Abstract
We propose a new approach to the study of (nonlinear) growth and instability for semilinear abstract evolution equations with compact nonlinearities. We show, in particular, that compact nonlinear perturbations of linear evolution equations can be treated as linear ones as far as the growth of their solutions is concerned. We obtain exponential lower bounds of solutions for initial values from a dense set in resolvent or spectral terms. The abstract results are applied, in particular, to the study of energy growth for semilinear backward damped wave equations.
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1 Introduction
The paper is devoted to the study of instability of solutions to semilinear evolution equations
on a Banach space X, where A generates a \(C_0\)semigroup on X and K is a nonlinear map on X subject to appropriate conditions ensuring the existence of global mild solutions to (1.1). While the problem of finding instability conditions for (1.1) in terms of A and K is of fundamental importance, very few results in this direction were obtained so far even when K is stationary, i.e., \(K(t,x)=K(x)\) for \(t\ge 0.\)
One of the basic and commonly used instability criteria is due to Shatah and Strauss, [65]. It says that the zero solution of the system governed by (1.1) is unstable if the spectral bound of A is greater than zero and K is small enough in a metric sense (that is, \(K(x)=\textrm{O}(\Vert x\Vert ^{1+\delta })\) with \(\delta >0\) as \(\Vert x\Vert \rightarrow 0\)) and continuous. The notion of smallness can be refined by replacing a polynomial bound for K with a bound of more general type, but the scheme of proof remains the same all the time. For bounded A such criteria go back to [22]. This linearized instability principle appeared to be very useful in a great number of applications. Other versions of such results are investigated in [33, 34], and [35].
Recently, the problem of keeping control over asymptotic behavior of trajectories of linear infinitedimensional systems under “small” nonlinear perturbations was revived in [29, 30], and [58]. There the opposite problem of stabilization by the nonlinearity was emphasized. Note that these papers treated the situation of discrete time, where a number of difficulties (e.g. failure of the spectral mapping theorem) is missing.
In this paper, we address the situation when K is a compact nonlinear map, i.e., K is small in a topological sense, but otherwise it can be large metrically. This type of perturbations allows for global instability results, instead of just local ones. Recall that the map \(K: X \rightarrow X\) is called compact if it is continuous (this assumption varies) and maps bounded sets to precompact sets. Assuming that K is a compact \(C^1\)map, the paper [72] considered the case that either the (Browder) essential spectrum of A intersects the open right halfplane \({{\mathbb {C}}}_+\) or that there are infinitely many eigenvalues of A in \(\varepsilon +{{\mathbb {C}}}_+\) for some \(\varepsilon >0\). Then for a residual set of initial values \(x_0\) the mild solutions \(\{x(t, x_0): 0\le t < T\}\) of (1.1) are unbounded on their maximal existence interval [0, T), where \(0< T \le \infty .\) The proof relied on a discrete version of this result: If A is a bounded linear operator on X with the essential spectral radius \(r_{e}(A)>1\) and \(T=A+K,\) then the trajectory \(\bigcup _{n \in {\mathbb {N}}} T^n(B)\) of the unit ball \(B \subset X\) is unbounded in X.
We substantially improve these results and, for the continuous time version (1.1), replace the spectral terms used in [72] by much weaker assumptions on resolvent bounds. While the results in [72] yield merely unboundedness (and then instability), we derive an optimal exponential lower bound. To the best of our knowledge, such results were absent in the literature. In fact, our approach goes much further and allows one to treat K depending on time. Moreover, the obtained results cannot be improved even for stationary linear compact perturbations, see Remark 7.12. To this aim we use techniques for the study of orbits of linear operators from [50] that, with certain modifications, work in the context of nonlinear maps as well. This approach is used for the treatment of nonlinear equations for the first time, and we hope it will be useful in a number of instances.
The following statement is an important partial case of our main results. See Theorem 7.6 combined with Theorem 2.2. (A linear version of this result is given in Corollary 7.10.)
Theorem 1.1
Let A generate a \(C_0\)semigroup \((T(t))_{t\ge 0}\) on an infinitedimensional Banach space X, with \(\alpha :=\limsup _{t \rightarrow 0}\Vert T(t)\Vert ,\) and let \(K:[0,\infty ) \times X \rightarrow X\) be jointly continuous and map bounded sets in precompact ones. Let \(B=\overline{B}(y,r):=\{x\in X: \Vert xy\Vert \le r\}.\) Assume that for every \(x_0\in B\) there exist a global mild solution \(x(t,x_0)\), \(t\in [0,\infty ),\) of (1.1) such that the set \(S(t_0,B)=\{x(t, x_0):\, 0\le t\le t_0, \,\, x_0 \in B\}\) is bounded for each \(t_0>0\). Let \(a:[0,\infty )\rightarrow [0,\infty )\) be nonincreasing with \(\lim _{t\rightarrow \infty }a(t)=0\), and let \(r>0\) and \(t_0=t_0(r)\ge 0\) be such that \(a(t_0)<\frac{r}{2\alpha }\).

(i)
Assume that the resolvent \((\omega +ibA)^{1}\) is welldefined and unbounded for \(b\in {{\mathbb {R}}}\) and some \( \omega \in {{\mathbb {R}}}\). Then there exist \(x_0 \in {B}\) and a mild solution \(x(t,x_0)\) of (1.1) such that
$$\begin{aligned} \Vert x(t,x_0)\Vert \ge a(t) e^{\omega t}, \qquad t\ge t_0. \end{aligned}$$ 
(ii)
In any case, there exist \(x_0 \in {B}\) and a mild solution \(x(t,x_0)\) of (1.1) such that
$$\begin{aligned} \Vert x(t,x_0)\Vert \ge a(t) e^{s_e(A) t}, \qquad t\ge t_0, \end{aligned}$$
where \(s_e(A)\) is the supremum of the real parts of \(\lambda \) from the essential spectrum of A.
In the above theorem one can replace the assumption on existence of global mild solutions \(x(t, x_0)\) with bounded sets \(S(t_0,B)\) by the (linear) growth assumption \(\Vert K(t,x)\Vert \le (1+c(t))\Vert x\Vert \) for some \(c\in L^1_{\textrm{loc}}([0,\infty ))\) and all \(t\ge 0\) and \(x\in X\). See Proposition 6.3 and Corollary 6.6. In the case of backward damped wave equations considered in Sect. 8, the condition of linear growth of K will be generalised by requiring that K satisfies a sign condition and possesses some extra regularity.
To the best of our knowledge, the result is genuinely new in three respects:

(a)
it is the first lower bound for global growth of solutions to a general class of nonlinear evolution equations in the literature (and in fact, it is new even in the setting of linear equations);

(b)
the result is formulated in explicit (a priori) spectral terms;

