Semigroup Forum DOI 10.1007/s00233-013-9534-3 R E S E A R C H A RT I C L E

Analyticity and Gevrey class regularity for a strongly damped wave equation with hyperbolic dynamic boundary conditions Philip Jameson Graber · Irena Lasiecka

Received: 22 April 2013 / Accepted: 13 September 2013 © Springer Science+Business Media New York 2013

Abstract We consider a linear system of PDEs of the form utt − cut − u = 0 in Ω × (0, T ) utt + ∂n (u + cut ) − Γ (cαut + u) = 0 on Γ1 × (0, T )

(1)

u = 0 on Γ0 × (0, T ) (u(0), ut (0), u|Γ1 (0), ut |Γ1 (0)) ∈ H on a bounded domain Ω with boundary Γ = Γ1 ∪ Γ0 . We show that the system generates a strongly continuous semigroup T (t) which is analytic for α > 0 and of Gevrey class for α = 0. In both cases the flow exhibits a regularizing effect on the data. In particular, we prove quantitative time-smoothing estimates of the form (d/dt)T (t)  |t|−1 for α > 0, (d/dt)T (t)  |t|−2 for α = 0. Moreover, when α = 0 we prove a novel result which shows that these estimates hold under relatively bounded perturbations up to 1/2 power of the generator. Keywords Gevrey’s semigroups · Analytic semigroups · Wave equation with dynamic boundary conditions · Pseudodifferential operators

Communicated by Jerome A. Goldstein. Author supported by NSF Grant DMS-0104305.

B

P.J. Graber ( ) Unité de Mathématiques Appliquées, Commands (ENSTA ParisTech, INRIA Saclay), 828, Boulevard des Marchaux, 91762 Palaiseau Cedex, France e-mail: [email protected] I. Lasiecka Department of Mathematics, University of Virginia, Charlottesville, VA 22901, USA

P.J. Graber, I. Lasiecka

1 Introduction 1.1 Description of the problem Let Ω be a bounded, open set in RN , with smooth boundary Γ = Γ0 ∪ Γ1 . Consider the following system of partial differential equations (PDEs): utt − cut − u = 0 in Ω × (0, T ) utt + ∂n (u + cut ) − Γ (cαut + u) = 0 on Γ1 × (0, T )

(2)

u = 0 on Γ0 × (0, T ) (u(0), ut (0), u|Γ1 (0), ut |Γ1 (0)) ∈ H The state space H is a Hilbert subspace of HΓ10 (Ω) × L2 (Ω) × H 1 (Γ1 ) × L2 (Γ1 ),  is the Laplace operator on RN , Γ is the Laplace-Beltrami operator on the N − 1dimensional submanifold Γ while ∂n is the derivative in the direction normal to Γ . (For more on our notation, see Sect. 2 below.) For simplicity we shall assume that Γ1 and Γ0 are disjoint. This is to avoid difficulties resulting from studying Zaremba type of problems. Remark 1.1 We note that ut |Γ1 (0) may not be the trace of ut (0)|Γ1 , even if the latter is sufficiently smooth. However, we will show that for all t > 0 the dynamics smoothes out and ut |Γ1 (t) agrees with ut (t)|Γ1 for t > 0. System (2) represents the dynamics of a wave in a bounded domain without neglecting the momentum on the boundary, hence the name dynamic boundary conditions. Such problems are of physical interest in the study of wave-structure interactions. In the above model, the boundary itself is governed by a wave equation coupled with the dynamics governing the acoustic pressure inside an acoustic chamber. System (2) can be easily generalized to a more abstract form where , Γ are replaced by some differential elliptic operators and the coupling between interior and boundary dynamics occurs through appropriate trace operator (e.g. plate or shell equation with free boundary conditions). However, such generalization is bound to introduce more complicated notations which will not add to the conceptual understanding of the problem. The benchmark problem presented has all the features needed. The goal of this study is to establish that the linear flow exhibits smoothing in time. Our main result states that system (2) generates an analytic semigroup for all α > 0, while for the case when α = 0 the semigroup is of Gevrey class. Gevrey class semigroups [30, 33] are not as smooth as analytic semigroups but more regular than a general differentiable semigroup. Such additional regularity can be applied to semilinear variations of the problem as for parabolic problems, though due to space constraints we will not consider this here. Smoothing estimates, in the context of strongly damped wave equations, have been a subject of long studies [3–5, 19, 27, 36]. It is known by now that the wave equation utt − cut − u = 0 equipped with either Dirichlet or Neumann-Robin boundary conditions is associated with the analytic semigroup defined on a variety of spaces

Analyticity and Gevrey class regularity for a strongly damped wave

based on a full scale of Lp (Ω) spaces, p ∈ [1, ∞) [24]. A more general result is available which says that for any self-adjoint positive operator A defined on a Hilbert space H the abstract strongly damped wave equation utt + cAα ut + Au = 0

(3)

generates an analytic semigroup eAt on H ≡ D(A1/2 ) × H whenever α ∈ [1/2, 1] [4]. For α ∈ (0, 1/2) the semigroup is of Gevrey type [5]. In fact, the result proved in [5] allows to consider the “noncommutative” situation where Aα is replaced by B with appropriate bounds from below and above relative to Aα . More recently Haraux and Otani [16] considered similar questions with the aim of deriving explicit time-dependent smoothing estimates. With reference to (3), in addition to proving the analytic estimate |AeAt |L(H) ≤

C t

for α ∈ [1/2, 1], the authors also show that for α ∈ (0, 1/2) one has |AeAt |L(H) ≤

C 1

(4)

t 2α

The above estimate is proved in [16] to be optimal. The estimate in (4) implies Gevrey regularity [5, 33]. It is also noteworthy to point out that some of the smoothing estimates obtained in [16] apply to non-Hilbertian setup including continuous functions and L∞ spaces. Lp theory has been also applied in [1] to the study of semilinear damped wave equations with nonlinearities of critical exponent. In this article we are interested in a strong damping added to the wave dynamics which possibly interacts with the boundary. This leads to the model with dynamic boundary conditions. Such models have received considerable attention. Recent work in [14] generalizes previous work (see references therein) on strongly damped wave equations with the acceleration term appearing on the boundary, showing how the analyticity of the underlying semigroup can be exploited to obtain solutions in the presence of highly nonlinear sources in the interior and on the boundary. See also [12] for related work on asymptotic behavior of solutions. A nice exposition of semigroup theory in the context of the models with either acoustic or Wentzell boundary conditions can be found in [11]. Non-autonomous wave equations with general Wentzell boundary conditions are discussed in [7, 9]. A rather complete Lp regularity theory pertinent to parabolic problems with dynamic boundary conditions has been recently presented in [15]. Problems involving abstract dynamic boundary conditions are studied in the works [25, 26, 28, 37, 38] and references therein. The goal of these works is to establish semigroup well-posedness and analyticity in an abstract Banach space setting, using the theory of semigroups generated by operator matrices [2, 6, 29]. The paradigm established by these authors is one of decoupling. Roughly speaking this means analyzing a coupled system by transforming the operator matrix generating the dynamics into a block diagonal operator matrix, from which spectral properties and

P.J. Graber, I. Lasiecka

regularity can be deduced. Depending on the coupling operators, one can show under quite general assumptions that good properties of the “decoupled” system (in our case, the strongly damped wave equation with homogeneous boundary conditions) are preserved. It is thus natural to ask whether the specific model considered in this paper can be treated from this more general point of view. However, the answer would appear to be negative. The main reason for the inapplicability of the abstract approach seems to be that it treats the operators appearing on the boundary like a perturbation, rather than a carrier of regularity as in our approach. Thus even in the case when α > 0, our problem (2) does not fit into the abstract framework considered in such works, cf. Assumptions 3.1 and 4.1 in [28]. Technical details are given in the remark below. Remark 1.2 In [28] Mugnolo studied wave equations with dynamic boundary conditions in an abstract setting: utt = Au(t) + Cut (t) wtt = B1 u(t) + B2 ut (t) + B3 w(t) + B4 wt (t)

(5)

where the operators A, C, Bi , i = 1 − 4 satisfy suitable conditions and variables u and w are connected via a “trace” type of operator L so that w = Lu. The basic framework presented by Mugnolo aims at proving generation of C0 semigroups (resp. analyticity) as being equivalent having the same properties for the blocks of operators     0 I 0 I and A C B3 B4 considered on Y × X and ∂Y × ∂X where Y ⊂ X, ∂Y ⊂ ∂X are Banach spaces (see Theorems 3.3, 3.8, 4.5, 4.12 [28]). The model under our consideration (2) can be recast in the form (5) by setting A = −, B1 = −

∂ , ∂ν

C = −c, B2 = −c

Lu = γ ≡ u|Γ1 , ∂ , ∂ν

B3 = −Γ ,

B4 = −cαΓ

with the corresponding spaces: X = L2 (Ω),

Y = HΓ10 (Ω),

∂X = L2 (Γ1 ),

∂Y = H 1 (Γ1 )

Results in this abstract setting depend critically on the “coupling” operator L. Here the relevant property is that L is bounded Y → ∂X but unbounded X → ∂X, which corresponds to the results of [28, Sect. 4]. However, both Theorems 4.5 and 4.12 are inapplicable to our model due to the severity of assumptions imposed on the operators Bi , i = 1 − 4. To wit, Part I of Theorem 4.5 assumes that ∂ B1 ∈ L(Y, ∂X), B2 ∈ L(X, ∂X). The above is never satisfied with B1 = ∂ν and the choices of spaces X, Y, ∂X. In addition, operators B3 , B4 do not comply with regularity requirements postulated in Theorem 4.5 unless they are bounded.

