The damped wave equation with unbounded damping
We analyze new phenomena arising in linear damped wave equations on unbounded domains when the damping is allowed to become unbounded at infinity. We prove the generation of a contraction semigroup, study the relation between the spectra of the semigroup generator and the associated quadratic operator function, the convergence of non-real eigenvalues in the asymptotic regime of diverging damping on a subdomain, and we investigate the appearance of essential spectrum on the negative real axis. We further show that the presence of the latter prevents exponential estimates for the semigroup and turns out to be a robust effect that cannot be easily canceled by adding a positive potential. These analytic results are illustrated by examples.
Key words and phrases:damped wave equation, unbounded damping, essential spectrum, quadratic operator function with unbounded coefficients, Schrödinger operators with complex potentials
2010 Mathematics Subject Classification:35L05, 35P05, 47A56, 47D06
We consider the spectral problem associated with the linearly damped wave equation
with non-negative damping and potential on an open (typically unbounded) subset of ; when is not all of we shall impose Dirichlet boundary conditions on its boundary . Here both the potential and the damping are allowed to be unbounded and/or singular.
The main goal of this paper is to analyze the new phenomena which arise when the damping term is allowed to grow to infinity on an unbounded domain. To this end, we formally rewrite (1.1) as a first order system
and realize the operator in a suitable Hilbert space without assuming that the damping is dominated by . Our main results show that, even under these weak assumptions on the damping, generates a contraction semi-group, but may have essential spectrum that covers the entire semi-axis . As a consequence, although the energy of solutions will still approach zero, this decay will now be polynomial and no longer exponential, cf.  and the discussion below. We further establish conditions for the latter and study the convergence of non-real eigenvalues in the asymptotic regime of diverging damping on a subdomain.
In most of the literature on linearly damped wave equations on unbounded domains only bounded damping terms were considered. This is a natural condition to allow for the exponential decay of solutions, while large damping terms in fact tend to weaken the decay giving rise to the phenomenon known as over-damping. More precisely, increasing the damping term past a certain threshold will cause part of the spectrum to approach the imaginary axis, thus producing a slower decay. This phenomenon may already occur in finite-dimensional systems and in equations like (1.1) with bounded damping where its effect on individual eigenvalues is well-understood. Unbounded accretive or sectorial damping terms of equal strength as were considered as an application of semigroup generation results and of spectral estimates for second order abstract Cauchy problems in [13, 12] which allow to control the spectrum, in particular, near the imaginary axis.
To the best of our knowledge, the only article where the damping has been allowed to become unbounded at infinity is the recent preprint . There the authors consider dampings on all of and, using methods different from ours, they prove the existence and uniqueness of weak solutions whose energy decays at least with . In fact, our result on the essential spectrum will show that one of the characteristics of such systems is that the essential spectrum covers the whole semi-axis , thus excluding exponential energy decay.
To illustrate this issue, consider the simple model case given by the generators , , of the wave equation on the real line with the family of damping terms
and a constant potential , . The formal limit as goes to leads to a simple problem (in a different space) with
and Dirichlet boundary conditions at . The spectrum of the generator is discrete and may be found explicitly. It consists of eigenvalues with all, but possibly finitely many, lying on the line , cf. Remark 6.2. Moreover, the energy of solutions of the corresponding wave equation is known to decay exponentially, cf. for instance . A natural question is to what extent the properties of are shared by . The non-real eigenvalues of , here given explicitly in terms of the eigenvalues of and located on rays of the form , cf. Proposition 6.1, do indeed converge to those of . However, while the spectrum of is discrete and does not contain , all , , have non-empty essential spectrum covering the entire negative semi-axis and thus is in the spectrum of . As a consequence, exponential decay of energy is lost, cf. for instance [6, Thm. 10.1.7].
The fundamental point here is that the essential spectrum can no longer be shifted away from by adding a positive potential , as might be done for bounded damping, to ensure that exponential energy decay still holds, cf. for instance [25, 18]. In fact, even a potential that is unbounded at infinity, but does not dominate the damping term, will not be enough to cancel this effect. On the other hand, a dominating potential can be used to shift the essential spectrum from , cf. Remark 3.3.
As we will see, (1.3) is not an isolated example and our results cover the much more general setting with an open (typically unbounded) domain , a potential with low regularity and a damping satisfying natural conditions allowing for a convenient separation property of the domain of the Schrödinger operator when belongs to , cf. Assumption I, Remark 2.1.i) and Theorem 2.4.
We emphasize that the unbounded damping at infinity can by no means be viewed as “small” when compared to and our results, even those which are qualitative, do not follow by standard perturbation techniques, traditionally used to handle bounded or relatively bounded damping terms.
