# On growth and instability for semilinear evolution equations: an abstract
approach

Vladimir Müller  V.M, Institute of Mathematics, Czech Academy of Sciences, ul.
Žitna 25, Prague, Czech Republic<EMAIL_ADDRESS>,  Roland Schnaubelt
R.S., Karlsruhe Institute of Technology, Department of Mathematics, 76128
Karlsruhe, Germany<EMAIL_ADDRESS>and  Yuri Tomilov  Y.T., Institute of
Mathematics, Polish Academy of Sciences, Śniadeckich str.8, 00-656 Warsaw,
Poland<EMAIL_ADDRESS>Faculty of Mathematics and Computer Science
Nicolas Copernicus University
ul. Chopina 12/18
87-100 Toruń, Poland

###### 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.

###### Key words and phrases:

Instability, exponential growth, resolvent, compact, semilinear

###### 1991 Mathematics Subject Classification:

35B40, 37L15, 47H08,47J35

The work of the first author was partially supported by grant No.20-22230L and
RVO:67985840. The second author was funded by the Deutsche
Forschungsgemeinschaft (DFG, German Research Foundation) – Project-ID
258734477 – SFB 1173. The third author was partially supported by the NCN
grant 2017/27/B/ST1/00078.

##  1\. Introduction

The paper is devoted to the study of instability of solutions to semilinear
evolution equations

(1.1) $x^{\prime}(t)=Ax(t)+K(t,x(t)),\quad t\geq 0,\qquad x(0)=x_{0}\in X,$

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 $t\geq 0.$

One of the basic and commonly used instability criteria is due to Shatah and
Strauss, [64]. 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, $K(x)={\rm O}(\|x\|^{1+\delta})$ with
$\delta>0$ as $\|x\|\to 0$) 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 infinite-dimensional systems under “small” nonlinear
perturbations was revived in [29], [30], and [57]. 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 $K:X\to X$ 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 [71]
considered the case that either the (Browder) essential spectrum of $A$
intersects the open right half-plane ${\mathbb{C}}_{+}$ 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 $\\{x(t,x_{0}):0\leq t<T\\}$ of (1.1) are unbounded on their maximal
existence interval $[0,T)$, where $0<T\leq\infty.$ 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 [71] by much weaker assumptions on
resolvent bounds. While the results in [71] 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 [49] 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 $(T(t))_{t\geq 0}$ on a Banach space $X,$
with $\alpha:=\limsup_{t\to 0}\|T(t)\|,$ and let $K:[0,\infty)\times X\to X$
be jointly continuous and map bounded sets in precompact ones. Let $B\subset
X$ be a bounded set. 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
$S(t_{0},B)=\\{x(t,x_{0}):\,0\leq t\leq t_{0},\,\,x_{0}\in B\\}$ is bounded
for each $t_{0}>0$. Let $a:[0,\infty)\to[0,\infty)$ be non-increasing with
$\lim_{t\to\infty}a(t)=0$, and let $r>0$, $y\in B$ and $t_{0}=t_{0}(r)\geq 0$
be such that $a(t_{0})<\frac{r}{2\alpha}$ and $\overline{B}(y,r)\subset B$.

  * (i)

Assume that the resolvent $(\omega+ib-A)^{-1}$ is well-defined and unbounded
for $b\in{\mathbb{R}}$ and some $\omega\in{\mathbb{R}}$. Then there exist
$x_{0}\in\overline{B}(y,r)$ and a mild solution $x(t,x_{0})$ of (1.1) such
that

$\|x(t,x_{0})\|\geq a(t)e^{\omega t},\qquad t\geq t_{0}.$

  * (ii)

In any case, there exist $x_{0}\in\overline{B}(y,r)$ and a mild solution
$x(t,x_{0})$ of (1.1) such that

$\|x(t,x_{0})\|\geq a(t)e^{s_{e}(A)t},\qquad t\geq t_{0},$

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 $\|K(t,x)\|\leq(1+c(t))\|x\|$ for some $c\in
L^{1}_{\mathrm{loc}}([0,\infty))$ and all $t\geq 0$ and $x\in X$. See
Proposition 6.3 and Corollary 6.6. In the case of backward damped wave
equations considered in Section 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)

(1.2) $x_{n+1}=Ax_{n}+K_{n}(x_{n}),\qquad x_{0}=x\in X,\qquad n\geq 0,$

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 a 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 non-increasing
sequence $(a_{n})_{n=1}^{\infty}\subset\mathbb{R}_{+}$ satisfying
$\lim_{n\to\infty}a_{n}=0$. Fix $y\in X$, $r>0$, and $n_{0}\in\mathbb{N}$ with
$a_{n_{0}}<\frac{r}{2}$. Then there exists $x_{0}\in B(y,r)$ such that

$\|x_{n}(x_{0})\|\geq a_{n}r_{e}(A)^{n},\qquad n\geq n_{0},$

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
[38, Theorem 5.1.5] or [29, Sections 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 [38]). Also our discrete
results are essentially optimal, cf. [48, Sections V.37 and V.39] for the case
$K=0$.

Using measures of non-compactness, 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 [71],
roughly speaking, the linear semigroup $(T(t))_{t\geq 0}$ is expanding along
infinitely many independent directions and the non-linear 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 infinite-dimensional
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

$Y_{n+1}=AY_{n}+KY_{n}^{2},\qquad Y_{0}\in B(X),\qquad n\in\mathbb{N}.$

If $X$ is finite-dimensional, then even if ${\rm dim}\,X=1$ the asymptotic
behavior of $\\{Y_{n}\\}$ could be extremely complicated (see e.g. [13]), and
any bounds for $\|Y_{n}\|$ can hardly put under control for _individual_
$Y_{0}.$ However, if ${\rm dim}\,X=\infty,$ and $K\in B(X)$ is compact, then
the quadratic part $Y\mapsto KY^{2}$, 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], [38],
[64] and [29], it is natural to consider also nonlinear perturbations $K+G$
with a compact operator $K$ and a metrically small map $G:B(0,r)\to X$
satisfying $G(x)=O(\|x\|^{1+\delta})$ as $\|x\|\to 0.$ However, at least in
the time discrete case given by (1.2) we show in Section 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 [45, Chapter 6] and [46, Chapters 10, 11] the survey [19], and the
papers [1], [5], [15], [20], [25], [40], [41], [42], [54], [56], [60], [61],
[65] 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 reaction-diffusion systems as in
[71]. However, they would require a separate treatment.

In Sections 2, 3, and 9 we provide tools for our analysis and discuss the
background. The main results are proved in Section 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 Section 6 and then used in Section 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 infinite-
dimensional. 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 $A,$
${\rm Ker}\,(A)$ for the kernel of $A,$ ${\rm Im}(A)$ 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 $K(X)$. For a subspace $M$ of a Banach space,
${\rm dim}\,M$ is the dimension of $M,$ and ${\rm codim}\,M$ its codimension.
For a subset $S$ of a topological space, $\partial S$ designates its boundary,
and ${\rm card}\,S$ the cardinality of an arbitrary set $S.$

##  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 a Banach space $X$ its
essential spectrum $\sigma_{e}(A)$ is defined as

$\sigma_{e}(A)=\\{\lambda\in\mathbb{C}:\lambda-A\,\,\text{is not
Fredholm}\\}.$

Clearly, $\sigma_{e}(A)\subset\sigma(A).$ Moreover, by [59, 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 [71]. A
crucial property of $\sigma_{e}(A)$ is that it is invariant is invariant under
relatively compact perturbations if $\rho(A)\neq\emptyset,$ see e.g. [23,
Theorem 11.2.6]. Moreover, by [23, Theorem 11.2.2], if there exists
$\mu\in\rho(A),$ then

(2.1)
$\sigma_{e}(R(\mu,A))\setminus\\{0\\}=\\{(\mu-\lambda)^{-1}:\,\,\lambda\in\sigma_{e}(A)\\}.$

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 non-empty 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 $\sigma(A)\setminus\\{\lambda:|\lambda|\leq
r_{e}(A)\\}$ consists of at most countably many isolated eigenvalues (of
finite multiplicity), see e.g. [48, 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 upper-Fredholm if
${\rm dim}\,{\rm Ker}\,(A)<\infty$ and ${\rm Im}\,(A)$ is closed. If
$\sigma_{e}(A)$ is large and $\rho(A)\neq\emptyset,$ 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
$\|(A-\lambda)u\|<\varepsilon.$ See Lemma 9.2.

The following useful proposition can be found in e.g. [59, 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 finite-dimensional
kernel (i.e., $A$ is upper Fredholm) if and only if ${\rm dim}\,{\rm
Ker}\,(A+K)<\infty$ 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 non-compactness

As far the essential spectrum is involved, measures of non-compactness
naturally come into play, although their role in our studies will rather be
supplementary in contrast to [71]. For $A\in B(X)$ let

$\|A\|_{\mu}=\inf\bigl{\\{}\|A\upharpoonright_{M}\|:M\subset X\,\,\text{is a
closed subspace},\,\,{\rm codim}\,M<\infty\bigr{\\}}.$

The mapping $A\mapsto\|A\|_{\mu}$ is called $\mu$-measure of non-compactness
on $B(X).$ It defines a semi-norm on $B(X),$ where $\|A\|_{\mu}=0$ if and only
if $A$ is compact. Moreover, we have
$\|AB\|_{\mu}\leq\|A\|_{\mu}\cdot\|B\|_{\mu}$ for all $A,B\in B(X).$

For $A\in B(X)$, we introduce the essential norm of $A$ by

$\|A\|_{e}:=\inf\\{\|A-K\|:K\in K(X)\\};$

i.e., $\|A\|_{e}$ 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

$\|A\|_{\mu}=\|A\|_{e}.$

In general, $\|A\|_{\mu}$ and $\|A\|_{e}$ are equivalent norms on $B(X)/K(X)$
if and only if $X$ has a so-called 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 well-known
Nussbaum formula

$r_{e}(A)=\lim_{n\to\infty}\|A^{n}\|_{\mu}^{1/n}=\lim_{n\to\infty}\|A^{n}\|_{e}^{1/n},$

valid for all definitions of the essential spectrum, the quantities
$\|\cdot\|_{\mu}$ and $\|\cdot\|_{e}$ are asymptotically equivalent in a
sense. Moreover, the limits above can be replaced with the infimums. Hence,
$\|A\|_{\mu}^{n}\geq r_{e}(A)^{n}$ and $\|A\|_{e}^{n}\geq r_{e}(A)^{n}$ 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 $(T(t))_{t\geq 0}$ a $C_{0}$-semigroup on a Banach space
$X,$ with generator $A,$ then

(2.2) $e^{t\sigma_{e}(A)}\subset\sigma_{e}(T(t))\qquad\text{for all}\quad
t\geq 0,$

see e.g. [50] 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 $(T(t))_{t\geq 0}$ by

$\displaystyle\omega_{0}(T):=\lim_{t\to\infty}\frac{\ln\|T(t)\|}{t},$

the spectral bound of its generator $A$ by

$s(A):=\sup\\{{\rm Re}\,\lambda:\lambda\in\sigma(A)\\},$

and the pseudo-spectral bound of $A$ (or abscissa of uniform boundedness of
the resolvent of $A$) by

$\displaystyle s_{0}(A):=\inf\\{\omega>s(A):R(\lambda,A)\,\,\text{is uniformly
bounded for }{\rm Re}\,\lambda\geq\omega\\}.$

The first two bounds possess “essential analogues” given by

$\omega_{e}(T):=\lim_{t\to\infty}\frac{\ln\|T(t)\|_{e}}{t}\qquad\text{and}\qquad
s_{e}(A):=\sup\\{{\rm Re}\,\lambda:\lambda\in\sigma_{e}(A)\\}.$

Note that the first limit exists, and $r_{e}(T(t))=e^{\omega_{e}(T)t}$ for
$t\geq 0,$ see e.g. [74]. 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
$\\{\lambda\in\sigma(A):{\rm Re}\,\lambda>s_{e}(A)\\}$ is an at most countable
set consisting of isolated eigenvalues of $A$ (with finite multiplicity).
Clearly,

$s_{e}(A)\leq s(A)\leq s_{0}(A)\leq\omega_{0}(T)\qquad\text{and}\qquad
s_{e}(A)\leq\omega_{e}(T)\leq\omega(T).$

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 $(T(t))_{t\geq 0}.$ Semigroups with $s(A)<\omega_{0}(T)$ also arise
from concrete partial differential equations, such as damped wave equations,
cf. Section 8.

On the other hand, if $X$ is a Hilbert space, then the well-known Gearhart-
Herbst-Prüss theorem guarantees that

(2.3) $\omega_{0}(T)=s_{0}(A),$

and the exponential decay of $(T(t))_{t\geq 0}$ 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 [53, Chapters 1-4].

