# Stability and Instability of Equilibria in Age-Structured Diffusive
Populations

Christoph Walker<EMAIL_ADDRESS>Leibniz Universität Hannover
Institut für Angewandte Mathematik
Welfengarten 1
D–30167 Hannover
Germany

###### Abstract.

The principle of linearized stability and instability is established for a
classical model describing the spatial movement of an age-structured
population with nonlinear vital rates. It is shown that the real parts of the
eigenvalues of the corresponding linearization at an equilibrium determine the
latter’s stability or instability. The key ingredient of the proof is the
eventual compactness of the semigroup associated with the linearized problem,
which is derived by a perturbation argument. The results are illustrated with
examples.

###### Key words and phrases:

Age structure, diffusion, semigroups, stability of equilibria, linearization.

###### 2010 Mathematics Subject Classification:

47D06, 35B35, 35M10, 92D25

##  1\. Introduction

The dynamics of a population structured by space and age is described by a
density function $u=u(t,a,x)\geq 0$, where $t\geq 0$ refers to time, $a\in
J:=[0,a_{m}]$ is the age variable with maximal age $a_{m}\in(0,\infty)$
(individuals may attain age greater than $a_{m}$ but are no longer tracked in
the model), and $x\in\Omega$ is the spatial position within a domain
$\Omega\subset\mathbb{R}^{n}$. Then

$\bar{u}(t,x):=\int_{0}^{a_{m}}\varrho(a,x)\,u(t,a,x)\,\mathrm{d}a$

is the weighted local overall population with weight $\varrho$ (i.e. the total
number of individuals at time instant $t$ and spatial position $x$ when
$\varrho\equiv 1$). Assuming that the death rate $m=m(\bar{u}(t,x),a)\geq 0$
and the birth rate $b=b(\bar{u}(t,x),a)\geq 0$ depend on this quantity and on
age, the governing equations for the density $u$ are

$\displaystyle\partial_{t}u+\partial_{a}u$
$\displaystyle=\mathrm{div}_{x}\big{(}d(a,x)\nabla_{x}u\big{)}-m(\bar{u}(t,x),a)u\
,$ $\displaystyle t>0\,,$ $\displaystyle a\in(0,a_{m})\,,$ $\displaystyle
x\in\Omega\,,$ (1.1a) $\displaystyle u(t,0,x)$
$\displaystyle=\int_{0}^{a_{m}}b(\bar{u}(t,x),a)u(t,a,x)\,\mathrm{d}a\,,$
$\displaystyle t>0\,,$ $\displaystyle x\in\Omega\,,$ (1.1b)
$\displaystyle\mathcal{B}u(t,a,x)$ $\displaystyle=0\ ,$ $\displaystyle t>0\,,$
$\displaystyle a\in(0,a_{m})\,,$ $\displaystyle x\in\partial\Omega\,,$ (1.1c)
$\displaystyle u(0,a,x)$ $\displaystyle=u_{0}(a,x)\ ,$ $\displaystyle
a\in(0,a_{m})\,,$ $\displaystyle x\in\Omega\,,$ (1.1d)

where

$\mathcal{B}u:=(1-\delta)u+\delta\partial_{\nu}u\,,\quad\delta\in\\{0,1\\}\,,$

means either Dirichlet boundary conditions $u|_{\partial\Omega}=0$ if
$\delta=0$ or Neumann boundary conditions $\partial_{\nu}u=0$ if $\delta=1$.
Note that the evolution problem (1.1) exhibits hyperbolic (due to the aging
term) and parabolic (due to the diffusion) features and involves a nonlocal
condition (1.1b) with respect to age.

Since many years linear and nonlinear age-structured populations with spatial
diffusion have been the focus of intensive research, see e.g. [30, 25] and the
references therein. In particular, problems of the form (1.1) [9, 17, 18, 21]
or variants thereof such as models including nonlocal diffusion [13] or
compartmental models for infectious diseases spreading [5, 6, 7, 11, 10, 14]
have been studied by various authors addressing questions related e.g. to
well-posedness or qualitative aspects under different assumptions (none of
these reference lists is close to being complete though). The present paper
contributes to the study of stability of equilibria to (1.1). While most
research so far on stability of equilibria in age-structured diffusive
populations apply the principle of linearized stability in an ad-hoc fashion,
the aim of the present paper is to provide a proof therefor.

The existence of (nontrivial) equilibria (i.e. time-independent solutions) to
(1.1) has been established under fairly general conditions by the author in a
series of papers using fixed point methods [22] or bifurcations techniques
[20, 22, 24]. A principle of linearized stability for age-structured
populations without spatial diffusion was established in [16], see also [29].
As for the case including spatial diffusion a criterion for linearized
stability was derived in a recent paper [28]. Herein, we shall refine and
simplify considerably this stability result and complement it with an
instability result. In particular, we show that the spectrum of the
linearization (as an unbounded operator) indeed consists of eigenvalues only
whose real parts determine stability and instability.

To give a first flavor of our findings we present a paraphrased version for
the particular case of the trivial equilibrium $\phi=0$. In the next section
we will state a more general version for an arbitrary equilibrium.

Writing the unique strong solution $v$ to the heat equation

$\displaystyle\partial_{a}v=\mathrm{div}_{x}\big{(}d(a,x)\nabla_{x}v\big{)}\,,\quad(a,x)\in
J\times\Omega\,,\qquad v(0,x)=v_{0}(x)\,,\quad x\in\Omega\,,$

subject to the boundary condition $\mathcal{B}v=0$ on $\partial\Omega$ and the
initial value $v_{0}\in L_{q}(\Omega)$ in the form $v(a)=\Pi_{*}(a,0)v_{0}$,
$a\in J$, we define by

$\displaystyle
Q_{0}:=\int_{0}^{a_{m}}b(0,a)\,\exp\left(-\int_{0}^{a}m(0,\sigma)\,\mathrm{d}\sigma\right)\,\Pi_{*}(a,0)\,\mathrm{d}a$

a (compact and positive) operator on $L_{q}(\Omega)$ and denote by $r(Q_{0})$
its spectral radius. Then the stability property of the trivial equilibrium is
determined according to:

######  Proposition 1.1.

Let $q>n$ and assume that

$d:J\times\bar{\Omega}\to(0,\infty)\,,\qquad b:\mathbb{R}\times
J\to(0,\infty)\,,\qquad m:\mathbb{R}\times J\to\mathbb{R}^{+}$

are (sufficiently) smooth functions.

  * (a)

If $r(Q_{0})<1$, then the trivial equilibrium to (1.1) is exponentially
asymptotically stable in $L_{1}\big{(}(0,a_{m}),W_{q}^{1}(\Omega)\big{)}$.

  * (b)

If $r(Q_{0})>1$, then the trivial equilibrium to (1.1) is unstable in
$L_{1}\big{(}(0,a_{m}),W_{q}^{1}(\Omega)\big{)}$.

In case of Neumann boundary conditions (i.e. $\delta=1$), the spectral radius
is

$r(Q_{0})=\int_{0}^{a_{m}}b(0,a)\,\exp\left(-\int_{0}^{a}m(0,s)\mathrm{d}s\right)\,\mathrm{d}a\,.$

Proposition 1.1 is a special case of Proposition 5.1 below. In fact, we can
prove a much more general result for an arbitrary equilibrium. For this
purpose, we shall consider problem (1.1) in an abstract setting and introduce
the notation111We suppress the $x$-variable consistently in the abstract
formulation by considering functions with values in function spaces on
$\Omega$. In particular, the data $m$, $b$, $\varrho$ may include an
$x$-dependence and $u(t,a)\in E_{0}$.

$A(a)w:=\mathrm{div}_{x}\big{(}d(a,\cdot)\nabla_{x}w\big{)}\,,\quad w\in
E_{1}\,,$

where e.g. $E_{1}:=W_{q,\mathcal{B}}^{2}(\Omega)$ with $q\in(1,\infty)$
denotes the Sobolev space of functions $w\in W_{q}^{2}(\Omega)$ satisfying the
boundary condition $\mathcal{B}w=0$ on $\partial\Omega$. Then $A(a)$ is for
each $a\in J$ the generator of an analytic semigroup on the Banach lattice
$E_{0}:=L_{q}(\Omega)$ with compactly and densely embedded domain $E_{1}$. The
abstract formulation of (1.1) now reads

$\displaystyle\partial_{t}u+\partial_{a}u\,$
$\displaystyle=A(a)u-m(\bar{u}(t),a)u\,,\qquad t>0\,,\quad a\in(0,a_{m})\,,$
(1.2a) $\displaystyle u(t,0)$
$\displaystyle=\int_{0}^{a_{m}}b(\bar{u}(t),a)\,u(t,a)\,\mathrm{d}a\,,\qquad
t>0\,,$ (1.2b) $\displaystyle u(0,a)$ $\displaystyle=u_{0}(a)\,,\qquad
a\in(0,a_{m})\,,$ (1.2c)

with

$\bar{u}(t):=\int_{0}^{a_{m}}\varrho(a)\,u(t,a)\,\mathrm{d}a\,.$

We then shall focus on (1.2) and present our main stability result for this
problem. Later we interpret our findings for the concrete equation (1.1) and
variants thereof. The regularizing effects from the diffusion, reflected in
(1.2) by the operator $A$, are of great importance since they will allow us to
handle the nonlinearities under mild assumptions (mainly on the regularity of
the vital rates $m$ and $b$).