(c)
the results is sharp, even in a linear context (see Remark 7.12).
To provide a necessary insight and because of independent interest, we first develop a similar theory for the discrete counterpart of (1.1)
where now A is a bounded linear operator on a Banach space X, and \(K_n\) are compact maps on X. In particular, we obtain an analogue of Theorem 1.1 for (1.2) only assuming compactness of each map \(K_n\), where the lower bounds depend on the essential spectral radius of the operator coefficient.
Theorem 1.2
Let A be a bounded linear operator on an infinitedimensional Banach space X, \((K_n)_{n=1}^\infty \) be a sequence of compact maps on X, and \((x_n(x_0))_{n=1}^\infty \) be given by (1.2). Take a nonincreasing sequence \((a_n)_{n=1}^{\infty }\subset {\mathbb {R}}_+\) satisfying \(\lim _{n\rightarrow \infty }a_n=0\). Fix \(y\in X\), \(r>0\), and \(n_0\in {\mathbb {N}}\) with \(a_{n_0}<\frac{r}{2}\). Then there exists \(x_0\in \overline{B}(y,r)\) such that
where \(r_{e}(A)\) is the essential spectral radius of A.
This result is proved in Theorem 4.6. If \(r_e(A)>1,\) then the orbits \(x_n(x_0)\) grow exponentially for a dense set of initial values \(x_0.\) Thus, if the nonlinear part K is small in a topological sense rather than in the sense of norm, Theorem 1.2 provides a global generalization of the classical (discrete) principle of linearized instability, as discussed e.g. in [39, Theorem 5.1.5] or [29, Sects. 1.2 and 4]. Note that the proofs of a number of statements on instability for continuous time are reduced to similar considerations in the discrete setting (as e.g. in [39]). Also our discrete results are essentially optimal, cf. [49, Sects. V.37 and V.39] for the case \(K=0\).
Using measures of noncompactness, we further obtain similar results producing residual sets of solutions to (1.1) and (1.2) with exponentially growing orbits. See Theorems 7.4 and 4.4, respectively.
To explain the general effects, we note that as remarked already in [72], roughly speaking, the linear semigroup \((T(t))_{t \ge 0}\) is expanding along infinitely many independent directions and the nonlinear perturbation K (being relatively compact) is not able to compensate this expansion, except perhaps in a finite number of directions. Our results say that up to a small multiplicative correction the expansion takes place as if the nonlinear part in (1.1) is absent.
The next toy example illustrates the specifics of the infinitedimensional setting very well. Let X be a separable Hilbert space. In the Banach space B(X) of bounded linear operators on X, consider the difference equation
If X is finitedimensional, then even if \(\textrm{dim} \, X = 1\) the asymptotic behavior of \(\{Y_n\}\) could be extremely complicated (see e.g. [13]), and any bounds for \(\Vert Y_n\Vert \) can hardly put under control for individual \(Y_0.\) However, if \(\textrm{dim} \, X = \infty ,\) and \(K \in B(X)\) is compact, then the quadratic part \(Y\mapsto K Y^2\), being compact in B(X), becomes “small” with respect to the linear part \(Y \mapsto AY\). Hence, by our Theorem 4.6 below, the (exponential) growth of trajectories \(\{Y_n\}\) for a residual set of initial values is determined by the essential spectral radius of the linear part \(L_A: Y\mapsto AY\) (i.e., by the essential spectral radius of A if one notes that \(\sigma _{e}(L_A)=\sigma (A)\) by e.g. [27, Theorem 3.1]).
Aiming at generalizations of local instability results as in e.g. [22, 39, 65] and [29], it is natural to consider also nonlinear perturbations \(K+G\) with a compact operator K and a metrically small map \(G:B(0,r)\rightarrow X\) satisfying \(G(x)=O(\Vert x\Vert ^{1+\delta })\) as \(\Vert x\Vert \rightarrow 0.\) However, at least in the time discrete case given by (1.2) we show in Sect. 5 that a local instability result analogous to Theorem 1.2 cannot hold for such perturbations \(K+G\).
As an illustration of our abstract results, we apply them to certain “excited” and backward damped wave equations in Examples 8.4, 8.5, 8.6, and 8.7. Here we allow for nonlinear forcing terms f(t, x, u) in which the scalar function f may grow superlinearly (if it has the right sign) and is only continuous in the last argument. In general, the study of damped wave equations is an extremely vast and challenging area of research with many open problems stemming from mathematical physics and control theory. One may consult, e.g., the books [46, Chapter 6] and [47, Chapters 10, 11] the survey [19], and the papers [1, 5, 15, 20, 25, 41,42,43, 55, 57, 61, 62, 66] for some of its developments in the linear setting, relevant for the nonlinear studies in the present paper. However, we are not aware of any results similar to Theorem 1.1 in the context of nonlinear damped wave equations.
We believe there are many other frameworks, where our instability criteria could be useful, e.g., in the context of reactiondiffusion systems as in [72]. However, they would require a separate treatment.
In Sects. 2, 3, and 9 we provide tools for our analysis and discuss the background. The main results are proved in Sects. 4 and 7 for discrete and continuous time, respectively. Section 5 contains counterexamples to local results in discrete time. The necessary information on nonlinear evolution equations is collected in Sect. 6 and then used in Sect. 8 for our examples.
Finally, we fix some notation used throughout the paper. All of the Banach (and Hilbert) spaces considered in this paper will be complex. To avoid trivialities, we will always assume that these spaces are infinitedimensional. For a densely defined closed operator A on a Banach space we denote by \(\rho (A)\) its resolvent set, by \(\sigma (A)\) its spectrum, and by \(\sigma _p(A)\) its point spectrum. We let D(A) stand for the domain of A, \(\textrm{Ker}\, (A)\) for the kernel of A, \(\textrm{Im}(A)\) for its range, and \(R(\lambda ,A)=(\lambda A)^{1}\) for the resolvent of A. The space of bounded linear operators on a Banach space X will be denoted by B(X) and that of compact linear operators by K(X). We let \(B(y,r):=\{x\in X: \Vert xy\Vert <r\}.\) For a subspace M of a Banach space, \(\textrm{dim}\, M\) is the dimension of M, and \(\textrm{codim}\, M\) its codimension. For a subset S of a topological space, \(\partial S\) designates its boundary, \(\textrm{card}\, S\) the cardinality of S, and \(\overline{S}\) the closure of S.
2 Preliminaries: a toolkit for getting (nonlinear) instability
In this section, we review several tools and techniques used for deriving instability. Some of them appear in the context of nonlinear evolution equations for the first time.
2.1 Fine spectral theory
We start with a short reminder of fine spectral theory. Recall that for a closed, densely defined linear operator A on an infinitedimensional Banach X its essential spectrum \(\sigma _{e}(A)\) is defined as
Clearly, \(\sigma _e(A)\subset \sigma (A).\) Moreover, by [60, Theorem 7.25], \(\sigma _e(A)\) is closed (but can be empty). There are many other “essential spectra” in the literature, e.g. Browder’s essential spectrum used in [72]. A crucial property of \(\sigma _{e}(A)\) is that it is invariant under relatively compact perturbations if \(\rho (A) \ne \emptyset ,\) see e.g. [23, Theorem 11.2.6]. Moreover, by [23, Theorem 11.2.2], if there exists \(\mu \in \rho (A),\) then
The property is a consequence of a more general spectral mapping theorem for essential spectrum ([31]), and it allows one to reduce many statements on essential spectrum for unbounded operators to their counterparts for bounded ones.
If A is bounded, then \(\sigma _{e}(A)\) is a nonempty compact subset of \({\mathbb {C}}.\) In this case, if \(r_{e}(A)\) denotes the essential spectral radius of A, then one has \(r_{e}(A^n)=r_{e}(A)^n\) for all \(n\in {{\mathbb {N}}}\), as a consequence of the spectral mapping theorem for the essential spectrum. Note that \(\sigma (A){\setminus } \{\lambda : \lambda \le r_e(A)\}\) consists of at most countably many isolated eigenvalues (of finite multiplicity), see e.g. [49, Theorem III.19.4].
Some generalizations of Fredholm operators will also play a role. Recall that a closed, densely defined operator A on X is called upperFredholm if \(\textrm{dim} \, \textrm{Ker}\, (A) < \infty \) and \(\textrm{Im}\, (A)\) is closed. If \(\sigma _e(A)\) is large and \(\rho (A)\ne \emptyset ,\) then the set of \(\lambda \in {\mathbb {C}}\) such that \(\lambda A\) is not upper Fredholm is large as well, since it contains the topological boundary \(\partial \sigma _e(A).\) More precisely, if \(\lambda \in \partial \sigma _{e}(A)\), then for every \(\varepsilon >0\) and every closed subspace \(M \subset X\) of finite codimension there exists a unit vector \(u \in M \cap D(A)\) such that \(\Vert (A\lambda )u\Vert <\varepsilon .\) See Lemma 9.2.
The following useful proposition can be found in e.g. [60, Theorem 9.43]. As several arguments below hold “up to compact perturbations”, the proposition allows one to deal with point spectrum rather than generic essential spectrum, and that is technically more convenient.
Proposition 2.1
Let \(A \in B(X).\) The operator A has closed range and finitedimensional kernel (i.e., A is upper Fredholm) if and only if \(\textrm{dim}\, \textrm{Ker}\, (A+K)<\infty \) for all compact operators K on X.
While there is a version of Proposition 2.1 for unbounded A, the version above will be sufficient for our purposes.
2.2 Measures of noncompactness
As far the essential spectrum is involved, measures of noncompactness naturally come into play, although their role in our studies will rather be supplementary in contrast to [72]. For \(A\in B(X)\) let
The mapping \(A \mapsto \Vert A \Vert _\mu \) is called \(\mu \)measure of noncompactness on B(X). It defines a seminorm on B(X), where \(\Vert A\Vert _\mu =0\) if and only if A is compact. Moreover, we have \(\Vert AB\Vert _\mu \le \Vert A\Vert _\mu \cdot \Vert B\Vert _\mu \) for all \(A,B\in B(X)\). See [49, Sect. III.24] for more details.
For \(A \in B(X)\), we introduce the essential norm of A by
i.e., \(\Vert A\Vert _{e}\) is the norm of the image of A in the Calkin algebra B(X)/K(X) under the corresponding quotient map. If X is a Hilbert space, then according to [78] (see also [75]) we have
In general, \(\Vert A\Vert _{\mu }\) and \(\Vert A\Vert _{e}\) are equivalent norms on B(X)/K(X) if and only if X has a socalled compact approximation property, see [6]. While substantial classes of Banach spaces possess this property, there are reflexive Banach spaces failing to satisfy it. However, by the wellknown Nussbaum formula (see e.g. [6, p. 393])
valid for all definitions of the essential spectrum, the quantities \(\Vert \cdot \Vert _{\mu }\) and \(\Vert \cdot \Vert _{e}\) are asymptotically equivalent in a sense. Moreover, the limits above can be replaced with the infimums. Hence, \(\Vert A\Vert _{\mu }^n \ge r_{e}(A)^n\) and \(\Vert A\Vert _{e}^n\ge r_{e}(A)^n\) for all \(n \in {\mathbb {N}}.\)
2.3 Spectral theory for operator semigroups and resolvent bounds
The spectral theory for \(C_0\)semigroups is rather involved due to the unboundedness of their generators, and one has to invoke the size of resolvents to partially remedy the situation.
First recall that if \((T(t))_{t \ge 0}\) a \(C_0\)semigroup on X, with generator A, then
see e.g. [51] for even finer versions of the above inclusion. This inclusion is strict in general. One can replace in (2.2) the essential spectrum by the spectrum, where again the inclusion can be strict.
The failure of the spectral mapping theorems for semigroups leads to a number of major difficulties in the semigroup theory. To discuss some of them define the exponential growth bound of \((T(t))_{t \ge 0}\) by
the spectral bound of its generator A by
where we set \(s(A)=\infty \) if \(\sigma (A)\) is empty, and the pseudospectral bound of A (or abscissa of uniform boundedness of the resolvent of A) by
The first two bounds possess “essential analogues” given by
where \(s_e(A):=\infty \) if \(\sigma _e(A)=\emptyset .\) Note that the first limit exists, and \(r_{e}(T(t))=e^{\omega _e(T) t}\) for \(t \ge 0,\) see e.g. [26, Corollary IV.2.10]. It is also crucial to observe that from [26, Corollary IV.2.11] it follows that \(\omega _0(T)=\max \{s(A), \omega _e(T)\},\) and moreover [37, Theorem XVII.2.1] implies that \(\{\lambda \in \sigma (A): \textrm{Re}\, \lambda > s_{e}(A)\}\) is an at most countable set consisting of isolated eigenvalues of A (with finite multiplicity). Clearly,
There are various examples of \(C_0\)semigroups making all or some of the above inequalities strict, see e.g. [4, Chapter 5.1]. So that, in general, neither the spectrum nor the resolvent of A determine the exponential norm bounds for \((T(t))_{t \ge 0}.\) Semigroups with \(s(A)<\omega _0(T)\) also arise from concrete partial differential equations, such as damped wave equations, cf. Sect. 8.
On the other hand, if X is a Hilbert space, then the wellknown Gearhart–Herbst–Prüss theorem guarantees that
and the exponential decay of \((T(t))_{t \ge 0}\) is equivalent to \(s_0(A)<0.\) For an exhaustive discussion of relations between these two and other related bounds see e.g. [4, Chapters 5.1\(\)5.3] or [54, Chapters 14].
To be able to obtain sharp lower bounds for the trajectories of (1.1) we need to introduce the new resolvent bound \(s_R(A)\) as the infimum of the set \(S_R\) of \(a >s_e(A)\) satisfying
This bound will play a crucial role in the sequel. Note that every vertical line \(a+i{\mathbb {R}}\) with \(a> s_e(A)\) contains at most countablly many eigenvalues. Using the discreteness of the set \(\{\lambda \in \sigma (A): \textrm{Re}\, \lambda > s_{e}(A)\}\) and the Neumann series expansion for the resolvent, one shows that the set \(S_R\) is open. Thus the infimum is not attained, and we have
Moreover, as the set \(\{\lambda \in \sigma (A): \textrm{Re}\, \lambda > s_{e}(A)\}\) is discrete, one may also define \(s_R(A)\) as the infimum of the set of \(a >s_e(A)\) such that
To explain the relevance of \(s_R(A)\) and to relate it to the spectral properties of \((T(t))_{t \ge 0}\), we introduce the notion of admissibility. We say that \(\omega \in {{\mathbb {R}}}\) is admissible if for every \(t_0>0\), every \(\varepsilon >0\) and every subspace \(M\subset X\) of the form \(M =\bigcap _{1\le j \le n}\textrm{Ker}\, y^*_j\) with \(y^*_j \in D(A^*)\) for \(j\in \{1,\dots n\}\), there exist \(x\in M\) with \(\Vert x\Vert =1\) and \(\mu \in {{\mathbb {C}}}\) with \(\textrm{Re}\,\mu =\omega \) such that
Note that \(\textrm{codim}\,M<\infty .\) Observe also that x and \(\mu \) depend on \(t_0,\) \(\epsilon \) and M, and so x is not an approximate eigenvalue of T(t), \(0\le t \le t_0,\) in general. However, the notion of admissibility will help us to “emulate” the approximate eigenvalues of T(t) to an extent that sufficices for the construction of growing solutions to nonlinear evolution equations.
Using Lemma 9.2 it is easy to show that if \(s_e(A)>\infty \), then \(s_e(A)\) is admissible. In this case, there exists \(\mu \) with \(\textrm{Re}\, \mu =s_e(A)\) such that for any \(\epsilon ,t_0>0\) and any closed subspace of finite codimension M, one can find a unit vector \(x \in M\) satisfying (2.5). However, Theorem 2.2 provides a more general statement showing that in fact \(s_R(A)\) is admissible. Thus the next result is one of the basic tools in this paper.
Theorem 2.2
Let \((T(t))_{t\ge 0}\) be a \(C_0\)semigroup on an infinitedimensional Banach space X with generator A. If \(s_R(A)>\infty ,\) then \(s_R(A)\) is admissible.
The proof of the theorem is given in the appendix. It is similar to the proof [50, Proposition 4.4], though it is technically more demanding.
3 Warmup: initial observations and comments
3.1 Spectrum does not suffice
It is wellknown that the spectral radius is not continuous on B(X). This leads to the fact that, in general, the instability of (1.1) is not preserved under small Lipschitz perturbations.
For a fixed \(a>0\) let \(X= L^2(0, 2a)\) and consider the selfadjoint operator \((A f)(s)=sf(s)\) on X. Clearly, \(\sigma (Aa)=[a,a]\) and thus there are initial values \(x_0 \in X\) for which the solutions of
grow exponentially in the sense that \(\Vert e^{(Aa)t} x_0\Vert \ge C_{x_0}e^{at/2}\) for \(t\ge 0\) with a constant \(C_{x_0}>0\).
By a classical result due to Herrero [40] (see [38] for a simple proof), if the spectrum of a bounded normal operator on X is connected and contains zero, then the operator is a limit of a sequence (or net) of bounded nilpotent operators on X in the uniform operator topology. Hence there exists a family of nilpotent operators \((A_\varepsilon )_{\varepsilon >0}\) on X such that \(A_\varepsilon \rightarrow A\) in L(X) as \(\varepsilon \rightarrow 0.\) Consider the perturbed sysem
Note that for any \(\delta >0\) there exists \(\varepsilon _0 >0\) such that
So, all solutions of (3.2) tend exponentially to 0 though (3.2) only differs by an arbitrarily small Lipschitz perturbation from the problem (3.1) possessing exponentially growing orbits. This general phenomena was observed in [68] and rediscovered recently in [58]. However, [58] went much further, see Sect. 3.3 below. We add that in [69] a Lipschitz perturbation K is constructed which destroys instability of a linear system \(x'(t)=Ax(t)\) and satisfies the strictly sublinear growth assumption \(\Vert K(x)\Vert /\Vert x\Vert \rightarrow 0\) as \(\Vert x\Vert \rightarrow 0.\) Our examples in Sect. 5 differ from these treatments.
3.2 Unstable orbits may coexist with very stable ones
On the Hilbert space \(X=L^2((0,\infty ), e^{2t} dt)\) let the operator \(Af=f'\) be defined on its maximal domain in X. Then A generates the \(C_0\)semigroup \((T(t))_{t \ge 0}\) of left shifts on X, and \(\sigma (A)=\{\lambda \in {\mathbb {C}}: \textrm{Re}\, \lambda \le 2\}.\) While \((T(t))_{t \ge 0}\) is hypercyclic, i.e., its trajectories are dense in X for a dense set of initial values (see e.g. [24]), the trajectories vanish eventually for every initial value with compact support. Thus, we have a dense set of initial values \(x_0\) for the abstract Cauchy problem
yielding “superstable” mild solutions and, at the same time, a dense set of initial values \(x_0\) whose trajectories have a totally unstable behavior. For more information on hypercyclic semigroups, we refer to e.g. [24].
3.3 Sublinear perturbations
Rodriguez and SoláMorales proved in [58] that in an infinitedimensional separable real Hilbert space X there exists a \(C^1\) map \(T: X \mapsto X\) of the form \(T = A+K,\) where A is a bounded linear operator on X with the spectral radius larger than one and the nonlinearity K satisfies \(\Vert K(x)\Vert /\Vert x\Vert \rightarrow 0\) as \(x \rightarrow 0\) and \(K(0)=0.\) Nevertheless the fixed point \(x = 0\) of T is (exponentially) stable. Moreover, given \(c_2>c_1>0\) there exists T as above such that
for x with \(\Vert x\Vert \) small enough.
3.4 Rankone perturbations
The instability properties for (1.2) become rather arbitrary if the essential spectrum of A is merely contained in the closed unit disc. A good illustration for that phenomena is provided by the fact that there exist a unitary operator A on a Hilbert space X and rankone operator K on X such that the operator \(A+K\) is hypercyclic, see e.g. [9, 32]. Clearly, \(r_{e}(A+K)=r_{e}(A)=1.\) On the other hand, it is wellknown that there exist a unitary operator \(A_1\) and a rankone operator \(K_1\) such that \(A_1+K_1\) is strongly stable, that is, \((A_1+K_1)^n x \rightarrow 0\) as \(n \rightarrow \infty \) for every \(x \in X.\) Indeed, if S is a unilateral shift on the Hardy space \(H^2({\mathbb {D}}),\) where \({\mathbb {D}}\) is the open unit disc, then [17] shows that certain unitary operators on \(H^2({\mathbb {D}})\) arise as rankone perturbations of compressions of S to the closed invariant subspaces of \(S^*.\) Moreover, clearly \(S^{*n} \rightarrow 0\) as \(n \rightarrow \infty ,\) strongly. A more general setup for such a construction can be found in [52].
Apparently, a similar example can be constructed in a continuous framework. Recall that for any selfadjoint operator A on a Hilbert space X the semigroup \((e^{iAt})_{t \ge 0}\) is unitary. It is plausible that there exist bounded selfadjoint operators A and \(A_1\) on X and rankone perturbation K and \(K_1\) such that semigroup \((e^{i (A+K)t})_{t \ge 0}\) is hypercyclic, while the semigroup \((e^{i(A_1+K_1)t})_{t \ge 0}\) is strongly stable. Although, no results have been published in this context.
4 Growth and instability for discrete time
In this section, we assume that X is an infinitedimensional Banach space, \(A\in B(X)\), and we let \((K_n)_{n=1}^{\infty }\) be a sequence of compact maps \(K_n: X\rightarrow X\) (in general, nonlinear). For \(x\in X\) and \(n\in {{\mathbb {N}}}\) let
and set \(f_0(x)=x\). In other words, \((f_n(x))_{n=0}^{\infty }\) is a solution of the difference equation
The next simple lemma replaces the difference equation (4.2) by a “difference inclusion”. In this way, arguing up to compact sets, we will able to reduce the study of asymptotic properties of \((f_n)_{n=1}^\infty \) to the study of such properties for linear iterations \((A^n)_{n=1}^\infty .\)
Lemma 4.1
Let \((f_n)_{n=1}^{\infty }\) be given by (4.1), \(n\in {{\mathbb {N}}}\) be fixed, and \(X_0\subset X\) be a bounded set. Then there exists a compact set \(C\subset X\) such that
for all \(x\in X_0\) and \(j\in \{1,\dots ,n\}\).
Proof
It is easy to see by induction on j that
for each \(x\in X\) and each \(j\in \{1, \dots , n\}\). Using induction again, we infer that for every \(s\in \{1, \dots , n\}\) the set \(f_{s}(X_0)\) is bounded, and thus \(K_{s+1}f_{s}(X_0)\) is precompact. Therefore, the set
is precompact, and we let C be its closure. Then \(f_j(x)\in A^jx+C\) for all \(x\in X_0\) and \(j\in \{1, \dots , n\}\). \(\square \)
Now we are able to give a lower bound for a finite piece of the trajectory in terms of the deviation of its linear part from a finitedimensional subspace. The next simple lemma provides an intuition behind the crucial Lemma 4.3.
Lemma 4.2
Let \((f_n)_{n=1}^{\infty }\) be given by (4.1). Let \(n\in {{\mathbb {N}}}\) be fixed and let \(X_0\subset X\) be a bounded set. Then for every \(\varepsilon >0\) there exists a finitedimensional subspace \(F\subset X\) such that
for all \(x\in X_0\).
Proof
Let \(\epsilon >0\) be fixed, and let C be the compact set constructed in Lemma 4.1. Since C is compact, there exists a finitedimensional subspace \(F\subset X\) such that \(\textrm{dist}\,\{c,F\}<\varepsilon \) for every \(c\in C\). For each \(x\in X_0\) we have \(f_n(x)\in A^nx+C\) implying
\(\square \)
Having obtained the estimate for a finite piece of trajectory, we can spread it out to a finitedimensional subspace of X, as the next lemma shows.
Lemma 4.3
Let \(A \in B(X),\) \(n\in {{\mathbb {N}}}\), \(\varepsilon > 0,\) and let \(F\subset X\) be a finitedimensional subspace. Then the following assertions hold.

(i)
There exists a unit vector \(u\in X\) such that \(\textrm{dist}\,\{A^n u, F\} \ge \frac{1\varepsilon }{2} \Vert A^n\Vert _\mu \).