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

(2.4) $\displaystyle{\rm
card}\,(\sigma_{p}(A)\cap(a+i\mathbb{R}))<\infty\quad\text{and}\quad\limsup_{|b|\to\infty}\|R(a+ib,A)\|<\infty.$

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 $\\{\lambda\in\sigma(A):{\rm
Re}\,\lambda>s_{e}(A)\\}$ 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

$\displaystyle s_{R}(A)=s_{e}(A)\qquad\text{or}\qquad{\rm
card}\,(\sigma_{p}(A)\cap(s_{R}(A)+i\mathbb{R}))=\infty$
$\displaystyle\text{or}\\!\quad{\rm
card}\,(\sigma_{p}(A)\cap(s_{R}(A)+i\mathbb{R}))<\infty\,\,\text{and}\,\,\limsup_{|b|\to\infty}\|R(s_{R}(A)+ib,A)\|=\infty.$

Moreover, as the set $\\{\lambda\in\sigma(A):{\rm Re}\,\lambda>s_{e}(A)\\}$ is
discrete, one may also define $s_{R}(A)$ as the infimum of the set of
$a>s_{e}(A)$ such that

$\exists\,\,\beta=\beta(a)>0:\,\,a+i(\mathbb{R}\setminus(-\beta,\beta))\subset\rho(A)\,\,\text{and}\,\,\sup_{|b|\geq\beta}\|R(a+i\beta,A)\|<\infty.$

To explain the relevance of $s_{R}(A)$ and to relate it to the spectral
properties of $(T(t))_{t\geq 0}$, 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 $M=\bigcap_{1\leq
j\leq n}{\rm Ker}\,y^{*}_{j}$ with $y^{*}_{j}\in D(A^{*})$ for $j\in\\{1,\dots
n\\}$, there exist $x\in M$ with $\|x\|=1$ and $\mu\in{\mathbb{C}}$ with ${\rm
Re}\,\mu=\omega$ such that

(2.5) $\|T(t)x-e^{\mu t}x\|<\varepsilon,\qquad 0\leq t\leq t_{0}.$

Note that ${\rm codim}\,M<\infty.$ Observe also that $x$ and $\mu$ depend on
$t_{0},$ $\epsilon$ and $M,$ and so $x$ is _not_ an approximate eigenvalue of
$T(t),$ $0\leq t\leq t_{0},$ 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 $s_{e}(A)$ is admissible. In this
case, there exists $\mu$ with ${\rm Re}\,\mu=s_{e}(A)$ 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 $(T(t))_{t\geq 0}$ be a $C_{0}$-semigroup on a Banach space $X$ with
generator $A,$ and let $s_{R}(A)$ be defined by (2.4). Then $s_{R}(A)$ is
admissible.

The proof of the theorem is given in the appendix. It is similar to the proof
[49, Proposition 4.4], though it is technically more demanding.

##  3\. Warm-up: initial observations and comments

###  3.1. Spectrum does not suffice

It is well-known 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
$(Af)(s)=sf(s)$ on $X.$ Clearly, $\sigma(A-a)=[-a,a]$ and thus there are
initial values $x_{0}\in X$ for which the solutions of

(3.1) $x^{\prime}(t)=(A-a)x(t),\qquad t\geq 0,\quad x(0)=x_{0},$

grow exponentially in the sense that $\|e^{(A-a)t}x_{0}\|\geq
C_{x_{0}}e^{at/2}$ for $t\geq 0$ with a constant $C_{x_{0}}>0$.

By a classical result due to Herrero [39] (see [37] 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
$A_{\varepsilon}\to A$ in $L(X)$ as $\varepsilon\to 0.$ Consider the perturbed
sysem

(3.2)
$x^{\prime}(t)=(A-a)x(t)+(A_{\varepsilon}-A)x(t)=(A_{\varepsilon}-a)x(t),\qquad
t\geq 0.$

Note that for any $\delta>0$ there exists $\varepsilon_{0}>0$ such that

$\sigma(A_{\varepsilon}-a)=\\{-a\\}\qquad\text{and}\qquad\|A_{\varepsilon}-A\|<\delta,\qquad\varepsilon\in(0,\varepsilon_{0}).$

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 [67] and rediscovered recently in [57]. However, [57] went much further,
see Subsection 3.3 below. We add that in [68] a Lipschitz perturbation $K$ is
constructed which destroys instability of a linear system
$x^{\prime}(t)=Ax(t)$ and satisfies the strictly sublinear growth assumption
$\|K(x)\|/\|x\|\to 0$ as $\|x\|\to 0.$ Our examples in Section 5 differ from
these treatments.

###  3.2. Unstable orbits may co-exist with very stable ones

On the Hilbert space $X=L^{2}((0,\infty),e^{-2t}dt)$ let the operator
$Af=f^{\prime}$ be defined on its maximal domain in $X$. Then $A$ generates
the $C_{0}$-semigroup $(T(t))_{t\geq 0}$ of left shifts on $X,$ and
$\sigma(A)=\\{\lambda\in\mathbb{C}:{\rm Re}\,\lambda\leq 2\\}.$ While
$(T(t))_{t\geq 0}$ 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

$x^{\prime}(t)=Ax(t),\qquad x_{0}=x_{0},$

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 Solá-Morales proved in [57] that in an infinite-dimensional
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
$\|K(x)\|/\|x\|\to 0$ as $x\to 0$ 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

$\left(\frac{1}{-\log\|x\|}\right)^{c_{2}}<\frac{\|Kx\|}{\|x\|}<4\left(\frac{1}{-\log(\|x\|)}\right)^{c_{1}}$

for $x$ with $\|x\|$ small enough.

###  3.4. Rank-one 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 rank-one operator $K$ on $X$
such that the operator $A+K$ is hypercyclic, see e.g. [32], [9]. Clearly,
$r_{e}(A+K)=r_{e}(A)=1.$ On the other hand, it is well-known that there exist
a unitary operator $A_{1}$ and a rank-one operator $K_{1}$ such that
$A_{1}+K_{1}$ is strongly stable, that is, $(A_{1}+K_{1})^{n}x\to 0$ as
$n\to\infty$ 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
rank-one perturbations of compressions of $S$ to the closed invariant
subspaces of $S^{*}.$ Moreover, clearly $S^{*n}\to 0$ as $n\to\infty,$
strongly. A more general set-up for such a construction can be found in [51].

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 $(e^{iAt})_{t\geq 0}$ is unitary. It is plausible that there exist
bounded selfadjoint operators $A$ and $A_{1}$ on $X$ and rank-one perturbation
$K$ and $K_{1}$ such that semigroup $(e^{i(A+K)t})_{t\geq 0}$ is hypercyclic,
while the semigroup $(e^{i(A_{1}+K_{1})t})_{t\geq 0}$ 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 a Banach space, $A\in B(X)$, and we let
$(K_{n})_{n=1}^{\infty}$ be a sequence of compact maps $K_{n}:X\to X$ (in
general, non-linear). For $x\in X$ and $n\in{\mathbb{N}}$ let

(4.1) $f_{n}(x)=(A+K_{n})\cdots(A+K_{1})x,$

and set $f_{0}(x)=x$. In other words, $(f_{n}(x))_{n=0}^{\infty}$ is a
solution of the difference equation

(4.2) $x_{n+1}=(A+K_{n})x_{n},\qquad x_{0}=x.$

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

$f_{j}(x)\in A^{j}x+C$

for all $x\in X_{0}$ and $j\in\\{1,\dots,n\\}$.

###### Proof.

It is easy to see by induction on $j$ that

$f_{j}(x)=A^{j}x+\sum_{s=0}^{j-1}A^{j-s-1}K_{s+1}f_{s}(x)$

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

$\bigcup_{j=1}^{n}\sum_{s=0}^{j-1}A^{j-s-1}K_{s+1}f_{s}(X_{0})$

is precompact, and we let $C$ be its closure. Then $f_{j}(x)\in A^{j}x+C$ for
all $x\in X_{0}$ and $j\in\\{1,\dots,n\\}$. ∎

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 finite-dimensional 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 finite-dimensional subspace $F\subset X$ such
that

$\|f_{n}(x)\|\geq{\rm dist}\,\\{A^{n}x,F\\}-\varepsilon$

for all $x\in X_{0}$.

###### Proof.

Let $C$ be the compact set constructed in Lemma 4.1. Since $C$ is compact,
there exists a finite-dimensional subspace $F\subset X$ such that ${\rm
dist}\,\\{c,F\\}<\varepsilon$ for every $c\in C$. For each $x\in X_{0}$ we
have $f_{n}(x)\in A^{n}x+C$ implying

$\|f_{n}(x)\|\geq{\rm dist}\,\\{A^{n}x,C\\}\geq{\rm
dist}\,\\{A^{n}x,F\\}-\varepsilon.\qed$

Having obtained the estimate for a finite piece of trajectory, we can spread
it out to a finite-dimensional 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 finite-dimensional subspace. Then the following assertions hold.

  * (i)

There exists a unit vector $u\in X$ such that ${\rm
dist}\,\\{A^{n}u,F\\}\geq\frac{1-\varepsilon}{2}\|A^{n}\|_{\mu}$.

  * (ii)

There exists a unit vector $u\in X$ such that ${\rm
dist}\,\\{A^{j}u,F\\}\geq\frac{1-\varepsilon}{2}r_{e}(A)^{j}$ 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

$\|f+m\|\geq(1-\varepsilon/2)\max\\{\|f\|,\|m\|/2\\}$

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
$\|A^{n}u\|\geq(1-\varepsilon/2)\|A^{n}\|_{\mu}$. Then

${\rm dist}\,\\{A^{n}u,F\\}=\inf\\{\|A^{n}u+f\|:f\in
F\\}\geq\frac{1-\varepsilon/2}{2}\|A^{n}u\|\geq\frac{1-\varepsilon}{2}\|A^{n}\|_{\mu},$

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 there exists a unit vector
$u\in L$ with $\|(A-\mu)u\|<\delta$. For each $j=1,\dots,n$ we have

$\|(A^{j}-\mu^{j})u\|=\Bigl{\|}\sum_{k=0}^{j-1}A^{j-k-1}\mu^{k}(A-\mu)u\Bigr{\|}\leq
j\|A\|^{j}\delta.$

We now choose $M$ as in the proof of (i) and set $L=\bigcap_{j=1}^{n}A^{-j}M$.
Then there is a unit vector $u\in L$ such that $\|(A-\mu)u\|$ is so small that
$\|(A^{j}-\mu^{j})u\|\leq r_{e}(A^{j})\varepsilon/2$ for all $j=1,\dots,n$.
Thus $\|A^{j}u\|\geq(1-\varepsilon/2)r_{e}(A^{j})$, and as above we deduce
that ${\rm dist}\,\\{A^{j}u,F\\}\geq\frac{1-\varepsilon}{2}r_{e}(A^{j})$ for
all $j=1,\dots,n$. ∎

The above lemmas allow us to obtain exponential lower bounds for the norms of
trajectories $\|f_{n}(x)\|$ 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 $K_{n}:X\to X$ be compact for $n\in{\mathbb{N}}$. Take a
sequence $(a_{n})_{n=1}^{\infty}$ in ${\mathbb{R}}_{+}$ such that
$\lim_{n\to\infty}a_{n}=0$, and let $L\subset{\mathbb{N}}$ be infinite. For
$x\in X$ define $f_{n}(x)$ by (4.1). Then the set

$\bigl{\\{}x\in X:\,\hbox{there are infinitely many}\,\,n\in
L\,\,\hbox{with}\,\,\|f_{n}(x)\|\geq a_{n}\|A^{n}\|_{\mu}\bigr{\\}}$

is residual in $X$.

###### Proof.

The statement is clear if $\|A^{n}\|_{\mu}=0$ for an infinite number of $n\in
L$. Thus, without loss of generality, we may assume that $\|A^{n}\|_{\mu}>0$
for all $n\in L$. For $k\in{\mathbb{N}}$ let

$M_{k}=\\{x\in X:\hbox{ there exists }\,\,n\in L\text{ \ with \ }n\geq
k\,\hbox{ and }\,\|f_{n}(x)\|>a_{n}\|A^{n}\|_{\mu}\\}.$

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 $n\geq k$ and
$a_{n}<\varepsilon/4$. By Lemma 4.1, there exists a compact set $C\subset X$
such that

$f_{n}(y+v)\in A^{n}(y+v)+C$

for all $v\in X$ with $\|v\|\leq\varepsilon$. Since $C$ is compact, there
exists a finite-dimensional subspace $F\subset X$ such that ${\rm
dist}\,\\{c,F\\}<\frac{\varepsilon\|A^{n}\|_{\mu}}{12}$ for all $c\in C$.
Lemma 4.3 (i) then yields a unit vector $u\in X$ with ${\rm
dist}\,\\{A^{n}u,F\\}>\frac{\|A^{n}\|_{\mu}}{3}$. Note that

${\rm dist}\,\\{A^{n}(y+\varepsilon u),F\\}+{\rm dist}\,\\{A^{n}(y-\varepsilon
u),F\\}\geq 2\varepsilon{\rm
dist}\,\\{A^{n}u,F\\}\geq\frac{2\varepsilon}{3}\|A^{n}\|_{\mu}.$

Indeed, passing to the quotient space $X/F$, the former inequality follows
from

$\|\pi(A^{n}y)+\varepsilon\pi(A^{n}u)\|_{X/F}+\|\pi(A^{n}y)-\varepsilon\pi(A^{n}u)\|_{X/F}\geq\frac{2\varepsilon}{3}\|\pi(A^{n}u)\|_{X/F}$

via the triangle inequality, where $\pi$ denotes the corresponding quotient
mapping. So $x:=y+\varepsilon u$ or $x:=y-\varepsilon u$ satisfy

$\|x-y\|\leq\varepsilon\quad\text{and}\quad{\rm
dist}\,\\{A^{n}x,F\\}\geq\frac{\varepsilon}{3}\|A^{n}\|_{\mu}.$

We conclude

$\|f_{n}(x)\|\geq{\rm
dist}\,\\{A^{n}x,F\\}-\frac{\varepsilon\|A^{n}\|_{\mu}}{12}\geq\frac{\varepsilon\|A^{n}\|_{\mu}}{4}\geq
a_{n}\|A^{n}\|_{\mu}.$

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. ∎

######  Corollary 4.5.

Let $A\in B(X)$ satisfy $\sup\\{\|A^{n}\|_{\mu}:n\in{\mathbb{N}}\\}=\infty$.
If $(f_{n})_{n=1}^{\infty}$ is as in Theorem 4.4, then the set

$\big{\\{}x\in X:\sup\nolimits_{n\in{\mathbb{N}}}\|f_{n}(x)\|=\infty\big{\\}}$

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
exponential growing 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 non-increasing
sequence $(a_{n})_{n=1}^{\infty}\subset\mathbb{R}_{+}$ satisfying
$\lim_{n\to\infty}a_{n}=0$. 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

$\|f_{n}(x)\|\geq a_{n}r_{e}(A)^{n},\qquad n\geq n_{0}.$

###### 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 $r_{e}^{-n}K_{n}(r_{e}^{n-1}\cdot)$ for
$n\geq 1$. 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 $A-I$ 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 $\dim{\rm Ker}\,(A-I-\widetilde{K})=\infty$. Thus, passing to
$A-\widetilde{K}$ and $K_{n}+\widetilde{K}$ if necessary, we may assume that
$\dim{\rm Ker}\,(A-I)=\infty.$

Take some $c_{1}\in(2a_{n_{0}},r)$. Fix integers $n_{0}<n_{1}<n_{2}<\cdots$
such that $a_{n_{k}}<2^{-(k+2)}(r-c_{1})$. Set

$c_{k}=2^{-(k-1)}(r-c_{1}),\qquad k\geq 2.$

Choose positive numbers $\varepsilon_{k}$ such that
$\varepsilon_{k+1}<\varepsilon_{k}$ and

$\frac{(1-\varepsilon_{k})^{2}}{2}c_{k}-\varepsilon_{k}\geq a_{n_{k-1}}$

for all $k\in\mathbb{N}$.