Analyticity and Gevrey class regularity for a strongly damped wave

What distinguishes our model is that the boundary dynamics contains the infinite dimensional PDE generator Γ . The interest and physical importance for the presence in the model of this boundary diffusive term has been elaborated in [13]. As pointed out in [13] this corresponds to the heat source located on the boundary and interacting with a heat flux in the surrounding area. On the mathematical side this changes the topological properties of the potential energy corresponding to the boundary evolution. The addition of the extra potential term on the boundary makes it more difficult to achieve the regularity required for analyticity the overall semigroup. This, in turn, undercuts the effectiveness of the strong interior damping. Our result quantifies this statement. It shows that while analyticity is achievable with an additional strong damping placed on the boundary, in the absence of this damping one still obtains a “smoothing effect” expressed by the following estimate with singularity at the origin: |AeAt |L(H) ≤

C , t2

t >0

(6)

The above regularity implies Gevrey regularity of type δ > 2 [33]. Qualitatively this means that the addition of Γ to the boundary dynamics corresponds to the deterioration of internal damping from α = 1 to α = 1/4 [5]. The above numerology is supported by the trace theory. Remark 1.3 The role of the presence of Γ in dynamic boundary conditions may be sometimes intriguing. It was shown in [35] that standard heat equation with dynamic boundary conditions and Neumann data of “a wrong sign” leads to the ill-posed dynamics with respect to any reasonable topology. However, an addition of Laplace Beltrami operator to the boundary data changes the scenario by making the problem well-posed for sufficiently smooth initial data. Thus, in this case the wellposedness theory benefits from the presence of Γ on the boundary. However the opposite holds for strongly damped wave equation. The addition of boundary diffusion reverts analyticity into Gevrey’s property only. 1.2 Main results We conclude this introduction with a statement of the main results. First we provide a definition of Gevrey class semigroups. Definition 1.4 A strongly continuous semigroup T (t) = eAt , defined on a Banach space X, is of Gevrey class δ > 0 for t > t0 iff T (t) is differentiable for t > t0 and for every compact K ⊂ (t0 , ∞) and each θ > 0 there exists a constant C = C(θ, K) such that T (n) (t)L(X) ≤ Cθ n (n!)δ ,

for all t ∈ K, n = 0, 1, 2 . . .

Our main result concerns the semigroup T (t) = eAα t generated by (2), with Aα given in (16) whose existence is proved in Sect. 2.2.

P.J. Graber, I. Lasiecka

Theorem 1.5 (i) If α > 0 then the semigroup eAα t obtained from Proposition 2.2 is analytic, and in particular it satisfies the estimate    d A t C  e α  (7) ≤ , t > 0,   dt t L (H ) where C > 0 is a fixed constant. The following resolvent estimate is satisfied: for a sufficiently large constant C > 0, we have R(iβ, Aα )L(H) ≤

C |β|

for all β ∈ R \ {0}.

(8)

(ii) If α = 0 then the semigroup eA0 t is of Gevrey class δ for all δ > 2, and it satisfies the estimate    d A t CT  e 0 ≤ 2 , t ∈ (0, T ], (9)   dt t L (H ) where CT is a constant depending on T > 0. The following resolvent estimate is also satisfied: for sufficiently large constants R, C > 0, we have that |β|1/2 R(iβ, A0 )L(H) ≤ C,

for all β ∈ R, |β| ≥ R.

(10)

We remark that the resolvent estimate (8) for analytic semigroups is equivalent to the existence of a time estimate of the form (7) [30]. However, to our knowledge no such equivalence exists between the time estimate (9), the resolvent estimate (10), and the Gevrey regularity of a semigroup. All the results cited in this paper to prove Gevrey regularity are sufficient conditions [33], and it is not clear whether they are necessary. The optimality of the value of δ > 2 can be inferred from numerical computations of the spectrum of the operator. The latter is delineated by a parabolic curve [10] which is consistent with Gevrey’s regularity of the type δ. However, we were unable to confirm the optimality analytically. Remark 1.6 Since the semigroup eA0 t is also exponentially stable [10], by the semigroup property and Theorem 1.5 one obtains global estimate for the derivative:    d A t  e 0  dt 

L (H )

≤

ce−ωt , t2

t ≥ 0, for some c, ω > 0

(11)

t ≥ 0, for some cn , ωn > 0

(12)

which is easily boot strapped to  n   d A t 0    dt n e 

L (H )

≤

cn e−ωn t , t 2n

Theorem 1.5 along with perturbation result in Sect. 4 imply the following:

Analyticity and Gevrey class regularity for a strongly damped wave

Corollary 1.7 The result of Theorem 1.5 applies to a perturbed operator A0 + P where P is a relatively bounded with respect to A0 with a relative bound < 1/2. In particular, time smoothing estimate is valid for any perturbed generator with P bounded on H. The result of Corollary 1.7 should be contrasted with the corresponding result for semigroups that are just differentiable. It turns out that differentiability is not stable under bounded perturbations [31]. On the other hand, it is well known that analyticity is stable under relatively bounded perturbations. Our perturbation result in Sect. 4 provides stability of time smoothing estimates (a stronger property than Gevrey regularity) for all relatively bounded perturbations with relative bound up to 1δ for a Gevrey semigroup of class greater than δ. However, to the best of our knowledge, it remains an open question whether Gevrey class semigroups are closed under (relatively) bounded perturbations of the generator. Remark 1.8 The model presented in (2) can be generalized to Riemannian setting where Ω is replaced by an N -dimensional manifold endowed with a Riemannian metric g(·, ·) = ·, · g and norm |X|2g ≡ g(X, X). The formulation in terms of Riemannian manifolds allows for a broad range of applications. For example, several authors have studied wave equations with variable coefficients in Euclidean space having the form utt = Au, where −A is an elliptic operator written in divergence form Au = div [a(x)∇u]. Then A can be rewritten as the Laplace-Beltrami operator for a Riemannian manifold with metric given by g(x) = a(x)−1 . See for instance [21, 22, 34, 39]. More generally, this sort of coordinate transformation applies to any second-order elliptic operator on a manifold M. The remainder of this paper is organized as follows. In Sect. 2 we provide some notation and standing assumptions, and then we show that System (2) generates a C0 semigroup of contractions on a well-defined Hilbert space H. Section 3 is then devoted to the proof of Theorem 1.5. Section 4 deals with some perturbation theory applicable to dynamics with a time smoothing estimate. Finally, we conclude with some open questions.

2 Preliminaries 2.1 Notation and assumptions We recall that Ω is a bounded, open set in RN , with smooth boundary Γ = Γ0 ∪ Γ1 , where Γ0 and Γ1 are disjoint. We furthermore assume this union is non-trivial, i.e. Γ0 and Γ1 both have nonempty interior in the N − 1-manifold Γ . The inner products (·, ·)Ω and ·, · Γ1 refer to the L2 inner products on Ω and Γ1 , respectively. That is,  (u, v)Ω =

 u(x)v(x)dx,

Ω

u, v Γ1 =

u(x)v(x)dΓ. Γ1

(13)

P.J. Graber, I. Lasiecka

The corresponding norms | · |Ω and | · |Γ1 are defined in the usual way. The Poincaré Inequality holds, that is there exists a constant C such that |u|Ω ≤ C|∇u|Ω . The space HΓ10 (Ω) = {u ∈ H 1 (Ω) : u|Γ0 = 0} is endowed with the norm |u|H 1

Γ0 (Ω)

=

|∇u|Ω . The vector ν = ν(x) is the unit normal on the boundary Γ . The operator ∂n := ∂ ∂ν = ν · ∇ is the normal derivative. The operator ∇Γ is the gradient on the manifold Γ , corresponding to tangential derivatives. The space H 1 (Γ1 ) is endowed with the norm |φ|H 1 (Γ1 ) = |∇Γ φ|Γ1 . Let {gij }i,j =1,...,N −1 denote local coordinates associated with the Riemannian metric endowed on Γ . The  natural volume element on Γ , denoted by dV , is given in local coordinates as dV = det(gij )dy1 dy2 . . . dyN −1 . Thus ∇Γ is the Riemannian gradient. The notation allows us to define Laplace-Beltrami operator by the variational formula   (Γ u)vdV = − (∇Γ u, ∇Γ v)dV Γ

for any u, v

∈ C ∞ (Γ ).

Γ

In local coordinates, the above expression reduces to Γ u = g −1/2

N −1  i,j =1

  ∂ ∂u g ij g 1/2 ∂yi ∂yj

where (g ij ) = (gij )−1 and g = det(gij ). We let H be the subset of HΓ10 (Ω) × L2 (Ω) × H 1 (Γ ) × L2 (Γ ) obtained by taking the closure of the subset of smooth functions together with their traces, i.e. H := {(u, v, u|Γ , v|Γ )T : (u, v) ∈ C 2 (Ω) × C 2 (Ω), u = 0, on Γ0 },

(14)

with closure taken in the norm (u, v, w, z)T 2H = |∇u|2Ω + |v|2Ω + |∇Γ w|2Γ + |z|2Γ .

(15)

Observe that for un → u in H 1 (Ω), we also have (by the trace theorem) that un |Γ → u|Γ . That means that if (u, v, w, z)T ∈ H, by taking un ∈ C 2 (Ω) such that (un , un |Γ ) → (u, w) in H 1 (Ω) × H 1 (Γ ) we see that, necessarily, u|Γ = w. This observation is summarized in the following remark: Remark 2.1 The state space H can be identified with H ≡ (u, v, w, z)T ∈ HΓ10 (Ω) × L2 (Ω) × H 1 (Γ1 ) × L2 (Γ1 ), w = z|Γ1 .