The proofs rely on a wider range of methods like elliptic estimates for Schrödinger operators with unbounded complex potentials, quadratic complements (quadratic operator functions associated with (1.2)), Fredholm theory, the use of suitable notions of essential spectra for non-self-adjoint operators, WKB expansions, convergence of sectorial forms acting in different spaces with -coefficients, spectral convergence of holomorphic operator families, and properties of solutions of second order ODE’s with polynomial potentials.
The crucial part of our analysis is the relation between the spectrum, and some of its subsets, of the generator and the associated quadratic operator function given by
While for bounded damping is defined for all and the equivalence of and is relatively straightforward, cf. for instance [23, Sec. 2.2, 2.3] for abstract results, the unboundedness of is a major challenge that requires a new approach; in particular, first has to be introduced as a closed operator with non-empty resolvent set acting in .
It is the precise description of , cf. Theorem 2.4, that enables us to prove both the generation of a contraction semigroup, cf. Theorem 2.2, and the spectral correspondence between and for the restricted range , cf. Theorem 3.2. Clearly, there are crucial differences between for and since the quadratic form of the latter is not semi-bounded. Nevertheless, for a general , convenient properties of with remain valid also for since the possibly negative real part of is compensated by the imaginary part of , cf. Section 2.1 for details.
When is unbounded at infinity, we show that the set , and hence the non-real spectrum of , consists only of discrete eigenvalues of finite multiplicity which may only accumulate at the semi-axis . Since the unboundedness of is not required for the equivalences in Theorem 3.2, also the non-real essential spectrum of can be analyzed by studying whether belongs to the essential spectrum of .
Because is not defined for , the negative real spectrum of is investigated directly for unbounded domains . We show that if grows to infinity in a channel in whose radius may shrink at at a rate controlled by the growth of , then belongs to the essential spectrum of . In fact, the whole real negative semi-axis belongs to the essential spectrum of even when is unbounded but does not dominate , cf. Theorem 4.2.
In Section 5, motivated by examples (1.3), (1.4) above, we prove a convergence result for non-real eigenvalues and corresponding eigenfunctions of a sequence of quadratic functions , , with dampings possibly diverging on a subset of , cf. Theorem 5.1. We thus justify the formal limit considered in the examples above.
In Section 6 we analyze two examples, the first is on the whole real line based on (1.3) and (1.4), while the second is on a horizontal strip in with damping and the corresponding discrete spectrum displaying the structure of a two–dimensional problem. Apart from showing what type of behavior one may now expect from isolated eigenvalues, more importantly both cases illustrate that having the discrete spectrum to the left of a line is, by itself, not enough to determine the type of decay of solutions in the presence of unbounded damping. Indeed, our results applied to both examples show that the essential spectrum covers the negative part of the real axis all the way up to , thus excluding the possibility of uniform exponential decay of solutions in general.
The following notations and conventions are used throughout the paper. The norm and inner product (linear in the first entry) in are denoted by and , respectively. The domain of a multiplication operator by a measurable function (here and ) in is always taken to be maximal, i.e.
The Dirichlet Laplacian on , introduced through the corresponding form, is denoted by , i.e.
When is viewed as an operator, the Dirichlet realization introduced through the form is meant, i.e.
For we view as a subspace of , , i.e. we use zero extensions. On the other hand, for , means . For consistency with earlier work, we denote the numerical range of a linear operator acting in a Hilbert space by
while may be more common in the operator theoretic literature; the numerical range of a quadratic form is introduced analogously, cf. [15, Sec. VI].
The essential spectrum of a non-self-adjoint operator may be defined in several, different and in general not equivalent, ways. Here we use the definition via Weyl singular sequences, denoted by in [7, Sec. IX],
2. Generation of a contraction semigroup
Throughout the paper, if not stated otherwise, we shall assume that the damping and the potential satisfy the following regularity conditions.
Assumption I (Regularity assumptions on the damping and the potential ).
Let , with . Suppose that can be decomposed into a regular and singular part as
with , and, for every , there exists a constant such that
Further assume that, for every , there exists a constant such that, for all , cf. (1.8),
Assumption II (Unboundedness of damping at infinity).
We are mostly interested in the situation when is not dominated by , and a typical potential being bounded (or even ). The case where dominates is discussed in Remark 3.3.
In order to find a suitable operator realization of the formal operator matrix , cf. (1.2), we denote by the completion of the pre-Hilbert space
the inner product of which is non-degenerate since is injective on , and we define the product Hilbert space
Here and equality holds if, for example, there is a positive constant such that , cf. [4, Thm. 1.8.1], or if has finite width and so Poincaré’s inequality applies, cf. for instance [1, Thm. 6.30]; then is uniformly positive and the space in (2.6) coincides with the usual choice of space for abstract operator matrices associated with quadratic operator functions in this case, cf. for instance , .