Set $x_{0}=y$, $F_{0}=\\{0\\}$ and $M_{0}=X$. Inductively, we construct
finite-dimensional subspaces $F_{1}\subset F_{2}\subset\cdots$, finite-
codimensional closed subspaces $M_{1}\supset M_{2}\supset\cdots$, and unit
vectors $x_{k}\in M_{k}\cap{\rm Ker}\,(A-I)$ for $k\geq 1$ such that (4.3)
below is true.

Let $k\geq 1$ and suppose that the vectors $x_{0},\dots,x_{k-1}$ and spaces
$F_{0},\dots,F_{k-1}$ and $M_{0},\dots,M_{k-1}$ have already been constructed.
Lemma 4.1 provides a compact set $C_{k}\subset X$ such that

$f_{n}(u)\in A^{n}u+C_{k}$

for all $n\leq n_{k}$ and $u\in X$ with $\|u\|\leq\|y\|+r.$ Using the
compactness of $C_{k},$ we find a finite-dimensional subspace $F_{k}\supset
F_{k-1}$ such that

(4.3) $\\{y,Ay,\dots,A^{n_{k}}y,x_{1},\dots,x_{k-1}\\}\subset F_{k}\quad\text{
and }\quad{\rm dist}\,\\{d,F_{k}\\}\leq\varepsilon_{k}$

for all $d\in C_{k}$. Lemma 9.1 yields a closed subspace $M_{k}\subset
M_{k-1}$ of finite codimension such that

$\|f+m\|\geq(1-\varepsilon_{k})\max\Bigl{\\{}\|f\|,\frac{\|m\|}{2}\Bigr{\\}}$

for all $f\in F_{k}$ and $m\in M_{k}.$ Since ${\rm Ker}\,(A-I)$ has infinite
dimension, there exists a unit vector $x_{k}\in M_{k}\cap{\rm Ker}\,(A-I)$.

For the elements $x_{k},$ $k\in{\mathbb{N}},$ constructed above, set

$x=y+\sum_{j=1}^{\infty}c_{j}x_{j}.$

We show that $x$ satisfies the assertions. First note that

$\|x-y\|\leq\sum_{j=1}^{\infty}c_{j}=c_{1}+\sum_{j=2}^{\infty}\frac{r-c_{1}}{2^{j-1}}=r,$

so that $x\in\overline{B}(y,r).$ Let $k\geq 1$ and $n_{k-1}\leq n\leq n_{k}$.
We estimate

$\displaystyle\|f_{n}(x)\|$ $\displaystyle\geq{\rm
dist}\,\\{A^{n}x,C_{k}\\}\geq{\rm dist}\,\\{A^{n}x,F_{k}\\}-\varepsilon_{k}$
$\displaystyle={\rm
dist}\,\Bigl{\\{}\sum_{j=k}^{\infty}c_{j}x_{j},F_{k}\Bigr{\\}}-\varepsilon_{k},$

employing that

$A^{n}x=A^{n}y+\sum_{j=1}^{\infty}c_{j}x_{j}\quad\text{and}\quad
A^{n}y,x_{1},\dots,x_{k-1}\in F_{k}.$

For $j\geq k$, the vector $x_{j}$ belongs to $M_{j}\subset M_{k}$, and so

$\|f_{n}(x)\|\geq\frac{1-\varepsilon_{k}}{2}\cdot\Bigl{\|}\sum_{j=k}^{\infty}c_{j}x_{j}\Bigr{\|}-\varepsilon_{k}.$

Since $x_{k}\in F_{k+1}$ and $x_{j}\in M_{j}\subset M_{k+1}$ for $j\geq k+1$,
we obtain

$\displaystyle\|f_{n}(x)\|$
$\displaystyle\geq\frac{(1-\varepsilon_{k})(1-\varepsilon_{k+1})}{2}\|c_{k}x_{k}\|-\varepsilon_{k}$
$\displaystyle\geq\frac{(1-\varepsilon_{k})^{2}}{2}\cdot
c_{k}-\varepsilon_{k}\geq a_{n_{k-1}}\geq a_{n}.$

Hence

$\|f_{n}(x)\|\geq a_{n}$

for all $n\geq n_{0}$, and the statement follows. ∎

######  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 $y\in X$, $n_{0}\geq 0$, and $a_{n_{0}}<r$. Then there
exists $x\in X$ with $\|x-y\|\leq r$ and $\|f_{n}(x)\|\geq a_{n}r_{e}(A)^{n}$
for all $n\geq n_{0}$.

######  Corollary 4.8.

Under the assumptions of Theorem 4.6, let
$(a_{n})_{n=1}^{\infty}\subset\mathbb{R}_{+}$ be a non-increasing sequence
satisfying $\lim_{n\to\infty}a_{n}=0$. Then

  * (i)

there exists $x\in X$ such that $\|f_{n}(x)\|\geq a_{n}r_{e}(A)^{n}$ for all
$n\in{\mathbb{N}}$;

  * (ii)

there exists a dense subset of vectors $x\in X$ such that $\|f_{n}(x)\|\geq
a_{n}r_{e}(A)^{n}$ 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 $\|x-y\|\leq\varepsilon$ and $\|f_{n}(x)\|\geq a_{n}r_{e}(A)^{n}$ for all
$n\geq n_{0}$, and assertion (ii) follows. ∎

######  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 [69].

##  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], [38], [64], [29], and “topological”
instability conditions as above, in order to obtain a local result showing
instability for the zero solution to the difference equation

(5.1) $x_{n+1}=(A+K+G)x_{n},\qquad n\in\mathbb{N},\quad x_{0}=x,$

in a Banach space $X.$ Here $A$ is a linear operator on $X$ with $r_{e}(A)>1,$
$K$ is a compact operator on $X$ with $K(0)=0$, and $G$ is a continuous map
defined on a ball $B(0,r)\subset X$ such that $\|G(x)\|=O(\|x\|^{1+\delta})$
as $\|x\|\to 0$ 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 $\|x_{0}\|<\delta$ one has
$\sup_{n}\|x_{n}\|<\varepsilon.$

We first note a simple lemma.

######  Lemma 5.1.

Let $\varepsilon\in(0,2^{-7})$. Then there exist continuous functions
$f:[0,\infty)\to[0,\varepsilon/2]$ and $g:[0,\infty)\times[0,\infty)\to[0,2]$
such that

  * (i)

$f(s)=0,\qquad 0\leq s\leq\varepsilon^{3}\quad\hbox{or}\quad
s\geq\varepsilon$;

  * (ii)

$f(s)=\varepsilon/2,\qquad 4\varepsilon^{3}\leq s\leq\frac{\varepsilon}{2}$;

and

  * (iii)

$g(s,t)=0,\qquad 0\leq s\leq\varepsilon^{3}\quad\hbox{or}\quad
s\geq\varepsilon,\,t\geq 0,$;

  * (iv)

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

  * (v)

$g(s,t)\leq s+\frac{t^{2}}{s},\qquad s>0,\,t\geq 0$.

###### Proof.

The existence of $f$ is obvious.

Let $\widetilde{g}:[0,\infty)\times[0,\infty)\to[0,2]$ 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 $t\geq
0,$ and $g(0,t)=0$ for $t\geq 0$. Clearly, $g$ fulfills (iii) and (v). If
$4\varepsilon^{3}\leq s\leq 16\varepsilon^{3}$ and $t=\frac{\varepsilon}{2},$
then

$s+\frac{t^{2}}{s}\geq\frac{t^{2}}{s}\geq\frac{\varepsilon^{2}}{4\cdot
16\varepsilon^{3}}=\frac{1}{64\varepsilon}>2=\tilde{g}(s,t).$

So $g$ satisfies (iv). ∎

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 infinite-dimensional 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 $A\in B(H),$ $K:H\to H,$ and
$G:H\to H$ defined by

$\displaystyle A(y,a)$ $\displaystyle:=(2y,0),\qquad(y,a)\in H;$
$\displaystyle K(y,a)$ $\displaystyle:=(0,f(\|y\|)),\qquad(y,a)\in H;$
$\displaystyle G(y,a)$
$\displaystyle:=\bigl{(}-g(\|y\|,|a|)y,0\bigr{)},\qquad(y,a)\in H.$

It is clear that $A\in B(H)$, $r_{e}(A)=2$, $K$ is compact, and $G$ satisfies

$\displaystyle\|G(y,a)\|$
$\displaystyle=g\bigl{(}\|y\|,|a|\bigr{)}\cdot\|y\|\leq\Bigl{(}\|y\|+\frac{|a|^{2}}{\|y\|}\Bigr{)}\|y\|$
$\displaystyle=\|y\|^{2}+|a|^{2}=\|(y,a)\|^{2}.$

Moreover,

$(A+K+G)(y,a)=\Bigl{(}(2-g\bigl{(}\|y\|,|a|)\bigr{)}y,f(\|y\|)\Bigr{)}.$

Let $x_{0}=(y_{0},a_{0})\in H$ be such that $\|y_{0}\|\leq\varepsilon^{3}$ and
$|a_{0}|\leq\varepsilon^{3}$, and let $(x_{n})_{n=1}^{\infty}$ be given by
(5.1). For $n\geq 1$ denote $x_{n}:=(y_{n},a_{n}).$ First observe that
$|a_{n}|\leq\frac{\varepsilon}{2}$ for all $n\geq 1,$ by the definition of
$f$. We prove that

(5.2) $\|y_{n}\|\leq\epsilon/2,\qquad n\in\mathbb{N}.$

Observe that $\|y_{n}\|\leq 2\|y_{n-1}\|,$ $n\in\mathbb{N}$. We distinguish
two cases.

A) There exists $n$ such that $\|y_{n}\|>4\varepsilon^{3}$ and
$\|y_{n+1}\|>4\varepsilon^{3}$. Let $n_{0}$ be the smallest integer with this
property. We then obain

$\|y_{n_{0}}\|\leq 2\|y_{n_{0}-1}\|\leq 8\varepsilon^{3},$

and so

$a_{n_{0}+1}=f(\|y_{n_{0}}\|)=\frac{\varepsilon}{2}\quad\text{and}\quad
4\varepsilon^{3}\leq\|y_{n_{0}+1}\|\leq 2\|y_{n_{0}}\|\leq 16\varepsilon^{3}.$

Hence, $g\bigl{(}\|y_{n_{0}+1}\|,|a_{n_{0}+1}|\bigr{)}=2$ and $y_{n_{0}+2}=0$.
Thus $y_{k}=0$ for all $k\geq n_{0}+2$. Moreover,
$\min\\{\|y_{m}\|,\|y_{m+1}\|\\}\leq 4\varepsilon^{3}$ for all $m\leq
n_{0}-1$. It follows

$\text{either}\qquad\|y_{m+1}\|\leq
4\varepsilon^{3}\qquad\text{or}\qquad\|y_{m+1}\|\leq 2\|y_{m}\|\leq
8\varepsilon^{3}.$

We have shown $\sup_{k}\|y_{k}\|\leq 16\varepsilon^{3}$ in this case.

B) Let $\min\\{\|y_{n}\|,\|y_{n+1}\\}\leq 4\varepsilon^{3}$ for all $n$. We
again have

$\text{either}\qquad\|y_{n+1}\|\leq
4\varepsilon^{3}\qquad\text{or}\qquad\|y_{n+1}\|\leq 2\|y_{n}\|\leq
8\varepsilon^{3},$

which yields $\sup_{n}\|y_{n}\|\leq 8\varepsilon^{3}$.

In both cases we have shown $\sup_{n}\|y_{n}\|\leq
16\varepsilon^{3}\leq\frac{\varepsilon}{2}$, and so
$\sup_{n}\|x_{n}\|\leq\varepsilon$.