(ii)
There exists a unit vector \(u\in X\) such that \(\textrm{dist}\,\{A^ju, F\}\ge \frac{1\varepsilon }{2}r_{e}(A)^j\) for all \(j=1,\dots ,n\).
Proof
To prove (i), note that by Lemma 9.1 in the appendix there exists a closed subspace \(M\subset X\) of a finite codimension such that
for all \(f\in F\) and \(m\in M\). Since \(L:= A^{n}(M)\) is a closed subspace of a finite codimension, there exists a unit vector \(u\in L\) with \(\Vert A^nu\Vert \ge (1  \varepsilon /2)\Vert A^n\Vert _\mu \). Then
which gives (i).
Let \(\mu \in \sigma _{e}(A)\) with \(\mu =r_{e}(A)\). Then \(\mu \in \partial \sigma _{e}(A)\), and by Lemma 9.2 given a closed subspace \(L\subset X, \textrm{codim}\, L<\infty ,\) and \(\delta >0\) there exists a unit vector \(u\in L\) with \(\Vert (A\mu )u\Vert <\delta \). For each \(j=1,\dots ,n\) we have
We now choose M as in the proof of (i) and set \(L=\bigcap _{j=1}^nA^{j}M\). Then there is a unit vector \(u\in L\) such that \(\Vert (A\mu )u\Vert \) is so small that \(\Vert (A^j\mu ^j)u\Vert \le r_{e}(A^j)\varepsilon /2\) for all \(j=1,\dots ,n\). Thus \(\Vert A^ju\Vert \ge (1\varepsilon /2)r_{e}(A^j)\), and as above we deduce that \(\textrm{dist}\,\{A^ju, F\}\ge \frac{1\varepsilon }{2}r_{e}(A^j)\) for all \(j=1,\dots ,n\). \(\square \)
The above lemmas allow us to obtain exponential lower bounds for the norms of trajectories \(\Vert f_{n}(x)\Vert \) for residual set of initial values x along some subsequences \(\{n_k\}.\) Apart from a usual Baire’s category theorem, the geometrical Lemma 4.3 is indispensable here.
Theorem 4.4
Let \(A \in B(X)\) and \(K_n:X\rightarrow X\) be compact for \(n\in {{\mathbb {N}}}\). Take a sequence \((a_n)_{n=1}^{\infty }\) in \({{\mathbb {R}}}_+\) such that \(\lim _{n\rightarrow \infty } a_n = 0\), and let \(L\subset {{\mathbb {N}}}\) be infinite. For \(x \in X\) define \(f_n(x)\) by (4.1). Then the set
is residual in X.
Proof
The statement is clear if \(\Vert A^n\Vert _\mu = 0\) for an infinite number of \(n\in L\). Thus, without loss of generality, we may assume that \(\Vert A^n\Vert _\mu > 0\) for all \(n\in L\). For \(k\in {{\mathbb {N}}}\) let
Clearly \(M_k\) is an open set. We show that it is dense in X. Let \(y\in X\) and fix \(\varepsilon > 0\). Choose \(n\in L\) with \(n\ge k\) and \(a_n < \varepsilon /4\). By Lemma 4.1, there exists a compact set \(C\subset X\) such that
for all \(v\in X\) with \(\Vert v\Vert \le \varepsilon \). Since C is compact, there exists a finitedimensional subspace \( F\subset X\) such that \(\textrm{dist}\,\{c, F\} < \frac{\varepsilon \Vert A^n\Vert _\mu }{12}\) for all \(c\in C\). Lemma 4.3 (i) then yields a unit vector \(u\in X\) with \(\textrm{dist}\,\{A^nu, F\} >\frac{\Vert A^n\Vert _\mu }{3}\). Note that
Indeed, passing to the quotient space X/F, the former inequality follows from
via the triangle inequality, where \(\pi \) denotes the corresponding quotient mapping. So \(x:= y + \varepsilon u\) or \(x:= y  \varepsilon u\) satisfy
We conclude that if x is chosen so to satisfy the inequalities above, then
Hence \(x\in M_k\), and \(M_k\) is dense since the choice of y and \(\varepsilon \) was arbitrary. By the Baire category theorem, \(\bigcap _{k=1}^\infty M_k\) is a dense \(G_\delta \) set and thus residual. \(\square \)
Corollary 4.5
Let \(A\in B(X)\) satisfy \(\sup \{\Vert A^n\Vert _\mu :n\in {{\mathbb {N}}}\}=\infty \). If \((f_n)_{n=1}^\infty \) is as in Theorem 4.4, then the set
is residual. In particular, this is true if \(r_{e}(A)>1\).
If we concentrate on merely dense sets of initial values rather than residual ones, then we can construct orbits \((f_n(x))_{n=1}^{\infty }\) with norm lower bounds for all large n.
Theorem 4.6
Let \(A \in B(X)\), \((K_n)_{n=1}^\infty \) be a sequence of compact maps on X, and \((f_n)_{n=1}^\infty \) be given by (4.1). Take a nonincreasing sequence \((a_n)_{n=1}^{\infty }\subset {\mathbb {R}}_+\) satisfying \(\lim _{n\rightarrow \infty }a_n=0\). Fix \(y\in X\), \(r>0\), and \(n_0\in {\mathbb {N}}\) with \(a_{n_0}<\frac{r}{2}\). Then there exists \(x\in {\overline{B}}(y,r)\) such that
Proof
Denote \(r_{e}:=r_{e}(A)\) for shorthand. The statement is clear if \(r_{e}=0\). Let \(r_{e}>0\). We start with several convenient reductions. Without loss of generality we may assume that \(r_{e}=1.\) If not, then consider the operator \(r_{e}^{1}A\) and the compact mappings \( r_{e}^{n} K_n(r_{e}^{n1}\cdot )\) for \(n \ge 1\). Replacing A with \(\lambda A\) for some \(\lambda =1\) and \(K_n\) with \(\lambda ^{n}K_n (\lambda ^{n+1}\cdot )\) if necessary, we may assume that \(1\in \sigma _{e}(A)\). The operator \(AI\) is not upper Fredholm since \(1\in \partial \sigma _{e}(A),\) see Lemma 9.2.
Proposition 2.1 thus gives a compact linear operator \({{\widetilde{K}}}\) on X with \(\dim \textrm{Ker}\, (AI{{\widetilde{K}}})=\infty \). Thus, passing to \(A{{\widetilde{K}}}\) and \(K_n+{{\widetilde{K}}}\) if necessary, we may assume that \(\dim \textrm{Ker}\, (AI)=\infty .\)
Take some \(c_1\in (2a_{n_0},r)\). Fix an increasing sequence of integers \((n_k)_{k=1}^\infty \subset (n_0,\infty )\) such that \(a_{n_k} < 2^{(k+2)}(r c_1) \) for \(k \ge 1\). Set
Choose positive numbers \(\varepsilon _k\) such that \(\varepsilon _{k+1}<\varepsilon _k\) and
for all \(k \in {\mathbb {N}}\).
Set \(x_0=y\), \(F_0=\{0\}\) and \(M_0=X\). Inductively, we construct finitedimensional subspaces \(F_1\subset F_2\subset \cdots \), finitecodimensional closed subspaces \(M_1\supset M_2\supset \cdots \), and unit vectors \(x_k\in M_k\cap \textrm{Ker}\,(AI)\) for \(k\ge 1\) such that (4.3) below is true.
Let \(k\ge 1\) and suppose that the vectors \(x_0,\dots , x_{k1}\) and spaces \(F_0,\dots ,F_{k1}\) and \(M_0,\dots , M_{k1}\) have already been constructed. Lemma 4.1 provides a compact set \(C_k\subset X\) such that
for all \(n\le n_k\) and \(u\in X\) with \(\Vert u\Vert \le \Vert y\Vert +r.\) Using the compactness of \(C_k,\) we find a finitedimensional subspace \(F_k\supset F_{k1}\) such that
for all \(d\in C_k\). Lemma 9.1 yields a closed subspace \(M_k\subset M_{k1}\) of finite codimension such that
for all \(f\in F_k\) and \(m\in M_k.\) Since \(\textrm{Ker}\, (AI)\) has infinite dimension, there exists a unit vector \(x_k\in M_k\cap \textrm{Ker}\,(AI)\).
For the elements \(x_k,\) \(k\in {{\mathbb {N}}},\) constructed above, set
We show that x satisfies the assertions. First note that
so that \(x \in {\overline{B}}(y,r).\) Let \(k\ge 1\) and \(n_{k1}\le n\le n_k\). We estimate
employing that
For \(j\ge k\), the vector \(x_j\) belongs to \(M_j\subset M_k\), and so
Since \(x_k\in F_{k+1}\) and \(x_j\in M_j\subset M_{k+1}\) for \(j\ge k+1\), we obtain
Hence
for all \(n\ge n_0\), and the statement follows. \(\square \)
Remark 4.7
If X is a Hilbert space, then in the proof of Theorem 4.6 one can take \(M_k=F_k^\perp \) in each step. It is then possible to obtain a slightly better estimate: Let \(y\in X\), \(n_0\ge 0\), and \(a_{n_0}<r\). Then there exists \(x\in X\) with \(\Vert xy\Vert \le r\) and \(\Vert f_n(x)\Vert \ge a_nr_{e}(A)^n\) for all \(n\ge n_0\).
Corollary 4.8
Under the assumptions of Theorem 4.6, let \((a_n)_{n=1}^{\infty }\subset {\mathbb {R}}_+\) be a nonincreasing sequence satisfying \(\lim _{n\rightarrow \infty }a_n=0\). Then

(i)
there exists \(x\in X\) such that \(\Vert f_n(x)\Vert \ge a_nr_{e}(A)^n\) for all \(n\in {{\mathbb {N}}}\);

(ii)
there exists a dense subset of vectors \(x\in X\) such that \(\Vert f_n(x)\Vert \ge a_nr_{e}(A)^n\) for all n sufficiently large.
Proof
To prove (i), it suffices to set \(y=0\) and to choose a big enough radius r in Theorem 4.6.
Let now \(y\in X\) and \(\varepsilon >0\) be fixed. Let \(n_0\) be such that \(a_{n_0}<\frac{\varepsilon }{2}\). By Theorem 4.6, there exists \(x\in X\) such that \(\Vert xy\Vert \le \varepsilon \) and \(\Vert f_n(x)\Vert \ge a_nr_{e}(A)^n\) for all \(n\ge n_0\), and assertion (ii) follows. \(\square \)
Remark 4.9
If one is interested in merely local instability properties of (4.2) assuming \(r_e(A)>1,\) then results close in spirit can be found in [70].
5 Local instability for discrete time: counterexample
As we mentioned in the introduction, it is natural to try to combine “metric” instability results, such as e.g. in [22, 29, 39, 65], and “topological” instability conditions as above, in order to obtain a local result showing instability for the zero solution to the difference equation
in a Banach space X. Here A is a linear operator on X with \(r_{e}(A)>1,\) K is a compact map on X with \(K(0)=0\), and G is a continuous map defined on a ball \(B(0,r)\subset X\) such that \(\Vert G(x)\Vert =O(\Vert x\Vert ^{1+\delta })\) as \(\Vert x\Vert \rightarrow 0\) for some \( \delta >0\). Unfortunately, this is not possible, in general, even if X is a Hilbert space, as Example 5.3 shows. Recall that the zero solution of (5.1) is called stable if for every \(\varepsilon > 0\) there is \(\delta > 0\) such for all \(x_0\) with \(\Vert x_0\Vert <\delta \) one has \(\sup _{n}\Vert x_n\Vert <\varepsilon .\)
We first note a simple lemma.
Lemma 5.1
Let \(\varepsilon \in (0,2^{7})\). Then there exist continuous functions \(f:[0,\infty )\rightarrow [0,\varepsilon /2]\) and \(g:[0,\infty )\times [0,\infty )\rightarrow [0,2]\) such that

(i)
\(f(s)=0, \qquad 0\le s\le \varepsilon ^3\quad \hbox {or} \quad s\ge \varepsilon \);

(ii)
\(f(s)=\varepsilon /2, \qquad 4\varepsilon ^3\le s\le \frac{\varepsilon }{2}\);
and

(iii)
\(g(s,t)=0, \qquad 0\le s\le \varepsilon ^3\quad \hbox {or} \quad s\ge \varepsilon ,\, t\ge 0,\);

(iv)
\(g(s,\frac{\varepsilon }{2})=2, \qquad 4\varepsilon ^3\le s\le 16\varepsilon ^3\);

