Asymptotics of semigroups generated by operator matrices

We survey some known results about operator semigroup generated by operator matrices with diagonal or coupled domain. These abstract results are applied to the characterization of well-/ill-posedness for a class of evolution equations with dynamic boundary conditions on domains or metric graphs. In particular, our ill-posedness results on the heat equation with general Wentzell-type boundary conditions complement those previously obtained by, among others, Bandle-von Below-Reichel and Vitillaro-V\'azquez.


Operator matrices with diagonal domain
While tackling abstract problems that are related to concrete initial-boundary value problems with dynamical boundary conditions and/or with coupled systems of PDE's, it is common that one has to check whether an operator matrix (1.1) A := A B C D generates a C 0 -semigroup on a suitable product Banach space, see e.g. [18], [17], and [16]. To fix the ideas, let us impose the following. Let us first deal with operator matrices of the form Then, it is an elementary exercise to check that A and D generate a semigroup on X and Y , respectively, if and only if the operator matrix generates a semigroup on the product space X × Y . Accordingly, taking into account standard perturbation results for generators of strongly continuous or analytic semigroups, the following can be proven using the techniques of [17, § 3]. Throughout the paper we define by for some α ∈ (0, 1), then also A and D generate semigroups of angle δ on X and Y , respectively. If any of the above assertions hold with B = 0, then is well-defined as a bounded operator from X to Y for all t ≥ 0 and there holds Likewise, if instead C = 0, then the semigroup generated by A has the form Proof. The assertions (1) and (2) and observe that the first addend on the right hand side has diagonal domain D(A)× D(C) and generates an analytic semigroup, so that the complex interpolation space is given by In the remainder of this section we are going to show that the matrix structure of our problem allows to prove better results. Recall that by the Datko-Pazy theorem a C 0 -semigroup on a Banach space E is uniformly exponentially stable if and only if it is of class L 1 (Ê + , L s (E)).
If the operator matrix A is upper or lower triangular, the form of (R(t)) t≥0 and (S(t)) t≥0 allows us to apply known results on convolutions of operator valued mappings. In the following we state most results in the case of B = 0 and C ∈ L(X, Y ), but of course analogous results hold whenever C = 0 and B ∈ L(Y, X). Proposition 1.4. Let Theorem 1.2 apply with B = 0 and C ∈ L(X, Y ). Assume (e tD ) t≥0 to be uniformly exponentially stable. Then the following hold.
(1) If for some x ∈ X the orbit (e tA x) t≥0 is bounded, then the orbit (R(t)x) t≥0 is bounded as well. (2) Under the assumptions of (1), if additionally the orbit (e tA x) t≥0 is asymptotically almost periodic, then the orbit (R(t)x) t≥0 is asymptotically almost periodic as well.
(4) If (e tA ) t≥0 is uniformly exponentially stable, then (e tA ) t≥0 is uniformly exponentially stable as well.
Proof. Observe that for all x ∈ X R(t)x can be seen as the convolution T * f , where (T (t)) t≥0 := (e tD ) t≥0 is a strongly continuous family of bounded linear operators on Y and for all x ∈ X the mapping f := Ce ·A x is of class L 1 loc (Ê + , Y ). Now it follows from the Young inequality for operator-valued functions, cf. [1,Prop. 1.3.5], that T * f ∈ L r (Ê + , Y ) whenever T ∈ L p (Ê + , L(Y )) and f ∈ L q (Ê + , Y ) for 1 ≤ p, q, r ≤ ∞ such that p −1 + q −1 = 1 + r −1 .
(2) If additionally (e tD ) t≥0 is asymptotically almost periodic, then (R(t)) t≥0 is asymptotically almost periodic. (3) If lim t→∞ e tD exists (resp., exists and is equal 0) in the strong operator topology, then lim t→∞ e tA exists (resp., exists and is equal 0) in the strong operator topology as well.
