 β,0(0, τ ;L2(Ω)), then the solution u belongs to the space C1([0, τ ]; (L2(Ω);D(A)) 1−β 2 ,2) ∩ C([0, τ ];D(A)). Remark 6.4. Note that for all β ∈ [0, 1[ we have (L2(Ω);D(A)) 1−β 2 ,2 = [L2(Ω);D(A)] 1−β 2 = [L2(Ω);H1(Ω)]1−β = H1−β(Ω). References [1] M. Achache; Maximal regularity for the damped wave equations, J. Elliptic Parabol. Equ., 6 (2020), 835-870. [2] M. Achache, E. M. Ouhabaz; Non-autonomous right and left multiplicative perturbations and maximal regularity, Studia Math., 242 (1) (2018), 1-30. [3] M. Achache, E. M. Ouhabaz; Lions’ maximal regularity problem with H1/2-regularity in time, J. Differential Equations., 266 (2019), 3654-3678. [4] W. Arendt, D. Dier, S. Fackler; J. L. Lions’ problem on maximal regularity, Arch. Math.(Basel)., 109 (2017), No. 1, 5972. [5] P. Auscher, A. Axelsson; Remarks on maximal regularity, Progress in Nonlinear Differential Equations and Their Applications, Vol. 80 (2011), 45-55. [6] P. Auscher, Ph. Tchamitchian; Square roots of elliptic second order divergence operators on strongly Lipschitz domains, J. Anal. Math., 90 (2003), 1-12. [7] P. Auscher, M. Egert; On non-autonomous maximal regularity for elliptic operators in di- vergence form, Arch. Math.(Basel)., 107 (2016), No. 3, 271-284. 24 A. MAHDI, T. HOSSNI EJDE-2020/124 [8] J. Bergh, J. Lofstrom; Interpolation spaces. An introduction. Grundlehren der Mathematis- chen Wissenschaften, Springer-Verlag, Berlin, (1976), No. 223, pp. 207. [9] M. Cowling, I. Doust, A. McIntosh, A. Yagi; Banach space operators with a bounded H∞ functional calculus, J. Austral. Math. Soc. Ser. A, 60 (1996), No. 1, 51-89. [10] M. Cuesta, L. Leadi, P. Nshimirimana; Maximum and antimaximum principles for the p- Laplacien with weighted Steklov boundary conditions, Electron. J. Differential Equations, Vol. 2020 (2020), No. 21, pp. 1-17. [11] J. Diestel, J. J. Uhl; Vector measures, American Mathematical Society, Providence, R.I., 1977. [12] S. N. Ethier, T. G. Kurtz; Markov processes, Wiley series in probability and mathematical statistics: probability and mathematical statistics, John Wiley and Sons, Inc., New York, (1986), Characterization and convergence. [13] S. Fackler; J. L. Lions’ problem concerning maximal regularity of equations governed by non- autonomous forms, Ann. Inst. H. Poincaré Anal., Non Linéaire 34 (2017), No 3, 699-709. [14] B. Haak, E. M. Ouhabaz; Maximal regularity for non-autonomous evolution equations, Math. Ann. 363 (2015), No. 3-4, 1117-1145. [15] T. Hytönen, J. V. Neerven, M. Veraar, L. Weis; Analysis in banach spaces Vol. I martingales and littlewood-paley theory, volume 63 ofergebnisse der mathematik undihrer grenzgebiete (3), Springer, 2016. [16] T. Kato; Fractional powers of dissipative operators, J. Math. Soc. Japan, 13 (1961), 246-274. [17] J. L. Lions; Équations différentielles opérationnelles et problèmes aux limites, Die Grundlehren der mathematischen Wissenschaften, Bd. 111, Springer-Verlag, Berlin, 1961. [18] J. L. Lions, E. Magenes; Non-homogeneous boundary value problems and applications, Springer, Vol. 1, 1972. [19] A. Lunardi; Interpolation theory. Second. Appunti. Scuola Normale Superiore di Pisa (Nuova Serie). Lecture Notes, Scuola Normale Superiore di Pisa. Edizioni della Normale, Pisa, 2009. [20] M. Meyries, R. Schnaubelt; Interpolation, Embeddings and traces of antisotropic fractional Sobolev spaces with temporal weights, Journal of Functional Analysis, Vol. 262, 1200-1229. [21] E. M. Ouhabaz; Analysis of heat equations on domains, London Mathematical society mono- graphs series, Princeton university press, Princeton, NJ, 31, 2005. [22] J. L. Rubio de Francia, F. J. Ruiz, J. L. Torrea; Calderón-Zygmund theory for operator-valued kernels, Adv. Math. 62 (1986), 7-48. [23] J. Simon; Sobolev, Besov and Nikolskii fractional spaces: imbeddings and comparisons for vector valued spaces on an interval, Ann. Mat. Pura Appl., 157 (4)(1990). [24] H. Triebel; Interpolation theory, function spaces, differential operators (second ed.), johann ambrosius barth, Heidelberg, (1995). Achache Mahdi Department of Mathematics, Univ. Bordeaux, Institut de Mathématiques (IMB). CNRS UMR 5251. 351, Cours de la Libération 33405 Talence, France Email address: Mahdi.Achache@math.ubordeaux.fr Tebbani Hossni Department of Mathematics, Univ. Sétif -1-, Algeria Email address: maths47@ymail.com 1. Introduction Notation 2. Properties of weighted spaces 3. Preliminaries 4. Maximal regularity for autonomous problems 5. Maximal regularity for non-autonomous problems 6. Applications 6.1. Elliptic operators in the divergence form 6.2. Robin boundary conditions References