(v)
\(g(s,t)\le s+\frac{t^2}{s}, \qquad s> 0, \, t \ge 0\).
Proof
The existence of f is obvious.
Let \({{\widetilde{g}}}:[0,\infty )\times [0,\infty )\rightarrow [0,2]\) be any continuous function satisfying (iii) and (iv). Set \(g(s,t)=\min \{{{\widetilde{g}}}(s,t), s+\frac{t^2}{s}\}\) for \(s> 0\) and \(t \ge 0,\) and \(g(0,t)=0\) for \(t\ge 0\). Clearly, g fulfills (iii) and (v). If \(4\varepsilon ^3\le s\le 16\varepsilon ^3\) and \(t=\frac{\varepsilon }{2},\) then
So g satisfies (iv). \(\square \)
Next we provide a counterexample for local instability of (5.1), where the nonlinear part is given by the sum of two nonlinear parts, each guaranteeing local instability when considered separately. To this aim, we will need an auxiliary example showing that there are a Hilbert space H and operators A, K and G on H as above such that the zero solution to (5.1) is stable for fixed \(\epsilon >0.\)
Example 5.2
Let \(H_0\) be an infinitedimensional Hilbert space, and set \(H=H_0\oplus {\mathbb {C}}\) with the Hilbert space norm. Let \(\varepsilon , f\) and g be given as above. Consider the mappings \(A\in B(H),\) \(K:H \rightarrow H,\) and \(G:H\rightarrow H\) defined by
It is clear that \(A\in B(H)\), \(r_{e}(A)=2\), K is compact, and G satisfies
Moreover,
Let \(x_0=(y_0,a_0)\in H\) be such that \(\Vert y_0\Vert \le \varepsilon ^3\) and \(a_0\le \varepsilon ^3\), and let \((x_{n})_{n=1}^\infty \) be given by (5.1). For \(n \ge 1\) denote \(x_{n}: =(y_n,a_n).\) First observe that \(a_n\le \frac{\varepsilon }{2}\) for all \(n \ge 1,\) by the definition of f. We prove that
Observe that \(\Vert y_n\Vert \le 2\Vert y_{n1}\Vert ,\) \(n \in {\mathbb {N}}\). We distinguish two cases.
(A) There exists n such that \(\Vert y_n\Vert > 4\varepsilon ^3\) and \(\Vert y_{n+1}\Vert > 4\varepsilon ^3\). Let \(n_0\) be the smallest integer with this property. We then obtain
and so
Hence, \(g\bigl (\Vert y_{n_0+1}\Vert ,a_{n_0+1}\bigr )=2\) and \(y_{n_0+2}=0\). Thus \(y_m=0\) for all \(m\ge n_0+2\). Moreover, \(\min \{\Vert y_m\Vert ,\Vert y_{m+1}\Vert \}\le 4\varepsilon ^3\) for all \(m\le n_01\). Hence, if \(m \le n_01,\) then
We have shown \(\sup _m\Vert y_m\Vert \le 16\varepsilon ^3\) in this case.
(B) Let \(\min \{\Vert y_n\Vert ,\Vert y_{n+1}\}\le 4\varepsilon ^3\) for all n. We again have
which yields \(\sup _n\Vert y_n\Vert \le 8\varepsilon ^3\).
In both cases we have shown \(\sup _n\Vert y_n\Vert \le 16\varepsilon ^3\le \frac{\varepsilon }{2}\), and so \(\sup _n\Vert x_n\Vert \le \varepsilon \).
Now we refine the construction in Example 5.2 and make it work for all \(\epsilon >0\) thus obtaining the desired counterexample.
Example 5.3
Let \((\varepsilon _j)_{j=1}^\infty \) be a fast decreasing sequence of positive numbers such that \(\varepsilon _1<2^{7}\) and \(\varepsilon _{j+1}<\frac{\varepsilon _j^3}{4}\) for all \(j\ge 1\). As above, let \(H=H_0\oplus \ell ^2\) where \(\dim H_0=\infty \). Take functions \(f_j\) and \(g_j\) as defined in Lemma 5.1 for \(\varepsilon =\varepsilon _j\). Define mappings A, K, and \(G:H\rightarrow H\) by
for all \(\bigl (y,(a_j)_{j=1}^{\infty }\bigr ) \in H.\) Note that for every \((y,(a_j)_{j=1}^\infty )\in H\) there is at most one j such that \(g_j\bigl (y,a_j\bigr )\ne 0\). So G is welldefined. Similarly, there is at most one j with \(f_j(\Vert y\Vert )\ne 0\).
Clearly, \(A\in B(H)\), \(r_{e}(A)=2\) and K is compact since
Furthermore,
We have
Now with A, G and K defined by (5.3)–(5.5), we consider equation (5.1), and prove that its zero solution is stable. Let \(x_0 \in H\) and \(k \in {\mathbb {N}}\) be such that \(\Vert x_0\Vert \le \frac{\varepsilon _k^3}{4},\) and let \((x_n)_{n=1}^\infty \) be given by (5.1).
Write \(x_n=\bigl (y_n, (a_{n,j})_{j=1}^\infty \bigr )\) for all \(n \ge 0.\) We show first that \(\Vert y_n\Vert \le \frac{\varepsilon _k}{2}\) for all n. Suppose the contrary. Let \(n_0\) be the smallest integer with \(\Vert y_{n_0}\Vert >\frac{\varepsilon _k}{2}\). Let \(n_1\) be the largest integer with \(0\le n_1<n_0\) and \(\Vert y_{n_1}\Vert <\frac{\varepsilon _k^3}{2}\). Note that
for all \(\bigl (y,(a_j)_{j=1}^\infty \bigr )\in H\), and hence
for all n. We infer that
for all \(m=n_1,\dots ,n_01\), implying
for all \(j\ne k\) and \(m=n_1,\dots ,n_01\). We further obtain
Consider the orthogonal projection \(P:H\rightarrow H\) onto \(H_0\oplus {{\mathbb {C}}}\) defined by
By the above observations, \(Px_{n_1+1}=(y_{n_1+1},0)\) satisfies
and \(Px_{n_1+1}, Px_{n_1+2},\dots ,Px_{n_0}\) are the iterations described in the previous example for \(\varepsilon =\varepsilon _k\). Using the estimate (5.2), we infer that \(\Vert y_{n_0}\Vert \le \frac{\varepsilon _k}{2},\) a contradiction. We have shown that
so that \(\sup _n\Vert x_n\Vert \le \varepsilon _k.\) Now to prove the stability of the zero solution to (5.1), given \(\epsilon >0,\) it suffices to consider \(\epsilon _k \in (0,\epsilon ),\) and to choose \(\delta =\epsilon _k^3/4.\)
We are short of a similar construction for continuous time. However, we suspect that an example similar to the above for (1.1) may not exist.
6 Wellposedness for continuous time and global existence
We are primarily interested in nonlinear evolution equations of the form
where \(T\in (0,\infty ]\), A generates a \(C_0\)semigroup \((T(t))_{t \ge 0}\) on a Banach space X, and \(K:[0,\infty )\times X \rightarrow X\) is a function such that \(x\mapsto K(t,x)\) is compact for fixed t and that satisfies some additional regularity assumptions. One can study classical solutions of (6.1), that is, maps \(x \in C^1([0,T), X)\cap C([0,T), D(A))\) solving (6.1). However, it is often convenient to deal with the integrated version of (6.1), namely
Recall that a continuous function \(x: [0,T)\rightarrow X\) is called a mild solution of (6.1) if it satisfies (6.2). Each classical solution of (6.1) is a mild solution, but not conversely. A mild solution is a classical one if \(x_0 \in D(A)\) and, for instance, \(K \in C^1([0,T)\times X,X).\) In the sequel we only treat mild solutions, which we will call solutions from now on, for simplicity.
If X has nice geometric properties, e.g., if it is reflexive, then mild solutions can be identified with socalled weak solutions of (6.1), that is, \(x \in L^1([0,T), X)\) satisfying (6.1) in a weak sense. This notion will not be used in this paper, however.
Let the solution \(x(t)=x(t,x_0)\) of (6.2) exist for all \(t\ge 0,\) i.e., one has \(T=\infty .\) It is called (nonlinearly) stable if for every \(\varepsilon > 0\) there is radius \(\delta > 0\) such for all \(y_0 \in B(x_0,\delta )\) all solutions y with \(y(0) = y_0\) are defined on \([0,\infty )\) and satisfy \(\Vert y(t)x (t)\Vert <\varepsilon \) for all \(t\ge 0\). Without loss of generality, one may assume here that \(x_0=0.\) If the solution x is not stable, then it is said to be unstable.
Instability is a local and rather weak property. It is often of interest to show global properties of x which are stronger than mere instability. Such properties are the main topic of this paper. To not overshadow the study of asymptotics of the solutions to (6.2) with assumptions on its wellposedness, we will just postulate the existence and minimal regularity properties of solutions that we need in the sequel. They can be satisfied in many situations of interest as we will make clear below. Aiming at the longterm behavior, it is natural to consider a setup when the solutions of (6.2) exist globally, i.e., on the whole of \([0,\infty ).\) However, we do not need uniqueness of solutions in our main Theorem 7.6. Below we will mention several statements which could be used as additional assumptions to our asymptotic results, so that the results could be formulated in a priori terms.
The next local existence result is wellknown, see [63, pp. 343–345, p. 350] or e.g. [16, Sect. 4], [56, Sect. 6].
Theorem 6.1
Let A generate the \(C_0\)semigroup \((T(t))_{t \ge 0}\) on a Banach space X and let \(K: [0,\infty ) \times X \rightarrow X\) be continuous and Lipschitz in x on bounded sets: for every \(T>0\) and every \(r>0\) there is \(C_{T,r}>0\) such that
for \(t \in [0,T]\) and \(x, y \in B(0, r).\) Then for every \(x_0 \in X\) there exists a maximal existence time \(T=T(x_0)\in (0,\infty ]\) such that the following holds.

(i)
There is a unique solution \(x=x(\cdot , x_0)\) of (6.1).

(ii)
If \(T<\infty ,\) then \(\lim _{t \rightarrow T}\Vert x(t, x_0) \Vert = \infty .\)

(iii)
For any \(T^* \in (0, T)\) there exists a radius \(\delta > 0\) such that \(T=T(y_0) > T^*\) for all \(y_0 \in B(x_0, \delta ).\) Moreover, the map \(B(x_0, \delta ) \rightarrow C([0, T^*], X),\) \(y_0 \rightarrow x(\cdot , y_0),\) is continuous.
Gronwall’s inequality and the above blowup condition (ii) imply the following global existence result.
Corollary 6.2
Besides the conditions of Theorem 6.1, assume that there exists \(c\in L^1_{\textrm{loc}}([0,\infty ))\) such that
Then for every \(x_0\in X\) there is a unique solution \(x(\cdot ,x_0)\) of (6.1) on \([0,\infty ).\)
See e.g. [56, Corollary 6.2.3], where the compactness of \((T(t))_{t \ge 0}\) is irrelevant under our assumptions.
In a similar way, the linear growth condition of the above corollary leads to bounds on the solution without assuming Lipschitz continuity of \(K(t,\cdot )\).
Proposition 6.3
Let A generate the \(C_0\)semigroup \((T(t))_{t \ge 0}\) on a Banach space X with \(\Vert T(t)\Vert \le M e^{\omega t}\) for \(t \ge 0\) and some \(\omega \ge 0\), and let \(K: [0,\infty ) \times X \rightarrow X\) be continuous separately in both variables.
(a) Assume that K satisfies (6.3). Let \(x:[0,T)\rightarrow X\) be a solution of (6.1) and \(t_0\in (0,T)\). Then \(\Vert x(t)\Vert \le C(1+\Vert x_0\Vert )\) for \(t\in [0,t_0]\) and a constant C depending only on \(t_0\), M, \(\omega \) and \(c(\cdot )\).
Let now (6.1) have a global solution \(x=x(t,x_0)\) for all \(x_0\in X_0\) and some subset \(X_0\subseteq X\).
(b) If (6.3) holds, then for every bounded set \(B \subset X_0\) the set \(\{x(t, x_0):\, 0\le t\le t_0, \,\, x_0 \in B\}\) is bounded for each \(t_0>0\).
(c) Assume that K is bounded on bounded sets and that there exist \(r_0 > 1\) and a nonincreasing function \(c: [r_0, \infty ) \rightarrow [0, \infty )\) such that \(\Vert K(t,x)\Vert \le c(\Vert x\Vert )\Vert x\Vert \) for all \(t\ge 0\) and \(x \in X\) with \(\Vert x\Vert \ge r_0.\) If \(\gamma > \omega \) and \(\lim _{r \rightarrow \infty } c(r) = 0,\) then there exists a constant \(M_{\gamma , x_0}\) such that \(\Vert x(t,x_0)\Vert \le M_{\gamma , x_0} e^{\gamma t}\) for \(t\ge 0\) and \(x_0\in X_0\).
Assertion (a) is a direct consequence of Gronwall’s inequality (as in Corollary 6.2), and part (b) follows immediately from (a). For the refinement in part (c), we refer to [75]. So, if K is sublinear as \(\Vert x\Vert \rightarrow \infty \), under the above assumptions the solutions of (6.2) have the same growth bound as those for its linear part.
If the mapping \(x\mapsto K(t,x)\) is compact on X, then one may drop (local) Lipschitz type assumptions on K at the price of losing the uniqueness of mild solutions to (6.2). (Roughly, one replaces Banach’s fixed point theorem, guaranteeing the uniquneess of a fixed point, with Schauder’s fixed point theorem, where the uniquenesss is hardly available.) However, as we have already remarked above, for our purposes mere existence suffices. The next result from [59] is an example of existence theorems based on compactness properties of the nonlinearity. We note that in this and related papers (such as [14] mentioned below) typically the concept of an integral solution is used. However, in our setting integral and mild solutions coincide, see the Corollary to Proposition 3 in [59].
Theorem 6.4
Let A be the generator of a \(C_0\)semigroup on a Banach space X, and \(K: [0, \infty )\times X \rightarrow X\) be jointly continuous and map bounded sets in \([0, \infty )\times X\) to relatively compact sets in X. Then for every \(x_0 \in X\) there exists a maximal existence time \(T=T(x_0)\) such that (6.1) admits a solution x on [0, T). If \(T <\infty \), then x is unbounded and \(\{K(t, x(t, x_0)): 0< t < T\}\) is not relatively compact. Thus, in particular, if the range of K is relatively compact, then T must be infinite.
Remark 6.5
The statement is formulated in [59, Theorem 2] for a semigroup of contractions. However, one can rescale the operators T(t) and \(K(t,\cdot )\) as in the proof of Theorem 7.6, thus reducing to a bounded semigroup \((T(t))_{t\ge 0}\). Passing to the equivalent norm \(\Vert x\Vert _c:=\sup _{t\ge 0} \Vert T(t)x\Vert \), the semigroup then becomes contractive, and we can apply [59, Theorem 2].
Combined with Proposition 6.3, the above theorem yields the following result which we already used after Theorem 1.1. For real Banach spaces, it was proved in [14] in a more general framework of differential inclusions and with weaker compactness and regularity assumptions on K.
Corollary 6.6
Assume the conditions of Theorem 6.4 and (6.3) hold. Then all solutions of (6.1) exist on the whole of \([0,\infty ).\)
Finally, we will need a result yielding the existence of a unique propagator to (6.1) in the linear setting, see e.g. [26, Corollary VI.9.20].
Proposition 6.7
Let A be the generator of a \(C_0\)semigroup \((T(t))_{t \ge 0}\) on Banach space X, and let \(K: [0,\infty )\rightarrow B(X)\) be strongly continuous. Then there exists a unique evolution family \((U(t,\tau ))_{t \ge \tau \ge 0}\subset B(X)\) such that
7 Growth and instability for continuous time
Following a similar strategy as in Sect. 4, we now obtain lower bounds for solutions of semilinear abstract differential equations with a linear part being a generator of a \(C_0\)semigroup. In view of the failure of the spectral mapping theorem for \(C_0\)semigroups, this task is more demanding than the one treated in Sect. 4. However, the ideas remain the same as for the systems with discrete time.
We start with proving auxiliary (and probably known) results on compactness. Below we will assume that the mapping \(K:[0,\infty )\times X\rightarrow X\) is separately continuous and collectively compact, i.e., the set \(K\bigl ( [0,t_0]\times B\bigr )\) is precompact in X for each bounded set \(B\subset X\) and each \(t_0\ge 0\). For this notion in the linear case one may consult e.g. [3].
A simple condition for collective compactness is provided by assuming continuity in t uniformly for x in bounded subsets.
Lemma 7.1
Let \(K:[0,\infty )\times X\rightarrow X\) be separately continuous such that \(K(s,\cdot ):X\rightarrow X\) is compact for all \(s\ge 0\). Suppose that for each \(t_0,\varepsilon >0\) and each bounded set \(B\subset X\) there exists a \(\delta _\varepsilon >0\) such that
for all \(s,t\in [0,t_0]\) with \(st\le \delta _\varepsilon \) and \(x\in B\). Then K is collectively compact.
Proof
Let \(t_0>0\) and \(B\subset X\) be a bounded set. Take \(\varepsilon >0\). Let \(\{t_1,\dots ,t_n\}\) be a finite \(\delta _{\varepsilon /2}\)net in the interval \([0,t_0]\). For each \(j=1,\dots ,n\) the set \(K(t_j)B\) is precompact and so is the set \(C=\{K(t_j,b): j\in \{1,\dots ,n\}, b\in B\}\). Let \(x_1,\dots ,x_m\) be a finite \(\frac{\varepsilon }{2}\)net for C.
Let \(0\le s\le t_0\) and \(u\in B\). There exist \(j\in \{1,\dots , n\}\) with \(st_j\le \delta _{\varepsilon /2}\) and \(i\in \{1,\dots , m\}\) with \(\Vert K(t_j,u)x_i\Vert <\varepsilon /2\). Using also the assumption, we obtain
So \(\{x_1,\dots ,x_m\}\) is a finite \(\varepsilon \)net for the set \(K\bigl ([0,t_0]\times B\bigr )\). Since \(\varepsilon >0\) was arbitrary, the set \(K\bigl ([0,t_0]\times B\bigr )\) is precompact. \(\square \)
Before stating a second compactness criterion, we collect our standing assumptions for semilinear evolution equations.

(A0)
Assume that X is an infinitedimensional Banach space, \(K:[0,\infty ]\times X \rightarrow X\) is collectively compact and continuous separately in both variables, and let \((T(t))_{t \ge 0}\) be a \(C_0\)semigroup on X with generator A.

(A1)
Let (A0) hold, \(X_0\subset X\) and assume that for every \(x _0 \in X_0\) there exists a unique global solution \(x(\cdot ,x_0)\in C([0,\infty ),X)\) of
$$\begin{aligned} x(t,x_0)=T(t)x_0+\int _0^t T(ts)K(s, x(s,x_0)) \, ds, \qquad t \ge 0. \end{aligned}$$(7.1)So the map \(x:[0,\infty )\times X_0\rightarrow X\) is welldefined, and we assume that it is continuous separately in both variables.

