ASYMPTOTIC BEHAVIOUR OF PARABOLIC PROBLEMS WITH DELAYS IN THE HIGHEST ORDER DERIVATIVES

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1 ASYMPTOTIC BEHAVIOUR OF PARABOLIC PROBLEMS WITH DELAYS IN THE HIGHEST ORDER DERIVATIVES ANDRÁS BÁTKAI AND ROLAND SCHNAUBELT Abstract. We use semigrou methods to investigate the artial functional differential equation u (t Au(t + db(θu(t + θ for a sectorial oerator A on a Banach sace r X and a function B : [ r, ] L(D(A, X of bounded variation having no mass at. Using a erturbation theorem due to Weiss and Staffans, we construct the solution semigrou on a roduct sace in order to solve the delay equation in a classical sense. Emloying the sectrum of the semigrou and its generator, we then study eonential dichotomy and stability of solutions. If X is a Hilbert sace, these roerties can be characterized by estimates on (λ A db(λ 1 L(X, D(A. Related results on stability also hold for general Banach saces. The case B ηa with scalar valued η is treated in some detail. 1. Introduction Perturbations of arabolic evolution equations may lead to a loss of regularity if the erturbing term is of the same order as the undelayed art and contains retardations. This already haens in the case of the simle delay equation u (t Au(t + βau(t r, t, u(t (t, t [ r, ], (1.1 where A generates an analytic C semigrou on a Banach sace X, β R\{}, r >, and : [ r, ] D(A. One can write this roblem as a Volterra equation by transforming the initial history into an inhomogeneity. In this framework, Prüss monograh [21] rovides a detailed study of the regularity of solutions of more general erturbation roblems with oerator valued kernels, see Sections of [21]. In articular, the solution oerator of the equation can be discontinuous at oints t > which are determined by the erturbation, [21, Thm.7.6]. In (1.1 this haens at t kr, k N, see [21, E.1.1]. Related investigations in a semigrou contet were carried out for the roblem u (t Au(t + A 1 u(t r + u(t (t, t [ r, ], r a(sa 2 u(t + s ds, t, (1.2 for instance in [7], [8], [9], [16], [17], [18], [19], [26, Ch.8]. (Here the oerators A 1 and A 2 are A-bounded and a L 2 [ r, ]. The very recent work [6] allows kernels K L 1 ([ r, ], L(D(A, X instead of a( A 2, but assumes further maing roerties of K 1991 Mathematics Subject Classification. 34K2, 34K3, 47D6. Key words and hrases. Partial functional differential equation, history function sace, maimal regularity, feedback, eonential dichotomy and stability, Gearhart s sectral maing theorem, Weis Wrobel theorem, Fourier tye. We like to thank Jan Prüss (Halle for stimulating discussions. A. B. was suorted by the OTKA grant Nr. F3484 and by the FKFP grant Nr. 49/21. 1

2 and A 1 with resect to etraolated norms. All these authors write (1.2 as a roblem on a suitable roduct sace X, as discussed in Chater 3, which allows to incororate the initial history function into the new state sace. On this sace one constructs a semigrou T L which gives the solution of (1.2. The semigrou corresonding to (1.1 turns out to be discontinuous in oerator norm at any t, cf. [8, Thm.3.1]. A different semigrou aroach for roblems arising in viscoelasticity was develoed in [24]. The lack of regularity of (1.1 leads to severe roblems in the study of the long term behaviour of solutions. In the semigrou framework, it is not clear whether T L satisfies the sectral maing theorem σ(t L (t \ {} e(tσ(a L, (1.3 (A L is the generator of T L since T L is not eventually norm continuous, cf. [11, IV.3]. We will indeed show in Eamle 5.8 that (1.3 is violated in the case of (1.1 for certain A A < on a Hilbert sace X rovided that β > 1. In this aer we study the more general roblem u (t Au(t + r db(θu(t + θ, t, u(t (t, t [ r, ], (1.4 for a ma B : [ r, ] L(D(A, X of bounded variation b with db([ t, ] as t. Imosing additional regularity assumtions on θ B(θ, the solution semigrou T L for (1.4 becomes eventually norm continuous. Then (1.3 holds so that the sectrum of the generator A L determines to a large etend the asymtotic behaviour of T L. This fact is eloited in [9, 5] or [17] for (1.2 with A 1, in [2], [3] for B defined on D((w A α with α < 1, and in [11, VI.6], [32] for B defined on X. On the other hand, the monograh [21] comrehensively treats the long term behaviour of a large class of roblems under various regularity assumtions on θ B(θ L(D(A, X. Thus we focus on functions B just being of bounded variation, where most of the results on asymtotics of [21] do not aly. In order to obtain ositive results for such irregular roblems, one may relace ure sectral criteria by estimates on the resolvent of the generator A L : On Hilbert saces Gearhart s sectral maing theorem characterizes in this way the eonential stability and dichotomy of a C semigrou. This fact was already used in [9] and [18] to deduce eonential stability for the equation (1.2. On general Banach saces theorems due to Weis, Wrobel and van Neerven give sufficient conditions for the eonential stability of classical solutions. See [2, 4.2] and the net section. On the other hand, Prüss has established a version of Gearhart s theorem for Volterra equations on Hilbert saces in [21, 11.2, 12.3] imosing norm estimates that etend Gearhart s conditions on the resolvent. In the resent aer we treat (1.4 by means of a semigrou aroach. We roceed in this way mainly for two reasons: First, we can aly semigrou results as the Weis Wrobel theorem not available for Volterra equations. Second, an eonential dichotomy on X cannot obtained for the Volterra equation since the initial data of (1.4 are X valued functions and not elements of X. However, the history function sace X allows to work with eonential dichotomies. Another motivation for the semigrou aroach can be seen in [24], [26, Ch.8]: It makes it ossible to transform (with some effort control roblems for delay equations into the standard setting of system theory. Having made 2