Analyticity and Gevrey class regularity for a strongly damped wave

2.2 Generation of a semigroup Define the operator Aα : D(Aα ) ⊂ H → H by ⎞ ⎛ 0 I 0 0 ⎜  c 0 0 ⎟ ⎟, Aα = ⎜ ⎝ 0 0 0 I ⎠ −∂n −c∂n Γ cαΓ ⎫ ⎧⎛ ⎞ v ∈ H 1 (Ω) u ⎪ ⎪ ⎪ ⎪ ⎬ ⎨⎜ v|Γ ∈ H 1 (Γ ) v ⎟ ⎜ ⎟ D(Aα ) = ⎝ . ∈ H : u|Γ ⎠ (u + cv) ∈ L2 (Ω) ⎪ ⎪ ⎪ ⎪ ⎭ ⎩ v|Γ ∂n (u + cv)|Γ − Γ (u|Γ + cαv|Γ ) ∈ L2 (Γ )

(16)

Our goal in this section is to show that Aα generates a strongly continuous semigroup of contractions on H, which we denote eAα t . This will give the solution operator for the system of PDEs (2). It bears mentioning that solutions of the form eAα t x0 for x0 ∈ H are solutions only in a generalized sense, and as such they do not possess all the properties of strong or classical solutions. We mention one important caveat: a priori, a generalized solution (u(t), v(t), w(t), z(t)) = eAα t x0 need not have the regularity sufficient to conclude that v(t)|Γ = z(t) in the sense of trace. Indeed, for a general v ∈ L2 (Ω) the trace v|Γ is not well-defined. One of the benefits of the smoothing effect proved later on is that the desired trace relationship is recovered in time even for arbitrary initial data in the state space H. We now prove the following. Proposition 2.2 For all α ≥ 0 the operator Aα : D(Aα ) ⊂ H → H is closed and densely defined, and it generates a C0 semigroup of contractions. Proof It is clear from the definition of H that D(Aα ) is dense in H, since (u, v, u|Γ , v|Γ ) ∈ D(Aα ) for u, v ∈ C 2 (Ω). We now investigate the resolvent of Aα . Take Φ = (φ1 , φ2 , φ1 |Γ , φ4 ) ∈ H and try to solve (λ − Aα )U = Φ, where U = (u, v, w, z)T . This means we want to solve λu − v = φ1 λv − u − cv = φ2 λw − z = φ1 λz + ∂n (u + cv) − Γ (w + cαz) = φ4

in Ω in Ω on Γ1 on Γ1

(17)

Take f ∈ HΓ10 (Ω) ∩ H 1 (Γ ) and use substitution and integration by parts in (17) to get the following variational form: 1 1 aλ (v, f ) = − (∇φ1 , ∇f )Ω + (φ2 , f )Ω − ∇Γ φ1 , ∇Γ f Γ + φ4 , f Γ λ λ

(18)

where aλ (v, f ) := λ(v, f )Ω +(1/λ+c)(∇v, ∇f )Ω +λv, f Γ +(1/λ+cα)∇Γ v, ∇Γ f Γ . (19)

P.J. Graber, I. Lasiecka

The critical fact here is that aλ (., .) is a continuous sesquilinear form on V := HΓ10 (Ω) ∩ H 1 (Γ ), endowed with the norm |f |2V := |∇f |2Ω + |∇Γ f |2Γ1 and moreover we have aλ (v, v) = λ|v|2Ω + (1/λ + c)|∇v|2Ω + λ|v|2Γ + (1/λ + cα)|∇Γ v|2Γ .

(20)

For λ > 0 real we thus have that aλ is coercive. By the Lax-Milgram Theorem, this proves that for each Φ there exists a unique solution v to the variational problem (18). We define, of necessity, z = v|Γ , u = λ−1 (v + φ1 ), w = u|Γ = λ−1 (z + φ1 |Γ ). It follows U = (u, v, w, z)T is the only possible solution to (λ − Aα )U = Φ. It remains to show that U ∈ D(Aα ) and that it satisfies (17). From (18) we have that 1 λ(v, f )Ω + (1/λ + c)(∇v, ∇f )Ω = − (∇φ1 , ∇f )Ω + (φ2 , f )Ω λ

(21)

for all f ∈ H01 (Ω). Using u = λ−1 (v + φ1 ) this becomes λ(v, f )Ω + (∇(u + cv), ∇f )Ω = (φ2 , f )Ω ,

for all f ∈ H01 (Ω).

(22)

Define (u + cv) as a distribution, then note that ((u + cv), f )Ω = −(∇(u + cv), ∇f )Ω = (λv − φ2 , f )Ω for all f ∈ H01 (Ω). (23) It follows that (u + cv) ∈ L2 (Ω) and that λv − u − cv = φ2 . Use this information in (18) to deduce that ∂n (u + cv), f Γ + λ2 v, f Γ + (1/λ + cα)∇Γ v, ∇Γ f Γ 1 = − ∇Γ φ1 , ∇Γ f Γ + φ4 , f Γ λ

(24)

for all f ∈ H 1 (Γ ). Substituting again u = λ1 (v + φ1 ) it follows that ∂n (u + cv), f Γ + λ2 v, f Γ + ∇Γ (u + cαv), ∇Γ f Γ = φ4 , f Γ

(25)

It follows that ∂n (u + cv) − Γ (u + cαv) ∈ L2 (Γ ) and that λv + ∂n (u + cv) − Γ (u + cαv) = φ4 on Γ . We have just proved, for λ > 0, that (17) has a unique solution in D(Aα ) for each Φ ∈ H. It follows that R(λ, Aα ) exists for λ > 0. This proves Aα is closed. We also have, by a simple calculation, −(Aα U, U ) = c|∇v|2Ω + cα|∇Γ v|2Γ ,

for all U ∈ D(Aα ).

(26)

This proves that Aα is dissipative. Thus we can appeal to the Lumer-Phillips Theorem to conclude that Aα generates a C0 semigroup of contractions. (Equivalently, we can conclude that R(λ, Aα ) ≤ 1/λ for λ > 0 and appeal to the Hille-Yosida Theorem.) 

Analyticity and Gevrey class regularity for a strongly damped wave

3 Proof of Theorem 1.5 3.1 Preliminaries The goal of this section is to prove that the C0 contraction semigroup eAα t satisfies the additional regularity claimed in Theorem 1.5. More precisely, we prove two different types of estimates, namely energy estimates which are singular in time at t = 0 and frequency domain estimates which give bounds on the resolvent. By semigroup theory [30, 33] both types of estimates are sufficient to prove the desired regularity: analyticity when α > 0, and Gevrey class when α = 0. We consider first the energy estimates. Our goal is to prove that, for some fixed constants C ≥ 0 and β ≥ 1, we have   d A t   e α Y0  ≤ C Y0 H , t > 0  dt  tβ H If β = 1, then the above estimate proves that eAα t is analytic [30, Theorem 5.2, p. 61]. If β > 1 and cannot be made equal to 1, then the semigroup is not analytic, but of Gevrey class δ for all δ > β [33, Corollary 7], see also [18, 23]. Let the energy functional E be given by 1 1 E(u; t) = E(t) = (u, ut , u|Γ1 , ut |Γ1 )T 2H = (|ut |2Ω + |∇u|2Ω + |ut |2Γ + |∇Γ u|2Γ ). 2 2 (27) A simple calculation using the multiplier ut applied first to regular solutions and then extended to all finite energy (or “generalized”) solutions shows that  t |∇ut |2Ω + α|∇Γ ut |2Γ dr = E(s). (28) E(t) + c s

The crucial term in our calculations is the dissipation term |∇ut |2Ω + α|∇Γ ut |2Γ . Based on (28) generalized solutions enjoy an a priori regularity of type ut ∈ L2 (0, ∞; H 1 (Ω)) and, provided α > 0, ut ∈ L2 (0, ∞; H 1 (Γ1 )). The general idea is to show that the additional regularity can be exploited to achieve energy estimates of the form 1 E(ut ; t)  2β E(u; 0), t which is equivalent to the desired semigroup estimates. Here and throughout the remainder of the article, the expression u  v means that u ≤ Cv for some constant C depending only on the domain Ω. The calculations in this section assume solutions have smooth initial data, hence we take derivatives freely. By a simple density argument, the final estimates can be shown to apply to all generalized semigroup solutions. Before we proceed with the different regimes α > 0 and α = 0, let us derive a preliminary estimate based on the “equi-partition of energy.” Using the multiplier u in Equation (2) one has t  c c (ut , u)Ω + ut , u Γ + |∇u|2Ω + α |∇Γ u|2Γ 2 2 0

P.J. Graber, I. Lasiecka



t

+ 0

(|∇u|2Ω − |ut |2Ω + |∇Γ u|2Γ − |ut |2Γ )ds = 0.

(29)

When we use the Poincaré inequality and the trace theorem, we are able to deduce that  t  t 2 2 (|∇u|Ω + |∇Γ u|Γ )ds  E(t) + E(0) + |∇ut |2Ω ds. (30) 0

0

N.B. This applies for any α ≥ 0, that is, we do not need any damping on the boundary for this to be true. Using the monotonicity of E(t) and the first energy identity (28) we deduce that  t E(s)ds  E(0), t > 0. (31) 0

From semigroup theory [30], Equation (31) allows us to conclude that E(t) decays exponentially, i.e. E(t)  e−ωt E(0) for some ω ≥ 0. This will be very helpful in the following and we formulate the corresponding result in the Proposition below Proposition 3.1 For all α ≥ 0 the semigroup generated by Aα is exponentially stable. This is to say: there exist constants M, ω > 0 such that eAα t L(H) ≤ Me−ωt , t > 0. 3.2 α > 0: Analytic semigroup estimates Using energy estimates, we now derive analyticity for the case α > 0, thus proving the first part of Theorem 1.5. In this case it is not surprising that the semigroup should be analytic, because in both interior and boundary dynamics independently we have a strong damping which provides smoothing. Of particular interest is the regime as α tends to zero, that is, as the strong damping on the boundary is weakened. Our estimates will show that in this regime the smoothing estimates blow up and no longer apply when α = 0. Using the multiplier tutt (as in [16]) one has  t  t s|utt |2Ω ds = s((u + cut ), utt )Ω ds 0



0

t

=

 s∂n (u + cut ), utt Γ ds −

0

t

s(∇(u + cut ), ∇utt )Ω ds.