Proof. The assertions follow from [1,Prop. 5.6.4], again by considering By the Young inequality we obtain that If moreover both e tA ≤ M 1 e ǫt and e tD ≤ M 2 e ǫt , t ≥ 0, hold, then for all t ≥ 0 and x ∈ X In particular Asymptotical results can also be obtained by imposing so-called non-resonance conditions, cf. [1, § 5.6].
If moreover for the vector x ∈ X the orbit (e tA x) t≥0 is bounded and (e tD ) t≥0 is bounded, too, then the following hold.
(2) Let the orbit (R(t)x) t≥0 be bounded. If (e tD ) t≥0 is asymptotically almost periodic and moreover the orbit (e tA x) t≥0 is asymptotically almost periodic, then (R(t)x) t≥0 is asymptotically almost periodic as well. (3) Let the orbit (R(t)x) t≥0 be bounded. If lim t→∞ e tD exists (resp., exists and is equal 0) in the strong operator topology, and if lim t→∞ e tA x exists (resp., exists and is equal 0), then lim t→∞ R(t)x exists (resp., exists and is equal 0) as well.
Finally, we are able to prove an asymptotical result for the semigroup generated by the complete (i.e., with B = 0 = C) operator matrix. The following result should be compared with Remark 1.3.
Let B and C be bounded operators and assume that M 1 M 2 B C < ǫ 1 ǫ 2 . Then the semigroup generated by the complete matrix A is uniformly exponentially stable.
Proof. The semigroup (e tA ) t≥0 generated by the complete matrix is given by the Dyson-Phillips series We are going to prove that the estimates x and e ·D y L 1 ≤ − M2 ǫ2 x , and by the Young inequality also R(·) x , and this proves that the above inequalities hold for k = 0. Let them now hold for k. Then for k + 1 one applies the Young inequality and obtains x .
The other three estimates can be proven likewise.
Let us now prove the proposition's claim. We can assume that C = 0, otherwise the claim follows directly by Proposition 1.4.(4). Let B C < ǫ1ǫ2 M1M2 . Then the series ∞ k=0 M1M2 B C ǫ1ǫ2 k converges, and by the dominated convergence theorem one has for all x ∈ X and y ∈ Y , By the theorem of Datko-Pazy this concludes the proof.

Operator matrices with non-diagonal domain
Motivated by applications to initial-boundary value problems (see, e.g., [3,14,4,10,5,15]), we want to deduce results similar to those of Section 1 for the same operator matrix A, defined however on a different, coupled domain where L is a boundary operator operator from X to ∂Y . Here ∂Y is a suitable Banach space continuously imbedded in the boundary space ∂X := Y . More precisely, in the remainder of this section we replace the Assumptions 1.1 by the following.

(5)
A L is closed (as an operator from X to X × ∂Y ). with the graph norm of the closed operator A L . In several situations one already knows that A L is closed as an operator from X to X × ∂X. It is clear that if A L is closed as an operator from X to X × ∂X, then it is also closed as an operator from X to X × ∂Y .
Let us consider the abstract eigenvalue Dirichlet problem The following is a slight modification of a result due to Greiner, cf.  An analogue of the above factorization is the starting point of the discussion in [7,8], and can be proven likewise, cf. also [12]. Unlike in the setting of [12], M λ is in general an unbounded operator on X. We are thus led to impose the following.
(1) By assumption D A,L λ = D A,L λ . Thus, we can decomposẽ Observe that the second operator on the right-hand side is bounded on X by (2) We decomposẽ Since C ∈ L(X, ∂X), by Lemma 2.3 the second operator on the right hand side is bounded on X. Hence, by the bounded perturbation theoremÃ λ generates an analytic semigroup on X if and only if generates an analytic semigroup on X. Since D A,L λ (D + CD A,L λ ) ∈ L([D(D) ∩ ∂X], X), the claim follows by Lemma 1.2.(2).