(A2)
Let (A0) hold, and \(B\subset X_0\) be a bounded set. Assume that for every \(x_0\in B\) there exist a global solution \(x=x(\cdot ,x_0)\) of (7.1) such that the set \(\{x(t, x_0):\, 0\le t\le t_0, \,\, x_0 \in B\}\) is bounded for each \(t_0>0\).
If (A0) holds and the solutions of (7.1) are unique, then (A1) is satisfied for all \(B \subset X_0\) provided that the map \(x:[0,\infty )\times X_0\rightarrow X\) maps bounded into bounded sets.
In condition (A2), the map \((t,x_0)\mapsto x(t,x_0)\) can be defined for each bounded subset of \(X_0,\) though it depends on this subset and the choice of corresponding solutions. Note that Corollary 6.6 provides an example of K satisfying (A2) for \(X=X_0\) and all bounded sets \(B \subset X_0.\) One can treat the nonunique case in a more systematic manner, for instance using “generalized semiflows”. See e.g. [8], where also nonlinear damped wave equations are treated. In this paper, we preferred however to avoid such concepts.
Proposition 7.2
Let assumption (A2) hold, and let \(t_0>0\) be fixed. Let C be the set whose elements are of the form
for all \(x_0\in B\) and \(t\in [0, t_0],\) where \(x(\cdot , x_0)\) and B are given by (A2). Then C is precompact in X.
Proof
Set \(k=\sup \{\Vert T(t)\Vert :t\in [0, t_0]\}<\infty \) and \(k'=\sup \{\Vert x(t)\Vert : t\in [0,t_0],\, x_0\in B\}<\infty \). Fix \(\varepsilon >0\).
By assumption, the set \(C_0=\bigcup _{0\le s\le t_0} \{K(s,y): y\in X, \,\Vert y\Vert \le k'\}\) is precompact. Let E be a finite \(\frac{\varepsilon }{3kt_0}\)net in \(C_0\). Since the set \(\bigcup _{s\le t_0} T(s)(E)\) is precompact, Mazur’s theorem yields the compactness of
where \(\overline{\textrm{conv}}\) stands for the closed convex hull. Choose a finite \(\frac{\varepsilon }{3}\)net \(E'\) in M. Let \(0\le t\le t_0\), \(x_0\in B\), and \(\varepsilon >0\). Then there exist \(N\in {{\mathbb {N}}},\) a partition \(0=t_0< t_1 <\cdots t_N=t\) of [0, t], and \(s_{j}\in [t_j, t_{j+1}]\) for \(0\le j \le N1\) such that
where
By the choice of E, for \(j=1,\dots ,N\) there are \(e_j\in E\) such that
Hence,
Since \(\sum _{j=1}^NT(ts_j)(t_{j}t_{j1})e_j\) belongs to M, we find \(e'\in E'\) with
It follows that
i.e., \(E'\) is a finite \(\varepsilon \)net for the set C. Since \(\varepsilon >0\) was arbitrary, C is precompact. \(\square \)
The next immediate corollary of (7.1) and Proposition 7.2 will play the role of Lemma 4.1 in the continuous setting.
Corollary 7.3
Under the assumptions of Proposition 7.2, there exists a compact set \(C_{0}=C_0(t_0,B)\subset X\) such that
for all \(x_0\in B\) and \(t \in [0,t_0]\).
The next statement is a counterpart of Theorem 4.4 providing an exponential growth bound on a residual set at the expense of stronger assumptions on K and a much smaller set where the bound holds.
Theorem 7.4
Assume that (A1) holds with \(X_0=X,\) and (A2) is satisfied for any bounded subset of X. Let \(a:[0,\infty )\rightarrow {{\mathbb {R}}}_+\) be a nonincreasing function with \(\lim _{t\rightarrow \infty } a(t) = 0\) and \(L\subset {\mathbb {R}}_+\) be unbounded. Then the set
is residual.
Proof
The proof follows that of Theorem 4.4, though we do not reduce it to this result. Without loss of generality we assume that \(\Vert T(t_0)\Vert _\mu \ne 0\) for all \(t_0\ge 0\) (since otherwise \(\Vert T(t)\Vert _\mu \le \Vert T(t_0)\Vert _\mu \cdot \Vert T(tt_0)\Vert _\mu =0\) for all \(t\ge t_0\).) For \(k\in {{\mathbb {N}}}\) set
Note that \(M_k\) is open since \((t,y)\rightarrow x(t,y)\) is continuous in y. We show that \(M_k\) is dense.
Let \(y\in X\) and fix \(\varepsilon > 0\). Choose \(t\in L\) with \(t\ge k\) and \(a(t) < \varepsilon /4\). By Corollary 7.3 there is a compact set \(C\subset X\) such that
for all \(y'\in X\) with \(\Vert y'y\Vert \le \varepsilon \). Since C is compact, there exists a finitedimensional subspace \(F\subset X\) such that \(\textrm{dist}\,\{c, F\} < \frac{\varepsilon \Vert T(t)\Vert _\mu }{12}\) for all \(c\in C\). Lemma 4.3 (i) (applied to \(A=T(t/n)\)) provides a unit vector \(u\in X\) with \(\textrm{dist}\,\{T(t)u, F\} >\frac{\Vert T(t)\Vert _\mu }{3}\). Arguing as in the proof of Theorem 4.4, we infer that
So \(x_0:= y + \varepsilon u\) or \(x_0:= y  \varepsilon u\) satisfy
It follows that if \(x_0\) is chosen so to satisfy the inequalities above, then
Hence \(x_0\in M_k\), and \(M_k\) is dense since \(y\in X\) and \(\varepsilon >0\) were arbitrary. The Baire category theorem shows that \(\bigcap _{k=1}^\infty M_k\) is a dense \(G_\delta \) set, and thus residual. \(\square \)
Corollary 7.5
Assume that all of the conditions of Theorem 7.4 hold. Let \(a:[0,\infty )\rightarrow {{\mathbb {R}}}_+\) be a nonincreasing function satisfying \(\lim _{t\rightarrow \infty } a(t) = 0\). Then there is a residual set \(M \subset X\) such that for every \(x_0 \in M\) there exist an unbounded sequence \((t_{n})_{n=1}^\infty \) with
We next prove a continuous analogue of Theorem 4.6 which is one of the main results of this paper. It provides global and sharp exponential lower bounds for solutions of semilinear differential equations with unbounded linear part. Recall here the notion of admissibility introduced in Sect. 2.
Theorem 7.6
Let (A2) hold for \(B={\overline{B}}(y,r)\subset X_0.\) Setting \(\alpha =\limsup _{t \rightarrow 0}\Vert T(t)\Vert ,\) let \(a:[0,\infty )\rightarrow {{\mathbb {R}}}_+\) be a nonincreasing function satisfying \(\lim _{t\rightarrow \infty }a(t)=0\), and \(t_0=t_0(r) \ge 0\) be such that with \(a(t_0)<\frac{r}{2\alpha }\). Assume that \(\omega \in {\mathbb {R}}\) is admissible. Then there exists \(z \in {\overline{B}}(y,r)\) and a solution \(x(\cdot , z)\) to (7.1) with \(x_0=z\) such that
Proof
Without loss of generality we may assume that \(\omega =0\). If not, then consider the semigroup \((e^{\omega t}T(t))_{t \ge 0}\) and compact perturbations \(e^{\omega s} K(s, e^{\omega s}\cdot )\). Define a new norm \(\Vert \cdot \Vert _1\) on X by
By (9.2), the norm \(\Vert \cdot \Vert _1\) is equivalent to \(\Vert \cdot \Vert \) with
for all \(x\in X\). Let a Banach space \(X_1\) be X equipped with \(\Vert \cdot \Vert _1.\)
Fix \(c_1\) such that \(2\alpha a(t_0)<c_1<r\). Pick an increasing sequence \((t_k)_{k=1}^\infty \subset (t_0,\infty )\) with \(a(t_k)<\frac{rc_1}{\alpha 2^{k+2}}\) for all \(k \in \mathbb N.\) Set
For each \(k \in {\mathbb {N}}\) choose \(\varepsilon _k>0\) such that \(\varepsilon _{k+1}<\varepsilon _k\) and
The required element z will be constructed as the sum of a series of appropriate approximate eigenvectors \(x_k\) for \((T(t))_{t \ge 0}\) and \(e^{\mu _k t}\). We set \(F_0=\{0\},\) \(M_0=X_1\), and \(x_0=y\). Inductively we construct unit vectors \((x_k)_{k=1}^\infty \subset X_1\), finitedimensional subspaces \(F_1\subset F_2\subset \cdots \), and closed finitecodimensional subspaces \(M_1\supset M_2\supset \cdots \) with \(x_k\in M_k\cap F_{k+1}\) of \(X_1\) which satisfy (7.2)–(7.5) below.
Let \(k\ge 1\) and suppose that the vectors \(x_0,\dots ,x_{k1}\) and subspaces \(F_0,\dots ,F_{k1}\) and \(M_0,\dots ,M_{k1}\) have already been constructed. Consider the family of solutions \(\{x(t, u): u\in B, t \ge 0\}\) to (7.1) given by (A2), and fix the notation \(x(t, \cdot )\) for this family. If \(B_1\) is the image of B under the identity embedding of X into \(X_1\), then Corollary 7.3 yields a compact set \(C_k\subset X\) such that \(x(t,u)\in T(t)u+C_k\) for all \(t\le t_k\) and \(u \in B_1.\) There thus exists a finitedimensional subspace \(F_k\supset F_{k1}\) such that \(x_{k1}\in F_k\) if \(k>1\) and
for all \(t\le t_k\) and
for all \(d\in C_k\), where \(\textrm{dist}\,\) stands for the distance in the new norm \(\Vert \cdot \Vert _1\). Lemma 9.1 yields a closed subspace \(M_k\subset M_{k1}\) of finite codimension such that
for all \(f\in F_k\) and \(m\in M_k\), where
for some \(y_j^* \in D(A^*)\) for \(1 \le j \le k.\) Since 0 is admissible, we can choose \(x_k\in M_k\) with \(\Vert x_k\Vert _1=1\) and \(\mu _k\in {{\mathbb {C}}}\) with \(\textrm{Re}\,\mu _k=0\) such that
for all \(t\le t_k\).
Suppose that the vectors \(x_k\), \(k\in {{\mathbb {N}}},\) have been constructed as above. Set
We show that z meets the requirements in the initial Banach space \((X, \Vert \cdot \Vert ).\) First, z belongs to \({\overline{B}}(y,r)\) since
Let \(x(\cdot , z)\) be a solution of (7.1) given by (A2). Fix \(k\ge 1\) and consider \(t \in [t_{k1},t_k]\). Properties (7.3) and (7.2) yield
By means of (7.5) we estimate
Taking into account (7.4) and that \(x_j\in M_j\subset M_k\) for all \(j\ge k\), we infer
Since \(x_k\in F_{k+1}\) and \(x_j\in M_j\subset M_{k+1}\) for \(j\ge k+1\), (7.4) also implies
Hence
for all \(t\ge t_0\), as required. \(\square \)
Corollary 7.7
(i) Let the assumptions of Theorem 7.6 hold with \(r> 2a(0)\alpha \). Then there exists \(y\in X_0\) with \(\Vert x(t, y)\Vert \ge a(t)e^{\omega t}\) for all \(t\ge 0\).
(ii) Assume that assumption (A2) is satisfied for all bounded subsets B of \(X_0.\) Then there exists a set \(S \subset \textrm{Int}\,X_0\) dense in \(\textrm{Int}\, X_0\) such that \(\Vert x(t, y)\Vert \ge a(t)e^{\omega t}\) for all \(y \in S\) and t sufficiently large.
Proof
To deduce (i) from Theorem 7.6, it suffices to set \(t_0=0,\) and to let y be the center of the ball in \(X_0\) of radius greater than \(2a(0)\alpha \).
To prove (ii), we may assume that \(\textrm{Int} X_0\ne \emptyset .\) Let \(y\in \textrm{Int}\,X_0\), \(\varepsilon =\textrm{dist}\,\{y,\partial X_0\}\), and \(\varepsilon _0 \in (0,\varepsilon ).\) Find \(t_0\ge 0\) such that \(a(t_0)<\frac{\varepsilon _0}{2\alpha }\). By Theorem 7.6, there exists \(x_0 \in X\) such that \(\Vert x_0y\Vert \le \varepsilon _0\) and \(\Vert x(t,x_0)\Vert \ge a(t)e^{\omega t}\) for all \(t\ge t_0\). It remains to note that the choice of \(\varepsilon _0\in (0,\varepsilon )\) is arbitrary. \(\square \)
Remark 7.8
If X is a Hilbert space, as in Sect. 4 one can obtain a better estimate. Let \(y\in X\), \(\{u:\Vert uy\Vert \le r\}\subset X_0\), \(t_0\ge 0\) and \(a(t_0)<r.\) Then there exists \(x_0 \in X\) with \(\Vert x_0y\Vert \le r\) and \(\Vert x(t, x_0)\Vert \ge a(t)e^{\omega t}\) for \(t\ge t_0\).
The next corollary is one of the main results of the paper. It is a direct consequence of Theorems 2.2 and 7.6.
Corollary 7.9
Under the conditions of Theorem 7.6, if \(s_R(A)>\infty ,\) then there exists \(x_0 \in {\overline{B}}(y,r)\) and \(t_0\ge 0\) such that
If \(X_0=X\) and the assumption (A2) is true for all bounded subsets B of X, then the set of \(x_0\) satisfying (7.6) is dense in X.
If in (7.6) the operator A is bounded and \(s_e(A)>0,\) then under appropriate local assumptions on K (involving compactness) the local instability of zero solution to (7.6) was shown in [71].
Our results on lower bounds are also new in the framework of linear equations (6.1), where \(K: [0,\infty )\rightarrow B(X)\) is strongly continuous and each operator K(t) is compact. In this case, K does not have to be collectively compact and so (A0) may be violated. (The result is new even for normcontinuous K, where (A0) holds by Lemma 7.1.) However, in the proof of Theorem 7.6 we need collective compactness only to apply Corollary 7.3, and this result is a direct consequence of Proposition 6.7 and [26, Theorem C.7] in the linear case. Hence, Theorem 2.2 and Corollary 7.7 yield the next estimate.
Corollary 7.10
Let A be the generator of a \(C_0\)semigroup on Banach space X and \(K: [0,\infty )\rightarrow B(X)\) be a strongly continuous function such that K(t) is a compact operator for each \(t \ge 0.\) Let \((U(t,\tau ))_{t \ge \tau \ge 0}\) be the evolution family given by Proposition 6.7. Assume that \(a:[0,\infty )\rightarrow [0,\infty )\) is a decreasing function satisfying \(\lim _{t \rightarrow \infty }a(t)=0\). Then there exists a dense set of vectors x such that
We finish this section with a discussion of our results. First, note that one may treat \(s_R(A)\) “up to compact perturbations”: In the above result one may consider \(s_R(A+S)\) for a compact perturbation \(S\in B(X)\) of A and substract S from the nonlinearity K in (7.1), without changing the assumptions on K.
Remark 7.11
Theorem 7.6 and Corollaries 7.7 and 7.9 in general do not hold for K being merely relatively compact with respect to A, i.e., such that \(K:D(A)\rightarrow X\) is compact, where D(A) is equipped with the graph norm. For instance, consider the setting of linear damped wave equations with the operator \(A_\) given by (8.6) below. Here we have
where D generates a unitary \(C_0\)group on an appropriate Hilbert space X, \(s_R(D)=0,\) K is relatively compact with respect to D. However, in view of [10], the operator \(A_\) generates an exponentially stable \(C_0\)semigroup on X if b is smooth on M and satisfies the socalled geometric control condition. (See also e.g. [1, 42], and [48] for a relevant discussion and some examples.) Thus, in this case, we have \(\omega _0(A_)<0,\) which excludes any results of the form of Theorem 7.6 and its corollaries with \(A=D\) and K as above. If K is compact, then the situation changes as we show in the next section.
Finally, we address the optimality of results stated in Theorem 7.6 and Corollary 7.9.
Remark 7.12
Let \((T(t))_{t \ge 0}\) be the left shift \(C_0\)semigroup on \(L^2({\mathbb {R}}_+)\) defined by \((T(t)f)(s)=f(s+t), t \ge 0,\) for \(f \in L^2({\mathbb {R}}_+).\) If A is the generator of \((T(t))_{t \ge 0},\) then \(\sigma (A)=\{\lambda \in {\mathbb {C}}: \textrm{Re}\, \lambda \le 0\},\) and \(s_e(A)=s_0(A)=s_R(A)=0.\) On the other hand, clearly \(T(t) \rightarrow 0\) as \(t \rightarrow \infty \) strongly. Thus Theorem 7.6 and Corollary 7.9 cannot be improved by removing the function a from their formulations, even for linear equations (6.1) with \(K=0.\) For other concrete examples of such semigroups one may consider the multiplication semigroups \((T(t)f)(z)=e^{tz}f(z)\), \(t \ge 0,\) on appropriate spaces \(L^2(\Omega )\) with \(\Omega \) from the closed left halfplane.
8 Backward damped wave equations and other applications
In this section, we illustrate our abstract results by applying them to the study of backward damped wave equations or, equivalently, “excited” wave equations, subject to nonlinear forcing terms of the form \(f(t,\cdot ,u)\). (The minus is chosen to simplify the sign condition in (8.16).) In this basic PDE setting a positive abscissa \(s_R(A)>0\) and compact nonlinear perturbations \(K(t,\cdot )\) occur naturally, including situations where positivity of \(s_R\) stems from the resolvent growth. In contrast to earlier work, as in e.g. in [72], we allow for timedependent and merely continuous f so that we cannot expect uniqueness in general. As we focus on asymptotic properties of solutions, our assumptions on nonlinearities are, of course, not best possible, and serve first of all to create the right framework to the study of lower bounds for solutions in spectral terms.