##  2\. Main Results

### General Assumptions and Notations

Set $J:=[0,a_{m}]$. Throughout the following, $E_{0}$ is a real Banach lattice
ordered by a closed convex cone $E_{0}^{+}$ (in the following we do not
distinguish $E_{0}$ from its complexification required at certain points) and

$E_{1}\stackrel{{\scriptstyle
d}}{{\mathrel{\lhook\joinrel\relbar\kern-3.44444pt\joinrel\lhook\joinrel\rightarrow}}}E_{0}\,,$

that is, $E_{1}$ is a densely and compactly embedded subspace of $E_{0}$. We
write $\mathcal{L}(E_{1},E_{0})$ for the Banach space of bounded linear
operators from $E_{1}$ to $E_{0}$, set
$\mathcal{L}(E_{0}):=\mathcal{L}(E_{0},E_{0})$, and denote by
$\mathcal{L}_{+}(E_{0})$ the positive operators. For a (possibly unbounded)
operator

$\mathcal{A}:\mathrm{dom}(\mathcal{A})\subset E_{0}\to E_{0}$

we mean by $D(\mathcal{A})$ its domain $\mathrm{dom}(\mathcal{A})$ endowed
with the graph norm. For $\theta\in(0,1)$ and an admissible interpolation
functor $(\cdot,\cdot)_{\theta}$ (see [3]), we put
$E_{\theta}:=(E_{0},E_{1})_{\theta}$ and equip it with the order naturally
induced by $E_{0}^{+}$. We use the notion

$\mathbb{E}_{\theta}:=L_{1}(J,E_{\theta})\,,\quad\theta\in[0,1]\,,$

and observe $\mathbb{E}_{\theta}\hookrightarrow\mathbb{E}_{0}$ for
$\theta\in[0,1]$. We assume that there is $\rho>0$ such that

$A\in C^{\rho}\big{(}J,\mathcal{H}(E_{1},E_{0})\big{)}$ (2.1a) and $A(a)\
\text{ is resolvent positive for each $a\in J$}\,,$ (2.1b)

where $\mathcal{H}(E_{1},E_{0})$ is the subspace of $\mathcal{L}(E_{1},E_{0})$
consisting of all generators of analytic semigroups on $E_{0}$ with domain
$E_{1}$. Then (2.1) and [3, II.Corollary 4.4.2] imply that $A$ generates a
positive, parabolic evolution operator

$\big{\\{}\Pi(a,\sigma)\in\mathcal{L}(E_{0})\,;\,a\in J\,,\,0\leq\sigma\leq
a\big{\\}}\,,$

on $E_{0}$ with regularity subspace $E_{1}$ in the sense of [3, Section
II.2.1] (see Appendix B for a summary of the most important properties of
parabolic evolution operators).

### Well-Posedness

Before stating our main stability result, let us recall the well-posedness of
the nonlinear problem (1.2) established in [21, 28]. In the following, let
$\alpha\in[0,1)$ be fixed. We assume for the birth and the death rate that

$\displaystyle\big{[}\bar{v}\to b(\bar{v},\cdot)]$ $\displaystyle\in
C_{b}^{1-}\big{(}E_{\alpha},L_{\infty}^{+}\big{(}J,\mathcal{L}(E_{\alpha},E_{0})\big{)}\big{)}\,,$
(2.2a) $\displaystyle\big{[}\bar{v}\to m(\bar{v},\cdot)]$ $\displaystyle\in
C_{b}^{1-}\big{(}E_{\alpha},L_{\infty}^{+}\big{(}J,\mathcal{L}(E_{\alpha},E_{0})\big{)}\big{)}\,,$
(2.2b) where $C_{b}^{1-}$ stands for locally Lipschitz continuous maps that
are bounded on bounded sets. The weight function $\varrho$ is such that
$\varrho\in
C\big{(}J,\mathcal{L}_{+}(E_{\theta})\big{)}\,,\quad\theta\in\\{0,\alpha,\vartheta\\}\,,$
(2.2c)

for some $\vartheta\in(0,1)$ (if $\alpha\in(0,1)$, then it suffices to take
$\vartheta=\alpha$). We use the notation

$\bar{v}:=\int_{0}^{a_{m}}\varrho(a)\,v(a)\,\mathrm{d}a\in E_{\theta}\,,\quad
v\in\mathbb{E}_{\theta}\,.$

Observe that integrating (1.2) formally along characteristics yields the
necessary condition that a solution
$u:\mathbb{R}^{+}\rightarrow\mathbb{E}_{0}$ with initial value
$u_{0}\in\mathbb{E}_{0}$ satisfies the fixed point equation

$u(t,a)\,=\,\left\\{\begin{aligned}
&\Pi(a,a-t)\,u_{0}(a-t)+G_{F(u)}(t,a)\,,&&a\in J\,,\ 0\leq t\leq a\,,\\\
&\Pi(a,0)\,B_{u}(t-a)+G_{F(u)}(t,a)\,,&&a\in J\,,\ t>a\,,\end{aligned}\right.$
(2.3a) where $F(u):=-m(\bar{u},\cdot)u$ and
$G_{v}(t,a):=\int_{(t-a)_{+}}^{t}\Pi(a,a-t+s)\,v(s,a-t+s)\,\mathrm{d}s$ (2.3b)
for $v:\mathbb{R}^{+}\rightarrow\mathbb{E}_{\alpha}$, and where $B_{u}$
satisfies the Volterra equation
$\begin{split}B_{u}(t)\,&=\,\int_{0}^{t}b(\bar{u}(t),a)\Pi(a,0)B_{u}(t-a)\,\mathrm{d}a+\int_{t}^{a_{m}}b(\bar{u}(t),a)\Pi(a,a-t)\,u_{0}(a-t)\,\mathrm{d}a\\\
&\quad+\int_{0}^{a_{m}}b(\bar{u}(t),a)G_{F(u)}(t,a)\,\mathrm{d}a\end{split}$
(2.3c) for $t\geq 0$ (we set $b(\bar{v},a):=0$ whenever $a\notin J$). That is,
$u(t,0)=B_{u}(t)$ for $t\geq 0$ by (2.3a), while (2.3c) implies
$\begin{split}B_{u}(t)\,&=\,\int_{0}^{a_{m}}b(\bar{u}(t),a)\,u(t,a)\,\mathrm{d}a\,,\quad
t\geq 0\,.\end{split}$

The following result was established in [28] (see also [21]):

######  Proposition 2.1.

Suppose (2.1) and (2.2). For every $u_{0}\in\mathbb{E}_{\alpha}$ there exists
a unique maximal solution $u=u(\cdot;u_{0})\in
C\big{(}I(u_{0}),\mathbb{E}_{\alpha}\big{)}$ to problem (1.2) on some maximal
interval of existence $I(u_{0})=[0,T_{max}(u_{0}))$; that is, $u(t;u_{0})$
satisfies (2.3) for $t\in I(u_{0})$. If

$\sup_{t\in I(u_{0})\cap[0,T]}\|u(t;u_{0})\|_{\mathbb{E}_{\alpha}}<\infty$

for every $T>0$, then the solution exists globally, i.e.,
$I(u_{0})=\mathbb{R}^{+}$. Finally, if $u_{0}\in\mathbb{E}_{\alpha}^{+}$, then
$u(t;u_{0})\in\mathbb{E}_{\alpha}^{+}$ for $t\in I(u_{0})$.

###### Proof.

This is [28, Proposition 2.1]. ∎

Note that assumptions (2.1) and (2.2) are not really restrictive and satisfied
for (sufficiently) smooth functions $m,b,\varrho$ and diffusion operators as
in the introduction, see Section 5.