0

Using Green’s formula we get  t s∂n (u + cut ), utt Γ ds 0

 = 0



t

sΓ u, utt Γ ds + cα



=− 0

t

 sΓ ut , utt Γ ds −

0 t



s∇Γ u, ∇Γ utt Γ ds − cα 0

0 t

t

s|utt |2Γ ds 

t

s∇Γ ut , ∇Γ utt Γ ds − 0

s|utt |2Γ ds.

Analyticity and Gevrey class regularity for a strongly damped wave

Rewrite the above to obtain  t  t  t s|utt |2Ω + s|utt |2Γ ds = − s(∇u, ∇utt )Ω ds − s∇u, ∇utt Γ ds 0

0

0



t

−c



t

s(∇ut , ∇utt )Ω ds − cα

0

s∇Γ ut , ∇Γ utt )Γ ds.

0

Integration by parts in time gives t   t 1 2 − s(∇u, ∇utt )Ω ds = −t (∇u(t), ∇ut (t))Ω − |∇u|Ω 2 0 0  t + s|∇ut |2Ω ds. 0



t

−

1 s∇Γ u, ∇Γ utt Γ ds = −t∇Γ u(t), ∇Γ ut (t) Γ − |∇Γ u|2Γ 2  t + s|∇Γ ut |2Γ ds.

0

 −c  −cα 0

0

t

s(∇ut , ∇utt )Ω ds =

0 t





−ct|∇ut (t)|2Ω

+c 0

t

t 0

|∇ut |2Ω ds.

s∇Γ ut , ∇Γ utt Γ ds = −cαt|∇Γ ut (t)|2Γ + cα

 0

t

|∇Γ ut |2Γ ds.

Assuming α is small (the regime which most interests us), we now deduce  t  t s|utt |2Ω + s|utt |2Γ ds  (cα)−1 tE(t) + E(0) + s(|∇ut |2Ω + |∇Γ ut |2Γ )ds. (32) 0

0

We calculate:  t  t s s(|∇ut |2Ω + |∇Γ ut |2Γ )ds = (|∇ut |2Ω + |∇Γ ut |2Γ )drds 0

0

=

0

 t 0

r

t

(|∇ut |2Ω + |∇Γ ut |2Γ )dsdr

−1



 (cα)

t

E(r)dr  (cα)−1 E(0).

0

Also, by the monotonicity of E(t) it follows that  t E(s)ds  E(0). tE(t) ≤

(33)

0

The monotonicity of E(ut ; t) can be deduced by considering ut in place of u in the PDE. We deduce from the above that  t 2 t E(ut ; t) ≤ 2 sE(ut ; s)ds  (cα)−1 E(u; 0). (34) 0

P.J. Graber, I. Lasiecka

This estimate, which depends explicitly on α and blows up as α → 0, shows the semigroup is analytic so long as there is some strong damping on the boundary. 3.3 α = 0: Gevrey class estimates in time Now we will consider the case α = 0 so that the smoothing on the boundary no longer applies: utt − cut − u = 0 in Ω × (0, T ) utt + ∂n (u + cut ) − Γ u = 0 on Γ × (0, T )

(35)

(u, ut , u|Γ , ut |Γ )(0) ∈ H In this case we seek a time-smoothing estimate of the form E(ut ; t)  or in other words,

1 E(u; 0), t4

   d A t  e 0  dt 

L (H )



t > 0,

1 . t2

(36)

(37)

This estimate immediately implies the second half of Theorem 1.5 by an appeal to the following result: Lemma 3.2 (Corollary 7 [33]) Let T (t) be a differentiable semigroup and β ≥ 1. Suppose that there exists constant C and  such that T  (t)L(H) ≤ Ct −β for 0 < t < . Then T (t) is of Gevrey class δ for t > 0, for every δ > β. We have already noted that the estimate (34) fails when α = 0, since the righthand side is proportional to α −1 . There is one particular term which we are unable to estimate directly when α = 0, namely  t s|∇Γ ut |2Γ ds. 0

Our priority is to recover the lost bounds on this term. We consider the fact that by the energy identity (28) we have ut ∈ L2 (0, ∞; H 1 (Ω)), which implies by the trace theorem the a priori regularity ut ∈ L2 (0, ∞; H 1/2 (Γ )). This is already “half a derivative” better than we are able to deduce from the hyperbolic boundary dynamics. However, we still need to recover another half derivative in order to obtain full L2 (0, ∞; H 1 (Γ )) regularity. In order to do this we shall localize solutions to the boundary layer. The key technical step is to introduce a pseudo-differential operator B which coincides with the Laplace-Beltrami operator on the boundary and allows us to scale derivatives (much the way one uses the Laplacian operator with homogeneous boundary conditions on a bounded domain). We will then introduce re-scaled energy space, push through the time-smoothing estimates on this new space, and then iterate the estimate to reach our goal.

Analyticity and Gevrey class regularity for a strongly damped wave

3.3.1 Interior analysis Since the loss of regularity occurs in the boundary layer, it is natural to localize the solution to the boundary collar denoted by C. This leads to the partition of solution into two parts u = φu + (1 − φ)u where φ(x) is a smooth function equal to one on the boundary Γ1 and with support in a boundary collar C of Γ1 . Since 1 − φ is supported away from the boundary, the function v := (1 − φ)u satisfies the standard strongly damped wave equation with zero Dirichlet data. This leads to consideration of a function v := ψu where ψ is supported away from a boundary collar C. Such v satisfies a classical strongly damped equation in the interior of Ω with the associated phase space H ≡ H01 (Ω) × L2 (Ω). vtt − cvt − v = f,

in Ω, v = 0 on Γ

(38)

where f := [, ψ](u + cut ),

t > 0.

Here the notation [D1 , D2 ] := D1 D2 − D2 D1 refers to the commutator of two PDOs D1 , D2 . Lemma 3.3 With reference to equation (38) and using the notation V := (v, vt )T the following time smoothing estimate holds:   d   V (t) ≤ C E(0), t > 0  dt  t H where E(0) = E(u; 0) is associated with the initial energy for the variable U = (u, ut )T . Proof Denote

 B :=

 0 I ,  c

F = (0, f )T

with the domain D(B) := {(v, z) ∈ H01 (Ω), cz + v ∈ L2 (Ω)} Equation (38) with V := (v, vt )T can be written as the evolution on the phase space H := H01 (Ω) × L2 (Ω) driven by the initial data V0 ∈ H and the forcing term F . d V = BV + F, dt

V (0) = V0

(39)

It is well known that B generates an analytic semigroup on H = H01 (Ω) × L2 (Ω). Moreover the following characterization of fractional powers given in [4] and [20, p. 290] is critical: D(B θ ) = {(v, z) ∈ H, z ∈ H02θ (Ω)},

θ ∈ [0, 1/2]

P.J. Graber, I. Lasiecka

and for θ ∈ (1/2, 1] one has D(B θ ) = {v, z ∈ H01 (Ω), cz + v ∈ H 2θ (Ω)},

θ ∈ [1/2, 1]

(40)

In particular D(B 1/2 ) = {(v, z) ∈ H, z ∈ H01 (Ω)}, By the commutator rules one easily obtains that f ∼ Dx (u + cut ) with Dx a first order differential operator, so that f (U ) can be regarded as a first order differential operator acting on U = (u, ut ). We thus obtain |F (U )|H  |u|H 1 (Ω) + |ut |H 1 (Ω)  |B 1/2 U |H By duality we also infer that and |B −1/2 F (t)|H  |U (t)|H  E(0)1/2 where we have used the fact that eA0 t generates a strongly continuous semigroup. The above regularity along with the variation of parameters formula leads to  t V (t) = eBt V (0) + eB(t−s) F (s)ds. 0

and with θ ∈ [0, 1] B θ V (t) = B θ eBt V (0) +



t

B θ+1/2 eB(t−s) B −1/2 F (s)ds.

0

Hence |B θ V (t)|H 

1 |V (0)|H + tθ



t

1 |B −1/2 F (s)|H ds. (t − s)1/2+θ

0

Applying the above with θ = 1/2 −  yields the first smoothing estimate: |B 1/2− (V (t))|H  (t −1/2+ + 1)E 1/2 (0)

(41)

Next we apply the same procedure but with cutoff function ψ such that supp ψ ⊂ (Ω − C1 ) with C1 ⊃ C. In view of this the new commutator F1 will inherit the additional, just obtained, regularity |B − F1 (t)|H  (t −1/2+ + 1)E 1/2 (0) This allows to write with θ ∈ (0, 1) B θ V (t) = B θ eBt V (0) +



t

B θ+ eB(t−s) B − F1 (s)ds.

0

Since

 0

t

|B

θ+ B (t−s) −

e

B

 F1 (s)|H ds  0

t

1 1 dsE(0)1/2 (t − s)θ+ s 1/2−

(42)

Analyticity and Gevrey class regularity for a strongly damped wave

the above can be applied now with θ = 1 − 2. Moreover for such θ we  t inequality 1 have 0 (t−s)θ+ ds  t 1/2−θ so that for all  > 0 s 1/2− |B 1− V (t)|H  (t −1+ + 1)E 1/2 (0)

(43)

The same estimate holds for (u, ut ) on a slightly smaller support Ω − C2 where u coincides with v. Moreover, the structure of the commutator F2 and the characterization of fractional powers of the generator (40) yield: [u + cut ]|Ω−C2 ∈ H 2−2 , hence ∇(u + cut )|Ω−C2 ∈ H 1−2 (Ω − C), which in turn implies |B 1/2− F2 (t)|H  (t −1+ + 1)E 1/2 (0) The latter allows to write 

Bt

t

|BV (t)|H  |Be V (0)|H +

|B 1/2+ eB(t−s) ||B 1/2− F (s)|H ds.