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 $j\geq 1$. 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 $G:H\to H$ by

(5.3) $\displaystyle A\bigl{(}y,(a_{j})_{j=1}^{\infty}\bigr{)}$
$\displaystyle=\bigl{(}2y,(0)\bigr{)},$ (5.4) $\displaystyle
K\bigl{(}y,(a_{j})_{j=1}^{\infty}\bigr{)}$
$\displaystyle=\bigl{(}0,(f_{j}(\|y\|))_{j=1}^{\infty}\bigr{)},$ (5.5)
$\displaystyle G\bigl{(}y,(a_{j})_{j=1}^{\infty}\bigr{)}$
$\displaystyle=\Bigl{(}-y\sum_{j=1}^{\infty}g_{j}\bigl{(}\|y\|,|a_{j}|\bigr{)},0\Bigr{)}$

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
$g_{j}\bigl{(}y,|a_{j}|\bigr{)}\neq 0$. So $G$ is well-defined. Similarly,
there is at most one $j$ with $f_{j}(\|y\|)\neq 0$.

Clearly, $A\in B(H)$, $r_{e}(A)=2$ and $K$ is compact since

$K(H)\subset\Bigl{\\{}(0,(a_{j})_{j=1}^{\infty}):|a_{j}|\leq\frac{\varepsilon_{j}}{2}\Bigr{\\}}.$

Furthermore,

$\displaystyle\|G(y,(a_{j})_{j=1}^{\infty})\|\leq$
$\displaystyle\|y\|\cdot\sum_{j=1}^{\infty}g_{j}\bigl{(}\|y\|,|a_{j}|\bigr{)}=\|y\|\cdot\max_{j}g_{j}\bigl{(}\|y\|,|a_{j}|\bigr{)}$
$\displaystyle\leq$
$\displaystyle\max_{j}\bigl{(}\|y\|^{2}+|a_{j}|^{2}\bigr{)}\leq\|(y,(a_{j})_{j=1}^{\infty})\|^{2}.$

We have

$(A+K+G)\bigl{(}y,(a_{j})_{j=1}^{\infty}\bigr{)}=\Bigl{(}(2-\sum_{j=1}^{\infty}g_{j}\bigl{(}\|y\|,|a_{j}|)\bigr{)}y,\bigl{(}f_{j}(\|y\|)\bigr{)}_{j=1}^{\infty}\Bigr{)}.$

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 $\|x_{0}\|\leq\frac{\varepsilon_{k}^{3}}{4}.$
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 $n\geq
0.$ We show first that $\|y_{n}\|\leq\frac{\varepsilon_{k}}{2}$ for all $n$.
Suppose the contrary. Let $n_{0}$ be the smallest integer with
$\|y_{n_{0}}\|>\frac{\varepsilon_{k}}{2}$. Let $n_{1}$ be the largest integer
with $0\leq n_{1}<n_{0}$ and $\|y_{n_{1}}\|<\frac{\varepsilon_{k}^{3}}{2}$.
Note that

$0\leq\sum_{j=1}^{\infty}g_{j}\bigl{(}\|y\|,|a_{j}|\bigr{)}\leq 2$

for all $\bigl{(}y,(a_{j})_{j=1}^{\infty}\bigr{)}\in H$, and hence

$\|y_{n+1}\|\leq 2\|y_{n}\|\ $

for all $n$. We infer that

$\|y_{n_{1}}\|\geq\frac{\varepsilon_{k}^{3}}{4},\quad\text{and
so}\quad\frac{\varepsilon_{k}^{3}}{4}\leq\|y_{m}\|\leq\frac{\varepsilon_{k}}{2}$

for all $m=n_{1},\dots,n_{0}-1$, implying

$f_{j}(\|y_{m}\|)=0=g_{j}(\|y_{m}\|,|a_{j}|)$

for all $j\neq k$ and $m=n_{1},\dots,n_{0}-1$. We further obtain

$\|y_{n_{1}+1}\|\leq 2\|y_{n_{1}}\|<\varepsilon_{k}^{3}\quad\text{and}\quad
a_{n_{1}+1,k}=f_{k}(\|y_{n_{1}}\|)=0.$

Consider the orthogonal projection $P:H\to H$ onto $H_{0}\oplus{\mathbb{C}}$
defined by

$P\big{(}y,(a_{j})_{j=1}^{\infty}\big{)}=(y,a_{k}).$

By the above observations, $Px_{n_{1}+1}=(y_{n_{1}+1},0)$ satisfies

$\|Px_{n_{1}+1}\|\leq\varepsilon_{k}^{3}$

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 $\|y_{n_{0}}\|\leq\frac{\varepsilon_{k}}{2},$ a
contradiction. We have shown that

$\sup_{n}\|y_{n}\|\leq\frac{\varepsilon_{k}}{2}\quad\text{and}\quad\sup_{j,n}f_{j}(\|y_{n}\|)\leq\frac{\varepsilon_{k}}{2},$

so that $\sup_{n}\|x_{n}\|\leq\varepsilon_{k}.$ 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\. Well-posedness for continuous time and global existence

We are primarily interested in nonlinear evolution equations of the form

(6.1) $x^{\prime}(t)=Ax(t)+K(t,x(t)),\qquad x(0)=x_{0},\quad t\in[0,T),$

where $T\in(0,\infty]$, $A$ generates a $C_{0}$-semigroup $(T(t))_{t\geq 0}$
on a Banach space $X$, and $K:[0,\infty)\times X\to X$ 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

(6.2) $x(t)=T(t)x_{0}+\int_{0}^{t}T(t-s)K(s,x(s))\,ds,\qquad t\geq 0.$

Recall that a continuous function $x:[0,T)\to X$ 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 so-called 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 $t\geq 0,$ 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
$\|y(t)-x(t)\|<\varepsilon$ for all $t\geq 0$. 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 well-posedness,
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 long-term behavior, it
is natural to consider a set-up 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 well-known, see [62, p. 343-345, p. 350] or
e.g. [16, Section 4], [55, Section 6].

######  Theorem 6.1.

Let $A$ generate the $C_{0}$-semigroup $(T(t))_{t\geq 0}$ on a Banach space
$X$ and let $K:[0,\infty)\times X\to X$ 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

$\|K(t,x)-K(t,y)\|\leq C_{T,r}\|x-y\|$

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 $\lim_{t\to T}\|x(t,x_{0})\|=\infty.$

  * (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
$B(x_{0},\delta)\to C([0,T^{*}],X),$ $y_{0}\to x(\cdot,y_{0}),$ is continuous.

Gronwall’s inequality and the above blow-up condition (ii) imply the following
global existence result.

######  Corollary 6.2.

Besides the conditions of Theorem 6.1, assume that there exists $c\in
L^{1}_{\mathrm{loc}}([0,\infty))$ such that

(6.3) $\|K(t,x)\|\leq c(t)(1+\|x\|)\qquad\text{ for all }x\in X,\ t\geq 0.$

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. [55, Corollary 6.2.3], where the compactness of $(T(t))_{t\geq 0}$ 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 $(T(t))_{t\geq 0}$ on a Banach space
$X$ with $\|T(t)\|\leq Me^{\omega t}$ for $t\geq 0$ and some $\omega\geq 0$,
and let $K:[0,\infty)\times X\to X$ be continuous separately in both
variables.

a) Assume that $K$ satisfies (6.3). Let $x:[0,T)\to X$ be a solution of (6.1)
and $t_{0}\in(0,T)$. Then $\|x(t)\|\leq C(1+\|x_{0}\|)$ 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
$\\{x(t,x_{0}):\,0\leq t\leq t_{0},\,\,x_{0}\in B\\}$ 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 non-increasing function $c:[r_{0},\infty)\to[0,\infty)$ such that
$\|K(t,x)\|\leq c(\|x\|)\|x\|$ for all $t\geq 0$ and $x\in X$ with $\|x\|\geq
r_{0}.$ If $\gamma>\omega$ and $\lim_{r\to\infty}c(r)=0,$ then there exists a
constant $M_{\gamma,x_{0}}$ such that $\|x(t,x_{0})\|\leq
M_{\gamma,x_{0}}e^{\gamma t}$ for $t\geq 0$ 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 $\|x\|\to\infty$, 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 [58] 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 [58].

######  Theorem 6.4.

Let $A$ be the generator of a $C_{0}$-semigroup on a Banach space $X,$ and
$K:[0,\infty)\times X\to X$ 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 [58, 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
$(T(t))_{t\geq 0}$. Passing to the equivalent norm $\|x\|_{c}:=\sup_{t\geq
0}\|T(t)x\|$, the semigroup then becomes contractive, and we can apply [58,
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 $(T(t))_{t\geq 0}$ on Banach
space $X,$ and let $K:[0,\infty)\to B(X)$ be strongly continuous. Then there
exists a unique evolution family $(U(t,\tau))_{t\geq\tau\geq 0}\subset B(X)$
such that

(6.4) $U(t,\tau)x=T(t-\tau)x+\int_{\tau}^{t}T(t-s)K(s)U(s,\tau)x\,ds,\qquad
t\geq\tau.$

##  7\. Growth and instability for continuous time

Following a similar strategy as in Section 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 Section 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 $K:[0,\infty)\times X\to X$ 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 $t_{0}\geq 0$. 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 $K:[0,\infty)\times X\to X$ be separately continuous such that
$K(s,\cdot):X\to X$ is compact for all $s\geq 0$. Suppose that for each
$t_{0},\varepsilon>0$ and each bounded set $B\subset X$ there exists a
$\delta_{\varepsilon}>0$ such that

$\|K(s,x)-K(t,x)\|\leq\varepsilon$

for all $s,t\in[0,t_{0}]$ with $|s-t|\leq\delta_{\varepsilon}$ 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 $0\leq s\leq t_{0}$ and $u\in B$. There exist $j\in\\{1,\dots,n\\}$ with
$|s-t_{j}|\leq\delta_{\varepsilon/2}$ and $i\in\\{1,\dots,m\\}$ with
$\|K(t_{j},u)-x_{i}\|<\varepsilon/2$. Using also the assumption, we obtain

$\|K(s,u)-x_{i}\|\leq\|K(s,u)-K(t_{j},u)\|+\|K(t_{j},u)-x_{i}\|\leq\frac{\varepsilon}{2}+\frac{\varepsilon}{2}=\varepsilon.$

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. ∎

Before stating a second compactness criterion, we collect our standing
assumptions for semilinear evolution equations.

  * (A0)

Assume that $K:[0,\infty]\times X\to X$ is collectively compact and continuous
separately in both variables, and let $(T(t))_{t\geq 0}$ be a
$C_{0}$-semigroup on a Banach space $X$ 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

(7.1) $x(t,x_{0})=T(t)x_{0}+\int_{0}^{t}T(t-s)K(s,x(s,x_{0}))\,ds,\qquad t\geq
0.$

So the map $x:[0,\infty)\times X_{0}\to X$ is well-defined, 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 solution $x=x(\cdot,x_{0})$ of (7.1) such that the
set $\\{x(t,x_{0}):\,0\leq t\leq t_{0},\,\,x_{0}\in B\\}$ 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 $x:[0,\infty)\times X_{0}\to X$
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 non-unique case in a more systematic manner, for instance
using “generalized semi-flows”. 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

$\int_{0}^{t}T(t-s)K(s,x(s,x_{0}))\,ds$

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 $k=\sup\\{\|T(t)\|:t\in[0,t_{0}]\\}<\infty$ and
$k^{\prime}=\sup\\{\|x(t)\|:t\in[0,t_{0}],\,x_{0}\in B\\}<\infty$. Fix
$\varepsilon>0$.

By assumption, the set $C_{0}=\bigcup_{0\leq s\leq t_{0}}\\{K(s,y):y\in
X,\,\|y\|\leq k^{\prime}\\}$ is precompact. Let $E$ be a finite
$\frac{\varepsilon}{3kt_{0}}$-net in $C_{0}$. Since the set $\bigcup_{s\leq
t_{0}}T(s)(E)$ is precompact, Mazur’s theorem yields the compactness of

$M:=[0,t_{0}]\cdot\overline{{\rm conv}}\,\Bigl{(}\bigcup_{s\leq
t_{0}}T(s)(E)\Bigr{)},$

where $\overline{{\rm conv}}$ stands for the closed convex hull. Choose a
finite $\frac{\varepsilon}{3}$-net $E^{\prime}$ in $M$. Let $0\leq t\leq
t_{0}$, $x_{0}\in B$, and $\varepsilon>0$. Then there exist
$N\in{\mathbb{N}},$ a partition $0=t_{0}<t_{1}<...t_{N}=t$ of $[0,t],$ and
$s_{j}\in[t_{j},t_{j+1}]$ for $0\leq j\leq N-1$ such that

$\Bigr{\|}\int_{0}^{t}T(t-s)K(s,x(s,x_{0}))\,ds-S\Bigr{\|}<\frac{\varepsilon}{3}.$

where

$S=\sum_{j=1}^{N}T(t-s_{j})K(s_{j},x(s_{j},x_{0}))(t_{j}-t_{j-1}).$

By the choice of $E$, for $j=1,\dots,N$ there are $e_{j}\in E$ such that

$\|K(s_{j},x(s_{j},x_{0}))-e_{j}\Bigr{\|}<\frac{\varepsilon}{3kt_{0}}.$

Hence,

$\Bigl{\|}S-\sum_{j=1}^{N}T(t-s_{j})(t_{j}-t_{j-1})e_{j}\Bigr{\|}<\frac{\varepsilon}{3},$

Since $\sum_{j=1}^{N}T(t-s_{j})(t_{j}-t_{j-1})e_{j}$ belongs to $M$, we find
$e^{\prime}\in E^{\prime}$ with

$\Bigl{\|}\sum_{j=1}^{N}T(t-s_{j})(t_{j}-t_{j-1})e_{j}-e^{\prime}\Bigr{\|}<\frac{\varepsilon}{3}.$

It follows that

$\Bigl{\|}\int_{0}^{t}T(t-s)K(s,x(s,x_{0}))\,ds-e^{\prime}\Bigr{\|}<\varepsilon,$

i.e., $E^{\prime}$ is a finite $\varepsilon$-net for the set $C$. Since
$\varepsilon>0$ was arbitrary, $C$ is precompact. ∎

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

$x(t,x_{0})\in T(t)x_{0}+C_{0}$

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 $a:[0,\infty)\to{\mathbb{R}}_{+}$ be a non-increasing
function with $\lim_{t\to\infty}a(t)=0$ and $L\subset\mathbb{R}_{+}$ be
unbounded. Then the set

$\bigl{\\{}y\in X:\,\hbox{there are infinitely many}\,\,t_{n}\in
L\,\,\hbox{with}\,\,\|x(t_{n},y)\|\geq a(t_{n})\|T(t_{n})\|_{\mu}\bigr{\\}}$

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 $\|T(t_{0})\|_{\mu}\neq 0$
for all $t_{0}\geq 0$ (since otherwise
$\|T(t)\|_{\mu}\leq\|T(t_{0})\|_{\mu}\cdot\|T(t-t_{0})\|_{\mu}=0$ for all
$t\geq t_{0}$.) For $k\in{\mathbb{N}}$ set

$M_{k}=\\{y\in X:\hbox{ there exists }t\in L\text{ with }t\geq
k,\,\|x(t,y)\|>a(t)\|T(t)\|_{\mu}\\}.$

Note that $M_{k}$ is open since $x$ is continuous in $y$. We show that $M_{k}$
is dense.