### Linearized Stability and Instability

In the following, an equilibrium (i.e. a time-independent solution) $\phi\in
C(J,E_{\alpha})$ to (1.2) is a mild solution (see (B.3b)) to

$\displaystyle\partial_{a}\phi\,$
$\displaystyle=A(a)\phi-m(\bar{\phi},a)\phi\,,\quad
a\in(0,a_{m})\,,\qquad\phi(0)=\int_{0}^{a_{m}}b(\bar{\phi},a)\,\phi(a)\,\mathrm{d}a\,.$
(2.4)

Clearly, $\phi\equiv 0$ is always an equilibrium. As pointed out above, fairly
general conditions sufficient for the existence of at least one positive non-
trivial equilibrium $\phi\in\mathbb{E}_{1}\cap C(J,E_{\alpha})$ were presented
in earlier works [20, 22, 24].

An equilibrium $\phi\in C(J,E_{\alpha})$ to (1.2) is said to be stable in
$\mathbb{E}_{\alpha}$ provided that for every $\varepsilon>0$ there exists
$\delta>0$ such that, if
$u_{0}\in\mathbb{B}_{\mathbb{E}_{\alpha}}(\phi,\delta)$, then
$T_{max}(u_{0})=\infty$ and
$u(t;u_{0})\in\mathbb{B}_{\mathbb{E}_{\alpha}}(\phi,\varepsilon)$ for every
$t\geq 0$, where $u(\cdot,u_{0})$ denotes the maximal solution to (1.2) from
Proposition 2.1. The equilibrium $\phi\in C(J,E_{\alpha})$ is asymptotically
exponentially stable in $\mathbb{E}_{\alpha}$, if it is stable and there are
$r>0$ and $M>0$ such that

$\|u(t;u_{0})-\phi\|_{\mathbb{E}_{\alpha}}\leq
Me^{-rt}\|u_{0}-\phi\|_{\mathbb{E}_{\alpha}}\,,\quad t\geq 0\,,$

for $u_{0}\in\mathbb{B}_{\mathbb{E}_{\alpha}}(\phi,\delta)$. Finally, an
equilibrium $\phi\in C(J,E_{\alpha})$ is unstable in $\mathbb{E}_{\alpha}$, if
it is not stable.

#### Assumptions

Let $\phi\in\mathbb{E}_{1}\cap C(J,E_{\alpha})$ be a fixed equilibrium to
(1.2). We assume that the death and the birth rate are continuously Fréchet
differentiable at $\bar{\phi}$. More precisely, for $\alpha\in[0,1)$ still
fixed, we assume that

$\displaystyle E_{\alpha}\rightarrow
L_{\infty}\big{(}J,\mathcal{L}(E_{\alpha},E_{0})\big{)}\,,\ z\mapsto
m(z,\cdot)\text{ is differentiable at $\bar{\phi}\in E_{1}$}\,,$ (2.5a)
$\displaystyle E_{\alpha}\rightarrow
L_{\infty}\big{(}J,\mathcal{L}(E_{\alpha},E_{0})\big{)}\,,\ z\mapsto
b(z,\cdot)\text{ is differentiable at $\bar{\phi}\in E_{1}$}\,,$ (2.5b) such
that for $v\in\mathbb{E}_{0}$ we can write (with $\partial$ indicating Fréchet
derivatives with respect to $\bar{\phi}$)
$m(\bar{v},\cdot)v-m(\bar{\phi},\cdot)\phi=m(\bar{\phi},\cdot)(v-\phi)+\partial
m(\bar{\phi},\cdot)[\bar{v}-\bar{\phi}]\phi+R_{m}(v-\phi)$ (2.5c) and
$b(\bar{v},\cdot)v-b(\bar{\phi},\cdot)\phi=b(\bar{\phi},\cdot)(v-\phi)+\partial
b(\bar{\phi},\cdot)[\bar{v}-\bar{\phi}]\phi+R_{b}(v-\phi)\,,$ (2.5d) where for
the reminder terms $R_{m}:\mathbb{E}_{\alpha}\to\mathbb{E}_{0}$ and
$R_{b}:\mathbb{E}_{\alpha}\to\mathbb{E}_{0}$ there exists an increasing
function $d_{o}\in C(\mathbb{R}^{+},\mathbb{R}^{+})$ with $d_{o}(0)=0$ such
that $d_{o}(r)>0$ for each $r>0$ with
$\|R_{m}(v)\|_{\mathbb{E}_{0}}+\|R_{b}(v)\|_{\mathbb{E}_{0}}\leq
d_{o}(r)\,\|v\|_{\mathbb{E}_{\alpha}}\,,\quad\|v\|_{\mathbb{E}_{\alpha}}\leq
r\,,$ (2.5e) and
$\|R_{m}(v_{1})-R_{m}(v_{2})\|_{\mathbb{E}_{0}}+\|R_{b}(v_{1})-R_{b}(v_{2})\|_{\mathbb{E}_{0}}\leq
d_{o}(r)\,\|v_{1}-v_{2}\|_{\mathbb{E}_{\alpha}}\,,\quad\|v_{1}\|_{\mathbb{E}_{\alpha}},\|v_{2}\|_{\mathbb{E}_{\alpha}}\leq
r\,.$ (2.5f) For technical reasons, we assume for the birth rate that (for
some $\vartheta\in(0,1)$, see (2.2c)) $\begin{split}b(\bar{\phi},\cdot)\in
C\big{(}J,\mathcal{L}(E_{0})\big{)}\cap
L_{\infty}\big{(}J,\mathcal{L}(E_{\theta})\big{)}\,,\quad\theta\in\\{0,\alpha,\vartheta\\}\,,\end{split}$
(2.5g) and $\big{[}z\mapsto\partial
b(\bar{\phi},\cdot)[z]\phi\big{]}\in\mathcal{L}\big{(}E_{\theta},\mathbb{E}_{\theta}\big{)}\,,\quad\theta\in\\{0,\alpha,\vartheta\\}\,,$
(2.5h) while for the death rate we impose that $m(\bar{\phi},\cdot)\in
C^{\rho}\big{(}J,\mathcal{L}(E_{\beta},E_{0})\big{)}$ (2.5i) for some
$\beta\in[0,1)$ and $\big{[}a\mapsto\partial
m(\bar{\phi},a)[\cdot]\phi(a)\big{]}\in
C\big{(}J,\mathcal{L}(E_{0})\big{)}\,.$ (2.5j)

The fact that we can handle nonlinearities $m$ and $b$ being defined on
interpolation spaces $E_{\alpha}$ guarantees great flexibility in concrete
applications. Indeed, the assumptions imposed above are rather easily checked
in problems such as (1.1) since they are mainly assumptions on the regularity
of the data (see Section 5 for details).

In [28] it was shown that the stability of an equilibrium $\phi$ can be
deduced from the (formal) linearization of (1.2) at $\phi$ given by

$\displaystyle\partial_{t}v+\partial_{a}v\,$
$\displaystyle=A(a)v-m\big{(}\bar{\phi},a\big{)}v-\partial
m\big{(}\bar{\phi},a\big{)}[\bar{v}(t)]\phi(a)\,,\qquad t>0\,,\quad
a\in(0,a_{m})\,,$ (2.6a) $\displaystyle v(t,0)$
$\displaystyle=\int_{0}^{a_{m}}b\big{(}\bar{\phi},a\big{)}\,v(t,a)\,\mathrm{d}a+\int_{0}^{a_{m}}\partial
b\big{(}\bar{\phi},a\big{)}[\bar{v}(t)]\,\phi(a)\,\mathrm{d}a\,,\qquad t>0\,,$
(2.6b) $\displaystyle v(0,a)$ $\displaystyle=v_{0}(a)\,,\qquad
a\in(0,a_{m})\,,$ (2.6c)

with $\partial$ indicating Fréchet derivatives with respect to $\bar{\phi}$.
More precisely, according to [28], an equilibrium $\phi$ is locally
asymptotically stable if the semigroup associated with the linearization (2.6)
has a negative growth bound. The characterization of the latter, however, was
left open. The aim of the present research now is to refine and improve this
stability result and complement it with an instability result. Concretely, we
prove that the real parts of the eigenvalues of the generator of the semigroup
associated with the linearization (2.6) determine stability or instability of
the equilibrium. Thus, we establish the classical principle of linearized
stability for (1.2).