0

Hence |BV (t)|H  t

−1



Bt

t

|e V (0)|H + E(0)

1/2 0

1 1 ds 1/2+ 1/2− (t − s) s

which gives the final conclusion applicable to any interior set Ω − C where C can be taken as any boundary collar.  3.3.2 Boundary analysis From the interior analysis of the previous section we already know that the semigroup solution has an analytic time estimate locally away from the boundary. Now we turn to a critical region—the boundary collar. Thus it suffices to restrict attention to a portion of the semigroup solution which is localized near the boundary. For this we introduce the technical tools needed for the estimates which follow. We shall introduce anisotropic Sobolev spaces [17] which distinguish normal and tangential directions in counting the derivatives. In a neighborhood C of Γ in Ω we introduce the special coordinates C = Γ × [0, 1] where [0, 1] stands for the normal direction. For any k ≥ 0, p ∈ R we define: |u|2(k,p) ≡

k 

|u|2H i (0,1;H p+k−i (Γ )) n

i=0

This means that we take i derivatives in the normal direction with the values in L2 (0, 1; H p+k−i (Γ )). The above tangential spaces can be augmented to the full interior spaces by defining |u|2k,p ≡ |u|2H p+k (Ω) + |u|2(k,p)

P.J. Graber, I. Lasiecka

These spaces, which we denote H k,p (Ω), are associated with the following classes of anisotropic classes of PDO symbols. Let x, ξ denote primal and dual variable with ξ = (ξ  , ξn ) representing normal and tangential dual variable. Let p(x, ξ ) denote a symbol of a pseudodifferential operator P (x, D). Following [32], we define the symbol class m Sρ,δ (Ω)

to consist of the set of p ∈ C ∞ (Ω × Rn ) with the property that, for any compact K ⊂ Ω, any multi-indices α, β, there exists a constant CK,α,β such that |Dxβ Dξα p(x, ξ )| ≤ CK,α,β (1 + |ξ |)m−ρ|α|+δ|β|

(44)

for all x ∈ K, ξ ∈ Rn . In particular, we consider the subclass p ∈ S m defined by |Dxβ Dξα p(x, ξ )| ≤ CK,α,β (1 + |ξ |)m−|α| for each multiindex α, β. We say that p ∈ S m,p provided that p(x, ξ ) =

m 

pk (x, ξ  )ξnk

k=0

where pk ∈ S p+m−k (T ∗ Γ ). With this notation, we define the operator B = pb (x,  D) as a pseudo-differential operator represented by a smooth symbol pb (x, ξ ) = n−1 j =1 bij (x)ξi ξj such that  2 pb (x, ξ ) ≥ mb(x)|ξ | with b(x) ≥ b0 , x ∈ C. For x ∈ Γ (i.e. when the normal coordinate of x is zero) B = pb (x, D) coincides with −Γ , the Laplace-Beltrami operator on Γ . We have that pb ∈ S 0,2 and B −1 ∈ OP S 0,−2 , B −α ∈ OP S 0,−2α . Let’s introduce the following re-scaled state space H−1/4 := {(u, v, w, z), B −1/4 w ∈ H 1 (Γ ), B −1/4 z ∈ L2 (Γ ), u ∈ H 1,−1/2 (Ω), v ∈ H 0,−1/2 (Ω)} with norms indicated by the membership. In what follows we shall analyze semigroup properties of the operator A0 on this re-scaled space H−1/4 . Lemma 3.4 The operator A0 generates a C0 semigroup on H−1/4 . In particular we have that there exist constants C ≥ 1, ω ≥ 0 such that with Y = (u, ut , w, wt ) where w = u|Γ eA0 t Y H−1/4 ≤ Ceωt Y H−1/4 Proof Since we already know that A0 generates a semigroup (of contractions) on H, it suffices to establish the energy estimates on H−1/4 . In particular we define the

Analyticity and Gevrey class regularity for a strongly damped wave

energy functional for a solution with values in H−1/4 as E−1/4 (u; t) = E−1/4 (t) 1 = [|B −1/4 ut |2Ω + |∇B −1/4 u|2Ω + |B −1/4 ut |2Γ + |∇Γ B −1/4 u|2Γ ]. 2 Our goal is to show that E−1/4 (t) ≤ CE−1/4 (0)eωt . In order to do this we localize solution u near the boundary. Let φ be a smooth function with support inside V and identically equal to 1 close enough to Γ . Performing the energy estimates on u = φu + (1 − φ)u amounts to calculations for φu, which we relabel u (note that they are equal in the relevant neighborhood of Γ ) in the following: utt − cut − u = F (u) utt +

∂ (u + cut ) − Γ u = 0 ∂ν

in Ω (45) on Γ

where F (u) = c[, φ]ut + c[, φ]u. Multiplying the first equation by B −1/2 ut and applying the commutator formula B −1/4  − B −1/4 = [B −1/4 , ] yields 1 d [|B −1/4 ut |2Ω + |∇B −1/4 u|2Ω + |B −1/4 ut |2Γ + |∇B −1/4 u|2Γ ] + |∇B −1/4 ut ||2Ω 2 dt = (F (u), B −1/2 ut )Ω + 2([B −1/4 , ]ut , B −1/4 ut )Ω + 2([B −1/4 , ]u, B −1/4 ut )Ω . (46) To estimate the right-hand side, we need to use techniques for estimating commutators of PDOs. We use the following representation of  in the collar neighborhood C of the boundary: − is represented by the symbol with principal part p(x  , xn , ξ  , ξn ) = ξn2 + a(x  , xn )|ξ  |2 where (x  , xn ) ∈ Γ × [0, 1] are the “tangential” and “normal” coordinates, respectively. The tangential operator B −1/4 is represented by b(ξ ) = (1 + |ξ  |)−1/2 . Following commutator rules (Poisson brackets) [32] we obtain the following approximation (principal part) for the symbol of the commutator. [p, b] ∼ Dx  a|ξ  |2

d b ∼ Dx  a(x  , xn )|ξ  |1/2 dξ 

This means that [, B −1/4 ] ∼ Dx  (half tangential derivative). Similarly, with φ considered as a zero order operator, we get 1/2

[p, φ] ∼ 2Dxn φξn + 2Dx  φξ  ,

P.J. Graber, I. Lasiecka

hence [, φ] is a first-order differential operator (in both normal and tangential directions). It follows that |[B −1/4 , ]v|Ω  |v|H 1,−1/2 (Ω)  |∇B −1/4 v|Ω |B −1/4 [, φ]v|Ω  |v|H 1,−1/2 (Ω)  |∇B −1/4 v|Ω .

and (47)

This yields the estimate d [|B −1/4 ut |2Ω + |∇B −1/4 u|2Ω + |B −1/4 ut |2Γ + |∇B −1/4 u|2Γ ] + |∇B −1/4 ut ||2Ω dt  |∇B −1/4 u|2Ω + |B −1/4 ut |2Ω .

(48)

We recall that the above estimate is applied to the localized near the boundary variable φ(u). On the other hand (1 − φ)u has its support away from the boundary, thus the original estimates performed for the case α > 0—for the first part of the theorem— (without the loss of 1/2 tangential derivative) apply. This implies that the estimate in (48) can be written for the entire solution u = φu + (1 − φ)u. Integrating in time and appealing to Gronwall’s inequality we realize E−1/4 (t) ≤  E−1/4 (0)eωt for some ω ≥ 0, as desired. Remark 3.5 From the proof of Lemma 3.4 we also deduce that  1 t E−1/4 (t) + |∇B −1/4 ut |2Ω ≤ E−1/4 (0)eωt 2 0

(49)

cf. (28) 3.3.3 Smoothing estimates in the re-scaled space Our next step is to obtain smoothing estimates in H−1/4 space. Lemma 3.6

 0

t

sE−1/4 (s)ds  E(0).

(50)

Proof We want to prove that  t s[|B −1/4 utt |2Ω + |∇B −1/4 ut |2Ω + |B −1/4 utt |2Γ + |∇Γ B −1/4 ut |2Γ ]ds  E(0). 0

Multiply the interior wave equation (2) by B −1/2 utt to obtain  t  t  t s|B −1/4 utt |2Ω ds = c s(ut , B −1/2 utt )Ω ds + s(u, B −1/2 utt )Ω ds 0

 = 0

0 t

0

s(B −1/4 (cut + u), B −1/4 utt )Ω ds

Analyticity and Gevrey class regularity for a strongly damped wave



t

+c 0 t



s([B −1/4 , ]u, B −1/4 utt )Ω ds

+  = 0

s([B −1/4 , ]ut , B −1/4 utt )Ω ds

0 t

sB −1/4 (Γ u − utt ), B −1/4 utt Γ ds



t

−

s(∇B −1/4 (cut + u), ∇B −1/4 utt )Ω ds

0



t

+c  +

s([B −1/4 , ]ut , B −1/4 utt )Ω ds

0 t

s([B −1/4 , ]u, B −1/4 utt )Ω ds.

0

Note that B and −Γ coincide on the boundary. Thus  t sB −1/4 Γ u, B −1/4 utt Γ ds 0

 =−

t

sB 1/4 u, B 1/4 utt Γ ds

0

1 1 = −tB 1/2 u(t), ut (t) + |B 1/4 u(t)|2Γ − |B 1/4 u(0)|2Γ + 2 2

 0

t

s|B 1/4 ut |2Γ ds.