Let $y\in X$ and fix $\varepsilon>0$. Choose $t\in L$ with $t\geq k$ and
$a(t)<\varepsilon/4$. By Corollary 7.3 there is a compact set $C\subset X$
such that

$x(t,y^{\prime})\in T(t)y^{\prime}+C$

for all $y^{\prime}\in X$ with $\|y^{\prime}-y\|\leq\varepsilon$. Since $C$ is
compact, there exists a finite-dimensional subspace $F\subset X$ such that
${\rm dist}\,\\{c,F\\}<\frac{\varepsilon\|T(t)\|_{\mu}}{12}$ for all $c\in C$.
Lemma 4.3 provides a unit vector $u\in X$ with ${\rm
dist}\,\\{T(t)u,F\\}>\frac{\|T(t)\|_{\mu}}{3}$. We compute

${\rm dist}\,\\!\\{T(t)(y+\varepsilon u),F\\}+{\rm
dist}\,\\!\\{T(t)(y-\varepsilon u),F\\}\geq 2\varepsilon{\rm
dist}\,\\!\\{T(t)u,F\\}\geq\tfrac{2\varepsilon}{3}\|T(t)\|_{\mu}.$

So $x_{0}:=y+\varepsilon u$ or $x_{0}:=y-\varepsilon u$ satisfy

$\|x_{0}-y\|\leq\varepsilon\quad\text{and}\quad{\rm
dist}\,\\{T(t)x_{0},F\\}\geq\frac{\varepsilon}{3}\|T(t)\|_{\mu}.$

It follows

$\|x(t,x_{0})\|\geq{\rm
dist}\,\\{T(t)x_{0},F\\}-\frac{\varepsilon\|T(t)\|_{\mu}}{12}\geq\frac{\varepsilon\|T(t)\|_{\mu}}{4}\geq
a(t)\|T(t)\|_{\mu}.$

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.
∎

######  Corollary 7.5.

Assume that all of the conditions of Theorem 7.4 hold. Let
$a:[0,\infty)\to{\mathbb{R}}_{+}$ be a non-increasing function satisfying
$\lim_{t\to\infty}a(t)=0$. 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

$\|x(t_{n},x_{0})\|\geq a(t_{n})e^{\omega_{e}(T)t_{n}},\qquad n\in\mathbb{N}.$

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 Section 2.

######  Theorem 7.6.

Let (A2) hold for $B=\overline{B}(y,r)\subset X_{0}.$ Setting
$\alpha=\limsup_{t\to 0}\|T(t)\|,$ let $a:[0,\infty)\to{\mathbb{R}}_{+}$ be a
non-increasing function satisfying $\lim_{t\to\infty}a(t)=0$, $t_{0}\geq 0$
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

$\|x(t,z)\|\geq a(t)e^{\omega t},\qquad t\geq t_{0}.$

###### Proof.

Without loss of generality we may assume that $\omega=0$. If not, then
consider the semigroup $(e^{-\omega t}T(t))_{t\geq 0}$ and compact
perturbations $e^{-\omega s}K(e^{\omega s}\cdot)$. Define a new norm
$\|\cdot\|_{1}$ on $X$ by

$\|x\|_{1}=\sup\\{|\langle x,x^{*}\rangle|:\,x^{*}\in
D(A^{*}),\,\|x^{*}\|=1\\}.$

By (9.2), the norm $\|\cdot\|_{1}$ is equivalent to $\|\cdot\|$ with

$\|x\|_{1}\leq\|x\|\leq\alpha\|x\|_{1}$

for all $x\in X$. Let a Banach space $X_{1}$ be $X$ equipped with
$\|\cdot\|_{1}.$

Fix $c_{1}$ such that $2a(t_{0})<c_{1}<r$ and $r-c_{1}<\frac{1}{2}$. Pick an
increasing sequence $(t_{k})_{k=0}^{\infty}$ with
$a(t_{k})<\frac{r-c_{1}}{\alpha 2^{k+2}}$ for all $k\in\mathbb{N}\cup\\{0\\}.$
Set

$c_{k}=\frac{r-c_{1}}{2^{k-1}},\qquad k\geq 1.$

For each $k\in\mathbb{N}$ choose $\varepsilon_{k}>0$ such that
$\varepsilon_{k+1}<\varepsilon_{k}$ and

$\frac{(1-\varepsilon_{k})^{2}}{2\alpha}c_{k}-2\varepsilon_{k}\geq a(t_{k-1})$

for all $k\geq 1$.

The required element $z$ will be constructed as the sum of a series of
appropriate approximate eigenvectors $x_{k}$ for $(T(t))_{t\geq 0}$ and
$e^{\mu_{k}t}$. We set $F_{0}=\\{0\\},$ $M_{0}=X$, and $x_{0}=y$. Inductively
we construct unit vectors $(x_{k})_{k=1}^{\infty}\subset X_{1}$, finite-
dimensional subspaces $F_{1}\subset F_{2}\subset\cdots$, and closed finite-
codimensional 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 $k\geq 1$ and suppose that the vectors $x_{0},\dots,x_{k-1}$ and subspaces
$F_{0},\dots,F_{k-1}$ and $M_{0},\dots,M_{k-1}$ have already been constructed.
Consider the family of solutions $\\{x(t,u):u\in B,t\geq 0\\}$ 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 $t\leq t_{k}$ and $u\in B_{1}.$ There thus exists a finite-dimensional
subspace $F_{k}\supset F_{k-1}$ such that $x_{k-1}\in F_{k}$ if $k>1$ and

(7.2) ${\rm
dist}\,\Bigl{\\{}T(t)\Bigl{(}y+\sum_{j=1}^{k-1}\alpha^{-1}c_{j}x_{j}\Bigr{)},F_{k}\Bigr{\\}}\leq\frac{\varepsilon_{k}}{2}$

for all $t\leq t_{k}$ and

(7.3) ${\rm dist}\,\\{d,F_{k}\\}\leq\varepsilon_{k}/2$

for all $d\in C_{k}$, where ${\rm dist}\,$ stands for the distance in the new
norm $\|\cdot\|_{1}$. Lemma 9.1 yields a closed subspace $M_{k}\subset
M_{k-1}$ of finite codimension such that

(7.4)
$\|f+m\|_{1}\geq(1-\varepsilon_{k})\max\Bigl{\\{}\|f\|_{1},\frac{\|m\|_{1}}{2}\Bigr{\\}}$

for all $f\in F_{k}$ and $m\in M_{k}$, where

$M_{k}=\bigcap_{1\leq j\leq k}{\rm Ker}\,y_{j}^{*}$

for some $y_{j}^{*}\in D(A^{*})$ for $1\leq j\leq k.$ Since $0$ is admissible,
we can choose $x_{k}\in M_{k}$ with $\|x_{k}\|_{1}=1$ and
$\mu_{k}\in{\mathbb{C}}$ with ${\rm Re}\,\mu_{k}=0$ such that

(7.5) $\|T(t)x_{k}-e^{\mu_{k}t}x_{k}\|_{1}\leq\varepsilon_{k}$

for all $t\leq t_{k}$.

Suppose that the vectors $x_{k}$, $k\in{\mathbb{N}},$ have been constructed as
above. Set

$z=y+\sum_{j=1}^{\infty}\alpha^{-1}c_{j}x_{j}.$

We show that $z$ meets the requirements in the initial Banach space
$(X,\|\cdot\|).$ First, $z$ belongs to $\overline{B}(y,r)$ since

$\|z-y\|\leq\alpha\|z-y\|_{1}\leq\sum_{j=1}^{\infty}c_{j}=c_{1}+\sum_{j=2}^{\infty}\frac{r-c_{1}}{2^{j-1}}=r.$

Let $x(\cdot,z)$ be a solution of (7.1) given by (A2). Fix $k\geq 1$ and
consider $t\in[t_{k-1},t_{k}]$. Properties (7.3) and (7.2) yield

$\displaystyle\|x(t,z)\|_{1}$ $\displaystyle\geq{\rm
dist}\,\\{T(t)z,C_{k}\\}\geq{\rm dist}\,\\{T(t)z,F_{k}\\}-\varepsilon_{k}/2$
$\displaystyle\geq{\rm
dist}\,\Bigl{\\{}\sum_{j=k}^{\infty}\alpha^{-1}c_{j}T(t)x_{j},F_{k}\Bigr{\\}}-\varepsilon_{k}.$

By means of (7.5) we estimate

$\displaystyle\|x(t,z)\|_{1}$ $\displaystyle\geq{\rm
dist}\,\Bigl{\\{}\sum_{j=k}^{\infty}\alpha^{-1}c_{j}e^{\mu_{j}t}x_{j},F_{k}\Bigr{\\}}-\sum_{j=k}^{\infty}\alpha^{-1}c_{j}\varepsilon_{j}-\varepsilon_{k}$
$\displaystyle\geq{\rm
dist}\,\Bigl{\\{}\sum_{j=k}^{\infty}\alpha^{-1}c_{j}e^{\mu_{j}t}x_{j},F_{k}\Bigr{\\}}-2\varepsilon_{k}.$

Taking into account (7.4) and that $x_{j}\in M_{j}\subset M_{k}$ for all
$j\geq k$, we infer

$\|x(t,z)\|_{1}\geq\frac{1-\varepsilon_{k}}{2}\Bigl{\|}\sum_{j=k}^{\infty}\alpha^{-1}c_{j}e^{\mu_{j}t}x_{j}\Bigr{\|}_{1}-2\varepsilon_{k}.$

Since $x_{k}\in F_{k+1}$ and $x_{j}\in M_{j}\subset M_{k+1}$ for $j\geq k+1$,
(7.4) also implies

$\displaystyle\|x(t,z)\|_{1}$
$\displaystyle\geq\frac{(1-\varepsilon_{k})(1-\varepsilon_{k+1})}{2}\|\alpha^{-1}c_{k}e^{\mu_{k}t}x_{k}\|_{1}-2\varepsilon_{k}$
$\displaystyle\geq\frac{(1-\varepsilon_{k})^{2}}{2}\cdot\alpha^{-1}c_{k}-2\varepsilon_{k}\geq
a(t_{k-1})\geq a(t).$

Hence

$\|x(t,z)\|\geq\|x(t,z)\|_{1}\geq a(t)$

for all $t\geq t_{0}$, as required. ∎

######  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 $\|x(t,y)\|\geq a(t)e^{\omega t}$ for all $t\geq 0$.

(ii) Assume that assumption (A2) is satisfied for all bounded subsets $B$ of
$X_{0}.$ Then there exists a set $S\subset{\rm Int}\,X_{0}$ dense in ${\rm
Int}\,X_{0}$ such that $\|x(t,y)\|\geq a(t)e^{\omega t}$ 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 ${\rm Int}X_{0}\neq\emptyset.$ Let $y\in{\rm
Int}\,X_{0}$, $\varepsilon={\rm dist}\,\\{y,\partial X_{0}\\}$, and
$\varepsilon_{0}\in(0,\varepsilon).$ Find $t_{0}\geq 0$ such that
$a(t_{0})<\frac{\varepsilon_{0}}{2\alpha}$. By Theorem 7.6, there exists
$x_{0}\in X$ such that $\|x_{0}-y\|\leq\varepsilon_{0}$ and
$\|x(t,x_{0})\|\geq a(t)e^{\omega t}$ for all $t\geq t_{0}$. It remains to
note that the choice of $\varepsilon_{0}\in(0,\varepsilon)$ is arbitrary. ∎

######  Remark 7.8.

If $X$ is a Hilbert space, as in Section 4 one can obtain a better estimate.
Let $y\in X$, $\\{u:\|u-y\|\leq r\\}\subset X_{0}$, $t_{0}\geq 0$ and
$a(t_{0})<r.$ Then there exists $x_{0}\in X$ with $\|x_{0}-y\|\leq r$ and
$\|x(t,x_{0})\|\geq a(t)e^{\omega t}$ for $t\geq t_{0}$.

The next corollary is one of the main results of the paper. It is a direct
consequence of Theorem 2.2 and Theorem 7.6.

######  Corollary 7.9.

Under the conditions of Theorem 7.6, there exists $x_{0}\in\overline{B}(y,r)$
and $t_{0}\geq 0$ such that

(7.6) $\|x(t,x_{0})\|\geq a(t)e^{s_{R}(A)t},\qquad t\geq t_{0}.$

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 [70].