In the following, an eigenvalue of the generator associated with (2.6) means a
number $\lambda\in\mathbb{C}$ for which there is a nontrivial mild solution
$w\in C(J,E_{0})$, $w\not\equiv 0$, to

$\displaystyle\lambda w+\partial_{a}w\,$
$\displaystyle=A(a)w-m(\bar{\phi},a)w-\partial
m(\bar{\phi},a)[\bar{w}]\phi(a)\,,\qquad a\in(0,a_{m})\,,$ (2.7a)
$\displaystyle w(0)$
$\displaystyle=\int_{0}^{a_{m}}b(\bar{\phi},a)\,w(a)\,\mathrm{d}a+\int_{0}^{a_{m}}\partial
b(\bar{\phi},a)[\bar{w}]\,\phi(a)\,\mathrm{d}a\,.$ (2.7b)

Here is the main result:

######  Theorem 2.2.

Let $\alpha\in[0,1)$. Assume (2.1), (2.2), and (2.5), where
$\phi\in\mathbb{E}_{1}\cap C(J,E_{\alpha})$ is an equilibrium to (1.2). The
following hold:

  * (a)

If $\mathrm{Re}\,\lambda<0$ for any eigenvalue $\lambda$ to (2.7), then $\phi$
is exponentially asymptotically stable in $\mathbb{E}_{\alpha}$.

  * (b)

If $\mathrm{Re}\,\lambda>0$ for some eigenvalue $\lambda$ to (2.7), then
$\phi$ is unstable in $\mathbb{E}_{\alpha}$.

Theorem 2.2 follows from Theorem 4.1 below and is an extension of [16, Theorem
2], [29, Theorem 4.13] to the case with spatial diffusion. We point out again
that the ability to work in the spaces $\mathbb{E}_{\alpha}$ (instead of only
in $\mathbb{E}_{0}$) allows us to treat general nonlinearities in the vital
rates $m$ and $b$, see Section 5.

Statement (a) of Theorem 2.2 refines the stability result of [28, Theorem
2.2]. Indeed, one of our main achievements herein is the characterization of
the growth bound of the semigroup associated with (2.6) in terms of the
spectral bound of the semigroup generator. Moreover, with the instability
statement (b) we complete [28, Theorem 2.2] and so provide a concise
characterization of stability or instability of an equilibrium to (1.2) via
the linear eigenvalue problem (2.7). Actually, we shall later see in Section 4
that the latter can be rephrased in a somewhat more accessible way for
applications.

The crucial key for the proof of Theorem 2.2 is to show that the semigroup on
$\mathbb{E}_{0}$ associated with the linear problem (2.6) (as well as its
restriction to $\mathbb{E}_{\alpha}$) is eventually compact. As a consequence
the growth bound of the semigroup and the spectral bound of the corresponding
generator coincide and further fundamental spectral properties of the
generator can be derived. We emphasize that, even though the semigroup
associated with (2.6) when $m\equiv 0$ is known to have this eventual
compactness property (see [26]), it is by no means obvious that this property
is inherited when including a nontrivial death rate $m$. This is due to the
fact that, on the one hand, a perturbation of the generator of an eventually
compact semigroup does not in general generate again an eventually compact
semigroup, and, on the other hand, that the perturbation $\partial
m(\bar{\phi},a)[\bar{w}]\phi(a)$ appearing in (2.6a) constitutes a nonlocal
perturbation with respect to $w$. In order to establish the fundamental
compactness property nonetheless, we make use of the particular form of this
perturbation, see (4.1d).

### Paper Outline

In Section 3 we first focus on the linear problem (2.6) and prove the eventual
compactness of the corresponding semigroup on $\mathbb{E}_{0}$ and of its
restriction to $\mathbb{E}_{\alpha}$. This and the implied spectral properties
of the generator are summarized in Theorem 3.3 and Corollary 3.5.

In Section 4 we use the crucial fact derived in [28] that the difference
$u(\cdot;u_{0})-\phi$ can be represented in terms of the linearization
semigroup associated with (2.6), see (4.13). Since the growth bound of the
linearization semigroup is determined by the spectral bound of the semigroup
generator as shown previously in Section 3, this yields statement (a) of
Theorem 2.2. Moreover, the construction of backwards solutions to problem
(1.2) in Lemma 4.5 under the assumptions of statement (b) of Theorem 2.2 then
leads to the instability result. This part of the proof is inspired by [29,
Theorem 4.13]. In Proposition 4.7 we give an alternative formulation of the
eigenvalue problem (2.7).

In Section 5 we consider concrete examples and prove, in particular,
Proposition 1.1 on the stability analysis of the trivial equilibrium to
problem (1.1). Moreover, we provide an instability result for a nontrivial
equilibrium, see Proposition 5.4.

Finally, two appendices are included. In Appendix A we provide the technical
and thus postponed proof of Proposition 3.4. In Appendix B we briefly
summarize the main properties of parabolic evolution operators which play an
important role in our analysis and are used throughout.

##  3\. The Linear Problem

In order to prepare the proof of Theorem 2.2 we focus our attention first on
the linear problem

$\displaystyle\partial_{t}u+\partial_{a}u\,$
$\displaystyle=A_{\ell}(a)u\,,\qquad t>0\,,\quad a\in(0,a_{m})\,,$ (3.1a)
$\displaystyle u(t,0)$
$\displaystyle=\int_{0}^{a_{m}}b_{\ell}(a)\,u(t,a)\,\mathrm{d}a\,,\qquad
t>0\,,$ (3.1b) $\displaystyle u(0,a)$ $\displaystyle=\psi(a)\,,\qquad
a\in(0,a_{m})\,,$ (3.1c)

where $A_{\ell}$ satisfies

$A_{\ell}\in C^{\rho}\big{(}J,\mathcal{H}(E_{1},E_{0})\big{)}$ (3.2)

for some $\rho>0$ and where we impose for the birth rate that there is
$\vartheta\in(0,1)$ with

$b_{\ell}\in
L_{\infty}\big{(}J,\mathcal{L}(E_{\theta})\big{)}\,,\quad\theta\in\\{0,\vartheta\\}\,.$
(3.3)

We then denote by

$\big{\\{}\Pi_{\ell}(a,\sigma)\in\mathcal{L}(E_{0})\,;\,a\in
J\,,\,0\leq\sigma\leq a\big{\\}}$

the parabolic evolution operator on $E_{0}$ with regularity subspace $E_{1}$
generated by $A_{\ell}$ (see Appendix B) and use the notation

$\|b_{\ell}\|_{\theta}:=\|b_{\ell}\|_{L_{\infty}(J,\mathcal{L}(E_{\theta}))}\,.$

###  3.1. Preliminaries

We first recall some of the results from [26]. Formal integrating of (3.1a)
along characteristics entails that the solution

$[\mathbb{S}(t)\psi](a):=u(t,a)\,,\qquad t\geq 0\,,\quad a\in J\,,$

to (3.1) for given $\psi\in\mathbb{E}_{0}=L_{1}(J,E_{0})$ is of the form

$\big{[}\mathbb{S}(t)\psi\big{]}(a)\,:=\,\left\\{\begin{aligned}
&\Pi_{\ell}(a,a-t)\,\psi(a-t)\,,&&a\in J\,,\ 0\leq t\leq a\,,\\\
&\Pi_{\ell}(a,0)\,\mathsf{B}_{\psi}(t-a)\,,&&a\in J\,,\
t>a\,,\end{aligned}\right.$ (3.4a) with $\mathsf{B}_{\psi}:=u(\cdot,0)$
satisfying the linear Volterra equation
$\begin{split}\mathsf{B}_{\psi}(t)\,=\,&\int_{0}^{t}\chi(a)\,b_{\ell}(a)\,\Pi_{\ell}(a,0)\,\mathsf{B}_{\psi}(t-a)\,\mathrm{d}a\,\\\
&+\,\int_{0}^{a_{m}-t}\chi(a+t)\,b_{\ell}(a+t)\,\Pi_{\ell}(a+t,a)\,\psi(a)\,\mathrm{d}a\,,\quad
t\geq 0\,,\end{split}$ (3.4b)

where $\chi$ is the characteristic function of the interval $(0,a_{m})$. Note
that $\mathsf{B}_{\psi}$ is such that

$\mathsf{B}_{\psi}(t)=\int_{0}^{a_{m}}b_{\ell}(a)\big{[}\mathbb{S}(t)\psi\big{]}(a)\,\mathrm{d}a\,,\quad
t\geq 0\,.$ (3.5)

It follows from [26] that there is a unique solution

$[\psi\mapsto\mathsf{B}_{\psi}]\in\mathcal{L}\big{(}\mathbb{E}_{0},C(\mathbb{R}^{+},E_{0})\big{)}$
(3.6)

to (3.4b) and that $(\mathbb{S}(t))_{t\geq 0}$ defines a strongly continuous
semigroup on $\mathbb{E}_{0}$ enjoying the property of eventual compactness
and exhibiting regularizing effects induced by the parabolic evolution
operator $\Pi_{\ell}$. Moreover, its generator can be characterized fully. We
summarize those properties which will be important for our purpose herein:

######  Theorem 3.1.

Suppose (3.2) and (3.3).