3 the decision to work with semigrous, we strive for arguments staying entirely inside this setting. Thus our aer gives an alternative aroach to some of the eistence results in Chater 7 of [21] which were obtained by more direct erturbation arguments. But certain drawbacks of the semigrou technique should be noted: On unbounded history intervals the semigrou aroach does not work well when investigating asymtotic roerties. It is further restricted to erturbation roblems (1.4 for a generator A and, as we see below, we will need an etra condition on A if X is not a Hilbert sace. In Section 3, we construct the solution semigrou directly by means of a erturbation theorem for semigrous, which was established by Weiss, [31], and Staffans, [25], for the study of feedback roblems in control theory. The resulting semigrou then yields the classical solutions of (1.4. To our knowledge, so far urely semigrou theoretic aroaches to (1.4 have been restricted to cases where B is more regular, see e.g. [2], [11], and the references therein. In [24] a somewhat different situation with scalar valued kernels defined on D(A was studied. In contrast to our method, in [7], [16], [26], [32] the roblem (1.2 is first solved by fied oint arguments and then the solutions are used to construct the semigrou. As ointed out in [21, 13.6], in the resent setting one can define the solution semigrou for (1.4 only on history function saces which enjoy the roerty of maimal regularity with resect to A. This works for instance if A generates an analytic semigrou and if one takes Hölder or Besov saces for the history functions. In this aer we use the history function sace L ([ r, ], X 1, 1 < <, and are thus forced to suose that A has maimal regularity of tye L, as introduced below. This roerty imlies that A generates an analytic semigrou. If X is a Hilbert sace, then each generator of an analytic semigrou has maimal regularity of tye L. (This is the framework of [7], [16], [26]. For X L q (Ω, q (1,, maimal L regularity has been established for large classes of oerators arising from des. These facts and other rerequisites are resented in the net section. Our main results are contained in Section 4, where we emloy the semigrou T L to study eonential stability and dichotomy of the solutions to (1.4. The erturbation aroach of Section 3 immediately yields a first robustness result for the eonential dichotomy of (1.4 valid in every Banach sace X, see Proosition 4.2. We further use the sectrum of the generator A L, which we determine in Proosition 4.3. Our formula for σ(a L seems to be a new result; even for (1.1 only certain inclusions were roven in the literature, see e.g. [9, Pro.4.2], [19, Thm.9]. In articular, the sectrum of A L coincides with the sectrum of (1.4 as defined in [21, (11.7], i.e., ρ(a L {λ C : H(λ : (λ A L λ 1 L(X, D(A}, where L λ db(θ e λθ. In [21] the norm H(λ in L(X, D(A is emloyed to determine asymtotic roerties of solutions. We show in Proosition 4.3 that the norm of R(λ, A L is essentially equivalent to the norm of H(λ in L(X, D(A. Combined with the results on the asymtotic behaviour of semigrous mentioned above, Proosition 4.3 thus yields conditions for eonential stability and dichotomy of (1.4 in terms of the growth of H(λ. Our criteria are necessary and sufficient in the Hilbert sace case, see Theorems 4.4 and 4.5. These theorems lead to refined robustness results in Corollary 4.6. So far we have dealt with the roerties of the solution u(t and its history u(t + at the same time. If one only considers u(t, one 3

4 can weaken the assumtions on H(λ, see Theorem 4.7. Our results for Banach saces X have no counterart in [21]. After Theorem 4.4 we discuss the relationshi between our theorems and those in [21] for the Hilbert sace case. In the final section, we then treat the secial case db(θ A dη(θ, where η is scalar valued and X is suosed to be a Hilbert sace. The robustness conditions of Section 4 now translate into various sector conditions for the range of the Lalace transform of dη which hence imly eonential stability or dichotomy of solutions. In some cases these conditions turn out to be shar: In Proosition 5.5 we show that if dη is ositive and A is sectorial with angle π, then growth and sectral bound of A L coincide and can be comuted quite elicitly. Nevertheless, the sectral maing theorem may be violated, as shown in Eamle 5.8. These discussions etend and imrove considerably the corresonding results from [9] and [18] for the eonential stability of roblem ( Assumtions and background We first secify our assumtions for the roblem (1.4. Let X be a Banach sace with norm, X 1 be the domain D(A of the oerator A on X with non emty resolvent set ρ(a, r >, 1 < <, and C([ r, ], X 1. Throughout we fi a number w in ρ(a of A and endow X 1 with the norm 1 (w A. We further set R(λ, A (λ A 1 for λ ρ(a and denote by σ(a C \ ρ(a the sectrum of A. Let L(Y, Z be the sace of bounded linear oerators between two Banach saces Y and Z, where L(Z L(Z, Z. The symbol means a continuous embedding. We designate by a +, a, a b, a b the ositive and negative art and the minimum and maimum of real numbers, resectively. Our main hyothesis reads as follows. (H A is a sectorial oerator in X of tye (φ, K, d, where φ (π/2, π], d R, K >, with dense domain X 1 D(A, A has maimal regularity of tye L q, and the oerator valued function B : [ r, ] L(X 1, X has bounded variation b with db([ t, ] as t. Sectoriality of tye (φ, K, d means that the sector Σ φ,d {λ C\{d} : arg(λ d < φ} belongs to ρ(a and R(λ, A K λ d 1 for λ Σ φ,d. It is well known that a sectorial, densely defined oerator A with φ > π/2 generates an analytic C semigrou T (T (t t on X. The oerator A has maimal regularity of tye L q if, for some q (1, and a > and every f L q ([, a], X, the mild solution u(t t T (t sf(s ds of the inhomogeneous roblem u (t Au(t + f(t, t, u(, (2.1 actually belongs to L q ([, a], X 1 W 1,q ([, a], X and solves (2.1 for a.e. t [, a]. If A has this roerty, then it holds in fact for all q (1, and a >. It can be seen that sectoriality of angle φ > π/2 is a necessary condition for maimal L q -regularity and that it is also sufficient if X is a Hilbert sace. Analytic semigrous on X L s (Ω, 1 < s <, arising from des tyically ossess maimal L q regularity, see e.g. [5]. Recently, Weis obtained a breakthrough in this subject, [27]. He characterized those oerators having maimal regularity of tye L q by a randomized version of the notion of sectoriality if X is a UMD-sace (e.g., X L s (Ω, 1 < s <. We refer the reader to [1], [5], [1], [21], [27] for these and related facts. 4

5 We further need the real interolation sace Y (X, X 1 1 1/, for some fied (1,. If X is a Hilbert sace and 2, then Y is again (isomorhic to a Hilbert sace (see (2.2. It is well known that X 1 Y X and that Y T ( L loc (R +, X 1, Y c T ( L ([,a],x 1 c Y, (2.2 where the constants may deend on a and, cf. [1, I.2]. Moreover, the arts of A in X 1 and Y (also denoted by A are again sectorial with the same d and φ. The analytic C semigrous generated by the arts of A are just the restrictions of T (t to Y and X 1 (thus also denoted by T (t. See [1, Thm.V.2.1.3] for these roerties. Moreover, the arts of A have the same sectrum as A by [11, Pro.IV.2.17], and hence the sectra of T (t and its restrictions also coincide due to the sectral maing theorem, see below in (2.7. We also need the continuous embedding W 1, ([, a], X L ([, a], X 1 C([, a], Y, a >, (2.3 roved in [1, Thm.III.4.1.2]. Our main interest is directed to the asymtotic behaviour of solutions, as described by the following concet (see e.g. Sections IV.2, IV.3, V.1 in [11]. Let C be the generator of a C semigrou U on a Banach sace Z. We say that U has an eonential dichotomy if there is a bounded linear rojection P on Z commuting with U(t, t, such that U(t : QZ QZ has the inverse U Q ( t and U(tP Ne δt, U Q ( tq Ne δt for t and some constants N, δ >, where Q I P. This roerty is equivalent to the fact that the unit circle T belongs to the resolvent set of U(t for some/all t >. Moreover, the dichotomy rojection is given by P 1 2πi T R(λ, U(t dλ, t >. (2.4 We say that P Z and QZ are the stable and unstable subsaces, resectively. If U has an eonential dichotomy, then it is easily verified that the oerator R(is e ist T (tp dt is the inverse of is A for s R. This shows that e ist T Q ( tq dt (2.5 U has an eonential dichotomy is ρ(c, su R(is, C <. (2.6 s R The converse imlication holds if X is a Hilbert sace due to Gearhart s theorem, [2, Thm.2.2.4], but not in general, see e.g. [11, V.1.12]. If the sectral maing theorem σ(u(t \ {} e(tσ(c, (2.7 is valid, then already ir ρ(c imlies the eonential dichotomy of U. If t U(t L(Z is continuous at some t >, then (2.7 is true. In articular, the analytic semigrou T generated by A satisfies the sectral maing theorem. Thus, if σ(a ir, then we have eonential dichotomies for T on X, Y, and X 1, and the corresonding rojections coincide on X 1 due to (2.4. Eonential stability is just the secial case of eonential dichotomy where P I or the sectral radius of U(t, t >, is strictly less than 1. We further set ω α (C inf{a R : U(t(γ C α M a e at, t } 5