Our results on lower bounds are also new in the framework of linear equations
(6.1), where $K:[0,\infty)\to B(X)$ 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 norm-continuous $K$,
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
$K:[0,\infty)\to B(X)$ be a strongly continuous function such that $K(t)$ is a
compact operator for each $t\geq 0.$ Let $(U(t,\tau))_{t\geq\tau\geq 0}$ be
the evolution family given by Proposition 6.7. Assume that
$a:[0,\infty)\to[0,\infty)$ is a decreasing function satisfying
$\lim_{t\to\infty}a(t)=0$. Then there exists a dense set of vectors $x$ such
that

$\|U(t,0)x\|\geq a(t)e^{s_{R}(A)t},\qquad t\geq t_{0}=t_{0}(x).$

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 $K:D(A)\to X$
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

$A_{-}=D+K,\qquad D:=\begin{pmatrix}0&I\\\ \Delta&0\end{pmatrix},\qquad
K:=\begin{pmatrix}0&0\\\ 0&-b\end{pmatrix},$

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 $b\in C^{\infty}(\mathcal{M})$ and $b$ satisfies
the so-called geometric control condition. (See also e.g. [1], [41], and [47]
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 $(T(t))_{t\geq 0}$ be the left shift $C_{0}$-semigroup on
$L^{2}(\mathbb{R}_{+})$ defined by $(T(t)f)(s)=f(s+t),t\geq 0,$ for $f\in
L^{2}(\mathbb{R}_{+}).$ If $A$ is the generator of $(T(t))_{t\geq 0},$ then
$\sigma(A)=\\{\lambda\in\mathbb{C}:{\rm Re}\,\lambda\leq 0\\},$ and
$s_{e}(A)=s_{0}(A)=s_{R}(A)=0.$ On the other hand, clearly $T(t)\to 0$ as
$t\to\infty$ 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 the multiplication semigroups $(T(t)f)(z)=e^{-tz}f(z)$,
$t\geq 0,$ on appropriate spaces $L^{2}(\Omega)$ with $\Omega$ from the closed
left half-plane.

##  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 [71], we allow for time-dependent 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

(8.1) $\displaystyle\partial_{tt}u(t,x)-\Delta u(t,x)-b(x)\partial_{t}u(t,x)$
$\displaystyle=-f(t,x,u(t,x)),\,\,x\in\mathcal{M},\ \ t\in\mathbb{R},$ (8.2)
$\displaystyle u(0,x)=u_{0}(x),\quad\partial_{t}u(t,0)$
$\displaystyle=u_{1}(x),\qquad x\in\mathcal{M},$

on a $d$-dimensional, 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 $\partial\mathcal{M}\neq\emptyset$, we additionally impose
Dirichlet boundary conditions in (8.1).

In order to apply the results from the previous section, we have to consider
complex-valued $u$ 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 ${\mathbb{R}}^{2}$ equipped with the Euclidean
scalar product $\zeta_{1}\cdot\zeta_{2}=\xi_{1}\xi_{2}+\eta_{1}\eta_{2}={\rm
Re}\,(\zeta_{1}\overline{\zeta_{2}})$ where
$\zeta_{j}=\xi_{j}+i\eta_{j}=(\xi_{j},\eta_{j})$. 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

(8.3) $\begin{split}&b\in L^{\infty}(\mathcal{M}),\qquad b\geq 0,\qquad
b>0\,\,\text{\ on an open subset of}\,\,\mathcal{M},\\\
&f:{\mathbb{R}}\times\mathcal{M}\times{\mathbb{C}}\to{\mathbb{C}}\text{ \ is
continuous in the $3$d variable and measurable},\\\
&|f(t,x,\zeta)|\leq\kappa(t)(1+|\zeta|^{\alpha}),\qquad|f(t,x,\zeta)-f(s,x,\zeta)|\leq\omega(|t-s|)(1+|\zeta|^{\alpha}),\end{split}$

for some $0\leq\alpha<d/(d-2)_{+}$ and all $t,s\in{\mathbb{R}}$,
$\zeta\in{\mathbb{C}}$, $x\in\mathcal{M}$, where
$\kappa:\mathbb{R}\to[0,\infty)$ is locally bounded and
$\omega:[0,\infty)\to[0,\infty)$ satisfies $\omega(\tau)\to 0$ as $\tau\to 0.$
(Here $y_{+}=\max\\{y,0\\}$ for $y\in{\mathbb{R}}$ and $\frac{d}{0}:=\infty$.)
The assumption $b\geq 0$ 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 Theorem 2.34 of [7]. On the state space
$X=V\times L^{2}(\mathcal{M})$ we introduce the operator matrix

(8.4) $A_{+}=\begin{pmatrix}0&I\\\ \Delta&b\end{pmatrix},\qquad
D(A_{+})=(H^{2}(\mathcal{M})\cap V)\times V.$

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, $K:{\mathbb{R}}\times X\to X$ is jointly continuous and collectively
compact.

We study (8.1) in forward time. Using standard properties of the wave equation
with the (continuous) right-hand side $t\mapsto h(t):=-f(t,\cdot,u(t))$, it
can be checked that a mild solution $w$ to

(8.5) $\displaystyle w^{\prime}(t)=A_{+}w(t)+K(t,w(t)),\quad t\geq 0,\quad
w(0)=w_{0}:=(u_{0},u_{1})\in X,$

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

(8.6) $A_{-}=\begin{pmatrix}0&I\\\ \Delta&-b\end{pmatrix},\qquad
D(A_{-})=D(A_{+}).$

It generates a $C_{0}$-group $(T_{-}(t))_{t\in{\mathbb{R}}}$, which is
contractive in forward time since $b\geq 0$, and it corresponds to the damped
wave equation

(8.7) $\displaystyle\partial_{tt}u(t,x)-\Delta u(t,x)+b(x)\partial_{t}u(t,x)$
$\displaystyle=-f(t,x,u(t,x)),\,\,x\in\mathcal{M},\ \ t\in\mathbb{R},$ (8.8)
$\displaystyle u(0,x)=u_{0}(x),\quad\partial_{t}u(t,0)$
$\displaystyle=u_{1}(x),\quad x\in\mathcal{M}.$

The above remarks on the solution also apply here. We study (8.7) _backwards_
in time, i.e., for $t\leq 0.$ To bring it to standard forward form, one looks
at $\tilde{u}(t)=u(-t)$ for $t\geq 0$ satisfying

(8.9)
$\displaystyle\partial_{tt}\tilde{u}(t,x)-\Delta\tilde{u}(t,x)-b(x)\partial_{t}\tilde{u}(t,x)$
$\displaystyle=-f(-t,x,\tilde{u}(t,x)),\,\,x\in\mathcal{M},\,t\geq 0,$ (8.10)
$\displaystyle\tilde{u}(0,x)=u_{0}(x),\quad\partial_{t}\tilde{u}(t,0)$
$\displaystyle=-u_{1}(x),\quad x\in\mathcal{M}.$

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 first-order problem

(8.11) $\displaystyle w^{\prime}(t)=Bw(t)-K(-t,w(t)),\qquad t\geq 0,\qquad
w(0)=(u_{0},-u_{1}).$

Moreover, if $J:X\to X$ 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 $R(\lambda,A_{+})=JR(\lambda,-A_{-})J^{-1}$
for all $\lambda\in\rho(A_{+}).$ Thus,

(8.12) $s_{R}(-A_{-})=s_{R}(A_{+}).$

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 well-known that $\sigma(A_{-})\subset\\{\lambda:-\|b\|_{\infty}\leq{\rm
Re}\,\lambda\leq 0\\}$ since $b\geq 0$, that $\sigma(A_{-})$ is invariant
under conjugation, and that it consists of a discrete set of eigenvalues since
$A_{-}$ has compact resolvent. Moreover if $b\geq 0$ is non-zero, we have
$\sigma(A_{-})\cap i\mathbb{R}=\emptyset$ if
$\partial\mathcal{M}\neq\emptyset$ and $\sigma(A_{-})\cap i\mathbb{R}=\\{0\\}$
(with constants as eigenfunctions) if $\mathcal{M}$ has no boundary. (See e.g.
[47, p.74], where the smoothness of $b$ assumed there does not play a role in
these statements, or [1, Section 4].)

In the latter case, let $P_{0}$ be the Riesz projection corresponding to $0$
and equip $X_{0}=(I-P_{0})X$ with the inner product norm

$\|(u_{0},u_{1})\|_{X_{0}}=\|(-\Delta)^{1/2}u_{0}\|_{L^{2}}+\|u_{1}\|_{L^{2}},\qquad(u_{0},u_{1})\in
X_{0}.$

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 $t\geq
0.$ Moreover, by e.g. [1, Section 4], we have
$\sigma(\dot{A}_{-})=\sigma(A)\setminus\\{0\\}\subset\\{\lambda:{\rm
Re}\,\lambda\leq 0\\}$ and there are $c_{1},c_{2}>0$ with

$c_{1}\|R(\lambda,A_{-})\|_{B(X)}\leq\|R(\lambda,\dot{A}_{-})\|_{B(X_{0})}\leq
c_{2}\|R(\lambda,A_{-})\|_{B(X)}$

for $\lambda\in\rho(A_{-})$ with $|\lambda|\geq\epsilon_{0}$ 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 [47, p.74]), and also [19] and [11]. We
will also use $\dot{A}_{-}$ in the sequel to setudy 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 $K:{\mathbb{R}}\times X\to X$
defined above is jointly continuous and collectively compact. If also

(8.13) $|f(t,x,\zeta)|\leq\kappa(t)(1+|\zeta|)$

for $\kappa$ from (8.3) and all
$(t,x,\zeta)\in{\mathbb{R}}\times\mathcal{M}\times{\mathbb{C}}$, then
$\|K(t,w)\|\leq c\kappa(t)(1+\|w\|)$ for all $w\in X$ 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 $t_{n}\to t$ in ${\mathbb{R}}$ and $w_{n}=(u_{n},v_{n})\to
w=(u,v)$ in $X$ as $n\to\infty.$ 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 $u_{n_{k}}\to u$ in
$L^{2\alpha}(\mathcal{M})$ and pointwise a.e. as $k\to\infty$ and
$|u_{n_{k}}|\leq g$ 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 $k\to\infty$ and
$|f(t_{n_{k}},\cdot,u_{n_{k}})|\leq m(1+|g|^{\alpha})\in L^{2}(\mathcal{M})$
by (8.3). It follows $f(t_{n_{k}},\cdot,u_{n_{k}})\to f(t,\cdot,u)$in
$L^{2}(\mathcal{M})$ as $k\to\infty$, 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 $u_{n_{k}}\to u$
in $L^{2\alpha}(\mathcal{M})$ and pointwise a.e. as $k\to\infty$ and
$|u_{n_{k}}|\leq g$ 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 $K(t,\cdot):X\to X$ is compact. To use Lemma 7.1, let $t_{0},r>0$,
$t,s\in[-t_{0},t_{0}]$ and $\|w\|\leq r$. Due to Sobolev’s embedding, we then
obtain $\|u\|_{L^{2\alpha}}\leq Cr$ for a constant $C>0$ independent of $r,$
and (8.3) yields

$\displaystyle\|f(t,\cdot,u)-f(s,\cdot,u)\|_{L^{2}(\mathcal{M})}\leq$
$\displaystyle\omega(|t-s|)\|1+|u|^{\alpha}\|_{L^{2}(\mathcal{M})}$
$\displaystyle\leq$
$\displaystyle\omega(|t-s|)(\operatorname{vol}(\mathcal{M})^{\frac{1}{2}}+C^{\alpha}r^{\alpha}).$

As $\omega(\tau)\to 0$ as $\tau\to 0$, the map $K$ is collectively compact by
Lemma 7.1. ∎

Due to this result, Corollary 6.6 and Proposition 6.3, if $\alpha\leq 1$ in
(8.3) then the operators $\pm K(\pm t,\cdot)$ fit to Theorem 7.6 and its
corollaries.

If $1<\alpha<d/(d-2)_{+}$, 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

(8.14) $\varphi(t,x,\zeta)=\int_{0}^{1}f(t,x,\tau\zeta)\cdot\zeta d\tau$

for $(t,x,\zeta)\in{\mathbb{R}}\times\mathcal{M}\times{\mathbb{C}}$. Notice
that

(8.15) $|\varphi(t,x,\zeta)|\leq
c\kappa(t)(1+|\zeta|^{\alpha+1}),\qquad(t,x,\zeta)\in{\mathbb{R}}\times\mathcal{M}\times{\mathbb{C}},$

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
$\psi:{\mathbb{R}}\times\mathcal{M}\times{\mathbb{R}}\to{\mathbb{R}}$ is
continuous in the third variable. Here we do _not_ need differentibility of
$f$ in $\zeta$ and obtain

$\varphi(t,x,\zeta)=\frac{1}{2}\int_{0}^{|\zeta|^{2}}\psi(t,x,r)dr.$

We note that in the next proposition we do no use the results discussed in
Section 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. [72, Section 4.4] for a similar approach
using a Galerkin approximation.)