(a) $(\mathbb{S}(t))_{t\geq 0}$ defined in (3.4) is a strongly continuous,
eventually compact semigroup on the space $\mathbb{E}_{0}=L_{1}(J,E_{0})$.

If $A_{\ell}(a)$ is resolvent positive for every $a\in J$ and if $b_{\ell}\in
L_{\infty}\big{(}J,\mathcal{L}_{+}(E_{0})\big{)}$, then the semigroup
$(\mathbb{S}(t))_{t\geq 0}$ is positive.

(b) In fact, given $\alpha\in[0,1)$, the restriction
$(\mathbb{S}(t)|_{\mathbb{E}_{\alpha}})_{t\geq 0}$ defines a strongly
continuous semigroup on $\mathbb{E}_{\alpha}$, and there are $M_{\alpha}\geq
1$ and $\varkappa_{\alpha}\in\mathbb{R}$ such that

$\|\mathbb{S}(t)\|_{\mathcal{L}(\mathbb{E}_{\alpha})}+t^{\alpha}\|\mathbb{S}(t)\|_{\mathcal{L}(\mathbb{E}_{0},\mathbb{E}_{\alpha})}\leq
M_{\alpha}\,e^{\varkappa_{\alpha}t}\,,\quad t\geq 0\,.$ (3.7)

(c) Denote by $\mathbb{A}$ the infinitesimal generator of the semigroup
$(\mathbb{S}(t))_{t\geq 0}$ on $\mathbb{E}_{0}$. Then
$\psi\in\mathrm{dom}(\mathbb{A})$ if and only if there exists
$\zeta\in\mathbb{E}_{0}$ such that $\psi\in C(J,E_{0})$ is the mild solution
to

$\partial_{a}\psi=A_{\ell}(a)\psi-\zeta(a)\,,\quad a\in
J\,,\qquad\psi(0)=\int_{0}^{a_{m}}b_{\ell}(a)\psi(a)\,\mathrm{d}a\,.$ (3.8)

In this case, $\mathbb{A}\psi=\zeta$.

Finally, the embedding $D(\mathbb{A})\hookrightarrow\mathbb{E}_{\alpha}$ is
continuous and dense for $\alpha\in[0,1)$.

###### Proof.

This follows from [26, Theorem 1.2, Corollary 1.3, Theorem 1.4]222The
positivity assumption [26, Assumption (1.5)] is in fact not needed in order to
prove [26, Theorem 1.4], since for $\mathrm{Re}\,\lambda>0$ large,
$(1-Q_{\lambda})^{-1}\in\mathcal{L}(E_{0})$ is ensured by the fact that
$\|Q_{\lambda}\|_{\mathcal{L}(E_{0})}<1$.. ∎

That $\psi\in C(J,E_{0})\subset\mathbb{E}_{0}$ is the mild solution to (3.8)
with $\zeta\in\mathbb{E}_{0}$ means that

$\psi(a)=\Pi_{\ell}(a,0)\psi(0)-\int_{0}^{a}\Pi_{\ell}(a,\sigma)\,\zeta(\sigma)\,\mathrm{d}\sigma\,,\quad
a\in J\,.$

The spectrum of the generator $\mathbb{A}$ has been investigated in [26] when
the generated semigroup $(\mathbb{S}(t))_{t\geq 0}$ is positive. In the
following, let

$s(\mathbb{A}):=\sup\left\\{\mathrm{Re}\,\lambda\,;\,\lambda\in\sigma(\mathbb{A})\right\\}$

be the spectral bound of the generator $\mathbb{A}$ and

$\omega_{0}(\mathbb{A}):=\inf\left\\{\omega\in\mathbb{R}\,;\,\sup_{t>0}\big{(}e^{-\omega
t}\|e^{t\mathbb{A}}\|_{\mathcal{L}(\mathbb{E}_{0})}\big{)}<\infty\right\\}$

be the growth bound of the corresponding semigroup
$\mathbb{S}(t)=e^{t\mathbb{A}}$, $t\geq 0$.

######  Proposition 3.2.

Suppose (3.2) and (3.3). Let $A_{\ell}(a)$ be resolvent positive for every
$a\in J$ and $b_{\ell}\in L_{\infty}\big{(}J,\mathcal{L}_{+}(E_{0})\big{)}$
such that $b_{\ell}(a)\Pi_{\ell}(a,0)\in\mathcal{L}_{+}(E_{0})$ is strongly
positive333Recall that if $E$ is an ordered Banach space, then
$T\in\mathcal{L}(E)$ is strongly positive if $Tz\in E$ is a quasi-interior
point for each $z\in E^{+}\setminus\\{0\\}$, that is, if $\langle
z^{\prime},Tz\rangle_{E}>0$ for every
$z^{\prime}\in(E^{\prime})^{+}\setminus\\{0\\}$. for $a$ in a subset of $J$ of
positive measure. Then

$s(\mathbb{A})=\omega_{0}(\mathbb{A})=\lambda_{0}\,,$

where $\lambda_{0}\in\mathbb{R}$ is uniquely determined from the condition
$r(Q_{\lambda_{0}})=1$ with $r(Q_{\lambda})$ denoting for
$\lambda\in\mathbb{R}$ the spectral radius of the strongly positive compact
operator $Q_{\lambda}\in\mathcal{L}(E_{0})$ given by

$Q_{\lambda}:=\int_{0}^{a_{m}}e^{-\lambda
a}\,b_{\ell}(a)\,\Pi_{\ell}(a,0)\,\mathrm{d}a\,.$

###### Proof.

This is [26, Corollary 4.3]. ∎

###  3.2. Eventual Compactness of the Perturbed Semigroup

If $\mathbb{B}\in\mathcal{L}(\mathbb{E}_{0})$, then
$\mathbb{G}:=\mathbb{A}+\mathbb{B}$ generates also a strongly continuous
semigroup on $\mathbb{E}_{0}$, which, however, is not necessarily eventually
compact even though the semigroup generated by $\mathbb{A}$ is. Nonetheless,
we next shall prove that for particular (nonlocal) perturbations of the form

$\displaystyle[\mathbb{B}\psi](a):=\int_{0}^{a_{m}}q(a,\sigma)\,\psi(\sigma)\,\mathrm{d}\sigma\,,\quad
a\in J\,,\quad\psi\in\mathbb{E}_{0}\,,$ (3.9a) with $q(a,\sigma)=q(a)(\sigma)$
satisfying $\displaystyle q\in
C\big{(}J,L_{\infty}(J,\mathcal{L}(E_{0}))\big{)}\,,$ (3.9b) the semigroup
generated by $\mathbb{G}=\mathbb{A}+\mathbb{B}$ is eventually compact. This
yields more information on the spectrum of $\mathbb{G}$ and implies, in
particular, that the spectral bound $s(\mathbb{G})$ and the growth bound
$\omega_{0}(\mathbb{G})$ coincide. This we shall apply later on to the
linearization of problem (1.2) in which the perturbation operator $\mathbb{B}$
has the particular form (3.9). Note that
$\mathbb{B}\in\mathcal{L}(\mathbb{E}_{0})$ with
$\|\mathbb{B}\|_{\mathcal{L}(\mathbb{E}_{0})}\leq
a_{m}\,\|q\|_{\infty}\,,\qquad\|q\|_{\infty}:=\|q\|_{C(J,L_{\infty}(J,\mathcal{L}(E_{0})))}\,.$
For the birth rate we impose additionally that $b_{\ell}\in
C\big{(}J,\mathcal{L}(E_{0})\big{)}\,.$ (3.9c)

We then shall prove the following theorem which is fundamental for our
purpose:

######  Theorem 3.3.

Suppose (3.2), (3.3), (3.9), and let $\mathbb{A}$ be the generator of the
semigroup $(\mathbb{S}(t))_{t\geq 0}$ defined in (3.4).Then the semigroup
$(\mathbb{T}(t))_{t\geq 0}$ on $\mathbb{E}_{0}$ generated by
$\mathbb{G}:=\mathbb{A}+\mathbb{B}$ is eventually compact. In particular,

$\displaystyle s(\mathbb{G})=\omega_{0}(\mathbb{G})\,$ (3.10)

and the spectrum $\sigma(\mathbb{G})=\sigma_{p}(\mathbb{G})$ is countable and
consists of poles of the resolvent $R(\cdot,\mathbb{G})$ of finite algebraic
multiplicities (in particular, $\sigma(\mathbb{G})$ is a pure point spectrum).
Moreover, for each $r\in\mathbb{R}$, the set
$\\{\lambda\in\sigma(\mathbb{G})\,;\,\mathrm{Re}\,\lambda\geq r\\}$ is finite.