6 for α [, 1] and some fied γ > ω (C. One has s(c : su{re λ : λ σ(c} ω 1 (C ω α (C ω β (C ω (C for 1 α β, and strict inequalities may occur, see e.g. Eamles and in [2]. Motivated by Gearhart s theorem, one also introduces s (C inf{a R : R(λ, C eists and is uniformly bounded for Re λ > a}. Similar as in (2.5 one sees that s (C ω (C, but again there are generators satisfying s (C < ω (C, [11, V.1.12]. In Hilbert saces, however, Gearhart s theorem imlies that ω (C s (C. This can be etended as follows: A Banach sace Z has Fourier tye q [1, 2] if the Fourier transform mas L q (R; Z continuously into L q (R, Z, where 1/q + 1/q 1. Clearly, any Banach sace has at least Fourier tye 1 and Hilbert saces have Fourier tye 2. If Z has Fourier tye q, then L s (Ω, Z also has Fourier tye q if q s q by [12, Pro.2.3] (Ω is a σ finite measure sace. The Weis Wrobel theorem now says that ω 1 q 1 (C s (C (2.8 q if Z has Fourier tye q. This covers both the stability case in Gearhart s theorem and the inequality ω 1 (C s (C valid for every Banach sace. The number 1/q 1/q in (2.8 cannot be imroved in general, see [2, E.4.2.9]. However, if (2.7 holds, then all quantities ω α (A, α, s(a, s (A coincide. Our eistence Theorem 3.6 is based on a erturbation result due to Weiss, [31], in Hilbert saces and due to Staffans, [25], in Banach saces; see also [23] for corresonding work on non autonomous roblems. These authors work in the framework of so called regular systems in control theory. For the reader s convenience, we collect here the concets and facts needed below. (In fact, [25] and [31] contain much more general results. Let Z and U be Banach saces, 1 < q <, and S be a C semigrou on Z generated by G. An outut system for S is a family of bounded linear oerators Ψ t : Z L q ([, t], U, t, satisfying [Ψ t+s z] (τ [Ψ t S(sz] (τ s (2.9 for τ [s, s + t], z Z, and t, s. Given Ψ t, there is a bounded outut oerator C : D(G U such that (Ψ t z(τ CS(τz for z D(G and τ t. Moreover, there eists the Yosida etension C of C defined by Cz lim s scr(s, Gz It can be shown that T (τz D( C and for z D( C {z Z : this limit eists in U}. (Ψ t z(τ CS(τz, z Z, a.e. τ [, t]. (2.1 These results are contained in [28], see in articular Theorem 4.5 and Proosition 4.7. An inut system for S is a family of bounded linear oerators Φ t : L q ([, t], U Z, t, such that Φ t+s u Φ t (u( + s [, t] + S(tΦ s (u [, s] (2.11 for u L q ([, s + t], U and t, s. To reresent Φ t, we need the etraolation sace Z 1 of Z for G which is the comletion of Z with resect to the norm R(w, Gz for some fied w ρ(g, see [11, II.5] for the definitions and roerties of these saces. One 6

7 can uniquely etend G to a bounded oerator G 1 : Z Z 1 which generates in Z 1 a C semigrou S 1 etending S. Every inut system can be written as Φ t u t S 1 (t sbu(s ds (2.12 where the integral eists in Z 1 and the inut oerator B L(U, Z 1 is given by 1 Bu lim Φ t t tu (in Z 1 (2.13 for u U (also denoting the corresonding constant function. These facts can be found in [29], in articular in Theorem 3.9. Given S(t, Ψ t and Φ t, we call bounded oerators F t on L q ([, t], U, t, inut outut oerators if [F t+s u] (τ [F t (u( + s [, t]] (τ s + [Ψ t Φ s (u [, s]] (τ s (2.14 for τ [s, s+t], t, s, and u L q ([, s+t], U. The system Σ (S(t, Φ t, Ψ t, F t ; t is called regular if 1 t lim [F t u ](τ dτ, (2.15 t t see [3]. It can be then shown that F t u(τ C τ T 1(τ σbu(σ dσ for u L q ([, t], U and a.e. τ [, t], but we will not use this fact. A bounded oerator K on U is called an admissible feedback for a regular system Σ if I F t K is invertible on L q ([, t], U on some (and hence all t >. Then G K G 1 + BK C with D(G K {z D( C : G K z Z} (the sum is defined in Z 1 generates the unique C semigrou S K on Z such that S K (tz S(tz + t S 1 (t sbk CS K (sz ds for z Z and t. Here one has CS K ( z (I F t K 1 Ψ t z for z Z. Observe that the erturbation BK C acts from an intermediate sace between D(G and Z into a sace larger than Z. This erturbation theorem was roved by Weiss, [31, 6+7], in the Hilbert sace setting; the above version is due to Staffans, [25, Cha.7]. 3. Eistence of solutions We want to recast (1.4 as a erturbation roblem for a semigrou on the roduct sace X (X, X 1 1 1, L ([ r, ], X 1 Y L ([ r, ], X 1 endowed with the norm ( X Y + L ([ r,],x 1. Throughout we suose that (H holds and fi some (1,. We first define { T (t + θ, θ [ r, ], t + θ, (T t (θ, θ [ r, ], t + θ <, { (t + θ, θ [ r, ], t + θ <, (S(t (θ, θ [ r, ], t + θ, 7

8 for X, L 1 ([ r, ], X, and t. Observe that S(t yields the left translation semigrou S on L ([ r, ], X 1 generated by D d dθ with domain D(D { W 1, ([ r, ], X 1 : ( }. Clearly, ρ(d C and R(λ, D (θ θ e λ(θ s (s ds θ e λs (θ s ds, θ [ r, ]. (3.1 We further set D d for W 1,1 ([ r, ], X. Emloying these maings, we dθ introduce the matri oerators ( ( T (t A T (t and A S(t D T t on X with D(A { ( X1 W 1, ([ r, ], X 1 : ( }. Finally, we set (e λ (θ e λθ for X, λ C, and θ [ r, ]. Lemma 3.1. Assume that A generates an analytic C semigrou T on X. Then A generates the C semigrou T (T (t t on X. Moreover, ω (A ω (A, σ(a σ(a, and ( R(λ, A R(λ, A : R e λ R(λ, A R(λ, D λ, λ ρ(a. Proof. Due to (2.2, T (t is a bounded oerator on X and T (t ( ( in X as t. It is easy to verify that T ( is a semigrou with growth bound ω (A, cf. [2]. Thus T is a C semigrou whose generator is denoted by Ã. Taking the Lalace transform of T, we see that R(λ, Ã R λ for Re λ > ω (A. Let λ ρ(a. One can check in a straightforward way that R λ X D(A, (λ AR λ I, and R λ (λ A ( ( ( for D(A, cf. [2]. Hence, λ ρ(a and R(λ, A Rλ for λ ρ(a. This fact also shows that A Ã. Assume that λ ρ(a. If (λ A for some X 1, then ( e λ D(A and (λ A ( ( e λ ; so that. If y Y is given, then there eists D(A such that (λ A ( ( y. In articular, X1 and (λ A y, and hence A Y. Therefore λ belongs to the resolvent set of the art of A in Y, so that λ ρ(a. Using the matri A, one can rewrite (1.4 as the evolution equation ( A u(t ( + L u(t (, t, u( ( (, (3.2 ( u (t v (t v(t v(t on X, where W 1, ([ r, ], X 1 C([ r, ], X 1, L r db(θ(θ, and L v( ( L, comare the roof of Theorem 3.6. Observe that the erturbation L mas Y C([ r, ], X 1 into X L ([ r, ], X 1 ; that is, it acts from an intermediate sace between D(A and X into a sace larger than X. In order to deal with this difficulty, we want to use the Weiss Staffans theory introduced in the revious section. The basic idea 8