We calculate in a similar fashion:  t −c s(∇B −1/4 ut , ∇B −1/4 utt )Ω ds 0

=−  −

t

c ct |∇B −1/4 ut (t)|2Ω + 2 2



t 0

|∇B −1/4 ut |2Ω ds,

s(∇B −1/4 u, ∇B −1/4 utt )Ω ds

0

1 = −t (∇B −1/4 u(t), ∇B −1/4 ut (t))Ω + |∇B −1/4 u(t)|2Ω 2  t 1 − |∇B −1/4 u(0)|2Ω + s|∇B −1/4 ut |2Ω ds. 2 0 The following inequalities are crucial and follow from the properties of Sobolev spaces and commutators, as well as trace theory: |B 1/4 ut |Γ = |ut |H 1/2 (Γ )  |ut |H 1 (Ω) = |∇ut |Ω , |∇B −1/4 ut |Ω = |ut |H 1,−1/2 (Ω)  |ut |H 1 (Ω) = |∇ut |Ω , |[B −1/4 , ]ut |Ω

 |∇B −1/4 ut |Ω

 |∇ut |Ω .

(51)

P.J. Graber, I. Lasiecka

Now we recall the fact that 

t

0

|∇ut |2Ω ds ≤ E(0)

and 

t

0

s|∇ut |2Ω ds

=

 t 0

=

0

 t 

0

t

r t

≤

s

|∇ut |2Ω drds |∇ut |2Ω dsdr

E(r)dr  E(0).

0

Combining all of these estimates with the calculations above, and bearing in mind that E−1/4 (t)  E(t), we arrive at the inequality  0

t

s|B −1/4 utt |2Ω + s|B −1/4 utt |2Γ ds  E(0).

(52)

Combining this inequality with 

t 0

s[|∇B −1/4 ut |2Ω + |∇Γ B −1/4 ut |2Γ ]ds 



t

0

s|∇ut |2Ω ds  E(0) 

finishes the proof. From Lemma 3.6,  0

t

   d A s 2 0  s  e Y0   ds

H−1/4

ds  Y0 2H

and from Lemma 3.4 applied to time derivative (using semigroup property) we have    d A s 2  e 0 Y0   ds 

H−1/4

≥ Ce

   d A t 2 0   dt e Y0 

−ωs 

,

s < t.

H−1/4

Combining the above gives our first smoothing result. Theorem 3.7

  d At   e 0 Y0   dt 

H−1/4

1  eωt/2 Y0 H . t

In the next step, we will attempt to reiterate this smoothing estimate to get an estimate on the derivative in the energy space.

Analyticity and Gevrey class regularity for a strongly damped wave

3.3.4 Re-iterating the smoothing estimate The above arguments can be repeated. In particular, we can define a space H−1/2 with energy given by 1 E−1/2 (t) := [|B −1/2 ut |2Ω + |∇B −1/2 u|2Ω + |B −1/2 ut |2Γ + |∇Γ B −1/2 u|2Γ ], (53) 2 and we have Lemma 3.8 A0 generates a C0 semigroup on H−1/2 . Proof The proof of Lemma 3.4 can be adapted without difficulty; the only real change is that [, B −1/2 ] is in fact a zero-order PDO. We omit the details.  Likewise, using the multiplier tB −1 utt and repeating the same argument as in Lemma 3.6, we obtain Lemma 3.9



t 0

sE−1/2 (s)ds  E−1/4 (0)eωt .

Proof For the proof of this lemma, we note that the identity obtained from the multiplier method is  t s|B −1/2 utt |2Ω + s|B −1/2 utt |2Γ ds 0

 1 2 t = −tB u(t), B ut (t) Γ + |u|Γ 2 0 t  1 ct −1/2 −1/2 −1/2 2 − t (∇B u, ∇B ut )Ω + |∇B u|Ω − |∇B −1/2 ut |2Ω 2 2 0  t  t  t c + s|ut |2Γ + s|∇B −1/2 ut |2Ω ds + |∇B −1/2 ut |2Ω ds 2 0 0 0  t s([B −1/2 , ](u + cut ), B −1/2 utt )Ω ds. + 1/4

−1/4



0

Keeping in mind the energy inequality (49) and using the estimate |∇B −1/2 ut |Ω , |ut |Γ  |∇B −1/4 ut |Ω we finish the proof just as before. Corollary 3.10 (Cf. Theorem 3.7)   d At  1  e 0 Y0   eωt/2 Y0 H−1/4 .   dt t H−1/2



P.J. Graber, I. Lasiecka

Hence, from Theorem 3.7, we have  2  d At  1 0    2 eωt Y0 H .  dt 2 e Y0  t H−1/2 As a consequence of Corollary 3.10, we have the estimate |B −1/2 uttt |2Ω + |∇B −1/2 utt |2Ω + |B −1/2 uttt |2Γ + |∇Γ B −1/2 utt |2Γ 

1 2ωt e E(0). t4

In particular, |utt |Ω + |utt |Γ  |∇B −1/2 utt |Ω + |∇Γ B −1/2 utt |Γ 

1 ωt e E(0)1/2 . t2

We have almost arrived at the final estimate. It remains to estimate the crucial term |∇Γ ut |Γ . One method is to differentiate the original system of equations in time, then multiply by t 4 ut . After integration by parts we obtain  t  t s 4 |∇ut |2Ω ds + s 4 |∇Γ ut |2 ds 0

0

= −t (utt , ut )Ω − t 4 utt , ut Γ  t  t c + 2t 3 |ut |2Ω + 2t 3 |ut |2Γ − 6s 2 |ut |2Ω − 6s 2 |ut |2Γ − t 4 |∇ut (t)|2Ω 2 0 0  t  t  t + s 4 |utt |2Ω + s 4 |utt |2Γ + 2c s 3 |∇ut |2Ω . 4

0

0

0

Combine previous energy estimates with the fact that |utt |Ω , |utt |Γ  t −2 eωt E(0)1/2 to get  t s 4 |∇Γ ut |2 ds  teωt E(0) 0

which, after adding the remaining energy terms, implies  t s 4 E(ut ; s)ds  teωt E(u; 0). 0

Since E(ut , utt ; t) is monotone decreasing, it follows that  t t 5 E(ut , utt ; t) ≤ 5s 4 E(ut , utt ; s)ds 0

and hence the critical estimate: E(ut , utt ; t) 

1 ωt e E(u, ut ; 0). t4

(54)

Analyticity and Gevrey class regularity for a strongly damped wave

We have just shown Theorem 3.11

  d At   e 0 Y0   1 eωt/2 Y0 H .   dt t2 H

By Lemma 3.2, this completes the proof of Theorem 1.5. 3.4 Resolvent estimates In this section we prove the resolvent estimates stated in Theorem 1.5. As a corollary, we have an alternative proof of the analyticity (when α > 0) or Gevrey class regularity (when α = 0) of the semigroup eAα t . Time smoothing estimates are closely related to Gevrey regularity of a semigroup [18, 23, 33]. However, unlike analytic semigroups, for Gevrey class semigroups it is not clear that time smoothing and resolvent estimates are equivalent. Thus the resolvent estimates for α = 0 constitute a new result independent of the time estimates of the previous sections. The resolvent estimates are also interesting due to their relative simplicity in comparison to the time estimates. In particular, we will see that the iterative approach used in the previous section will not be necessary here. To infer Gevrey class regularity from the resolvent estimates, we need only appeal to the following theorem: Lemma 3.12 (Theorem 4 [33], cf. Theorem 4.9 p. 57 of [30]) Let T (t) = eAt be a strongly continuous semigroup satisfying T (t) ≤ Meωt , and let R(λ, A) denote the resolvent of A. Suppose that, for some μ ≥ ω and γ satisfying 0 < γ ≤ 1, lim sup |τ |γ R(μ + iτ, A) = C < ∞. |τ |→∞

Then T (t) is of Gevrey class δ for t > 0, for every δ > 1/γ . For the convenience of the reader, we now restate what we intend to prove from Theorem 1.5. Theorem 3.13 (a) Suppose α > 0. For a sufficiently large constant C > 0, we have R(iβ, Aα )L(H) ≤

C |β|

for all β ∈ R \ {0}.

This implies that the semigroup eAα t is analytic. (b) Suppose α = 0. For sufficiently large constants R, C > 0, we have that |β|1/2 R(iβ, A0 )L(H) ≤ C

for all β ∈ R, |β| ≥ R.

This implies that the semigroup eA0 t is Gevrey class of order δ > 2 for all t > t0 = 0. As a preliminary, we state the following lemma:

P.J. Graber, I. Lasiecka

Lemma 3.14 For any α ≥ 0, R(iβ, Aα )L(H)  1 for all β ∈ R. Proof This follows from the fact that eAα t is a uniformly stable semigroup—see Remark 3.1.  Let us now prove the first part of Theorem 3.13, with remarks to follow. Proof of Theorem 3.13(a) Let Φ = (φ1 , φ2 , φ1 |Γ1 , φ4 )T ∈ H and let R(iβ, A0 )Φ = U = (u, v, u|Γ1 , v|Γ1 )T ∈ D(A0 ) be the solution to (iβ − A0 )U = Φ. We recall from Sect. 2.2 that this means iβu − v = φ1 iβv − u − cv = φ2 iβv + ∂n (u + cv) − Γ (u + αv) = φ4

in Ω ∪ Γ1 in Ω on Γ1

(55)

where the equations on Γ1 hold in the sense of trace. We see from the proof of Proposition 2.2, Equation (18), that for all f ∈ HΓ10 (Ω) ∩ H 1 (Γ1 ) we have aiβ (v, f ) =

i i (∇φ1 , ∇f )Ω + (φ2 , f )Ω + ∇Γ φ1 , ∇Γ f Γ1 + φ4 , f Γ1 β β

(56)

where aiβ (v, f ) := iβ(v, f )Ω + (c − i/β)(∇v, ∇f )Ω + iβv, f Γ1 + (cα − i/β)∇Γ v, ∇Γ f Γ1 .