We first look at the excited wave equation
on a ddimensional, compact, smooth and connected Riemannian manifold \({\mathcal {M}}\) without or with boundary \(\partial {\mathcal {M}}.\) Here \(\Delta \) is the Laplace–Beltrami operator on \({\mathcal {M}},\) depending in general on a metric on \({\mathcal {M}}.\) We do not indicate this dependence since it will not be relevant. If \(\partial M \ne \emptyset ,\) then equation (8.1) (along with (8.9)) and related function spaces are considered on \(M \setminus \partial M,\) denoted in these two settings, with some abuse of notation, again by M. Additionally, we impose Dirichlet boundary conditions.
In order to apply the results from the previous section, we have to consider complexvalued u in (8.1) and related equations. Since we do not want to restrict ourselves to holomorphic maps \(\zeta \mapsto f(t,x,\zeta )\), we identify \({{\mathbb {C}}}\) with \({{\mathbb {R}}}^2\) equipping it with the Euclidean scalar product \(\zeta _1\cdot \zeta _2=\xi _1\xi _2+\eta _1\eta _2=\textrm{Re}\,(\zeta _1\overline{\zeta _2})\) where \(\zeta _j=\xi _j+i\eta _j=(\xi _j,\eta _j), j=1,2\). Differentiability is then understood in the real sense and derivatives are only \({{\mathbb {R}}}\)linear. Fortunately, this does not affect the basic rules from calculus that we use here. Throughout we assume that
for some \(1\le \alpha < d/(d2)_+\) and all \(t,s\in {{\mathbb {R}}}\), \(\zeta \in {{\mathbb {C}}}\), \(x\in {\mathcal {M}}\), where \(\kappa : {\mathbb {R}} \rightarrow [0,\infty )\) is locally bounded and \(\omega :[0,\infty )\rightarrow [0,\infty )\) satisfies \(\omega (\tau )\rightarrow 0\) as \(\tau \rightarrow 0.\) (Here \(y_+=\max \{y,0\}\) for \(y\in {{\mathbb {R}}}\) and \(\frac{d}{0}:=\infty \).) The assumption \(b\ge 0\) fits to the interpretation of (8.1) as excited wave equation. It is crucial to observe that (8.3) implies the continuity of f in \((t,\zeta ).\)
To formulate (8.1), (8.2) as an evolution equation of the form (1.1), we set \(V=H^1({\mathcal {M}})\) if \({\mathcal {M}}\) has no boundary and \(V=H^1_0({\mathcal {M}})\) otherwise. Recall that V is compactly embedded into \(L^{2\alpha }({\mathcal {M}})\), see [7, Theorem 2.34]. On the state space \(X=V\times L^2({\mathcal {M}})\) we introduce the operator matrix
It is well known that \(A_+\) generates a \(C_0\)group \((T_+(t))_{t\in {{\mathbb {R}}}}\) on X, i.e., \(\pm A_+\) are generators of \(C_0\)semigroups. We write elements of X as \(w=(u,v)\). The forcing term is expressed by \(K(t,w)= (0,f(t,\cdot , u))\). As we will prove in Proposition 8.1 below, \(K:{{\mathbb {R}}}\times X\rightarrow X\) is jointly continuous and collectively compact.
We study (8.1) in forward time. Using standard properties of the wave equation with the (continuous) righthand side \(t\mapsto h(t):=f(t,\cdot ,u(t))\), it can be checked that a mild solution w to
on a time interval \(J=[0,T)\) is of the form \(w=(u,\partial _t u)\) for a function u in \(C^2(J,H^{1}({\mathcal {M}}))\cap C^1(J, L^2({\mathcal {M}}))\cap C(J,V)\) which solves (8.1). (Cf. [16, Lemma 6.2.1]).
For the damped case, we replace \(+b\) by \(b\) obtaining the operator
It generates a \(C_0\)group \((T_(t))_{t\in {{\mathbb {R}}}}\), which is contractive in forward time since \(b\ge 0\), and it corresponds to the damped wave equation
The above remarks on the solution also apply here. We study (8.7) backwards in time, i.e., for \(t \le 0.\) To bring it to standard forward form, one looks at \({\tilde{u}}(t)=u(t)\) for \(t \ge 0\) satisfying
This system coincides with (8.1), (8.2) except for the additional minus in f and before \(u_1\). We drop the tilde. To rewrite this problem as (1.1), we use the operators \(B=A_\) and \(w\mapsto K(t,w)=(0,f(t,\cdot ,u))\). Then (8.9), (8.10) can be reformulated as the firstorder problem
Moreover, if \(J: X \rightarrow X\) is defined by \(J(u,v)=(u,v),\) then \(A_+=J(A_)J^{1}\) (as unbounded operators, see [26, p. 59]), so that \(\sigma (A_+)=\sigma (A_)\) and \(R(\lambda , A_+)=J R(\lambda , A_)J^{1}\) for all \(\lambda \in \rho (A_+).\) Thus,
To clarify the relations between \(A_+\) and \(A_,\) note that by e.g. [77, Lemma 1, p. 75] we have \((A_+)^*=A_,\) and thus one obtains (8.12) once again.
It is wellknown that \(\sigma (A_)\subset \{\lambda : \Vert b\Vert _\infty \le \textrm{Re}\, \lambda \le 0\}\) since \(b\ge 0\), that \(\sigma (A_)\) is invariant under conjugation, and that it consists of a discrete set of eigenvalues since \(A_\) has compact resolvent. Moreover if \(b\ge 0\) is nonzero, we have \(\sigma (A_)\cap i{\mathbb {R}}=\emptyset \) if \(\partial {\mathcal {M}} \ne \emptyset \) and \(\sigma (A_)\cap i{\mathbb {R}}=\{0\}\) (with constants as eigenfunctions) if \({\mathcal {M}}\) has no boundary. (See e.g. [48, p. 74], where the smoothness of b assumed there does not play a role in these statements, or [1, Sect. 4].)
In the latter case, let \(P_0\) be the Riesz projection corresponding to 0 and equip \(X_0=(IP_0)X\) with the inner product norm
Then \(\dot{T}_{}(t):=T_{}(t){\upharpoonright }_{X_0}, t \in {\mathbb {R}},\) is a \(C_0\)group on the Hilbert space \(X_0\) generated by \(\dot{A}_{}:=A_ \upharpoonright _{X_0},\) which is contractive for \(t \ge 0.\) Moreover, by e.g. [1, Sect. 4], we have \(\sigma (\dot{A}_{})=\sigma (A){\setminus } \{0\} \subset \{\lambda : \textrm{Re}\, \lambda \le 0\}\) and there are \(c_1,c_2>0\) with
for \(\lambda \in \rho (A_)\) with \(\lambda  \ge \epsilon _0\) for an appropriate \(\epsilon _0>0.\) This construction allows one, in particular, to study the energy asymptotics for (8.7) in a unified manner, see e.g. [1] (assumption [1, (2–6)] holds due to [48, p. 74]), and also [19] and [11]. We will also use \(\dot{A}_{}\) in the sequel to study the resolvent of \(A_\), see Example 8.7.
Before we consider spectrum and resolvent of \(A_\pm \) in specific cases, we first establish the required properties of K,
Proposition 8.1
Let f and b satisfy (8.3). Then the map \(K:{{\mathbb {R}}}\times X\rightarrow X\) defined above is jointly continuous and collectively compact. If also
for \(\kappa \) from (8.3) and all \((t,x,\zeta )\in {{\mathbb {R}}}\times {\mathcal {M}}\times {{\mathbb {C}}}\), then \(\Vert K(t,w)\Vert \le c\kappa (t)(1+\Vert w\Vert )\) for all \(w \in X, t \in \mathbb R,\) and some \(c>0\).
Proof
The last claim is clear. For the first, let \((t_n)_{n=1}^{\infty }\subset {\mathbb {R}}\) and \((w_n)_{n=1}^\infty \subset X\) be such that \(t_n\rightarrow t\) in \({{\mathbb {R}}}\) and \(w_n=(u_n,v_n)\rightarrow w=(u,v)\) in X as \(n \rightarrow \infty .\) Since V is compactly embedded in \(L^{2\alpha }({\mathcal {M}})\), there is a subsequence \((n_k)\) and a map \(g\in L^{2\alpha }({\mathcal {M}})\) such that \(u_{n_k}\rightarrow u\) in \(L^{2\alpha }({\mathcal {M}})\) and pointwise a.e. as \(k\rightarrow \infty \) and \(u_{n_k}\le g\) a.e. for all k. As noted above, f is jointly continuous in \((t,\zeta ),\) and since \(\kappa \) is locally bounded we have \(m:=\sup _n \kappa (t_n)<\infty \). Hence, \(f(t_{n_k},\cdot , u_{n_k})\) tends pointwise a.e. to \(f(t,\cdot ,u)\) as \(k \rightarrow \infty \) and \(f(t_{n_k},\cdot ,u_{n_k})\le m(1+g^\alpha )\in L^2({\mathcal {M}})\) by (8.3). It follows \(f(t_{n_k},\cdot ,u_{n_k})\rightarrow f(t,\cdot ,u)\)in \(L^2({\mathcal {M}})\) as \(k \rightarrow \infty \), and so K is continuous.
To prove compactness, take \(t\in {{\mathbb {R}}}\) and a bounded sequence \((w_n)_{n=1}^\infty \) in X. Again, there is a subsequence \((n_k)\) and maps \(g\in L^{2\alpha }({\mathcal {M}})\) and \(u\in V\) such that \(u_{n_k}\rightarrow u\) in \(L^{2\alpha }({\mathcal {M}})\) and pointwise a.e. as \(k\rightarrow \infty \) and \(u_{n_k}\le g\) a.e. for all k. As above, we infer that \(f(t,\cdot ,u_{n_k})\) tends to \(f(t,\cdot ,u)\) in \(L^2({\mathcal {M}})\), and thus \(K(t,\cdot ):X\rightarrow X\) is compact. To use Lemma 7.1, let \(t_0,r>0\), \(t,s\in [t_0,t_0]\) and \(\Vert w\Vert \le r\). Due to Sobolev’s embedding, we then obtain \(\Vert u\Vert _{L^{2\alpha }}\le Cr\) for a constant \(C>0\) independent of r, and (8.3) yields for all \(t,s \in \mathbb R,\)
As \(\omega (\tau )\rightarrow 0\) as \(\tau \rightarrow 0\), the map K is collectively compact by Lemma 7.1. \(\square \)
Due to this result, Corollary 6.6 and Proposition 6.3, if \(\alpha =1\) in (8.3) then the operators \(\pm K(\pm t,\cdot )\) fit to Theorem 7.6 and its corollaries.
If \(1<\alpha <d/(d2)_+\), we need a sign condition and some extra time regularity of f to show global existence by means of a standard energy estimate, cf. e.g. [16, Chapter 6]. We define the potential of \(\zeta \mapsto f(t,x,\zeta )\) by the (real) line integral
for \((t,x,\zeta )\in {{\mathbb {R}}}\times {\mathcal {M}}\times {{\mathbb {C}}}\). Notice that
for a constant \(c>0\), by (8.3). In the next result it is assumed that \(\varphi \) is differentiable in \(\zeta \) with \(\nabla _\zeta \varphi =f\). We first discuss this assumption.
Remark 8.2
If f is \(C^1\) in \(\zeta =(\xi ,\eta )\), then the same is true for \(\varphi \). We then have \(\nabla _\zeta \varphi = f\) if and only if \(\partial _\eta f_1=\partial _\xi f_2\). However, \(\nabla _\zeta \varphi \) also exists and is equal to f for the standard example \(f(t,x,\zeta )=\psi (t,x,\zeta ^2)\zeta \), where \(\psi :{{\mathbb {R}}}\times {\mathcal {M}}\times {{\mathbb {R}}}\rightarrow {{\mathbb {R}}}\) is continuous in the third variable. Here we do not need differentibility of f in \(\zeta \) and obtain
We note that in the next proposition we do no use the results discussed in Sect. 6, besides a variant of the basic Theorem 6.1. Instead we construct the solutions as weak* limits of (subsequences of) solutions to regularized problems. This standard method is based on uniform bounds for the energy and the compactness of K (see e.g. [73, Sect. 4.4] for a similar approach using a Galerkin approximation).
Proposition 8.3
Let b and f satisfy (8.3) with \(1\le \alpha <d/(d2)_+\) and \(\varphi \) be given by (8.14). Assume that \(\varphi \) is differentiable in the third variable,
for \(t_0>0\), \(t,s\in [t_0,t_0]\), \(\zeta \in {{\mathbb {C}}}\), \(x\in {\mathcal {M}}\), and a locally bounded map \(\ell :[0,\infty )\rightarrow [0,\infty )\). Then (8.5) and (8.11) have global solutions w such that \(\Vert w(t)\Vert \le c(t_0,r)\) for \(0\le t\le t_0\) and \(\Vert (u_0,u_1)\Vert \le r\) and every \(r,t_0>0\).
Proof
We only treat (8.5) since (8.11) is completely analogous. For the energy estimate we need classical solutions of (8.1) so that we approximate f and \(w_0=(u_0,u_1)\in X\) by more regular functions. Let \(t\in {{\mathbb {R}}}, \) \(x\in {\mathcal {M}}\), \(\zeta \in {{\mathbb {C}}}\), \(t_0>0\), and \(n \in {\mathbb {N}}.\) Identifying \(\mathbb C\) with \(\mathbb R^2,\) take smooth mollifiers \(\rho _n: \mathbb R^2 \rightarrow [0,\infty )\) with support in \(\overline{B}(0,\frac{1}{n})\) and \(\int _{\mathbb R^2} \rho _n(\zeta ) d\zeta =1\) and functions \(\chi _n\in C^1({{\mathbb {R}}}^2,{{\mathbb {R}}}^2)\) with range in \(B(0,n+1)\), \(\chi _n(\zeta )=\zeta \) for \(\zeta \le n\), \(\chi '_n(\zeta )\le c\), so that \(\chi _n(\zeta )\le c_1\zeta ,\) for absolute constants \(c,c_1 \ge 1.\) We set
It is straightforward to check that \(\nabla _\zeta \varphi _n=f_n\) and \(\varphi _n\ge 0\) for all n. Moreover, for each \(n \in {\mathbb {N}},\) the maps \({\tilde{f}}_n,\) \(f_n,\) \(\partial _\zeta f_n\), \(\partial _t f_n\) are bounded on \([t_0,t_0]\times {\mathcal {M}}\times {{\mathbb {C}}}\) (in general not uniformly in n) and there exist \(c_{n,k}=c_{n,k}(t_0) \ge 0\) such that \(\partial _\zeta ^k\varphi _n(t,x,\zeta )\le c_{n,k}(1+\zeta ^{1+\alpha })\) for \((t,x,\zeta ) \in [t_0,t_0]\times {\mathcal {M}}\times {{\mathbb {C}}}\) and \(k \ge 0\). Finally, \(f_n\) and \(\varphi _n\) satisfy (8.3) respectively (8.15) with constants uniform in n, as well as
for \(n \in {\mathbb {N}}\), \(x \in \mathcal M, \zeta \in \mathbb C,\) and a.e. \(t\in [t_0, t_0]\). Here and below in this proof we denote by c and C constants which may vary from line to line and depend only on \(t_0>0\) and quantities given by (8.3) and (8.16). In particular, they do not depend on n, \(t\in [0,t_0]\), or \(w_0\). Set \(K_n(t,w):=(0,f_n(t,\cdot ,u))\). Observe that \(K_n:[t_0,t_0]\times X\rightarrow X\) is (globally) Lipschitz for each \(n \in {\mathbb {N}}\) by the properties of \(f_n\).
Let \(w_0=(u_0,u_1)\in X\) and consider a sequence \((w_{0,n})_{n=1}^\infty \) in \(D(A_+)\) with \(w_{0,n}=(u_{0,n},u_{1,n})\) such that \(w_n \rightarrow w_0\) in X as \(n \rightarrow \infty \). Due to [56, Theorem 6.1.6], problem (8.5) for \(K_n\) and \(w_{0,n}\) has a (unique) solution \(w_n\in C^1([0, \infty ),X)\cap C([0,\infty ),D(A_+))\), where \(D(A_+)\) is endowed with the graph norm. It is now easy to see that \(w_n=(u_n,\partial _t u_n),\) where
solves (8.1) for \(f_n\) and \(w_{0,n}\).
We show that the solutions \(w_n\) are bounded in X uniformly in \(n\in {{\mathbb {N}}}\) and then pass to the limit as \(n\rightarrow \infty \). To this aim, we look at the energies
for \(w=(u,v)\in X\). The above observations imply that for all \(w \in X\) and \(t>0\) one has \(E^0(w)\le E_n(t,w)\le c(1+\Vert w\Vert _X^{1+\alpha })\), \(E_n(t, \cdot ), E^0(\cdot )\in C^1(X,{{\mathbb {R}}})\), and \(E_n(\cdot ,w)\in W^{1,\infty }_{\textrm{loc}}({{\mathbb {R}}})\), where we also use that \(H^1({\mathcal {M}})\hookrightarrow L^{\alpha +1}({\mathcal {M}})\) in view of \(\alpha +1 < 2d/(d2)_+\). For a.e. \(t\in [0, t_0 ]\), we compute
We can bound the \(L^2\)norm of \(u_n(t)\) by means of \(u_n(s)=u_{0,n}+\int _0^s \partial _t u_n(\tau )\, d\tau \), so that
Gronwall’s inequality then yields
and hence
for \(0\le t\le t_0\) and constants not depending on n and \(w_0\).
Using (8.17), we can find a subsequence of \((w_n)_{n=1}^\infty \), denoted by the same symbol, such that \(u_n\) and \(\partial _t u_n\) tend to functions u and v weakly* in each \(L^\infty ([0,t_0], V)\) respectively \(L^{\infty }([0,t_0], L^2({\mathcal {M}}))\), and \(w=(u,v)\) satisfies (8.17) for a.e. \(t\in [0,t_0]\). It is straightforward to check that \(v=\partial _t u\), and so \(w_n\) converges to \(w=(u,\partial _t u)\) weakly* in \(L^\infty ([0,t_0], X)\) as \(n \rightarrow \infty \). Moreover, by the uniform bound (8.17) and the compact embedding from [64, Corollary 4] the sequence \((u_n)\) is precompact in \(C([0,t], L^{2\alpha }(\mathcal M))\), hence in \(L^{2\alpha }([0,t_0] \times \mathcal M)\). There thus exists a subsequence, also denoted by \((u_{n})\), such that \(u_{n}\rightarrow u\) in \(L^{2\alpha }([0,t_0]\times {\mathcal {M}})\) and pointwise a.e., as well as \(u_{n}\le g\) a.e. for a function \(g\in L^{2\alpha }([0,t_0]\times {\mathcal {M}})\).