9 is to factorize L BC via the sace U X 1 L ([ r, ], X 1 and the continuous oerators ( w A B : U X L ([ r, ], X 1, ( R(w, AL C : X C([ r, ], X 1 U (recall that w ρ(a was fied. Observe that C L(D(A, U if we endow D(A with the grah norm. For technical reasons, we work rimarily with the corresonding inut and outut mas defined by ( ( t u T (t s(w Au(s ds Φ t : f t T, t s(w Au(s ds [ ( ] ( ( R(w, ALg, (s Ψ t (s : CT (s, ( u L ([, t], U, t, (3.3 f ( D(A, s t, (3.4 where [g, (s](θ T (s + θ if s + θ and [g, (s](θ (s + θ if s + θ <, for θ [ r, ] and s [, t]. Let t, s and θ [ r, ]. Observe that [T t s (w Au(s](θ if t s + θ, and [T t s (w Au(s](θ T (t + θ s(w Au(s if t s + θ. Hence, the second comonent of Φ t ( u f is equal to (t+θ + T ((t + θ + s(w Au(s ds, θ [ r, ]. In what follows, we do not distinguish in notation between a function defined on an interval and its restrictions to subintervals. Lemma 3.2. Assume that (H holds. Then ( Ψ t can be etended to a bounded oerator from X into L ([, t], U such that Ψ t L ([,t],u C 1 B BV ( X for ( X, t [, r], and a constant C 1 only deending on, w, r, and the tye of A. Moreover Φ t and Ψ t, t, are inut and outut systems for T. Proof. 1 By maimal regularity and the embedding (2.3, Φ t mas contiuously into X. Let ( u f L loc (R +, U and t, s. Then ( ( s u T (t + s τ(w Au(τ dτ T (tφ s, where f I(t, s { s T (t + θ + s τ(w Au(τ dτ, t + θ, θ [ r, ], I(t, s (θ (t+θ+s + T ((t + θ + s + τ(w Au(τ dτ, t + θ <, θ [ r, ]. On the other hand, ( ( s+t u( + s T (t + s τ(w Au(τ dτ s Φ t f( + s s+(t+θ + T ((t + θ + + s τ(w Au(τ dτ so that (2.11 is verified. s 9

10 2 Identity (2.9 is clear on D(A. Let ( D(A, t r, and 1/ + 1/ 1. Using Hölder s inequality, Fubini s theorem, and (2.2, we estimate ( [ Ψ t t L [,t],u R(w, AL(T s + S(s X 1 ds [ t [ ] ds ] 1 [ t [ s T (s + θ X1 db(θ + B B B 1 BV 1 BV 1 BV s ([ t ([ t ([ t s t θ ] 1 T (s + θ X 1 db(θ ds + ] 1 T (s + θ X 1 ds db(θ + c Y db(θ ] 1 + ] 1 r s [ t [ r ] ds ] 1 (s + θ X1 db(θ r θ t [ ] 1 L ([ r,],x 1 db(θ c B BV ( X. The assertions for Ψ t can now easily be deduced, cf. [23, Lem. 2.3]. r ] 1 (s + θ X 1 db(θ ds ] 1 (s + θ X 1 ds db(θ In the net lemma we comute the Yosida etension C of C for a class of vectors which is large enough for our uroses, namely D C { ( X1 C([ r, ], X 1 : ( }. Lemma 3.3. Assume that (H holds. If ( DC, then ( D( C and C( ( C (R(w, AL, T. Moreover, T (td C D C for t. ( Proof. By aroimation, we see that Ψ t (R(w, ALg,, T for ( DC, cf. (3.4. ( Moreover, the function s g, (s C([ r, ], X 1 is continuous so that s Ψ t (s U is a continuous for s. This imlies the first assertion thanks to [28, Pro.4.7]. The second one is clear. In order to define the corresonding inut outut oerator, we use the sace X 2 D(A 2 and introduce ( [F u ( t f ] (s u CΦs for ( u f f L loc (R +, X 2 L ([ r, ], X 1 and s t. Observe that for these inuts we have Φ s ( u f DC. Lemma 3.4. Assume that (H holds. Then F t, t, can be etended to bounded oerators F t : L ([, t], U L ([, t], U, where F t C 2 db([ t, ] for t r and a constant C 2 only deending on A,, w, r. These etensions are regular inut outut oerators for S(t, Φ t and Ψ t. Proof. Identity (2.14 is an easy consequence of (2.1 and Lemmas 3.2 and 3.3 if ( u f L loc (R +, X 2 L ([ r, ], X 1. For t r and using Hölder s inequality, Fubini s theorem, and the maimal regularity of A, we further estimate ( F u t s t f L ([,t],u L T s τ (w Au(τ dτ ds X t [ s+θ T (s + θ τ(w Au(τ dτ db(θ] ds X1 s 1

11 t b( s 1 b( t 1 t c b( t 1 s t+θ t t+θ s+θ σ c b( t u L ([,t],x 1. T (s + θ τ(w Au(τ dτ db(θ ds X 1 T (σ τ(w Au(τ dτ dσ db(θ X 1 u(τ X 1 dτ db(θ Thus F t can continuously be etended to a bounded oerator in L ([, t], U if t r. Due to (2.14 and Lemma 3.2, this etension can be carried out iteratively for all t and (2.14 remains valid. To check the regularity roerty (2.15, we first comute ( ( s ( ( (w A T (τ dτ R(w, A T (s Φ s (w A (s+ wφ + s + T (τ dτ T ((s + + for ( U (also denoting the corresonding constant function. Aroimating in X1 by n X 2, we see that ( ( (s+ + ( wr(w, AL T (τ dτ F t (s R(w, AL( T ((s for s t. Therefore F t ( (s in U as s, and Ft is regular. So we have established that (T (t, Φ t, Ψ t, F t is a regular system. But we still have to identify the oerator defined in (2.13, which we temorarily denote by B : U X 1. Besides the etraolation sace X 1 for A, we need the sace Z X L 1 ([ r, ], X 1 endowed with its natural norm. Note that B mas U into Z. Lemma 3.5. Assume that (H holds. Then X Z X 1 and B B. Proof. 1 One can etend R(w, A to a continuous and injective oerator R : Z X (having the same reresentation due to Lemma 3.1. Let v Z. Then there are v n X converging to v in Z. Since v n v m X 1 R(v n v m X c v n v m Z, (v n is a Cauchy sequence in X 1. We set Jv (v n + N, where N is the sace of null sequences in (X, X 1. Observe that J : Z X 1 is well defined, linear, and bounded. If Jv for some v Z, then v n in X 1, i.e., Rv n in X. Hence, Rv which shows that v. So we have embedded Z into X 1. 2 It remains to comute the limit defining B. For t (, r] and ( U, one has ( ( ( t R(w, A T (t s(w A ds R(w, AΦ t e w R(w, A R(w, D (t+ + T ((t + + s(w A ds ( t T (s ds t e w T (s ds + I, where t 11