Finally, if $A_{\ell}(a)$ is resolvent positive for every $a\in J$, if
$b_{\ell}\in L_{\infty}\big{(}J,\mathcal{L}_{+}(E_{0})\big{)}$, and if
$\mathbb{B}\in\mathcal{L}_{+}(\mathbb{E}_{0})$, then the semigroup
$(\mathbb{T}(t))_{t\geq 0}$ is positive, and if $s(\mathbb{G})>-\infty$, then
$s(\mathbb{G})$ is an eigenvalue of $\mathbb{G}$.

Theorem 3.3 relies on the following crucial observation:

######  Proposition 3.4.

Suppose (3.2), (3.3), and (3.9). Set

$\displaystyle\mathcal{V}\mathbb{S}(t)\psi:=\int_{0}^{t}\mathbb{S}(t-s)\,\mathbb{B}\,\mathbb{S}(s)\psi\,\mathrm{d}s\,,\quad
t>0\,,\quad\psi\in\mathbb{E}_{0}\,.$

Then $\mathcal{V}\mathbb{S}:(0,\infty)\to\mathcal{K}(\mathbb{E}_{0})$, i.e.
$\mathcal{V}\mathbb{S}(t)$ is a bounded compact operator on $\mathbb{E}_{0}$
for every $t\in(0,\infty)$.

The proof of Proposition 3.4 uses the explicit form of the semigroup
$(\mathbb{S}(t))_{t\geq 0}$ in (3.4) and the particular form of the
perturbation $\mathbb{B}$ in (3.9a), but is rather technical and thus
postponed to Appendix A.

### Proof of Theorem 3.3

Recall that $\mathbb{S}(t)=e^{t\mathbb{A}}$ and
$\mathbb{T}(t)=e^{t(\mathbb{A}+\mathbb{B})}$. Theorem 3.1 ensures that the
semigroup $(\mathbb{S}(t))_{t\geq 0}$ is eventually compact. Therefore, since
$\mathbb{B}\in\mathcal{L}(\mathbb{E}_{0})$ and since $\mathcal{V}\mathbb{S}$
is compact on $(0,\infty)$ according to Proposition 3.4, we infer from [12,
III.Theorem 1.16 (ii)] (with $k=1$) that also $(\mathbb{T}(t))_{t\geq 0}$ is
eventually compact. This implies (3.10) due to [12, IV.Corollary 3.11] while
the remaining statements regarding the spectrum of
$\mathbb{G}=\mathbb{A}+\mathbb{B}$ now follow from [12, V.Corollary 3.2].

Finally, if $A_{\ell}(a)$ is resolvent positive for every $a\in J$ and
$b_{\ell}\in L_{\infty}\big{(}J,\mathcal{L}_{+}(E_{0})\big{)}$, then Theorem
3.1 entails that $\mathbb{S}(t)=e^{t\mathbb{A}}$ is positive. Since
$\mathbb{B}\in\mathcal{L}_{+}(\mathbb{E}_{0})$, it is well-known that the
semigroup $\mathbb{T}(t)=e^{t(\mathbb{A}+\mathbb{B})}$ is positive as well,
e.g. see [4, Proposition 12.11]. Since $\mathbb{E}_{0}$ is a Banach lattice,
this implies that $s(\mathbb{G})$ is an eigenvalue of $\mathbb{G}$ if
$s(\mathbb{G})>-\infty$ according to [4, Corollary 12.9]. This yields Theorem
3.3.∎

We can derive now also properties of the semigroup $(\mathbb{T}(t))_{t\geq 0}$
restricted to $\mathbb{E}_{\alpha}$.

######  Corollary 3.5.

Suppose (3.2), (3.3), (3.9), and let $\mathbb{A}$ be the generator of the
semigroup $(\mathbb{S}(t))_{t\geq 0}$ defined in (3.4). Let
$(\mathbb{T}(t))_{t\geq 0}$ be the strongly continuous semigroup on
$\mathbb{E}_{0}$ generated by $\mathbb{G}=\mathbb{A}+\mathbb{B}$. For every
$\alpha\in[0,1)$, the restriction
$\left(\mathbb{T}(t)|_{\mathbb{E}_{\alpha}}\right)_{t\geq 0}$ is a strongly
continuous, eventually compact semigroup on $\mathbb{E}_{\alpha}$. Its
generator $\mathbb{G}_{\alpha}$ is the $\mathbb{E}_{\alpha}$-realization of
$\mathbb{G}$. For the corresponding (point) spectra it holds that
$\sigma(\mathbb{G})=\sigma(\mathbb{G}_{\alpha})$. Moreover, for every
$\omega>s(\mathbb{G})$ there is $N_{\alpha}\geq 1$ such that

$\displaystyle\|\mathbb{T}(t)\|_{\mathcal{L}(\mathbb{E}_{0})}+\|\mathbb{T}(t)\|_{\mathcal{L}(\mathbb{E}_{\alpha})}+t^{\alpha}\|\mathbb{T}(t)\|_{\mathcal{L}(\mathbb{E}_{0},\mathbb{E}_{\alpha})}\leq
N_{\alpha}e^{\omega t}\,,\quad t\geq 0\,.$ (3.11)

###### Proof.

(i) It was shown in [26, Theorem 1.2] that the semigroup
$(\mathbb{T}(t))_{t\geq 0}$ on $\mathbb{E}_{0}$ generated by
$\mathbb{G}=\mathbb{A}+\mathbb{B}$ is given by

$\mathbb{T}(t)\phi=\mathbb{S}(t)\phi+\int_{0}^{t}\mathbb{S}(t-s)\,\mathbb{B}\,\mathbb{T}(s)\,\phi\,\mathrm{d}s\,,\quad
t\geq 0\,,\quad\phi\in\mathbb{E}_{0}\,,$ (3.12)

and that there are $C_{\alpha}\geq 1$ and $\varsigma_{\alpha}\geq 0$ such that

$\|\mathbb{T}(t)\|_{\mathcal{L}(\mathbb{E}_{0},\mathbb{E}_{\alpha})}\leq
C_{\alpha}\,e^{\varsigma_{\alpha}t}t^{-\alpha}\,,\quad t>0\,.$ (3.13)

We then infer from (3.7) and (3.12) that

$\|\mathbb{T}(t)\phi\|_{\mathbb{E}_{\alpha}}\leq
M_{\alpha}e^{\varkappa_{\alpha}t}\|\phi\|_{\mathbb{E}_{\alpha}}+M_{\alpha}\|\mathbb{B}\|_{\mathcal{L}(\mathbb{E}_{\alpha},\mathbb{E}_{0})}\int_{0}^{t}e^{\varkappa_{\alpha}(t-s)}\,(t-s)^{-\alpha}\,\|\mathbb{T}(s)\phi\|_{\mathbb{E}_{\alpha}}\,\mathrm{d}s\,,\quad
t\geq 0\,,$

for $\phi\in\mathbb{E}_{\alpha}$ so that Gronwall’s inequality implies

$\|\mathbb{T}(t)\|_{\mathcal{L}(\mathbb{E}_{\alpha})}\leq
c_{\alpha}e^{\omega_{1}t}\,,\quad t\geq 0\,,$ (3.14)

for some $c_{\alpha}\geq 1$ and $\omega_{1}>0$. Moreover, since

$\|\mathbb{T}(t)\phi-\phi\|_{\mathbb{E}_{\alpha}}\leq\|\mathbb{S}(t)\phi-\phi\|_{\mathbb{E}_{\alpha}}+\int_{0}^{t}\|\mathbb{S}(t-s)\|_{\mathcal{L}(\mathbb{E}_{0},\mathbb{E}_{\alpha})}\,\|\mathbb{B}\|_{\mathcal{L}(\mathbb{E}_{\alpha},\mathbb{E}_{0})}\,\|\mathbb{T}(s)\phi\|_{\mathbb{E}_{\alpha}}\,\mathrm{d}s$

for $t\geq 0$ and $\phi\in\mathbb{E}_{\alpha}$, the strong continuity of
$\left(\mathbb{T}(t)|_{\mathbb{E}_{\alpha}}\right)_{t\geq 0}$ on
$\mathbb{E}_{\alpha}$ follows from the strong continuity of
$(\mathbb{S}(t)|_{\mathbb{E}_{\alpha}})_{t\geq 0}$ on $\mathbb{E}_{\alpha}$
guaranteed by Theorem 3.1 and from (3.7) and (3.14). Therefore,
$\left(\mathbb{T}(t)|_{\mathbb{E}_{\alpha}}\right)_{t\geq 0}$ is a strongly
continuous semigroup on $\mathbb{E}_{\alpha}$. Writing

$\mathbb{T}(t)|_{\mathbb{E}_{\alpha}}=\mathbb{T}(t-t_{0})\mathbb{T}(t_{0})|_{\mathbb{E}_{\alpha}}$

and noticing that $\mathbb{T}(t_{0})\in\mathcal{L}(\mathbb{E}_{0})$ is compact
for $t_{0}$ large due to Theorem 3.3 while
$\mathbb{T}(t-t_{0})\in\mathcal{L}(\mathbb{E}_{0},\mathbb{E}_{\alpha})$ by
(3.13) for $t>t_{0}$, we deduce that
$\left(\mathbb{T}(t)|_{\mathbb{E}_{\alpha}}\right)_{t\geq 0}$ is eventually
compact on $\mathbb{E}_{\alpha}$.