(3) We decomposẽ The first addend on the right-hand side generates an analytic semigroup on X and for α ∈ (0, 1) the corresponding complex interpolation space is , ∂X] α . Thus, by assumption the second addend on the right-hand side is bounded from [D(Ã λ )] to [D(Ã λ ), X] α , while the third one is bounded on X. Hence, by the Desch-Schappacher perturbation theorem (see [6]) the operator matrixÃ λ generates an analytic semigroup on X.
Let 0 ∈ ρ(A 0 ) and let C = 0. Then by Lemma 2.7 the operator A is similar to where (S(t)) t≥0 is a suitable strongly continuous family of convolution operators. In order to apply the results obtained in Section 1 for triangular operator matrices, for the remainder of this section we impose the following. Proposition 2.12. Under the Assumptions 2.1 and 2.10, let A 0 generate a uniformly exponentially stable C 0 -semigroup and C = 0. If also D generates a C 0semigroup, then the following hold.
(1) If for some y ∈ Y the orbit (e tD y) t≥0 is bounded, then the orbit (S(t)y) t≥0 is bounded as well. (2) Under the assumptions of (1), if additionally the orbit (e tD y) t≥0 is asymptotically almost periodic, then the orbit (S(t)y) t≥0 is asymptotically almost periodic as well. (4) If (e tD ) t≥0 is uniformly exponentially stable, then (e tA ) t≥0 is uniformly exponentially stable as well.
Such assertions can be directly proved by observing that the assumptions of Proposition 1.4 are satisfied, since in particular the upper-right entry of (2.3) is bounded from ∂X to X. Similarly, from Proposition 1.5 we obtain the following. Proposition 2.13. Under the Assumptions 2.1 and 2.10, let A 0 generate a bounded C 0 -semigroup and C = 0. If D generates a uniformly exponentially stable semigroup, then the following hold.
(2) If additionally (e tA0 ) t≥0 is asymptotically almost periodic, then (R(t)) t≥0 is asymptotically almost periodic. (3) If lim t→∞ e tA0 exists (resp., exists and is equal 0) in the strong operator topology, then lim t→∞ e tA exists (resp., exists and is equal 0) in the strong operator topology as well. (4) If (e tA0 ) t≥0 is uniformly exponentially stable, then (e tA ) t≥0 is uniformly exponentially stable as well.
(2) Let the orbit (S(t)y) t≥0 be bounded. If (e tA0 ) t≥0 is asymptotically almost periodic and moreover the orbit (e tD y) t≥0 is asymptotically almost periodic, then (S(t)y) t≥0 is asymptotically almost periodic as well. (3) Let the orbit (S(t)y) t≥0 be bounded. If lim t→∞ e tA0 exists (resp., exists and is equal 0) in the strong operator topology, and if lim t→∞ e tD y exists (resp., exists and is equal 0), then lim t→∞ S(t)y exists (resp., exists and is equal 0) as well. In several concrete applications it is important to allow abstract boundary feedback operators C = 0. The following is analogue to Proposition 1.8. If C is a bounded operator and then the semigroup generated by A is uniformly exponentially stable, too.
Observe that (2.4) can be interpreted as a sufficient condition for stabilizability of the system associated with A, if we regard B as a feebdack control.

Two applications
Example 3.1. Let Ω be an open, bounded domain of Ê n with smooth boundary ∂Ω. We first show how the generation result of Section 2 can be applied in order to discuss the Laplacian on Ω equipped with so-called Wentzell-Robin (or generalized Wentzell) boundary conditions, i.e., This problem has been tackled and already solved in three papers ([9], [2], and [19]) by quite different methods. We are going to prove the generation result by means of the abstract technique of operator matrices with coupled domain: in fact, the onedimensional case has already been considered, also by means of operator matrices, by Kramar, Nagel, and the author in [12, § 9], thus we now focus on the case n ≥ 2 (see also [4] for yet another approach to a similar, non-dissipative system).