(57)

Take f = v. Note that aiβ (v, v) = c|∇v|2Ω + cα|∇Γ v|2Γ1 , βaiβ (v, v) = β 2 [|v|2Ω + |v|2Γ1 ] − |∇v|2Ω − |∇Γ v|2Γ1 . Taking real parts in Equation (56) with f = v and using the fact that v = iβu − φ1 we derive the estimate c|∇v|2Ω + cα|∇Γ v|2Γ1  U 2H + Φ2H  Φ2H ,

(58)

where the last inequality comes from Lemma 3.14. Taking imaginary parts and multiplying by β, we obtain the estimate β 2 [|v|2Ω + |v|2Γ1 ]  |∇v|2Ω + |∇Γ v|2Γ1 + Φ2H  (1 + (cα)−1 )Φ2H .

(59)

Again using the fact that v = iβu − φ1 we add in the rest of the terms in U 2H to get β 2 U H  |∇v|2Ω + |∇Γ v|2Γ1 + (1 + (cα)−1 )Φ2H  (1 + (cα)−1 )Φ2H . (60) This completes the proof.



Analyticity and Gevrey class regularity for a strongly damped wave

Before proving the second part of Theorem 3.13, let us make a few remarks. In the proof of Theorem 3.13(a) above we have noted the dependence of the estimates on α > 0 to highlight precisely where it fails when α = 0. We note in particular the role of |∇Γ v|Γ1 as a smoothing term. Equation (58) says that |∇Γ v|Γ1  ΦH , which followed by Equation (59) means that we have a bound |v|Ω , |v|Γ1  |β|−1 ΦH . With α = 0 the term |∇Γ v|Γ1 changes its role entirely, becoming the missing term in our estimates. Estimates on |v|Ω , |v|Γ1 must be obtained indirectly through the sole source of smoothing |∇v|Ω . The main point is that the estimate |v|Ω , |v|Γ1  |∇v|Ω is not sharp, that is, we can still afford another half derivative. This fact is exploited by way of the PDO B 1/4 , revealing finally that |v|Ω , |v|Γ1  |β|−1/2 ΦH , which is essentially the last step of the proof. We will first need a lemma showing that the lower level energy terms can be estimated in a way that resembles analytic smoothing estimates. We will then make up for the loss of half derivative (manifested in a loss of half-power of |β|) in the argument to follow. Lemma 3.15 R(iβ, A0 )L(H,H−1/4 )  |β|−1 for all β sufficiently large. Proof As before, it suffices to consider φu, φv where φ localizes the solution near the boundary. With abuse of the notation we shall use u, v instead. In Equation (55) multiply by B −1/2 v and integrate by parts. After some rearranging, we get iβ[|B −1/4 v|2Ω + |B −1/4 v|2Γ1 − |∇B −1/4 u|2Ω − |B 1/4 u|2Γ ] + c|∇B −1/4 v|2Ω = (∇B −1/4 u, ∇B −1/4 φ1 )Ω + B 1/4 u, B 1/4 φ1 Γ1 + (B −1/4 φ2 , B −1/4 v)Ω + ([B −1/4 , ](u + cv), B −1/4 v)Ω + B −1/4 φ4 , B −1/4 v Γ1 .

(61)

From Equation (58), we know that c|∇v|Ω  ΦH . Using iβu = v + φ1 , we can write |∇u|Ω  |β|−1 ΦH . Then by a combination of the theory of pseudo-differential operators and trace theory, we have |∇B −1/4 u|Ω , |B 1/4 u|Γ , |[B −1/4 , ]u|Ω  |∇u|Ω  |β|−1 ΦH as well as |[B −1/4 , ]v|Ω  |∇v|Ω  ΦH .

P.J. Graber, I. Lasiecka

(In particular, recall that [B −1/4 , ] ∼ Dx  , where x  is the tangential direction.) Using the above estimates in Equation (61) we deduce that 1/2

β 2 [|B −1/4 v|2Ω + |B −1/4 v|2Γ1 ]  Φ2H + Φ2H−1/4  Φ2H .

(62)

The other terms U 2H−1/4 have already been likewise estimated, hence it follows that U H−1/4  |β|−1 ΦH . 

This finishes the proof. We will now finish the proof of Theorem 3.13.

Proof of Theorem 3.13(b) Assume that β is sufficiently large (that is, |β| ≥ R for some fixed R > 0 large enough). By Lemma 3.15 and previous estimates, in particular |∇v|Ω  ΦH , we have |v|2Γ1 = B −1/4 v, B 1/4 v Γ1 ≤ U H−1/4 |∇v|Ω  |β|−1 Φ2H and likewise |v|2Ω = (B −1/4 v, B 1/4 v)Ω ≤ U H−1/4 |∇v|Ω  |β|−1 Φ2H . (Remark: this latter estimate is not sharp, whereas the former is. The boundary estimates prove once again the most crucial.) From the above, we have β 2 [|v|2Ω + |v|2Γ1 ]  |β|Φ2H . Returning to the proof of Theorem 3.13(a), that is, taking imaginary parts in Equation (56) with f = v and multiplying by β, we obtain an estimate on the previously elusive term: |∇Γ v|2Γ1  β 2 [|v|2Ω + |v|2Γ1 ] + Φ2H  |β|Φ2H .

(63)

Putting together the above estimates, and using the fact that iβu = v + φ1 , we obtain, finally, |∇u|2Ω + |∇Γ u|2Γ1 + |v|2Ω + |v|2Γ1  |β|−1 Φ2H , from which equation (10) follows.



4 Perturbation theory In applications it is often useful to know that smoothing estimates are preserved under perturbations of the generator. It is known that for general differentiable semigroups even a bounded perturbation may cause the semigroup to lose differentiability (example due to Renardy [31]). However, our smoothing estimate is stronger than for

Analyticity and Gevrey class regularity for a strongly damped wave

general differentiable semigroups. To the best of our knowledge, there is currently no robust perturbation theory for Gevrey semigroups in the literature (S. Taylor, private communication). The following result will demonstrate that, at least, the class of semigroups with time smoothing estimates that imply Gevrey regularity is closed under a certain class of perturbations. As a corollary of the two lemmas in this section, the results of this paper (Theorem 1.5) hold also for some relatively bounded perturbations of the generator A0 . Lemma 4.1 Let T (t) = eAt be a strongly continuous semigroup satisfying T (t) ≤ Meωt , and let R(λ, A) denote the resolvent of A. Suppose that, for some μ ≥ ω and γ satisfying 0 < γ ≤ 1, lim sup |τ |γ R(μ + iτ, A) = C < ∞. |τ |→∞

Suppose, moreover, that P is an Aγ0 -bounded operator, where γ0 < γ . Then A + P satisfies the same resolvent estimate. As a result, by Lemma 3.12 A + P generates a semigroup S(t) which is also of Gevrey class δ for t > 0, for every δ > 1/γ . Proof Start with the formula R(λ, A + P ) = [I + R(λ, A)P ]−1 R(λ, A).

(64)

Now using interpolation theory we get R(λ, A)P   R(λ, A)Aγ0   R(λ, A)1−γ0 AR(λ, A)γ0 .

(65)

Then notice that for |λ| large enough and for λ ≥ μ, we have AR(λ, A) = λR(λ, A) − I  ≤ |λ|R(λ, A) + 1  |λ|1−γ ,

(66)

using the resolvent estimate given in the hypotheses. By (65) and (66) we have R(λ, A)P   |λ|γ0 (1−γ ) R(λ, A)1−γ0 = |λ|γ0 −γ |λ|γ R(λ, A)1−γ0 ,

(67)

which can be made as small as we like by setting λ = μ + iτ for |τ | large. In particular, for |τ | sufficiently large we have that R(μ + iτ, A)P  <

1 2

⇒

[I + R(μ + iτ, A)P ]−1  ≤ 2.

(68)

Thus |τ |γ R(μ + iτ, A + P ) ≤ [I + R(μ + iτ, A)P ]−1 |τ |γ R(μ + iτ, A)  1 (69) for |τ | large. This completes the proof.



Lemma 4.2 Let T (t) = eAt be a differentiable semigroup on a Banach space X and β ≥ 1. Suppose that there exists constant C and  such that T  (t) ≤ Ct −β for 0 < t < . Suppose moreover that P is an Aγ0 -bounded operator, where γ0 < γ := 1/β. Then the semigroup S(t) generated by A1 = A + P satisfies the same estimate, and thus by Lemma 3.2 it is of Gevrey class δ for t > 0, for every δ > β.

P.J. Graber, I. Lasiecka

Proof Without loss of generality, we will assume T (t) is an exponentially decaying semigroup, in particular iR ⊂ ρ(A). Set c = γγ0 = γ0 β < 1. Consider the initial value problem y(t) ˙ = (A + P )y(t),

y(0) = y0 ∈ X.

We seek a variation of parameters solution of the form  t y(t) = T (t)y0 + T (t − s)P y(s)ds.

(70)

(71)

0

Using the fact that P is an Aγ0 -bounded operator, and the estimate Aγ0 T (t)  AT (t)γ0  |t|−βγ0 = |t|−c

(72)

where c < 1, a simple contraction-mapping argument yields a solution (fixed-point) y ∈ C([0, T ]; X). This is because the kernel  t (t − s)−c ds (73) 0

converges. To prove the additional desired regularity of the semigroup S(t), set Y (t) = t c Aδ y(t) for δ ≤ γ0 , which must satisfy the equation  t c δ c Y (t) = t A T (t)y0 + t s −c Aδ T (t − s)P A−δ Y (s)ds. (74) 0

We would like to show that Y ∈ C([0, T ]; X) for small enough T > 0. Since δβ ≤ γ0 β = c, the first term is bounded: t c Aδ T (t)y0   t c−βδ . As for the integral, rearrange powers to get  t tc s −c Aγ0 T (t − s)Aδ−γ0 P A−δ Y (s)ds. 0

Since Aδ−γ0 P A−δ is bounded, this term is bounded by  t  1 tc s −c (t − s)−γ0 β ds = t 1−γ0 β r −c (1 − r)−c dr. 0

(75)

0

Since c < 1, this is a converging kernel. The claim that Y is continuous now follows from a standard fixed point argument. We have shown that t c Aδ S(t)  1

(76)

for all δ ≤ γ0 . Since P is γ0 -bounded it clearly follows that t c P S(t)  1. Finally, let A1 := A + P . From the above, γ

t c A10 S(t)  1.