(ii) Denote by $\mathbb{G}_{\alpha}$ the generator of the restricted semigroup
to $\mathbb{E}_{\alpha}$ so that
$\mathbb{T}(t)|_{\mathbb{E}_{\alpha}}=e^{t\mathbb{G}_{\alpha}}$ for $t\geq 0$.
We prove that $\mathbb{G}_{\alpha}$ is the $\mathbb{E}_{\alpha}$-realization
of $\mathbb{G}$. To this end, denote the latter by
$\mathbb{G}_{\mathbb{E}_{\alpha}}$ and let
$\zeta\in\mathrm{dom}(\mathbb{G}_{\alpha})$. Then

$\frac{1}{t}\big{(}\mathbb{T}(t)\zeta-\zeta\big{)}=\frac{1}{t}\left(e^{t\mathbb{G}_{\alpha}}\zeta-\zeta\right)\rightarrow\mathbb{G}_{\alpha}\zeta\
\ \text{ in }\ \mathbb{E}_{\alpha}\hookrightarrow\mathbb{E}_{0}\ \text{ as }\
t\to 0\,,$

hence $\zeta\in\mathrm{dom}(\mathbb{G})$ and

$\displaystyle\mathbb{G}\zeta=\mathbb{G}_{\alpha}\zeta\in\mathbb{E}_{\alpha}\,.$
(3.15)

Consequently, $\zeta\in\mathrm{dom}(\mathbb{G}_{\mathbb{E}_{\alpha}})$ and
$\mathbb{G}_{\mathbb{E}_{\alpha}}\zeta=\mathbb{G}_{\alpha}\zeta$.

Conversely, let $\psi\in\mathrm{dom}(\mathbb{G}_{\mathbb{E}_{\alpha}})$ and
$\lambda\in\rho(\mathbb{G})\cap\rho(\mathbb{G}_{\alpha})$ (this is possible
since both $\mathbb{G}$ and $\mathbb{G}_{\alpha}$ are semigroup generators).
Then

$\psi\in\mathrm{dom}(\mathbb{G})=\mathrm{dom}(\mathbb{A})\subset\mathbb{E}_{\alpha}$

by Theorem 3.1 and $\mathbb{G}\psi\in\mathbb{E}_{\alpha}$. Thus, since
$(\lambda-\mathbb{G})\psi\in\mathbb{E}_{\alpha}$ and since
$\lambda\in\rho(\mathbb{G}_{\alpha})$, there is a unique
$\zeta\in\mathrm{dom}(\mathbb{G}_{\alpha})$ such that
$(\lambda-\mathbb{G}_{\alpha})\zeta=(\lambda-\mathbb{G})\psi$. Since
$(\lambda-\mathbb{G}_{\alpha})\zeta=(\lambda-\mathbb{G})\zeta$ by (3.15) and
since $\lambda\in\rho(\mathbb{G})$, we conclude
$\psi=\zeta\in\mathrm{dom}(\mathbb{G}_{\alpha})$. Therefore,
$\mathbb{G}_{\alpha}=\mathbb{G}_{\mathbb{E}_{\alpha}}$; that is,
$\mathbb{G}_{\alpha}$ coincides with the $\mathbb{E}_{\alpha}$-realization of
$\mathbb{G}$.

(iii) We claim that $\sigma(\mathbb{G})=\sigma(\mathbb{G}_{\alpha})$, where we
recall that both $\sigma(\mathbb{G})=\sigma_{p}(\mathbb{G})$ and
$\sigma(\mathbb{G}_{\alpha})=\sigma_{p}(\mathbb{G}_{\alpha})$ are point
spectra since both $\mathbb{G}$ and $\mathbb{G}_{\alpha}$ generate eventually
compact semigroups by Theorem 3.3 respectively (i). Let
$\lambda\in\sigma(\mathbb{G})$. Then there is
$\psi\in\mathrm{dom}(\mathbb{G})\setminus\\{0\\}$ with

$\mathbb{G}\psi=\lambda\psi\in\mathrm{dom}(\mathbb{G})=\mathrm{dom}(\mathbb{A})\subset\mathbb{E}_{\alpha}$

by Theorem 3.1. From (ii) we deduce that
$\psi\in\mathrm{dom}(\mathbb{G}_{\alpha})$ with
$\mathbb{G}_{\alpha}\psi=\lambda\psi$. Therefore,
$\lambda\in\sigma(\mathbb{G}_{\alpha})$.

Conversely, let $\mu\in\sigma(\mathbb{G}_{\alpha})$. Then there is
$\zeta\in\mathrm{dom}(\mathbb{G}_{\alpha})\setminus\\{0\\}$ with
$\mathbb{G}_{\alpha}\zeta=\mu\zeta$. Since
$\mathbb{G}_{\alpha}\zeta=\mathbb{G}\zeta$, we conclude that
$\mu\in\sigma(\mathbb{G})$.

(iv) Consider now $\omega>s(\mathbb{G})$. Since $\mathbb{G}$ and
$\mathbb{G}_{\alpha}$ both generate eventually compact semigroups, it follows
from (iii) and [12, IV. Corollary 3.11] that

$\omega_{0}(\mathbb{G})=s(\mathbb{G})=s(\mathbb{G}_{\alpha})=\omega_{0}(\mathbb{G}_{\alpha})\,.$

Thus, for $\omega-\varepsilon>s(\mathbb{G})$ with $\varepsilon>0$ there is
$c_{\alpha}\geq 1$ such that

$\displaystyle\|\mathbb{T}(t)\|_{\mathcal{L}(\mathbb{E}_{0})}+\|\mathbb{T}(t)\|_{\mathcal{L}(\mathbb{E}_{\alpha})}\leq
c_{\alpha}e^{(\omega-\varepsilon)t}\,,\quad t\geq 0\,.$ (3.16)

On the one hand, it follows from (3.13) and (3.16) for $t>1$ that

$\displaystyle\|\mathbb{T}(t)\|_{\mathcal{L}(\mathbb{E}_{0},\mathbb{E}_{\alpha})}$
$\displaystyle\leq\|\mathbb{T}(1)\|_{\mathcal{L}(\mathbb{E}_{0},\mathbb{E}_{\alpha})}\,\|\mathbb{T}(t-1)\|_{\mathcal{L}(\mathbb{E}_{0})}\,\leq
C_{\alpha}\,e^{\varsigma_{\alpha}}\,c_{\alpha}\,e^{(\omega-\varepsilon)(t-1)}$
$\displaystyle\leq
c_{\alpha}\,C_{\alpha}\,e^{-\omega+\varepsilon}\,\left(\sup_{s>0}s^{\alpha}e^{-\varepsilon
s}\right)\,t^{-\alpha}\,e^{\omega t}\,.$

On the other hand, due to (3.13) we have, for $0<t\leq 1$,

$\displaystyle\|\mathbb{T}(t)\|_{\mathcal{L}(\mathbb{E}_{0},\mathbb{E}_{\alpha})}$
$\displaystyle\leq C_{\alpha}\,e^{\varsigma_{\alpha}}t^{-\alpha}\leq
C_{\alpha}\,e^{\varsigma_{\alpha}}\,e^{|\omega|}\,t^{-\alpha}\,e^{\omega
t}\,.$

Consequently, there is $N_{\alpha}\geq 1$ such that

$\|\mathbb{T}(t)\|_{\mathcal{L}(\mathbb{E}_{0},\mathbb{E}_{\alpha})}\leq
N_{\alpha}\,t^{-\alpha}\,e^{\omega t}\,,\quad t>0\,,$

and the assertion follows. ∎

######  Remark 3.6.

One can show that $\mathbb{A}+\mathbb{B}$ has compact resolvent for any
$\mathbb{B}\in\mathcal{L}(\mathbb{E}_{0})$ (not necessarily satisfying (3.9)).
We refer to a forthcoming paper [27].