12 I t (θ t θ t θ t+τ e w(θ τ (w A I t (θ X1 ct (t X1 + su σ t As a result, R(w, A 1 ( t Φ t T (s ds dτ t+τ e [w w(θ τ T (s ds + T (t + τ T (σ X1. ( R(w, A e w ] dτ, ( ( (w A R(w, AB in X as t, which means that 1 t Φ t( B ( in X 1. We now come to the main result in this section, where we construct the solution semigrou for (1.4 using the above set-u. Theorem 3.6. Assume that (H holds. Then the oerator ( A L A L with D D(A L { ( X1 W 1, ([ r, ], X 1 : (, L + A Y } (3.5 generates the C semigrou T L on X satisfying ( ( t ( T L (t T (t + T 1 (t sbct L (s ds (3.6 ( ( t T (t T (t slv(sds + T t + S(t t T (3.7 t slv(s ds for v(t [T L (t ( ]2, ( D(AL, and t. Moreover, CT L ( ( L ([,t],u 2C 1 B BV ( X (3.8 for t [, t ], ( X, the constant C1 of Lemma 3.2, and a number t (, r] such that db([ t, ] (2C 2 1, where C 2 is given by Lemma 3.4. If W 1, ([ r, ], X 1, then u [T L ( ( ( ]1 belongs to C 1 (R +, Y C(R +, X 1 W 1, loc (R +, X 1 and solves (1.4. Moreover, [v(t] (θ u(t + θ where we set u(t (t for r t. The solution of (1.4 is unique in the class L loc (R +, X 1 W 1, loc (R +, X. Proof. Lemmas 3.1, 3.2, 3.4, and 3.5 show that Σ (T (t, Φ t, Ψ t, F t is a regular well osed system with inut and outut oerators B and C. Lemma 3.4 also yields that I F t is invertible on L ([, t], U and (I F t 1 2 if t t. Therefore we can aly Theorems 7.1.2, 7.1.8, 7.5.3(iii, and Remark of [25] to deduce that the oerator à L : A 1 + B C with D(ÃL : { ( D( C : ÃL ( X } (3.9 (the sum is defined in X 1 generates a C semigrou T L on X satisfying (3.6 with C relaced by C and that CT L ( ( ( (I Ft 1 Ψ t (3.1 The feedthrough oerator D in [25] equals in our setting. 12

13 for ( X. Estimate (3.8 thus follows from Lemma 3.2. Using the reresentation of D(ÃL established below and Lemma 3.3, we also see that we can relace C by C in (3.6. This equation further yields (3.7 because of (2.12 and Lemma 3.5. We have to verify that A L ÃL. Take ( D(AL. Due to Lemma 3.1 and (3.1, we obtain ( ( λar(λ, A AλR(λ, A De λ λr(λ, A + λdr(λ, D ( ( λar(λ, A A λe λ (λr(λ, A + λr(λ, D D D ( in Z as λ. This limit also eists in X 1 by Lemma 3.5 so that A ( 1 A D Z on D(A L. Moreover, D(A L D( C and C C on D(A L by Lemma 3.3. Thus the oerator AL defined by (3.9 etends A L given in (3.5. The two oerators coincide if µ A L is surjective for some µ > w (ÃL. So let ( y ψ X and set eµ + R(µ, D ψ for X 1 and µ > w (ÃL d to be determined. Using (µ D ψ, we see that ( D(AL, (µ A L ( ( y ψ R(µ, ALeµ R(µ, A(LR(µ, D ψ + y. (3.11 We have to find µ and satisfying the right hand side of this equivalence. Observe that R(µ, A L(X,X1 1+2K for µ µ and a sufficiently large µ. We further estimate Le µ X ε e µθ db(θ X1 + ε r e µθ db(θ X1 (db([ ε, ] + B BV e µε X1 1 2(1 + 2K X 1 (3.12 for sufficiently small ε (, r and large µ µ (here we use (H. We can thus invert I R(µ, ALe µ in L(X 1 for a fied µ to obtain X 1 satisfying (3.11. So we arrive at the asserted reresentation for ÃL. Finally, take ( D(AL. Then the function ( u v TL ( ( belongs to C 1 (R +, X, ( u(t v(t D(AL, and we have ( ( u (t v (t u(t AL v(t. In articular, u C 1 (R +, Y and v C(R +, W 1, ([ r, ], X 1 fulfill the equations u (t Au(t + Lv(t (the sum is taken in X and v (t Dv(t. Thus u C(R +, X 1 and the equation for v yields [v(t] (θ { u(t + θ, t + θ, θ [ r, ], (t + θ, t + θ <, θ [ r, ], (3.13 because [v(t]( u(t and v(. As a result, we have solved (1.4 in the required regularity class for the initial history. To rove uniqueness, assume that u L loc (R +, X 1 W 1, loc (R +, X solves (1.4 with and etend u to [ r, ] by. Let τ (, r]. Then we estimate τ τ [ Lu s ] ds X ds db(θ u(s + θ X1 (db[ τ, ] s 13 τ u(s X 1 ds

14 using Young s inequality for measures. The maimal regularity of A and (1.4 thus imly τ τ u(s X 1 ds c (db[ τ, ] u(s X 1 ds for a constant c > only deending on A and r. As a result, u on [, τ] for some small τ >. Iterating this argument, we obtain u on R +. Theorems in [21] imly similar results on a.e. solutions of (1.4 assuming less regularity for the initial data. In fact, the roblem is studied in [21] in full detail. The main difference is that Prüss achieves (with much effort to work on X itself, whereas we only come arbitrarily close to X letting 1. For later use we further establish a result on the inhomogeneous roblem with more general initial data using the above setting. Proosition 3.7. Assume that (H holds. Let ( X and f L (J, Y for J [, T ] and some T >. Then there is a unique u L (J, X 1 W 1, (J, X 1 C(J, Y solving u (t Au(t+f(t+ r which is given by (u(t ( T L (t + u t db(θu(t+θ, a.e. t, u(t (t, a.e. t [ r, ], (3.14 t ( f(s T L (t s ds, t. (3.15 Proof. The uniqueness of solutions can be roved as in Theorem 3.6. Denote by ( u(t v(t the right hand side of (3.15 for given ( X and f L (J, Y, and etend u by setting u(t (t for t [ r, ]. First take regular data ( D(AL and f C 1 (J, Y. In this case ( u v C(R +, D(A L C 1 (R +, X solves ( u(t ( AL + f(t (, t, u( ( u (t v (t v(t v( (, due to standard semigrou theory. Therefore v(t u t and u C 1 (R +, Y C(R +, X 1 satisfies (3.14. This shows that u(t T (t + t T (t s(f(s + Lu s ds, t. (3.16 Second, we aroimate given ( X and f L (J, Y in these saces by ( n n D(A L and f C 1 (J, Y. Denote by ( ( u u and un v n the right hand side of (3.15 for these given and aroimating data, resectively. Then (3.15 imlies that u n converges to u in L ([ r, T ], X 1 C([ r, T ], Y and v(t u t. Observe that s Lu s belongs to L (J, X. Thus we can ass to the limit also in (3.16 so that (3.16 holds for general data, too. The assertion then follows from the maimal regularity of A. 4. Asymtotic behaviour of solutions We start with a robustness result for eonential dichotomy based on (3.6 and the following lemma. Lemma 4.1. Assume that (H holds and that σ(a ir. Then T has an eonential dichotomy on X with the same eonent, and the dimension of its unstable subsace coincides with that of T. 14