It has been shown both in [9] and [2] that the correct L 2 -realization of the Laplacian equipped with (WBC) is the operator matrix In order to apply the abstract results of Section 2, consider A as an operator matrix Here we have assumed that γ ∈ L ∞ (Ω). Finally, we define which is known to be a surjective operator from D(A) to ∂Y whenever ∂Ω is smooth enough, cf. [13, Vol. I, Thm. 2.7.4]. By standard boundary regularity results we now obtain that A | ker(L) is in fact the Laplacian with homogeneous Dirichlet boundary conditions, the generator of an analytic semigroup of angle π 2 on L 2 (Ω). Moreover, the closedness of A L holds by interior estimates for general elliptic operators, (a short proof of this can be found in [4, § 3]), and D is bounded whenever γ ∈ L ∞ (∂Ω). This shows that the Assumptions 2.6 are satisfied.
In particular, by Lemma 2.3 there exists for all λ ∈ ρ(A 0 ) the Dirichlet operator D A,L λ associated with (A, L), a bounded operator from ∂Y to X. In fact, it is known from [13,Vol. I,7] that for all λ ≥ 0 the operator D A,L λ has a bounded extension D A,L λ from ∂X to X. Moreover, it also follows from [13, Vol. I, Thm. 2.7.4] that C is a bounded operator from [D(A) L ] to ∂X, so that we can apply Theorem 2.9.
We still need to take a closer look to D 0 := D + CD A,L 0 : such an operator maps H 1 (∂Ω) into L 2 (∂Ω) by Such an operator often occurs in the contexts of PDE's and control theory, and it is sometimes called Dirichlet-Neumann operator. It is known that D 0 is the operator associated to the sesquilinear form Example 3.2. As a simple application of the stability results obtained in the paper, let us now consider the initial boundary-value problem, (t, x) = ∆u(t, x) − p(x)u(t, x), t ≥ 0, x ∈ Ω, w(t, z) = ∆w(t, z) − q(z)w(t, z), t ≥ 0, z ∈ ∂Ω, w(t, z) = ∂u ∂ν (t, z), t ≥ 0, z ∈ ∂Ω, u(0, x) = f (x), x ∈ Ω, w(0, z) = h(z), z ∈ ∂Ω, which has already been discussed in [4] and [15].
Here Ω is a bounded open domain of Ê n with smooth boundary ∂Ω, and 0 ≤ p ∈ L ∞ (Ω), 0 ≤ q ∈ L ∞ (∂Ω). Then, A 0 = A |ker(L) is (up to a bounded perturbation) the Laplacian with Neumann boundary conditions, and one sees that the Assumptions 2.6 are satisfied, hence Theorem 2.9.(2) applies and we conclude that (3.1) is governed by an analytic semigroup on L 2 (Ω) × H 1 2 (∂Ω) (in fact, as shown in [4, § 3] and [15, § 5], the problem is well-posed on the whole space L 2 (Ω) × L 2 (∂Ω)). Observe that a direct computation shows that the generator A of such semigroup is not dissipative.
However, if n ≥ 2 it is known (see [13, Vol. I, Thm. 2.7.4]) that the operator D A,L λ extends to an operator that is bounded from H − 3 2 (∂Ω) to L 2 (Ω). Moreover, the Laplace-Beltrami operator D maps H 1 2 (∂Ω) into H − 3 2 (∂Ω), so that the Assumptions 2.10 are satisfied. Since both A 0 and D are dissipative and self-adjoint, the non-resonance condition of Proposition 2.14 is clearly satisfied and we conclude that the semigroup generated by A on L 2 (Ω) × H 1 2 (∂Ω) is bounded. It is asymptotically almost periodic as well, since A 0 and D have compact resolvent. Now, observe that A 0 and D are invertible (hence generate uniformly exponentially stable semigroups) if (and only if) p = 0 = q. Summing up, we can apply Propositions 2.12.(4) and obtain that if p = 0 = q, then the semigroup generated by A is uniformly exponentially stable.