######  Proposition 8.3.

Let $b$ and $f$ satisfy (8.3) with $1\leq\alpha<d/(d-2)_{+}$ and $\varphi$ be
given by (8.14). Assume that $\varphi$ is differentiable in the third
variable,

(8.16) $\varphi\geq
0,\qquad\nabla_{\zeta}\varphi=f,\qquad|f(t,x,\zeta)-f(s,x,\zeta)|\leq\ell(t_{0})|t-s|(1+|\zeta|)$

for $t_{0}>0$, $t,s\in[-t_{0},t_{0}]$, $\zeta\in{\mathbb{C}}$,
$x\in\mathcal{M}$, and a locally bounded map $\ell:[0,\infty)\to[0,\infty)$.
Then (8.5) and (8.11) have global solutions $w$ such that $\|w(t)\|\leq
c(t_{0},r)$ for $0\leq t\leq t_{0}$ and $\|(u_{0},u_{1})\|\leq r$ 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 $n\in\mathbb{N}.$
Take standard mollifiers $\rho_{n}:{\mathbb{R}}^{2}\to[0,\infty)$ with support
in $\overline{B}(0,\frac{1}{n})$ 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 $|\zeta|\leq n$, $|\chi^{\prime}_{n}(\zeta)|\leq
c$, and $|\chi_{n}(\zeta)|\leq c|\zeta|$ for a constant $c\geq 1$. We set

$\displaystyle\tilde{\varphi}_{n}(t,x,\zeta)$
$\displaystyle=\varphi(t,x,\chi_{n}(\zeta)),\qquad\varphi_{n}(t,x,\cdot)=\rho_{n}\ast\tilde{\varphi}_{n}(t,x,\cdot),$
$\displaystyle\tilde{f}_{n}(t,x,\zeta)$
$\displaystyle=\chi_{n}^{\prime}(\zeta)^{\top}f(t,x,\chi_{n}(\zeta)),\qquad
f_{n}(t,x,\cdot)=\rho_{n}\ast\tilde{f}_{n}(t,x,\cdot).$

It is straightforward to check that $\nabla_{\zeta}\varphi_{n}=f_{n}$ and
$\varphi_{n}\geq 0$ 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 $c_{n,k}=c_{n,k}(t_{0})\geq 0$ such that
$|\partial_{\zeta}^{k}\varphi_{n}(t,x,\zeta)|\leq
c_{n,k}(1+|\zeta|^{1+\alpha})$ for
$(t,x,\zeta)\in[-t_{0},t_{0}]\times\mathcal{M}\times{\mathbb{C}}$ and
$k\in{\mathbb{N}}_{0}$. Finally, $f_{n}$ and $\varphi_{n}$ satisfy (8.3)
respectively (8.15) with constants uniform in $n$, as well as

$|\partial_{t}f_{n}(t,x,\zeta)|\leq c\ell(t_{0})(1+|\zeta|)\quad\text{ and
}\quad|\partial_{t}\varphi_{n}(t,x,\zeta)|\leq c\ell(t_{0})(1+|\zeta|^{2})$

for $n\in\mathbb{N}$ 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 $K_{n}:[-t_{0},t_{0}]\times
X\to X$ 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 $w_{n}\to w_{0}$ in $X$ as $n\to\infty$. Due to [55, 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

$u_{n}\in C^{2}([0,\infty),L^{2}(\mathcal{M}))\cap C^{1}([0,\infty),V)\cap
C([0,\infty),H^{2}(\mathcal{M}))$

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 $n\to\infty$. To this aim, we
look at the energies

$E^{0}(w)=\frac{1}{2}\int_{\mathcal{M}}(|\nabla
u|^{2}+|\partial_{t}u|^{2})dx,\qquad
E_{n}(t,w)=E^{0}(w)+\int_{\mathcal{M}}\varphi_{n}(t,x,u)dx$

for $w=(u,v)\in X$. The above observations imply that for all $w\in X$ and
$t>0$ one has $E^{0}(w)\leq E_{n}(t,w)\leq c(1+\|w\|_{X}^{1+\alpha})$,
$E_{n}(t,\cdot),E^{0}(\cdot)\in C^{1}(X,{\mathbb{R}})$, and $E_{n}(\cdot,w)\in
W^{1,\infty}_{\mathrm{loc}}({\mathbb{R}})$, where we also use that
$H^{1}(\mathcal{M})\hookrightarrow L^{\alpha+1}(\mathcal{M})$ in view of
$\alpha+1<2d/(d-2)_{+}$. For a.e. $t\in[0,t_{0}]$, we compute

$\displaystyle\tfrac{d}{dt}E_{n}(t,w_{n}(t))$
$\displaystyle=\int_{\mathcal{M}}\big{[}\partial_{t}u_{n}(t)\cdot[\partial_{tt}u_{n}(t)-\Delta
u_{n}(t)+f_{n}(t,\\!\cdot,u_{n}(t))]$
$\displaystyle\qquad+(\partial_{t}\varphi_{n})(t,\cdot,u_{n}(t))\big{]}dx$
$\displaystyle=\\!\int_{\mathcal{M}}\big{[}b|\partial_{t}u_{n}(t)|^{2}+(\partial_{t}\varphi_{n})(t,\cdot,u_{n}(t))\big{]}dx$
$\displaystyle\leq c(\|(u_{n}(t),\partial_{t}u_{n}(t))\|_{L^{2}}^{2}+1),$
$\displaystyle E_{n}(t,w_{n}(t))$ $\displaystyle\leq
E_{n}(0,w_{0,n})+c+c\int_{0}^{t}\big{(}\|u_{n}(s)\|_{L^{2}(\mathcal{M})}^{2}+E_{n}(s,w_{n}(s))\big{)}ds.$

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

$E_{n}(t,w_{n}(t))\leq
cE_{n}(0,w_{0,n})+c+c\int_{0}^{t}E_{n}(s,w_{n}(s)\big{)}ds.$

Gronwall’s inequality then yields

$E^{0}(t,w_{n}(t))\leq E_{n}(t,w_{n}(t))\leq
c(\|w_{0,n}\|_{X}^{1+\alpha}+1)e^{ct}\leq
C(t_{0})(\|w_{0}\|_{X}^{1+\alpha}+1),$

and hence

(8.17) $\|w_{n}(t)\|_{X}^{2}=\|u_{n}(t)\|_{L^{2}}^{2}+2E^{0}(t,w_{n}(t))\leq
C(t_{0})(\|w_{0}\|_{X}^{1+\alpha}+1)$

for $0\leq t\leq t_{0}$ 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 $n\to\infty$. Moreover, by the uniform bound
(8.17) and the compact embedding from [63, Corollary 4] the sequence $(u_{n})$
is precompact in $C([0,t],L^{2\alpha}(M))$, hence in
$L^{2\alpha}([0,t_{0}]\times M)$. There thus exists a subsequence, also
denoted by $(u_{n})$, such that $u_{n}\to u$ in
$L^{2\alpha}([0,t_{0}]\times\mathcal{M})$ and pointwise a.e., as well as
$|u_{n}|\leq g$ 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 $|u(t,x)|\leq n-1$. There is
a number $\delta\in(0,1)$ such that
$|f(t,x,\zeta)-f(t,x,u(t,x))|\leq\varepsilon$ if $|\zeta-u(t,x)|\leq\delta$
since $f$ is continuous in $\zeta$. So there exists an index $n_{\varepsilon}$
such that for all $n\geq n_{\varepsilon}$ and $|\eta|\leq 1/n$ we have
$|u_{n}(t,x)-\eta|<n$ and

$\displaystyle|\tilde{f}_{n}(t,x,u_{n}(t,x)-$
$\displaystyle\eta)-f(t,x,u(t,x))|$
$\displaystyle=|f(t,x,u_{n}(t,x)-\eta)-f(t,x,u(t,x))|\leq\varepsilon.$

It follows that $|f_{n}(t,x,u_{n}(t,x))-f(t,x,u(t,x))|\leq\varepsilon$, and
hence $f_{n}(t,\cdot,u_{n}(t))$ tends pointwise a.e. to $f(t,\cdot,u(t))$ as
$n\to\infty$, for a.e. $t\in[0,t_{0}]$.

Since $f_{n}$ satisfies (8.3) uniformly in $n$, we have
$|f_{n}(t,\cdot,u_{n}(t))|\leq c\kappa(t)(1+g(t,\cdot)^{\alpha})$ and thus
$f_{n}(t,\cdot,u_{n}(t))\to f(t,\cdot,u(t))$ in $L^{2}(\mathcal{M})$ for a.e.
$t\in[0,t_{0}]$. Moreover, by (8.3) and (8.17) the map
$t\mapsto\|f_{n}(t,\cdot,u_{n}(t))\|_{L^{2}(\mathcal{M})}$ is locally bounded
independent of $n$. Therefore the right-hand side in the formula for mild
solutions

$w_{n}(t)=T_{+}(t)w_{0,{n}}+\int_{0}^{t}T_{+}(t-s)K_{n}(s,w_{n}(s))ds$

converges in $C([0,t_{0}],X)$ as $n\to\infty$. As seen above, the left-hand
side tends to $w=(u,\partial_{t}u)$ weakly* in $L^{\infty}([0,t_{0}],X)$ so
that

$w(t)=T_{+}(t)w_{0}+\int_{0}^{t}T_{+}(t-s)K(s,w(s))ds$

holds for a.e. $t\in[0,t_{0}]$. Since the right-hand side is continuous and
$t_{0}>0$ is arbitrary, $w$ solves (8.5) and satisfies (8.17) for all $0\leq
t\leq t_{0}$. ∎

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 [71] in various
respects: We can treat time-depending $f$, 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 $d$-dimensional, 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 $s_{R}(-A_{-})\geq\frac{b}{2}$.
By Corollary 7.9 and Propositions 8.1 and 8.3, given a function
$a:[0,\infty)\to[0,\infty)$ decreasing to $0,$ there are a dense set of
initial values $w_{0}\in X$ and $t_{0}\geq 0$ such that the corresponding
solutions $w(t,w_{0})$ of backward damped wave equation (8.11) admit the lower
bound $\|w(t,w_{0})\|\geq a(t)e^{bt/2}$ for $t\geq t_{0}.$ 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 non-stationary 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 $b\in{\rm BV}([0,1])$ 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 ${\rm
Re}\,\lambda_{n}\to-\frac{\beta}{2}$ as $n\to\infty$ 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,
$s_{R}(-A_{-})\geq\frac{\beta}{2}.$ As above, using Corollary 7.9 along with
Propositions 8.1 and 8.3, for a given function $a:[0,\infty)\to[0,\infty)$
decreasing to $0$ we find a dense set of initial values $w_{0}$ such the
solutions of (8.11) satisfy $\|w(t,w_{0})\|\geq a(t)e^{\beta t/2}$ for $t\geq
t_{0}$ and some $t_{0}\geq 0$. Moreover, by [28, Theorem 3.4], the same result
holds without assuming $b>0$ if $\|b\|_{\infty}$ 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 [44, Appendix B], see also [12, Section
2.1]. Write

$\rho_{t}=(x_{t},\xi_{t})=g^{t}(\rho_{0}),\qquad\rho_{0}=(x_{0},\xi_{0})\in
S^{*}\mathcal{M},\,t\in\mathbb{R},$

and let $\pi:S^{*}\mathcal{M}\to\mathcal{M}$ be a canonical projection. Given
a damping $b\in C^{\infty}(\mathcal{M})$ with $b\geq 0,$ 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

$\displaystyle b_{-}$ $\displaystyle:=\sup_{t>0}\inf_{\rho_{0}\in
S^{*}\mathcal{M}}\langle
b\rangle_{t}(\rho_{0})=\lim_{t\to\infty}\inf_{\rho_{0}\in
S^{*}\mathcal{M}}\langle b\rangle_{t}(\rho_{0}),$ $\displaystyle b_{+}$
$\displaystyle:=\inf_{t>0}\sup_{\rho_{0}\in S^{*}\mathcal{M}}\langle
b\rangle_{t}(\rho_{0})=\lim_{t\to\infty}\sup_{\rho_{0}\in
S^{*}\mathcal{M}}\langle b\rangle_{t}(\rho_{0}).$

Note that by the Birkhoff ergodic theorem, $\langle
b\rangle_{\infty}(\rho_{0}):=\lim_{t\to\infty}\langle b\rangle_{t}(\rho_{0})$
exists almost everywhere with respect to the flow invariant (normalised)
Liouville measure on $S^{*}\mathcal{M}$, and setting

$b^{-}_{\infty}:={\rm ess\,inf}_{\rho_{0}\in S^{*}\mathcal{M}}\,\langle
b\rangle_{\infty},\qquad b^{+}_{\infty}:={\rm ess\,sup}_{\rho_{0}\in
S^{*}\mathcal{M}}\,\langle b\rangle_{\infty},$

we have

$b_{-}\leq b^{-}_{\infty}\leq b^{+}_{\infty}\leq b_{+},$

where all of the inequalities can in general be strict. If the geodesic flow
is ergodic, then one has

$b^{-}_{\infty}=b^{+}_{\infty}=\frac{1}{{\rm
vol}\,(\mathcal{M})}\int_{\mathcal{M}}b(x)\,dx:=b^{*}_{\infty}.$

It was proved in [47] 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 $s_{R}(-A_{-})\geq\frac{b_{-}}{2}.$ We also
refer to [65] for comments on the generality of this result and an alternative
proof.

The result was improved in [65] 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 [65], 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 $s_{R}(-A_{-})\geq\frac{b^{-}_{\infty}}{2},$ or, if
$(g^{t})_{t\in\mathbb{R}}$ is ergodic,
$s_{R}(-A_{-})\geq\frac{b^{*}_{\infty}}{2}$, 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 $(\dot{T}_{-}(t))_{t\geq 0}$
satisfies $\|\dot{T}_{-}(t)R(\mu_{0},\dot{A})^{s}\|\leq M_{s}e^{-\omega_{s}t}$
for $t\geq 0$ and some $s,\omega_{s},M_{s}>0$ as well as, at the same time,
$\omega_{0}(\dot{T}_{-})\geq 0.$ In other words, $(\dot{T}_{-}(t))_{t\geq 0}$
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 $(\dot{T}_{-}(t))_{t\geq 0}$ 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 $\\{\lambda:{\rm
Re}\,\lambda\geq-a\\}$ 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
$\|R(\lambda,\dot{A}_{-})\|\leq({\rm Re}\,\lambda)^{-1}$ for ${\rm
Re}\,\lambda>0,$ 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 $s_{R}(-A_{-})\geq\beta>0$. 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 [61, Theorem 1] that each such an orbit gives rise to a smooth
damping $b$ such that $(\dot{T}_{-}(t))_{t\geq 0}$ 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 [61]
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 [40, 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 so-called 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
$(\dot{T}_{-}(t))_{t\geq 0},$ sometimes with explicit rates, can also be found
in [15, Section 4], [54] and [60, Theorem 3 and the subsequent Remark]. We
avoid their discussion to keep our exposition within reasonable limits.