(77)

Analyticity and Gevrey class regularity for a strongly damped wave

Let n ∈ N be such that β ≤ nc = nγ0 β, and let Y0 ∈ D(A1 ). We have 1/n

t β A1 S(t)Y0  = (t β/n A1 S(t/n))Y0  1

β

≤ t n A1n S(t/n)n Y0  γ

≤ t c A10 S(t/n)n Y0   t c Aγ0 S(t/n)n + t c P S(t/n)n Y0   Y0 . By density this estimate extends to all Y0 ∈ H, thus proving that t β A1 S(t)L(H)  1. 

This completes the proof.

5 Open questions We conclude this article with some open questions. Optimality We have shown that eA0 t is of Gevrey class δ > 2. Is this optimal? From numerical computations of the spectrum of the operator [10] we expect this to be true. To our knowledge, however, there is no rigorous proof of optimality currently available. As a first step toward a proof, one might try to prove that the resolvent estimate (10) is optimal. This is suggested by the sharpness of the estimate |B 1/4 v|Γ1  |∇v|Ω . Lp theory It will be interesting to obtain smoothing estimates expressed in terms of the Lp norms. In fact, such theory for parabolic problems, with dynamic boundary conditions, is already in place—see [8, 15] and references therein. In the case of strongly damped wave equation with homogenous boundary data some Lp estimates (including L∞ ) are derived in [16]. When α > 0 (the analytic case for p = 2) it may be possible to adapt rather general techniques developed in [8]. Damping on different scales For the wave equation with homogeneous boundary conditions, the damping need not be so strong. In particular, as mentioned in the introduction, for any self-adjoint positive operator A defined on a Hilbert space H the abstract strongly damped wave equation utt + cAα ut + Au = 0 generates an analytic semigroup on H ≡ D(A1/2 ) × H whenever α ∈ [1/2, 1] [4]. It would be interesting to see if there is some way to formulate appropriate dynamic boundary conditions corresponding to, for example, fractional powers of the Laplacian, and then imitate the results of this paper. Note, however, that in absence of homogeneous boundary conditions the fractional powers of the Laplacian on a bounded domain are technically undefined. Thus one would need a significant technical step in order to achieve the appropriate definition of the damping operator.

P.J. Graber, I. Lasiecka

References 1. Carvalho, A.N., Cholewa, J.W., Dlotko, T.: Strongly damped wave problems: bootstrapping and regularity of solutions. J. Differ. Equ. 244, 2310–2333 (2008) 2. Casarino, V., Engel, K.-J., Nickel, G., Piazzera, S.: Decoupling techinques for wave equations with dynamic boundary conditions. Discrete Contin. Dyn. Syst. 12, 761–772 (2005) 3. Chen, G., Russell, D.L.: A mathematical model for linear elastic systems with structural damping. Tech. rep., DTIC Document (1980) 4. Chen, S., Triggiani, R.: Proof of extensions of two conjectures on structural damping for elastic systems. Pac. J. Math. 136 (1989) 5. Chen, S.P., Triggiani, R.: Gevrey class semigroups arising from elastic systems with gentle dissipation: the case 0 < α < 1/2. Proc. Am. Math. Soc. 110, 401–415 (1990) 6. Engel, K.-J.: Spectral theory and generator property for one-sided coupled operator matrices. Semigroup Forum 58, 267–295 (1999). doi:10.1007/s002339900020 7. Favini, A., Gal, C.G., Goldstein, J., Romanelli, S., Goldstein, G.R.: The non-autonomous wave equation with general Wentzell boundary conditions. In: Proceedings of the Royal Society of Edinburgh— A-Mathematics, vol. 135, pp. 317–330. Cambridge Univ. Press, Cambridge (2005) 8. Favini, A., Goldstein, G., Goldstein, J., Romanelli, S.: The heat equation with generalized Wentzell boundary conditions. J. Evol. Equ. 2, 1–19 (2002) 9. Favini, A., Goldstein, G.R., Goldstein, J.A., Obrecht, E., Romanelli, S.: Elliptic operators with general Wentzell boundary conditions, analytic semigroups and the angle concavity theorem. Math. Nachr. 283, 504–521 (2010) 10. Fourrier, N.: Regularity, stability and numerical analysis of a wave equation with strong damping and dynamic boundary conditions. PhD thesis, University of Virginia (2013) 11. Gal, C., Goldstein, G., Goldstein, J.: Oscillatory boundary conditions for acoustic wave equations. J. Evol. Equ. 3, 623–635 (2003) 12. Gerbi, S., Said-Houari, B.: Asymptotic stability and blow up for a semilinear damped wave equation with dynamic boundary conditions. Nonlinear Anal., Theory Methods Appl. 74, 7137–7150 (2011) 13. Goldstein, G.: Derivation and physical interpretation of general boundary conditions. Adv. Differ. Equ. 11, 457–480 (2006) 14. Graber, P.J., Said-Houari, B.: Existence and asymptotic behavior of the wave equation with dynamic boundary conditions. Appl. Math. Optim. 42 pp. doi:10.1007/s00245-012-9165-1 15. Guidetti, D.: Abstract elliptic problems depending on a parameter and parabolic problems with dynamic boundary conditions. Preprint (2013) 16. Haraux, A., Otani, M., et al.: Analyticity and regularity for a class of second order evolution equations. Evol. Equ. Control Theory 2, 101–117 (2013) 17. Hörmander, L.: The Analysis of Linear Partial Differential Operators III: Pseudo-Differential Operators vol. 3. Springer, Berlin (2007) 18. Horn, M.A., Littman, W.: Boundary control of a Schrödinger equation with nonconstant principal part. In: Control of partial differential equations and applications, Laredo, 1994. Lecture Notes in Pure and Appl. Math, vol. 174, pp. 101–106 (1994) 19. Huang, F.: On the mathematical model for linear elastic systems with analytic damping. SIAM J. Control Optim. 26, 714–724 (1988) 20. Lasiecka, I., Triggiani, R.: Control Theory for Partial Differential Equations. Cambridge University Press, Cambridge (2000) 21. Lasiecka, I., Triggiani, R., Yao, P.-F.: Inverse/observability estimates for second-order hyperbolic equations with variable coefficients. J. Math. Anal. Appl. 235, 13–57 (1999) 22. Lasiecka, I., Triggiani, R., Yao, P.-F.: An observability estimate in L2 (Ω) × H −1 (Ω) for secondorder hyperbolic equations with variable coefficients. Control Distribut. Parameter Stoch. Syst. 13, 71 (1999) 23. Littman, W., Taylor, S.: Smoothing evolution equations and boundary control theory. J. Anal. Math. 59, 117–131 (1992) 24. Lunardi, A.: Analytic Semigroups and Optimal Regularity in Parabolic Problems, vol. 16. Birkhäuser, Basel (2003) 25. Mugnolo, D.: Second order abstract initial-boundary value problems. PhD thesis, Eberhard-KarlsUniversität Tübingen (2004) 26. Mugnolo, D.: Abstract wave equation with acoustic boundary conditions. Math. Nachr. 279, 299–318 (2006)

Analyticity and Gevrey class regularity for a strongly damped wave 27. Mugnolo, D.: A variational approach to strongly damped wave equations. In: Functional Analysis and Evolution Equations, pp. 503–514. Springer, Berlin (2008) 28. Mugnolo, D.: Damped wave equations with dynamic boundary conditions. J. Anal. Appl. 17, 241 (2011) 29. Nagel, R.: Toward a “matrix theory” for unbounded operator matrices. Math. Z. 201, 57–68 (1989) 30. Pazy, A.: Semigroups of Linear Operators and Applications to Partial Differential Equations. Applied Mathematical Sciences, vol. 44. Springer, New York (1983) 31. Renardy, M.: On the stability of differentiability of semigroups. Semigroup Forum 51, 343–346 (1995) 32. Taylor, M.E.: Pseudodifferential Operators. Princeton Mathematical Series. Princeton University Press, Princeton (1981) 33. Taylor, S.: Gevrey semigroups (Chapter). PhD thesis, Thesis, School of Mathematics, University of Minnesota (1989) 34. Triggiani, R., Yao, P., Lasiecka, I.: Exact controllability for second-order hyperbolic equations with variable coefficient-principal part and first-order terms. Nonlinear Anal., Theory Methods Appl. 30, 111–122 (1997) 35. Vazquez, J., Vitillaro, E.: Heat equation with dynamical boundary conditions of reactive-diffusive type. J. Differ. Equ. 250, 2143–2161 (2011) 36. Xiao, T.-J., Liang, J.: The Cauchy Problem for Higher Order Abstract Differential Equations. Springer, Berlin (1998) 37. Xiao, T.-J., Liang, J.: Complete second order differential equations in Banach spaces with dynamic boundary conditions. J. Differ. Equ. 200, 105–136 (2004) 38. Xiao, T.-J., Liang, J.: Second order parabolic equations in Banach spaces with dynamic boundary conditions. Trans. Am. Math. Soc. 356, 4787–4810 (2004) 39. Yao, P.-F.: On the observability inequalities for exact controllability of wave equations with variable coefficients. SIAM J. Control Optim. 37, 1568–1599 (1999)