15 Proof. By assumtion, T has an eonential dichotomy on X and Y with rojections P and Q I P and constants N, δ >. Moreover, Q T (1T Q ( 1Q so that Q L(X, X 1 and QX QY. We define ( Q Q. T Q ( Q Clearly, Q is a bounded rojection on X, dim QX dim QY dim QX, and T (tq QT (t. The inverse of T (t, t, on QX is given by ( TQ ( t T Q ( t. T Q ( t + Q This formula imlies that T Q ( tq N 1 e δt for t. For t > r and P I Q, it holds ( ( ( T (t P T (tp T (tp T t T Q ( Q I T t P so that T (tp N 2 e δt for t and a constant N 2. We remark that one can rove that T is eventually norm continuous using (2.2 so that T satisfies the sectral maing theorem (2.7. Combined with Lemma 3.1, this fact also shows that T inherits the eonential dichotomy of T ; but this argument does not yield the reservation of the dimension of the unstable subsace. The following results describe the asymtotic behaviour of T L. Note that the eonential stability of T L imlies that u(t Y + u t L ([ r,],x 1 Ne δt ( ( Y + L ([ r,],x 1, t, for the solutions of (1.4. Further, the eonential dichotomy of T L leads to an dichotomy on the level of the history functions u t L ([ r, ], X 1. Proosition 4.2. Assume that (H holds and that σ(a ir. If B BV is sufficiently small, then T L has an eonential dichotomy on X and the dimension of its unstable subsace coincides with that of T. If, in addition, T is eonentially stable, then also T L is eonentially stable. Proof. Due to (3.6, (3.8, (2.12, and Lemma 3.2 we have T L (t T (t c B BV where c does not deend on B and t > is given by Theorem 3.6. Thus Lemma 4.1 and [22, Pro.2.3] show the assertion rovided that B BV is sufficiently small. In the above roof we have used Proosition 2.3 of [22] which deals with non autonomous roblems. In the resent situation one can in fact relace this result by simler arguments based on standard sectral theory, cf. Corollary 4.6. We have not secified the smallness condition for B BV in Proosition 4.2. Insecting the roofs, one sees that it involves r,, w, the dichotomy constants N, δ of T, the tye of A, a constant related with the maimal regularity of A, and the decay of db([ t, ] as t. But the aroach of Proosition 4.2 does not give very shar conditions and is of course restricted to oerators A having an eonential dichotomy. To obtain further results, we now emloy the sectrum of A L. We first comute σ(a L thereby imroving and etending various facts established in [9]. See also [19] for an investigation of the (generalized 15

16 eigenfunction saces of A L for (1.2, and [2], [11], [32] for the case of more regular B. We set L λ Le λ : X 1 X and C a {λ C : Re λ a}. Proosition 4.3. Assume that (H holds. Then ρ(a L {λ C : H(λ : (λ A L λ 1 L(X, X 1 }, ( H(λ H(λLR(λ, D R(λ, A L : R e λ H(λ (e λ H(λL + IR(λ, D λ, R(λ, A L L(X c a H(λ L(X,X1, for λ ρ(a L with Re λ a for some a R. Moreover, if H(λ L(X,X1 is unbounded on a closed subset of C a ρ(a L for some a R, then R(λ, A L L(X is unbounded on the same set. Proof. 1 Assume that H(λ (λ A L λ 1 L(X, X 1 eists for some λ C. Then it is clear that R λ is a bounded oerator on X. More recisely, since e λ and the norm of R(λ, D : L ([ r, ], X 1 C([ r, ], X 1 are uniformly bounded for Re λ a, one has the estimate R λ L(X c a H(λ L(X,X1 if Re λ a. Take ( ( X. Then y ( ψ Rλ belongs to X 1 W 1, ([ r, ], X 1, ψ( y, and Ay + Lψ (A + L λ H(λ + (A + L λ H(λLR(λ, D + LR(λ, D λh(λ + λh(λlr(λ, D λy Y. ( This shows that R λ D(AL and (λ A L R λ I (notice that (λ De λ. Thus λ A L is surjective. Let (λ A L ( ( for some D(AL. Then e λ and hence (λ A L λ, so that and. As a result, λ ρ(a L and R λ R(λ, A L. 2 Conversely, let λ ρ(a L. We etend A L to an oerator on W X L ([ r, ], X 1 by setting ( A L à L, D(ÃL { ( D X1 W 1, ([ r, ], X 1 : ( }. Observe that A L is the art of à L in X. As in (3.12 we can choose a sufficiently large µ > ω (A L d such that L µ R(µ, A L(X 1/2. Hence µ A L µ has the inverse H(µ R(µ, A(I L µ R(µ, A 1 L(X, X 1. (4.1 As in the first ste, one sees that µ ρ(ãl and R(µ, ÃL R µ. Since R µ : W X is bounded, we also obtain that the grah norm of à L in W dominates the norm of X. Consequently, ρ(ãl ρ(a L λ by [11, Pro.IV.2.17]. If (λ A L λ for some X 1, then ( e λ belongs to the kernel of λ à L, so that. Given y X, there eists ( D( à L with (λ ÃL ( ( y. This imlies that e λ and (λ A L λ y. Summing u, λ A L λ : X 1 X is bijective and continuous, and thus invertible. 3 Assume that a n : H(λ n L(X,X1 as n for some λ n in a closed subset of C a ρ(a L. Then λ n and thus λ n Σ φ,d for n n and some n 1. Moreover, H(λ L(X, X 1 is uniformly bounded on a half lane C γ for a sufficiently large γ a due to (4.1. Hence, Re λ n is uniformly bounded. Take unit vectors n X such that 16

17 H(λ n n X1 a n 1/n. Define y n R(λ n, A n and b n a /2 n for 1/ + 1/ 1. Observe that the vectors y n are uniformly bounded in X 1 for n n. We further take n 1 n such that b n 1/r for n n 1 and set n (θ b n e λnθ 1 [ b 1 n,](θ, θ [ r, ], for n n 1 n. We then obtain n c b n and, using (3.1, { bn θ, θ 1/b n, R(λ n, D n y n e λn ψ n y n, where ψ n (θ : 1, b 1 n θ r. This equality yields [R(λ n, A L ( ny n ]2 e λn (H(λ n L λn y n + y n + e λn H(λ n Le λn (ψ n 1y n + e λn (ψ n 1y n : S 1 + S 2 + S 3. First, note that S 1 e λn H(λ n n. Using ψ n 1 1 [ b 1 n,], we estimate S 2 L ([ r,],x 1 c 1 db([ b 1 n, ] a n and S 3 L ([ r,],x 1 c 2 b 1 n for some constants c k indeendent of n. Combining these facts, we arrive at 1 c 3 b n R(λ n, A L L(X R(λ n, A L ( 1 ny n X c 4 (a n 1/n c 1 db([ b 1 n, ] a n c 2 b 1 n c 5 a n for sufficiently large n and constants c k >. As a result, R(λ n, A L c a n. Our following main theorems are immediate consequences of the above roosition and the results from sectral theory of semigrous recalled in Section 2. Theorem 4.4. Assume that (H holds and that X is Hilbert sace. Take 2. (a T L has an eonential dichotomy on X if and only if H(is eists and is uniformly bounded in L(X, X 1 for s R. (b T L is eonentially stable on X if and only if H(λ eists and is uniformly bounded in L(X, X 1 for Re λ. Proof. Observe that X is a Hilbert sace if X is a Hilbert sace and 2. Thus Gearhart s theorem, [2, Thm.2.2.4], characterizes eonential dichotomy and stability of T L by the uniform boundedness of R(λ, A L for λ ir and Re λ, resectively. The assertions hence follow from Proosition 4.3. We comare the above result with Prüss monograh [21]. Assume that A L has an eonential dichotomy. Then for every f L 2 (R, Y there is a unique ( u v L 2 (R, X C(R, X such that ( ( u(t u(s t ( f(τ T L (t s + T L (t τ dτ, t s, v(t v(s see e.g. [4, Thm.4.33]. Proosition 3.7 further shows that u L 2 loc (R, X 1 W 1,2 loc (R, X L 2 (R, Y and u (t Au(t + Lu t + f(t 17 s