To show convergence of \((f_{n}(t,\cdot ,u_{n}(t)))\) and thus of \((K_n (t,w_n))\), let \(\varepsilon >0\) and take \(t\in [0,t_0]\) such that \((u_{n}(t,\cdot ))\) tends to \(u(t,\cdot )\) a.e. and is bounded a.e. by \(g(t,\cdot )\in L^{2\alpha }({\mathcal {M}})\). Choose \(x\in {\mathcal {M}}\) outside this null set. For sufficiently large n we have \(u(t,x)\le n1\). There is a number \(\delta \in (0,1)\) such that \(f(t,x,\zeta )f(t,x,u(t,x))\le \varepsilon \) if \(\zeta u(t,x)\le \delta \) since f is continuous in \(\zeta \). So there exists an index \(n_\varepsilon \) such that for all \(n\ge n_\varepsilon \) and \(\eta \le 1/n\) we have \(u_{n}(t,x)\eta < n\) and
It follows that \(f_{n}(t,x,u_{n}(t,x))f(t,x,u(t,x)) \le \varepsilon \), and hence \(f_{n}(t,\cdot ,u_{n}(t))\) tends pointwise a.e. to \(f(t,\cdot ,u(t))\) as \(n\rightarrow \infty \), for a.e. \(t\in [0,t_0]\).
Since \(f_n\) satisfies (8.3) uniformly in n, we have \(f_{n}(t,\cdot ,u_{n}(t))\le c\kappa (t)(1+g(t,\cdot )^\alpha )\) and thus \(f_{n}(t,\cdot ,u_{n}(t))\rightarrow f(t,\cdot , u(t))\) in \(L^2({\mathcal {M}})\) for a.e. \(t\in [0,t_0]\). Moreover, by (8.3) and (8.17) the map \(t\mapsto \Vert f_n(t,\cdot ,u_n(t))\Vert _{L^2({\mathcal {M}})}\) is locally bounded independent of n. Therefore the righthand side in the formula for mild solutions
converges in \(C([0,t_0],X)\) as \(n\rightarrow \infty \). As seen above, the lefthand side tends to \(w=(u,\partial _t u)\) weakly* in \(L^\infty ([0,t_0],X)\) as \(n \rightarrow \infty \) so that
holds for a.e. \(t\in [0,t_0]\). Since the righthand side is continuous and \(t_0>0\) is arbitrary, w solves (8.5) and satisfies (8.17) for all \(0\le t\le t_0\). \(\square \)
Combined with the above analysis and known spectral properties, Theorem 7.6 now shows growth of orbits and thus global instability in concrete examples. One could also use Theorem 7.4, adding more assumptions on f to satisfy (A1). We avoid doing so since the modifications are easy. Note that in view of (8.12), we can switch freely between the two equations (8.5) and (8.11). So we concentrate just on (8.11).
As the first (toy) example, we improve Theorem III in [72] in various respects: We can treat timedepending f, reduce the regularity requirements in \(\zeta \) from \(C^2\) to continuity, and obtain exponential growth instead of mere unboundedness. In all examples we assume that f satisfies (8.3) and either (8.13) or (8.16).
Example 8.4
Let \(b>0\) be constant and \({\mathcal {M}}\) be a ddimensional, compact, smooth and connected Riemannian manifold without or with boundary. It is straightforward to check that the (point) spectrum of \(A_\) is given by an unbounded sequence of eigenvalues on \(\frac{b}{2}+i{{\mathbb {R}}}\) and at most finitely many points in \((b,0]\). It follows \(s_R(A_)\ge \frac{b}{2}\). By Corollary 7.9 and Propositions 8.1 and 8.3, given a function \(a:[0,\infty )\rightarrow [0,\infty )\) decreasing to 0, there are a dense set of initial values \(w_0\in X\) and \(t_0 \ge 0\) such that the corresponding solutions \(w(t,w_0)\) of backward damped wave equation (8.11) admit the lower bound \(\Vert w(t,w_0)\Vert \ge a(t)e^{bt/2}\) for \(t\ge t_0.\) In other words, the energy of solutions to (8.9) grows exponentially in backward time for a dense of initial values.
The next example generalizes the preceding one by allowing the damping b to be nonstationary and far from being smooth. At the same time, it concerns only the case \(d=1.\)
Example 8.5
Let now \({\mathcal {M}}=[0,1]\) and \(b \in \textrm{BV}([0,1])\) with \(b > 0.\) (Recall that we then impose Dirichlet boundary conditions.) It was proved in [21, Theorem 5.3] that there is a sequence of eigenvalues \((\lambda _n)_{n=1}^\infty \subset \sigma (A_)\) with \(\textrm{Re}\, \lambda _n \rightarrow \frac{\beta }{2}\) as \(n \rightarrow \infty \) where \(\beta :=\int _{0}^{1}b(s) ds.\) So either \(\frac{\beta }{2}+i{\mathbb {R}}\) contains an infinite number of the eigenvalues of \(A_,\) or the resolvent of \(A_\) is unbounded on \(\frac{\beta }{2}+i{\mathbb {R}},\) and therefore, \(s_R(A_)\ge \frac{\beta }{2}.\) As above, using Corollary 7.9 along with Propositions 8.1 and 8.3, for a given function \(a:[0,\infty )\rightarrow [0,\infty )\) decreasing to 0 we find a dense set of initial values \(w_0\) such the solutions of (8.11) satisfy \(\Vert w(t,w_0)\Vert \ge a(t)e^{\beta t/2}\) for \(t \ge t_0\) and some \(t_0\ge 0\). Moreover, by [28, Theorem 3.4], the same result holds without assuming \(b > 0\) if \(\Vert b\Vert _\infty \) is sufficiently small.
We proceed with more involved frameworks, relying on quite subtle results from the spectral theory of damped wave equations.
Example 8.6
Let \({\mathcal {M}}\) be a manifold as in Example 8.4, having no boundary. (One may also consider manifolds with boundary and the corresponding generalised geodesic flows, but this setting leads to technical complications, and is thus omitted for simplicity.) To explain our next example, we need to introduce several auxiliary notions pertaining to dynamics of the (Hamiltonian) geodesic flow \((g^t)_{t \in {\mathbb {R}}}\) on a Riemannian cosphere bundle \({\mathcal {S}}^* {\mathcal {M}}\) over \({\mathcal {M}}.\) A relevant discussion of geodesic flows can be found e.g. in [45, Appendix B], see also [12, Sect. 2.1]. Write
and let \(\pi : S^* {\mathcal {M}} \rightarrow {\mathcal {M}}\) be a canonical projection. Given a damping \(b \in C^\infty ({\mathcal {M}})\) with \(b \ge 0,\) define its Birkhoff ergodic average over the geodesic curve in \({\mathcal {M}}\) as \(\langle b \rangle _t (\rho _0): = \frac{1}{2t} \int ^t_{t} (b\circ \pi \circ g^s)(\rho _0)\, ds, t >0,\) and let
Note that by the Birkhoff ergodic theorem, \(\langle b \rangle _\infty (\rho _0):=\lim _{t \rightarrow \infty } \langle b \rangle _t (\rho _0)\) exists almost everywhere with respect to the flow invariant (normalised) Liouville measure on \(S^*{\mathcal {M}}\), and setting
we have
where all of the inequalities can in general be strict. If the geodesic flow is ergodic, then one has
It was proved in [48] that for every \(\epsilon >0\) there are at most finite number of the eigenvalues of \(A_\) outside the strip \([\frac{b_+}{2} \epsilon ,\frac{b_}{2}+\epsilon ]+i{\mathbb {R}}\) (where, in particular \({\mathcal {M}}\) may have a boundary). So that there is a sequence of the eigenvalues clustering at \(\beta +i{\mathbb {R}}\) for some \(\beta \in [\frac{b_{+}}{2}, \frac{b_}{2}]+i{\mathbb {R}},\) and then arguing as in Example 8.5 one concludes that \(s_R(A_)\ge \frac{b_}{2}.\) We also refer to [66] for comments on the generality of this result and an alternative proof.
The result was improved in [66] by showing that for every \(\epsilon >0\) an infinite number of the eigenvalues of \(A_\) belong to the strip \([\frac{b^+_{\infty }}{2}\epsilon , \frac{b^{}_{\infty }}{2}+\epsilon ]+i{\mathbb {R}}.\) Moreover, as proved in [66], if \((g^t)_{t \in {\mathbb {R}}}\) is ergodic, then the eigenvalues of \(A_\) cluster at \(\frac{b^*_\infty }{2}+i{\mathbb {R}}.\) (See also [5] for an illuminating discussion of these results.) Thus, observing that \(s_R(A_) \ge \frac{b^_{\infty }}{2},\) or, if \((g^t)_{t \in {\mathbb {R}}}\) is ergodic, \(s_R(A_)\ge \frac{b^*_\infty }{2}\), we get an exponential lower norm bound for \(w(t,w_0)\) given by (8.11). Clearly, the two estimates for \(s_R(A_)\) considered in this example may produce \(s_R(A_)>0,\) and thus yield a dense set of initial values for exponentially growing solutions to (8.11).
So far, our examples depended on the properties of the spectrum of \(A_.\) However, there are interesting situations when one has to invoke the resolvent of \(A_\) and thus to use a full strength of our Corollary 7.9.
Example 8.7
There are many examples in the literature where \((\dot{T}_{}(t))_{t\ge 0}\) satisfies \(\Vert \dot{T}_{}(t)R(\mu _0, \dot{A})^s\Vert \le M_s e^{\omega _s t}\) for \(t \ge 0\) and some \(s,\omega _s, M_s>0\) as well as, at the same time, \(\omega _0(\dot{T}_{})\ge 0.\) In other words, \((\dot{T}_{}(t))_{t\ge 0}\) decays exponentially in the operator norm as a map from \(D((\dot{A}_)^s)\) for some \(s>0\) to X, but not in B(X). By interpolation, it then follows that \((\dot{T}_{}(t))_{t \ge 0}\) enjoys such a decay for all \(s>0,\) and for us it suffices to fix \(s=1.\)
In view of e.g. [76, Theorem 1.4], for some \(a,c>0\) the resolvent \(R(\lambda , \dot{A}_{})\) then extends analytically to \(\{\lambda : \textrm{Re}\,\lambda \ge a\}\) and is norm bounded there by \(c(1+\lambda ).\) Note that \(R(\lambda , \dot{A}_)\) is unbounded on \(i{{\mathbb {R}}}\). Indeed, if this was wrong, using the Neumann’s series expansion for \(R(is, \dot{A}_{})\), \(s \in {\mathbb {R}},\) and the resolvent estimate \(\Vert R(\lambda , \dot{A}_{})\Vert \le (\textrm{Re}\, \lambda )^{1}\) for \(\textrm{Re}\, \lambda >0,\) we would infer that \(s_0(\dot{A}_{})<0,\) and then \(\omega _0(\dot{T}_{})<0\) by (2.3). By a standard application of Phragmen–Lindelöf’s theorem, \(R(\lambda , \dot{A}_{})\) also has to be unbounded on a vertical line \(\beta +i{{\mathbb {R}}}\) for some \(\beta \in (0,a]\). As a result, the resolvent of \(A_\) is unbounded on \(\beta +i{{\mathbb {R}}},\) so that \(s_R(A_)\ge \beta >0\). Hence, as above for a dense set of initial values we infer that the norms of solutions to the backward damped wave equation (8.11) grow exponentially and thus show a global instability result.
To describe concrete situations when such an effect can happen recall that if \({\mathcal {M}}\) is as in Example 8.6 and is negatively curved, then by a classical result due to Anosov (see e.g. [2]) a geodesic flow on \(\mathcal {S^*{\mathcal {M}}}\) admits countably many periodic orbits. It was revealed in [62, Theorem 1] that each such an orbit gives rise to a smooth damping b such that \((\dot{T}_{}(t))_{t \ge 0}\) does not decay exponentially, while its orbits \(\dot{T}_{}(t)x\) decay exponentially for sufficiently smooth initial data x. More precisely, it was shown in [62] that, for any periodic geodesic \(\gamma \) in \({\mathcal {M}}\) and \(b_0\in C^{\infty }({\mathcal {M}}),\) there exists \(\epsilon >0\) such that if \(b_0\) vanishes in an \(\epsilon \)neighborhood of \(\gamma \) and is positive everywhere else, then such a decay takes place for \(b=cb_0\) for all sufficiently large \(c>0.\)
If \({\mathcal {M}}\) is a hyperbolic surface with constant negative curvature, then as proved in [41, Theorem 1.1], the decay takes place for all smooth dampings b. This property can in fact be generalized to all surfaces \({\mathcal {M}}\) whose geodesic flow has the socalled Anosov property, see [25, Theorem 6], though such a generalisation is very deep and demanding. Other instances of the exponential decay for only smooth enough orbits of \((\dot{T}_{}(t))_{t \ge 0},\) sometimes with explicit rates, can also be found in [15, Sect. 4], [55] and [61, Theorem 3 and the subsequent Remark]. We avoid their discussion to keep our exposition within reasonable limits.
Note that the assumption \(b \ge 0\) was chosen just to fit in the framework of the existing work, and it can be avoided in many cases (e.g. in Examples 8.5, 8.4, and 8.6). In this case, \(s_R\) provides just a lower bound, not necessarily growing exponentially.
Finally, we show nonstabilizability of certain nonlinear infinitedimensional control systems. In this way, we generalize the corresponding results in [36] or [74], for instance, where the operators B, F, and C used below are linear and bounded. The literature on stabilization of control systems is enormous (and, thus, we skipped a discussion of asymptotics for damped wave equations as a stabilization problem, see e.g. [46] and [47] concerning that). We just refer to [44] and [67] as sample works on nonlinear stabilization and to [18] for general concepts of nonlinear control. For basics of linear theory one may consult [26, Sect. VI.8].
Example 8.8
Let \((T(t))_{t \ge 0}\) be a \(C_0\)semigroup on a Banach space X with generator A, and let the control map \(B:U\rightarrow X\), the feedback \(F:Y\rightarrow U\) and the observation map \(C:X\rightarrow Y\) be all (possibly) nonlinear and continuous with linear growth (i.e., satisfy (6.3) with a constant c), where U and Y are Banach spaces. Combined with Corollary 6.6 and Proposition 6.3, Theorems 4.6 and 7.6 imply the following results:
If \(s_R(A)=0,\) then the system \(x'=Ax+ BFCx\) is not exponentially stabilizable by a compact nonlinear feedback F, i.e., it will always have solutions not decaying exponentially. If \(s_R(A)>0,\) the system is not strongly stabilizable by a compact nonlinear feedback F, i.e., some of its solutions will not converge to zero. (In fact, they will grow exponentially.)
The analogous statement holds for timediscrete feedback systems \(x_{n+1}=Ax_n+ BFCx_n, n \ge 0,\) for bounded A and continuous B, F, and C mapping bounded sets into bounded sets, if one replaces \(s_R(A)\) with \(r_e(A).\)
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The work of the first author was partially supported by grant No.2022230L and RVO:67985840. The second author was funded by the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation)—ProjectID 258734477—SFB 1173. The third author was partially supported by the NCN grant 2017/27/B/ST1/00078.
Appendix
Appendix
In this section we prove several results on geometric properties of Banach spaces and fine spectral theory of semigroups and their generators which are crucial for our lower estimates. Some of them, e.g. Theorem 2.2, are of independent interest. The exposition here follows [50, Sect. 4] with appropriate changes and improvements, which warrant an independent treatment. The section makes the paper essentially selfcontained.
First, we note a geometric statement from Banach space theory. Its versions are often used in iterative constructions arising in the study of orbits of linear operators, and it are important in our studies too. To make our presentation selfcontained and to provide a better understanding of our constructions, we give its proof below. Let X be a Banach space. Recall that a subset \(\Lambda \subset X^*\) is norming if
for all \(x\in X\). Part (a) of the next result is [49, Lemma V.37.6]. We add two related statements whose proof is a variation of the arguments in [49].
Lemma 9.1
Let F be a finitedimensional subspace of a Banach space X and \(\varepsilon >0\).