In we introduce the densely defined operator
The following theorem states the fundamental property that
generates a contraction semigroup; the proof is given at the end of Section 2.1 after all necessary ingredients have been derived.
2.1. The associated quadratic operator function
Employing sectorial forms, we introduce the family , , cf. (1.5), of closed operators in . Although the operator function resembles one of the quadratic complements of , cf. [23, Sec. 2.2], however, here is considered as an operator from to and not from to .
We shall introduce as
via the one-parameter family of operators
which will be defined rigorously below, using the first representation theorem for a rotated version of . In fact, the numerical range of is contained in a sector with semi-angle smaller then which need not lie in the right half-plane; however, after multiplication of by , we obtain a sectorial operator . We mention that the operator family , , is not uniformly sectorial, cf. (2.12) below.
Note that here we have included on purpose, although the domains of for and for are very different. Clearly, for , no rotation is needed since is self-adjoint and bounded from below. For convenience, we set in what follows.
In the definition of as well as in several auxiliary results, it suffices to assume less regularity of than in Assumption I.
Let and . Then the following hold.
For fixed , the form
is closed in and sectorial with
is a core of and determines a unique m-sectorial operator in .
If Assumption II holds, then , , has compact resolvent.
The operator family
is a holomorphic family of closed operators.
i) We denote . Since
the sectoriality of follows from
iii) The resolvent of is compact if and only if the resolvent of is compact, cf. [15, Thm. VI.3.3.]. The operator , induced by the form , is self-adjoint and has compact resolvent if is compactly embedded in , cf. [20, Thm. XIII.67]. If is bounded, then , and hence , is compactly embedded in by the Rellich-Kondrachov Theorem, cf. [1, Thm. 6.3]. For unbounded , let . The zero extension of belongs to , cf. [1, Lem. 3.27], and to where
Moreover, there exists a non-negative constant , independent of , such that
iv) We verify that is holomorphic (in the sense of [15, Sec. VI.1.2]) in a neighborhood of any . The strategy is to use the analyticity of the associated quadratic form. Nonetheless, we first note that, in a neighborhood of , is equal to the operator
where is the m-sectorial operator introduced through the sectorial form
the sectoriality and closedness of can be verified as in the proof of i), cf. (2.14)–(2.15), and the equality of the operators and follows from [15, Cor. VI.2.4]. Since the rotation in is independent of , the form associated with is obviously an analytic family of type (a) in a neighborhood of , cf. [15, Sec. VII.4.2]. Thus , and hence , are holomorphic in a neighborhood of . ∎
In case of higher regularity of as required in Assumption I, we obtain the following separation property of which ensures that is defined as a sum of unbounded operators. The strategy of the proof is similar to , but the different type of potentials used here requires new estimates.
Similarly, for , introduced in the same way as through the form , we have
Below we prove that is a core of and that there exist positive constants and such that, for all ,
from which it follows that .
By (2.3) in Assumption I and (2.25), is a relatively bounded perturbation of with relative bound , thus . Moreover, and a standard perturbation argument shows that, for sufficiently large positive , we have . Hence is m-sectorial, and , cf. [15, Sec. V.3].
To prove (2.20), it therefore remains to be shown that is a core of and that (2.25) holds. Take and notice that by Assumption I, thus as well. We first prove the core property by a suitable cut-off, cf. [5, Proof of Thm. 8.2.1]. Let be a function taking on non-negative values such that if and if . For define
From the derived regularity of and the compactness of , we conclude that . Moreover, by the dominated convergence theorem, as , and
since and .
note that the second step is justified since it can be verified that . Straightforward manipulations with the last term yield that
Hence, for every ,
where we used Young’s inequality in the last step. Since and satisfies (2.2), we see that, for every ,
where is independent of . Combining the estimates above, we obtain
It remains to consider the term in (2.32). Clearly, we have
for any . On the other hand,
Hence, we can successively select such that the coefficients of the first three terms are positive. Then a standard argument shows the existence of , cf. for instance [3, Proof of Lem. 2.9], as required in (2.25).
If satisfies certain regularity assumptions similar to those for , then also
The latter holds e.g. if there is a decomposition with , , and, for each , there are constants and such that
and, for all ,
The proof is a simpler version of the proof of Theorem 2.4.
Proof of Theorem 2.2.
Using integration by parts, it is straightforward to check that, for all ,
Thus and so is closable by [15, Thm. V.3.4].
Let be the core of defined in (2.21). We prove that . To this end, we take an arbitrary and find a solution of , i.e. of the system
Solving the first equation for and inserting this into the second equation, we get
Note that the left hand side equals with defined in Section 2.1, cf. (2.9). Moreover, for , , cf. Theorem 2.4, and since is uniformly positive. Thus is a bounded operator in and hence we obtain the solution ,