18 for a.e. t R. This roerty is called admissibility of L 2 (R, Y. Admissibility of L 2 (R, X was obtained in [21, Pro.12.1] (with slightly less local regularity also assuming that H(is : X X 1 is bounded for s R and that X is a Hilbert sace. Conversely, admissibility of BUC(R, X imlies boundedness of H(is : X X 1 for s R by [21, Pro.11.5]. Prüss results are in fact more general and fleible than ours. He can allow for sectrum on ir and unboundedness of H(is L(X,X1 imosing regularity and sectral conditions on the class of inhomogeneities f. On the other hand, it is not clear whether admissibility of L 2 (R, X or L 2 (R, Y in turn imlies the eonential dichotomy of A L on X. (Roughly seaking, admissibility and eonential dichotomy are equivalent for non retarded roblems, [4], and for retarded equations with B BV ([ r, ], L(X, [14]. In the following two theorems we etend art (b of Theorem 4.4 to general Banach saces. These results have no counterart in [21]. Theorem 4.5. Assume that (H holds and that X has Fourier tye q [1, 2]. Take [q, q ] (1,. If H(λ eists and is uniformly bounded in L(X, X 1 for Re λ, then there are constants N, δ > such that T L (t ( ( X Ne δt (γ A L 1/q 1/q for 1/q + 1/q 1 and ( D((γ AL 1/q 1/q (where γ > ω (A L is fied. Proof. It is clear that X 1 has Fourier tye q. Let us check that also Y has Fourier tye q. We may assume that q > 1 since every Banach sace has Fourier tye 1. For 1 < q q, the estimate in (2.2 and the integral version of Jessen s inequality (cf. [15, 22, 23] imly [ [ 1 Ff L q (R,Y c T (τ(ff(t X 1 dτ R [ 1 [ c [ 1 [ c R R [ 1 [ c R ] q ] 1 q dt ] ] T (τ(ff(t q q X 1 dt 1 dτ ] ] [F(T (τf](t q q X 1 dt 1 dτ T (τf(s q X 1 ds ] q ] 1 dτ [ [ 1 ] q ] 1 c T (τf(s q X 1 dτ ds R c f L q (R,Y, where the F denotes the Fourier transform and f is an X 1 valued Schwartz function, say. Moreover, L ([ r, ], X 1 and thus W 1, ([ r, ], X 1 have Fourier tye q by [12, Pro.2.3]. Therefore X has Fourier tye q. On the other hand, s (A L < by the assumtion and Proosition 4.3. The theorem is now a consequence of the Weis Wrobel theorem (2.8. We can now artly imrove the robustness result Proosition 4.2. Corollary 4.6. Assume that (H holds, that A is invertible, and that X has Fourier tye q [1, 2]. Take [q, q ] (1, and w, i.e., X1 A X. (a If s(a < and B BV < 1 1+K, then the assertion of Theorem 4.5 holds. In articular, T L is eonentially stable if X is a Hilbert sace. 18

19 (b If X is a Hilbert sace (i.e., q 2, ir ρ(a, and B BV < inf s R (1 + sr(is, A L(X 1, then T L has an eonential dichotomy and the unstable subsaces of T L and T have the same dimension. Proof. (a For s(a < and Re λ, we have L λ R(λ, A L(X B BV R(λ, A L(X,X1 B BV (1 + K : c < 1 by the assumtion. Thus there eists H(λ R(λ, A [I L λ R(λ, A] 1 L(X, X 1. and H(λ (1 + K/(1 c. Therefore assertion (a follows from Theorem 4.5. (b Let σ(a ir. As above one verifies that, if the estimate for B holds, L is R(is, A L(X c < 1 and H(is L(X,X1 c. for s R. Thus A L has an eonential dichotomy by Theorem 4.4. In order to check the assertion concerning the dimensions, we introduce the erturbations αb for α [, 1]. Since the hyotheses hold for αb with the same constants, we obtain the corresonding eonentially dichotomic semigrous T αl : T α, where T T and T 1 T L. In view of [13, Lem.II.4.3] and Lemma 4.1, we have to show that the rojections Q α for T α deend continuously on α in oerator norm. This is the case if α T α (t L(X is continuous for some t >, because of formula (2.4. We now emloy more results from feedback theory in order to establish the continuity of this ma. We fi the inut oerator B and the outut oerator C. Then we obtain the semigrou T α from T by the admissible feedback αi : U U, cf. Theorem 3.6. In fact, there eists another regular system Σ α (T α (t, Φ α t, Ψ α t, F α t with Ψ α t (I αf t 1 Ψ t, see [25, Thm.7.1.2]. Moreover, (β αi : U U, β [, 1], is an admissible feedback for Σ α roducing Σ β due to [25, Lem.7.1.7]. In articular, T β (t T α (t (β αφ α t Ψ β t for the number t > given in Theorem 3.6. This eression tends to in norm as β α by Lemma 3.4. If we restrict ourselves to decay of u(t in Y, then it suffices that H is bounded in a weaker norm. As a samle of ossible consequences of Sections in [2], we resent a result on eonential stability not involving geometric roerties of X. We denote by Π 1 : X Y the rojection onto the first comonent of X. Theorem 4.7. Assume that (H holds and that H(λ : X Y is bounded for Re λ > ε and some ε >. Then Π 1 T L (t ( Y Ne δt (w A L ( X for ( D(AL, t, and some constants N, δ >. Proof. Due to Proosition 4.3 the oerator Π 1 R(λ, A L + ε : X Y has a bounded analytic etension to the halflane Re λ >. Thus Theorem of [2] shows that Π 1 e εt T L (t M(1 + t, which yields the assertion. The reader will have noticed that our results on eonential dichotomy in the non Hilbert sace case are restricted to Proosition 4.2. This is due to the fact that, so far, we have no theorem of Weis Wrobel tye for eonential dichotomy. 19