(a)
There exists a closed subspace \(M\subset X\) of finite codimension such that
$$\begin{aligned} \Vert f+m\Vert \ge (1\varepsilon )\max \bigl \{\Vert f\Vert ,{\Vert m\Vert /2}\bigr \} \end{aligned}$$(9.1)for all \(f\in F\) and \(m\in M\).

(b)
Let M be given by (a) and E be a finitedimensional subspace with \(F \subset E\). Then there exists \(L\subset M\) of finite codimension such that (9.1) holds for \(f \in E\) and \(m \in L\).

(c)
If \(\Lambda \subset X^*\) is a norming set, then there exists a subspace \(M\subset X\) satisfying (9.1) such that \(M=\bigcap _{j=1}^k\textrm{Ker} \, y^*_j\) for some \(y^*_1,\dots ,y^*_k\in \Lambda \).
Proof
We can assume that \(\varepsilon <1\). The unit sphere in F is compact, therefore there exists a finite subset \(D\subset \{f\in F:\Vert f\Vert =1\}\) satisfying \(\textrm{dist}\,\{f,D\}\le \varepsilon /2\) for all \(f\in F\) with \(\Vert f\Vert =1\). Let \(\Lambda \subset X^*\) be norming. For each \(d\in D\) there is a functional \(y^*_d\in \Lambda \) such that \(\bigl \bigl \langle d,y^*_d/\Vert y^*_d\Vert \bigr \rangle \bigr >1\varepsilon /2\). Set \(M=\bigcap _{d\in D}\textrm{Ker}\, y^*_d\). Clearly, M is closed with finite codimension.
To prove the required inequality, let \(f\in F\) and \(m\in M\). We can assume that \(\Vert f\Vert \ne 0\) since the assertion is clear for \(f=0\). Choose \(d\in D\) with \(\bigl \Vert df/\Vert f\Vert \bigr \Vert \le \frac{\varepsilon }{2}\). As in [49, Lemma V.37.6], we then estimate
One can now conlcude as in [49, Lemma V.37.6] to show (a) and (c). Part (b) is is direct consequence of the construction above. \(\square \)
If X is a Hilbert space, we can take \(M=F^\perp \) in the lemma. Thus the subspace \(M\subset X\) constructed above plays a role of the orthogonal complement of a finitedimensional subspace for general Banach spaces.
Next we turn to the spectral theory of semigroups related to constrution of approximate eigenvectors with some additional geometric properties. First we prove a property of the boundary essential spectrum of unbounded operators, wellknown in the bounded case.
Lemma 9.2
Let A be a closed, densely defined operator on a Banach space X, such that \(\rho (A)\ne \emptyset \), and \(\lambda \in \partial \sigma _e(A)\). Let \(M\subset X\) be a closed subspace of finite codimension and \(\epsilon >0\). Then there exists a unit vector \(x\in M\cap D(A)\) such that \(\Vert (A\lambda )x\Vert <\epsilon \).
Proof
We will rely on the fact, that the statement is true if A is bounded, see e.g. [49, Proposition III.19.1 and Theorem III.16.8]. Without loss of generality we may assume that \(\lambda =0\).
Let \(\mu \in \rho (A)\) and let \(T:=A(A\mu )^{1}=I+\mu (A\mu )^{1}\). Then T is bounded. Moreover, \(\sigma _e(T){\setminus }\{1\}=\{1+\frac{\mu }{z\mu }:z\in \sigma _e(A)\}\) by (2.1). So \(0\in \partial \sigma _e(T)\). Let \(M'=\textrm{Im}\, ((A\mu ){\upharpoonright }_M)\). Then \(\textrm{codim}\, M'<\infty \), and as \(T \in B(X)\) there exists a sequence \((x_n)_{n=1}^\infty \subset M'\) such that \(\Vert x_n\Vert =1\) for all n and \(Tx_n\rightarrow 0\) as \(n \rightarrow \infty \). Set \(y_n=(A\mu )^{1}x_n\) for \(n \in {{\mathbb {N}}}\). Then \(y_n\in M\cap D(A)\) for all n and \(Ay_n\rightarrow 0\) as \(n \rightarrow \infty .\) Moreover, since
we have \(\liminf _{n \rightarrow \infty }\Vert y_n\Vert =1/\mu >0.\) It remains to choose \(a y_n\) for appropriate \(a>0\) and n. \(\square \)
The proof of Theorem 2.2 is based on the next lemma.
Lemma 9.3
Let A generate the \(C_0\)semigroup \((T(t))_{t\ge 0}\) on a Banach space X. Let either \(\omega >s_e(A)\) or \(\omega =s_e(A)\) and \((\omega +i{\mathbb {R}})\cap \sigma _{e}(A)=\emptyset \). Assume that either
or
Then there exist \((\mu _n)_{n=1}^\infty \) from \(\omega +i{\mathbb {R}}\) and \((u_n)_{n=1}^\infty \subset D(A)\) such that \(\Vert u_n\Vert =1\) for all \(n \in {\mathbb {N}},\) \(\Vert (\mu _nA)u_n\Vert \rightarrow 0\) as \(n \rightarrow \infty \), and for every \(y^* \in D(A^*)\) we have
In particular, if X is reflexive, then \((u_n)_{n=1}^\infty \) tends weakly to 0 (since then \(D(A^*)\) is dense in \(X^*\)).
Proof
By our assumptions, there exist a sequence \((b_n)_{n=1}^\infty \) with \(b_n\rightarrow \infty \) and unit vectors \(u_n\in D(A)\) for \(n \ge 1\) such that \(\Vert (\omega +ib_n A)u_n\Vert \rightarrow 0\) as \(n \rightarrow \infty \). Set \(\mu _n=\omega +ib_n\). We show that
for each \(y^*\in D(A^*)\). Let \(y^*\in D(A^*)\subset X^*\) have norm 1. Pick a vector \(y\in D(A)\) with \(\langle y,y^*\rangle > \frac{1}{2}\). Let \(M=\textrm{Ker}\, y^*\). Write \(u_n=m_n+\alpha _n y\) for some \(m_n\in M\) and \(\alpha _n\in {\mathbb {C}}\). Then the sequences \((m_n)_{n=1}^\infty \) and \( (\alpha _n)_{n=1}^\infty \) are bounded. Furthermore,
and
Since the last two terms are uniformly bounded and \(\mu _n\rightarrow \infty \), we have \(\alpha _n\rightarrow 0\) as \(n \rightarrow \infty \). It follows \(\langle u_n,y^*\rangle =\alpha _n \langle y,y^*\rangle \rightarrow 0\). \(\square \)
Let \((T(t))_{t\ge 0}\) be a \(C_0\)semigroup on a Banach space X, and set
for \(x\in X\). Then \(\Vert \cdot \Vert _1\) is an equivalent norm satisfying
for all \(x\in X\). See e.g. [53, Theorems 1.3.1 and 1.3.5]. Hence, renorming X with \(\Vert \cdot \Vert _1,\) we can make \(D(A^*)\) a norming set for the Banach space \((X,\Vert \cdot \Vert _1)\).
Recall the definitions of the resolvent bound \(s_R\) and of the notion of admissibility given in Sect. 2. Now we are ready to prove Theorem 2.2 stated there and describing one of admissible \(\omega \) in resolvent terms. More precisely, we show that if A generates a \(C_0\)semigroup \((T(t))_{t\ge 0}\) on a Banach space X, then the bound \(s_R(A)\) is admissible, if it is finite.
Proof of Theorem 2.2
Let M be a closed subspace of finite codimension in X given by \(M=\bigcap _{j=1}^k\textrm{Ker}\, y^*_j\) for some functionals \(y^*_1,\dots ,y^*_k \in D(A^*).\) Let \(\varepsilon >0\) and \(t_0 >0\) be fixed, and set \(K:=\sup \{\Vert T(t)\Vert :0\le t\le t_0\}\). Since the admissibility does not depend on equivalent renormings of the underlying space, in view of (9.2), we can assume that \(D(A^*)\) is a norming set for X.
Let \(s_R(A)=s_e(A)\) and \(\mu \in \sigma _{e}(A)\cap (s_R(A)+i{\mathbb {R}})\). Lemma 9.2 then yields \(x \in D(A)\cap M\) with \(\Vert x\Vert =1\) and \(\Vert (A\mu )x\Vert \le \varepsilon .\) Hence
for all \(t\in [0, t_0]\).
Let \(s_R(A)\ge s_e(A)\) and \(\sigma _{e}(A)\cap (s_R(A)+i{\mathbb {R}}) =\emptyset \). Employing Lemma 9.3, we find sequences \((\mu _n)_{n=1}^\infty \subset {\mathbb {C}}\) with \(\mu _n=s_R(A)+i b_n\) for \(n \in {\mathbb {N}}\) and \((u_n)_{n=1}^\infty \subset D(A)\) with \(\Vert u_n\Vert =1\) such that
By [60, Lemma 7.4], there exists a finitedimensional subspace \(F\subset D(A)\) such that \(X=M\oplus F\). Let P be the projection onto F with \(\textrm{Ker}\,P= M\). By the choice of \(u_n\) we have \(\Vert Pu_n\Vert \rightarrow 0\) so that \(\Vert (IP)u_n\Vert \rightarrow 1\) as \(n \rightarrow \infty \) and
Choose \(n_0\in {{\mathbb {N}}}\) such that
Set
For every \(0\le t\le t_0,\) we have
and
This finishes the proof. \(\square \)
Remark 9.4
If X is a reflexive Banach space, then the proof of the previous theorem is simpler. In this case it is not necessary to do the renormalization since \(\Vert \cdot \Vert _1=\Vert \cdot \Vert \) by the density of \(D(A^*)\), and the subspace M in the statement of Theorem 2.2 can be any close subspace of finite codimension.
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Müller, V., Schnaubelt, R. & Tomilov, Y. On growth and instability for semilinear evolution equations: an abstract approach. Math. Ann. 389, 3885–3933 (2024). https://doi.org/10.1007/s00208023027334
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DOI: https://doi.org/10.1007/s00208023027334