Note that the assumption $b\geq 0$ 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 non-stabilizability of certain nonlinear infinite-dimensional
control systems. In this way, we generalize the corresponding results in [36]
or [73], 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. [45] and [46] concerning that).
We just refer to [43] and [66] as sample works on nonlinear stabilization and
to [18] for general concepts of nonlinear control. For basics of linear theory
one may consult [26, Section VI.8].

######  Example 8.8.

Let $(T(t))_{t\geq 0}$ be a $C_{0}$-semigroup on a Banach space $X$ with
generator $A,$ and let the control map $B:U\to X$, the feedback $F:Y\to U$ and
the observation map $C:X\to Y$ 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 $x^{\prime}=Ax+BFCx$ 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 time-discrete feedback systems
$x_{n+1}=Ax_{n}+BFCx_{n},n\geq 0,$ for bounded $A$ and continuous $B$, $F$,
and $C$ mapping bounded sets into bounded sets, if one replaces $s_{R}(A)$
with $r_{e}(A).$

##  9\. 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 [49, Section 4] with
appropriate changes and improvements, which warrant an independent treatment.
The section makes the paper essentially self-contained.

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 self-contained 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

$\|x\|=\sup\Bigl{\\{}\frac{|\langle x,y\rangle|}{\|y\|}:y\in\Lambda,y\neq
0\Bigr{\\}}$

for all $x\in X$. Part (a) of the next result is [48, Lemma V.37.6]. We add
two related statements whose proof is a variation of the arguments in [48].

######  Lemma 9.1.

Let $F$ be a finite-dimensional subspace of a Banach space $X$ and
$\varepsilon>0$.

  * (a)

There exists a closed subspace $M\subset X$ of finite codimension such that

(9.1) $\|f+m\|\geq(1-\varepsilon)\max\bigl{\\{}\|f\|,{\|m\|/2}\bigr{\\}}$

for all $f\in F$ and $m\in M$.

  * (b)

Let $M$ be given by (a) and $E$ be a finite-dimensional 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 $M=\bigcap_{j=1}^{k}{\rm
Ker}\,y^{*}_{j}$ 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 $D\subset\\{f\in F:\|f\|=1\\}$
satisfying ${\rm dist}\,\\{e,D\\}\leq\varepsilon/2$ for all $f\in F$ with
$\|f\|=1$. Let $\Lambda\subset X^{*}$ be norming. For each $d\in D$ there is a
functional $y^{*}_{d}\in\Lambda$ such that
$\bigl{|}\bigl{\langle}d,y^{*}_{d}/\|y^{*}_{d}\|\bigr{\rangle}\bigr{|}>1-\varepsilon/2$.
Set $M=\bigcap_{d\in D}{\rm Ker}\,y^{*}_{d}$. Clearly, $M$ is closed with
finite codimension.

To prove the required inequality, let $f\in F$ and $m\in M$. We can assume
that $\|f\|\neq 0$ since the assertion is clear for $f=0$. Choose $d\in D$
with $\bigl{\|}d-f/\|f\|\bigr{\|}\leq\frac{\varepsilon}{2}$. As in [48, Lemma
V.37.6], we then estimate

$\displaystyle\|f+m\|$
$\displaystyle\geq\Bigl{|}\Bigl{\langle}f+m,\frac{y^{*}_{d}}{\|y^{*}_{d}\|}\Bigr{\rangle}\Bigr{|}=\Bigl{|}\Bigl{\langle}f-\|f\|d,\frac{y^{*}_{d}}{\|y^{*}_{d}\|}\Bigr{\rangle}+\Bigl{\langle}\|f\|d,\frac{y^{*}_{d}}{\|y^{*}_{d}\|}\Bigr{\rangle}\Bigr{|}$
$\displaystyle\geq\|f\|(1-\varepsilon/2)-\bigl{\|}f-\|f\|d\bigr{\|}\geq\|f\|(1-\varepsilon).$

One can now conlcude as in [48, Lemma V.37.6] to show (a) and (c). Part (b) is
is direct consequence of the construction above. ∎

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 finite-dimensional 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,
well-known in the bounded case.

######  Lemma 9.2.

Let $A$ be a closed, densely defined operator on a Banach space $X,$ such that
$\rho(A)\neq\emptyset$, 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 $\|(A-\lambda)x\|<\epsilon$.

###### Proof.

We will rely on the fact, that the statement is true if $A$ is bounded, see
e.g. [48, 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 $M^{\prime}={\rm
Im}\,((A-\mu)\upharpoonright_{M})$. Then ${\rm codim}\,M^{\prime}<\infty$, and
as $T\in B(X)$ there exists a sequence $(x_{n})_{n=1}^{\infty}\subset
M^{\prime}$ such that $\|x_{n}\|=1$ for all $n$ and $Tx_{n}\to 0$ as
$n\to\infty$. 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 $Ay_{n}\to 0$ as $n\to\infty.$ Moreover,
since

$Tx_{n}=x_{n}+\mu(A-\mu)^{-1}x_{n}=x_{n}+\mu y_{n},$

we have $\liminf_{n\to\infty}\|y_{n}\|=1/|\mu|>0.$ It remains to choose
$ay_{n}$ for appropriate $a>0$ and $n.$ ∎

The proof of Theorem 2.2 is based on the next lemma.

######  Lemma 9.3.

Let $A$ generate the $C_{0}$-semigroup $(T(t))_{t\geq 0}$ 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

$\sigma_{p}(A)\cap(\omega+i\mathbb{R})\,\,\text{is infinite}$

or

$\sigma_{p}(A)\cap(\omega+i\mathbb{R})\text{ \ is at most finite \quad
and}\quad\limsup_{|b|\to\infty}\|R(\omega+ib,A)\|=\infty.$

Then there exist $(\mu_{n})_{n=1}^{\infty}\subset\omega+i\mathbb{R}$ and
$(u_{n})_{n=1}^{\infty}\subset D(A)$ such that $\|u_{n}\|=1$ for all
$n\in\mathbb{N},$ $\|(\mu_{n}-A)u_{n}\|\to 0$ as $n\to\infty$, and for every
$y^{*}\in D(A^{*})$ we have

$\langle u_{n},y^{*}\rangle\to 0,\quad n\to\infty.$

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
$|b_{n}|\to\infty$ and unit vectors $u_{n}\in D(A)$ for $n\geq 1$ such that
$\|(\omega+ib_{n}-A)u_{n}\|\to 0$ as $n\to\infty$. Set
$\mu_{n}=\omega+ib_{n}$. We show that

$\langle u_{n},y^{*}\rangle\to 0,\quad n\to\infty,$

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
$M={\rm Ker}\,y^{*}$. 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,

$\langle(\mu_{n}-A)u_{n},y^{*}\rangle\to 0,\qquad n\to\infty,$

and

$\displaystyle\langle(\mu_{n}-A)u_{n},y^{*}\rangle$
$\displaystyle=\langle(\mu_{n}-A)m_{n},y^{*}\rangle+\alpha_{n}\langle(\mu_{n}-A)y,y^{*}\rangle$
$\displaystyle=\alpha_{n}\mu_{n}\langle y,y^{*}\rangle-\langle
m_{n},A^{*}y^{*}\rangle-\alpha_{n}\langle Ay,y^{*}\rangle.$

Since the last two terms are uniformly bounded and $|\mu_{n}|\to\infty$, we
have $\alpha_{n}\to 0$ as $n\to\infty$. It follows $\langle
u_{n},y^{*}\rangle=\alpha_{n}\langle y,y^{*}\rangle\to 0$. ∎

Let $(T(t))_{t\geq 0}$ be a $C_{0}$-semigroup on a Banach space $X$, and set

$\|x\|_{1}:=\sup\\{|\langle x,x^{*}\rangle|:x^{*}\in D(A^{*}),\|x^{*}\|\leq
1\\}$

for $x\in X$. Then $\|\cdot\|_{1}$ is an equivalent norm satisfying

(9.2) $\|x\|_{1}\leq\|x\|\leq\alpha\|x\|_{1},\qquad\text{where \
}\alpha:=\limsup_{t\to 0}\|T(t)\|,$

for all $x\in X$. See e.g. [52, Theorems 1.3.1 and 1.3.5]. Hence, renorming
$X$ with $\|\cdot\|_{1},$ we can make $D(A^{*})$ a norming set for the Banach
space $(X,\|\cdot\|_{1})$.

Recall the definitions of the resolvent bound $s_{R}$ and of the notion of
admissibility given in Section 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 $(T(t))_{t\geq
0}$ on a Banach space $X,$ then the number $s_{R}(A)$ given by (2.4) is
admissible.

_Proof of Theorem 2.2._ Let $M$ be a closed subspace of finite codimension in
$X$ given by $M=\bigcap_{j=1}^{k}{\rm Ker}\,y^{*}_{j}$ for some functionals
$y^{*}_{1},\dots,y^{*}_{k}\in D(A^{*}).$ Let $\varepsilon>0$ and $t_{0}>0$ be
fixed, and set $K:=\sup\\{\|T(t)\|:0\leq t\leq t_{0}\\}$. 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 $\|x\|=1$ and
$\|(A-\mu)x\|\leq\varepsilon.$ Hence

$\|T(t)x-e^{\mu
t}x\|=\Bigl{\|}\int_{0}^{t}e^{\mu(t-s)}T(s)(\mu-A)xds\Bigr{\|}\leq\varepsilon
t_{0}e^{s_{R}(A)t_{0}}K$

for all $t\in[0,t_{0}]$.

Let $s_{R}(A)\geq s_{e}(A)$ 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
$\mu_{n}=s_{R}(A)+ib_{n}$ for $n\in\mathbb{N}$ and
$(u_{n})_{n=1}^{\infty}\subset D(A)$ with $\|u_{n}\|=1$ such that

$\langle u_{n},y^{*}\rangle\to 0\quad\text{for every}\,\,y^{*}\in
D(A^{*})\quad\text{and}\qquad\|(\mu_{n}-A)u_{n}\|\to 0,\quad n\to\infty.$

By [59, Lemma 7.4], there exists a finite-dimensional subspace $F\subset D(A)$
such that $X=M\oplus F$. Let $P$ be the projection onto $F$ with ${\rm
Ker}\,P=M$. By the choice of $u_{n}$ we have $\|Pu_{n}\|\to 0$ so that
$\|(I-P)u_{n}\|\to 1$ as $n\to\infty$ and

$\Bigl{\|}u_{n}-\frac{u_{n}-Pu_{n}}{\|u_{n}-Pu_{n}\|}\Bigr{\|}\to 0,\qquad
n\to\infty.$

Choose $n_{0}\in{\mathbb{N}}$ such that

$\displaystyle\Bigl{\|}u_{n_{0}}-\frac{u_{n_{0}}-Pu_{n_{0}}}{\|u_{n_{0}}-Pu_{n_{0}}\|}\Bigr{\|}$
$\displaystyle\leq\min\left\\{\frac{\varepsilon}{4K},\frac{\varepsilon}{4e^{s_{R}(A)t_{0}}}\right\\},$
$\displaystyle\|(\mu_{n_{0}}-A)u_{n_{0}}\|$
$\displaystyle\leq\frac{\varepsilon}{4t_{0}K\max\\{1,e^{t_{0}s_{R}(A)}\\}}.$

Set

$\mu=\mu_{n_{0}}\quad\text{and}\quad
x=\frac{u_{n_{0}}-Pu_{n_{0}}}{\|u_{n_{0}}-Pu_{n_{0}}\|}.$

For every $0\leq t\leq t_{0},$ we have

$\|T(t)u_{n_{0}}-e^{\mu
t}u_{n_{0}}\|=\Bigl{\|}\int_{0}^{t}e^{\mu(t-s)}T(s)(\mu_{n_{0}}-A)u_{n_{0}}ds\Bigr{\|}\leq\varepsilon/4$

and

$\displaystyle\|T(t)x-e^{\mu t}x\|$
$\displaystyle\leq\|T(t)x-\\!T(t)u_{n_{0}}\|+\|T(t)u_{n_{0}}\\!-e^{\mu
t}u_{n_{0}}\|+\|e^{\mu t}u_{n_{0}}\\!-e^{\mu t}x\|$ $\displaystyle\leq
K\|x-u_{n_{0}}\|+\varepsilon/4+e^{ts_{R}(A)}\|x-u_{n_{0}}\|$
$\displaystyle<\varepsilon.$

This finishes the proof. ∎

######  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
$\|\cdot\|_{1}=\|\cdot\|$ 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.

##  10\. Acknowledgements

We are grateful to the referee for valuable suggestions and remarks which led
to improvements of this paper.

##  11\. Conflict of interests statement

On behalf of all authors, the corresponding author states that there is no
conflict of interest.

##  12\. Data availability statement

Data sharing not applicable to this article as no datasets were generated or
analysed during the current study.

##  13\. Competing interests statement

The authors have no competing interests to declare that are relevant to the
content of this article.

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