20 5. Scalar valued kernels In this section we treat the case that B ηa for a scalar valued function η in the sace BV ([ r, ] of functions with total variation b satisfying db([ t, ] as t. We normalize η by η(. Throughout we assume that X is Hilbert sace and that A is sectorial with φ > π/2 and dense domain X 1 X. Thus (H holds. We set C a {λ C : Re λ a}, Σ φ Σ φ,, and dη(λ r e λθ dη(θ, λ C. Observe that λ A L λ λ (1 + dη(λa. Hence, λ ρ(a L if and only if 1 + dη(λ and µ(λ : λ(1 + dη(λ 1 ρ(a, and then H(λ (1+ dη(λ 1 R(µ(λ, A. We start with an easy case where the sectrum of A and the range of dη do not interfere. Proosition 5.1. Let X be a Hilbert sace. Let A be sectorial with d and φ > π/2, and let η BV ([ r, ]. If there are φ (, φ π/2 and a > such that 1 + dη(c a Σ φ, (5.1 then ω(a L < a. If d < and (5.1 is valid for a, then T L is eonentially stable. Condition (5.1 holds in articular, if φ π and su dη(λ < 1. (5.2 Re λ a Proof. Let Re λ a >. Then (5.1 imlies 1 + dη(λ and 1 + dη(λ 1 c for some constant c. Observe that µ(λ Σ φ by (5.1. Hence H(λ eists and H(λ L(X,X1 1 + dη(λ 1 (w AR(µ(λ, A L(X wk λ 1 + c(k + 1 c. Due to Proosition 4.3, the resolvent of A L is bounded on C a and thus it is bounded on C a for some a < a. Therefore Gearhart s theorem [2, Thm.2.2.4] shows the first assertion. If d <, this argument also works for a so that second assertion is also roved. The last assertion is clearly true. Condition (5.2 was already used in [9, Thm.4.5] and [18, Thm.2] for secial classes of η and A. The estimates for ω (A L become more comlicated if we have d > or a <. Proosition 5.2. Let X be a Hilbert sace, A be sectorial with φ > π/2 and d, and let η BV ([ r, ]. Assume that there are φ (, φ π/2 and a such that (5.1 holds. Hence Re(1 + dη(λ 1 r for Re λ a and some r >. If ar > d, then ω (A L < a. Proof. Let λ ρ + iτ for ρ a > and τ R, and let (1 + dη(λ 1 + iy for r, y R. Observe that (1 + dη(λ 1 c and y tan φ by (5.1. For y, τ, we have Re(µ(λ ar ar ρ + τy a(r + τy τy (τ + ρy tan φ tan(φ Im(µ(λ ar, 2

21 i.e., µ(λ ar Σ φ +π/2. This fact can be roved in a similar way if y and τ have other signs. Using ar > d, the assertion can now be shown as in the revious roosition. Proosition 5.3. Let X be a Hilbert sace, A be sectorial with φ π and d, and let η BV ([ r, ]. Assume that there are φ (, π/2 and a such that (5.1 holds and that If as > d, then ω (A L < a. s : su{ + y 2 / : + iy (1 + dη(c a 1,, y R} <. Proof. We use the notation of the revious roof. In view of Proosition 5.1 there is nothing to rove if Re λ. So let > ρ a. Since µ(λ ρ τy + i(τ + ρy, it is easy to see that µ(λ Σ φ for some φ < π and large τ. Moreover, if Im µ(λ, then τ ρy/ and thus µ(λ ρ( + y 2 / as. Combining these facts, we see that µ(λ Σ ψ,as for some ψ < π. The assertion now follows as above. Corollary 5.4. Let X be a Hilbert sace. Let A be sectorial with φ π and let η BV ([ r, ]. Assume that there is a R such that su dη(λ q < 1. Re λ a (a If a, d and a/(1 + q > s(a, then ω (A L < a. (b If d and a/(1 q > s(a, then ω (A L < a. Proof. Observe that (1 + dη(c a 1 is contained in the ball in C which is symmetric with resect to R and whose boundary intersects R at the oints 1/(1 + q and 1/(1 q. In Proositions 5.2 and 5.3 we thus have r 1/(1 + q and s 1/(1 q, which yields both assertions. The net rosition shows that the seemingly rough smallness condition (5.2 gives a recise equality for ω (A L if η is decreasing, i.e., the measure dη is ositive. (See also [18, Thm.2] for (1.2 with A A 1 A 2. In fact, we even obtain ω (A L s(a L. We oint out that standard results on ositive semigrous, see e.g. [2], are not alicable here since we require no order roerties for X and T (t; and even if we did so: Y and X 1 would not be Banach lattices in most alications. Using η BV ([ r, ], one checks that the function R R +, a dη(a, is strictly decreasing and bijective if dη and dη. Proosition 5.5. Let X be a Hilbert sace, η BV ([ r, ], and A be sectorial with d, φ π, and X 1 X. Assume that dη is a ositive measure and not equal to. Then eists a unique ω R such that dη(ω 1. Denote by a the infimum of the numbers a > ω such that α(1 + dη(α 1 > s(a for all α a. Then s(a L ω (A L ma{ω, a }. Proof. Let Re λ a > ma{ω, a }. Then dη(λ r e aθ dη(θ : q < dη(ω 1. 21

22 Thus condition (5.2 holds and a 1 q a > s(a. 1 + dη(a Corollary 5.4(b now yields ω (A L ma{ω, a }. On the other hand, ω A L ω ω since dη(ω 1. Observe that a (1 + dη(a 1 s(a if a > ω, so that in this case a A L a (1 + dη(a (s(a A. This shows that H(ω and H(a either do not eist or do not belong to L(X, X 1, so that ma{ω, a } s(a L ω (A L by Proosition 4.3. Eamle 5.6. Consider equation (1.1 on a Hilbert sace X with β R \ {}, i.e., η β1 { r}, dη β δ r, and dη(λ βe λr. Assume that A is sectorial with φ π and X X 1. We denote by a (res., a 1 the infimum of the numbers a > 1 log β such that r α(1 + βe αr 1 > s(a (res., α(1 + β e αr 1 > s(a for α a. (a Let d and β 1. Then s(a L ω (A L 1 log β. In fact, Proosition 5.5 r shows these equalities if β 1. If β 1, then dη(λ c a < 1 for Re λ a > 1 log β. r Thus, ω (A L 1 log β by Proosition 5.1. On the other hand, dη( 1 (iπ + log β 1 r r so that s(a L 1 log β by Proosition 4.3. r (b Let d and 1 < β <. Then Proosition 5.5 directly imlies that s(a L ω (A L ma{ 1 log β, a r }. (c If d, then ω (A L ma{a 1, 1 log β } by Corollary 5.4. Further, if β > we r have s(a L ω (A L ma{a 1, 1 log β} since ω A L r ω ω for ω : 1 log β and r a 1 A L a1 (1 + βe a 1r (s(a A if a 1 > ω. We now turn our attention to eonential dichotomy. For simlicity we concentrate on the case φ π. Proosition 5.7. Let X be a Hilbert sace. Let A be invertible and sectorial with angle φ π, and let η BV ([ r, ]. If 1 + dη(ir : {λ C \ {} : arg λ < ψ or arg( λ < ψ} (5.3 for some ψ < π/2, then T L has an eonential dichotomy. Condition (5.3 holds in articular, if su s R dη(is < 1. Proof. From (5.3 we deduce that 1 + dη(is 1 c and that µ(is belongs to the rotated double sector i, for s R. This easily imlies that (w AH(is (1 + dη(is 1 (w AR(µ(is, A is uniformly bounded for s R. The assertion is then a consequence of Theorem 4.4. Eamle 5.8. Let dη βδ r for some β R. (a Let A be sectorial with φ > π/2. Then A L does not have an eonential dichotomy if β 1 or if β > 1 and σ(a R is unbounded. (This fact indeed holds for every Banach sace X. (b Let β > 1. Then there eists a self adjoint and negative definite oerator A on X l 2 such that ir belongs to ρ(a L. 22