##  4\. Linearized Stability for the Nonlinear Problem: Proof of Theorem 2.2

The development of this section is based upon the treatment of linearized
stability in [28], which, in turn, is based on the treatment of the case
without spatial diffusion in [16]. In fact, we follow the exquisite exposition
in [29, Section 4.5] of this case.

For the reminder of this section, let $\phi\in\mathbb{E}_{1}\cap
C(J,E_{\alpha})$ be a fixed equilibrium to the nonlinear problem (1.2) and
assume (2.1), (2.2), and (2.5). As pointed out in Section 2 we shall derive
statements on the stability or instability of $\phi$ from information on the
(formally) linearized problem (2.6), that is, from information on

$\displaystyle\partial_{t}v+\partial_{a}v\,$
$\displaystyle=A(a)v-m\big{(}\bar{\phi},a\big{)}v-\partial
m\big{(}\bar{\phi},a\big{)}[\bar{v}(t)]\phi(a)\,,\qquad t>0\,,\quad
a\in(0,a_{m})\,,$ $\displaystyle v(t,0)$
$\displaystyle=\int_{0}^{a_{m}}b\big{(}\bar{\phi},a\big{)}\,v(t,a)\,\mathrm{d}a+\int_{0}^{a_{m}}\partial
b\big{(}\bar{\phi},a\big{)}[\bar{v}(t)]\,\phi(a)\,\mathrm{d}a\,,\qquad t>0\,,$
$\displaystyle v(0,a)$ $\displaystyle=v_{0}(a)\,,\qquad a\in(0,a_{m})\,.$

In the following, we demonstrate that this linear problem fits into the
framework of Section 3. More precisely, the solution $v$ is given by a
semigroup $(\mathbb{T}_{\phi}(t))_{t\geq 0}$ generated by an (unbounded)
operator of the form $\mathbb{A}_{\phi}+\mathbb{B}_{\phi}$ as in Theorem 3.3
with perturbation $\mathbb{B}_{\phi}w=-\partial
m(\bar{\phi},a)[\bar{w}]\phi(a)$. Moreover, if $u(\cdot;u_{0})$ is the
solution to the nonlinear problem (1.2) provided by Proposition 2.1, then the
difference $u(\cdot;u_{0})-\phi$ can be represented in terms of this semigroup
$(\mathbb{T}_{\phi}(t))_{t\geq 0}$, see Proposition 4.4 below. This will be
the key for our stability and instability results stated in Theorem 2.2 (see
also Theorem 4.1 below).

We focus our attention on the linearization (2.6). Regarding the linearized
age boundary conditions (2.6b) we point out that

$\int_{0}^{a_{m}}b(\bar{\phi},a)\,\zeta(a)\,\mathrm{d}a+\int_{0}^{a_{m}}\partial
b(\bar{\phi},a)[\bar{\zeta}]\phi(a)\,\mathrm{d}a=\int_{0}^{a_{m}}b_{\phi}(a)\zeta(a)\,\mathrm{d}a\,,\quad\zeta\in\mathbb{E}_{0}\,,$

where we set

$b_{\phi}(a)v:=b(\bar{\phi},a)v+\int_{0}^{a_{m}}\partial
b(\bar{\phi},\sigma)[\varrho(a)v]\,\phi(\sigma)\,\mathrm{d}\sigma\,,\quad a\in
J\,,\quad v\in E_{0}\,,$ (4.1a) so that $b_{\phi}\in
C\big{(}J,\mathcal{L}(E_{0})\big{)}\cap
L_{\infty}\big{(}J,\mathcal{L}(E_{\theta})\big{)}\,,\quad\theta\in\\{0,\alpha,\vartheta\\}\,,$
according to (2.5g), (2.5h), and (2.2c). We also introduce
$A_{\phi}(a)v:=A(a)v-m(\bar{\phi},a)v\,,\quad v\in E_{1}\,,\quad a\in J\,,$
(4.1b) and infer from (2.1), (2.5i), and [3, I.Theorem 1.3.1] that
$A_{\phi}\in C^{\rho}\big{(}J,\mathcal{H}(E_{1},E_{0})\big{)}\,.$ Therefore,
$A_{\phi}$ and $b_{\phi}$ satisfy (3.2), (3.3), and (3.9c). Moreover, we
define $\mathbb{B}_{\phi}\in\mathcal{L}(\mathbb{E}_{0})$ by
$[\mathbb{B}_{\phi}\zeta](a):=-\partial
m(\bar{\phi},a)[\bar{\zeta}]\phi(a)\,,\quad a\in
J\,,\quad\zeta\in\mathbb{E}_{0}\,,$ (4.1c) and infer from (2.5a), (2.5j), and
(2.2c) that $[\mathbb{B}_{\phi}\zeta](a)=-\partial
m(\bar{\phi},a)\left[\int_{0}^{a_{m}}\varrho(\sigma)\zeta(\sigma)\,\mathrm{d}\sigma\right]\phi(a)=\int_{0}^{a_{m}}q(a,\sigma)\zeta(\sigma)\,\mathrm{d}\sigma\,,\quad
a\in J\,,$ (4.1d) for $\zeta\in\mathbb{E}_{0}$, where $q\in
C\big{(}J,C(J,\mathcal{L}(E_{0}))\big{)}$ is given by $q(a,\sigma)v:=-\partial
m(\bar{\phi},a)[\varrho(\sigma)v]\phi(a)\,,\quad a,\sigma\in J\,,\quad v\in
E_{0}\,.$ Hence, (3.9a) and (3.9b) also hold, and we are in a position to
apply the results from the previous section with $A_{\ell}$ and $b_{\ell}$
replaced by $A_{\phi}$ respectively $b_{\phi}$. Throughout the reminder of
this section we thus assume (see Theorem 3.1) that
$\begin{split}&\textit{$\mathbb{A}_{\phi}$ is the generator of the semigroup
associated with~{}\eqref{PP} and the data}\\\ &\textit{$A_{\phi}$ and
$b_{\phi}$ introduced in~{}\eqref{bell}-\eqref{Aell}. Moreover,
$(\mathbb{T}_{\phi}(t))_{t\geq 0}$ is the strongly}\\\ &\textit{continuous
semigroup on~{}$\mathbb{E}_{0}$ generated by
$\mathbb{G}_{\phi}:=\mathbb{A}_{\phi}+\mathbb{B}_{\phi}$}.\end{split}$ (4.1e)

Theorem 3.3 implies that $(\mathbb{T}_{\phi}(t))_{t\geq 0}$ is eventually
compact and the spectrum consists of eigenvalues only, i.e.
$\sigma(\mathbb{G}_{\phi})=\sigma_{p}(\mathbb{G}_{\phi})$. Moreover, the
characterization of $\mathbb{A}_{\phi}$ (and thus of $\mathbb{G}_{\phi}$) in
Theorem 3.1 (c) and (4.1) imply that the eigenvalue problem for
$\mathbb{G}_{\phi}$ corresponds exactly to (2.7).

We shall then prove the following reformulation of Theorem 2.2 regarding the
stability of equilibria to the nonlinear problem (1.2):

######  Theorem 4.1.

Let $\alpha\in[0,1)$ and let $\phi\in\mathbb{E}_{1}\cap C(J,E_{\alpha})$ be an
equilibrium to (1.2). Assume (2.1), (2.2), (2.5), and use the notation (4.1).
The following hold:

  * (a)

If $\mathrm{Re}\,\lambda<0$ for any $\lambda\in\sigma_{p}(\mathbb{G}_{\phi})$,
then $\phi$ is exponentially asymptotically stable in $\mathbb{E}_{\alpha}$.

  * (b)

If $\mathrm{Re}\,\lambda>0$ for some
$\lambda\in\sigma_{p}(\mathbb{G}_{\phi})$, then $\phi$ is unstable in
$\mathbb{E}_{\alpha}$.

### Proof of Theorem 4.1 (a): Stability

Assumptions (2.1), (2.2), (2.5) ensure that we are in a position to apply [28,
Theorem 2.2], where it was shown that the equilibrium $\phi$ is exponentially
asymptotically stable in $\mathbb{E}_{\alpha}$ provided that there is
$\omega_{\alpha}(\phi)>0$ such that

$\displaystyle\|\mathbb{T}_{\phi}(t)\|_{\mathcal{L}(\mathbb{E}_{\alpha})}+t^{\alpha}\|\mathbb{T}_{\phi}(t)\|_{\mathcal{L}(\mathbb{E}_{0},\mathbb{E}_{\alpha})}\leq
Me^{-\omega_{\alpha}(\phi)t}\,,\quad t>0\,.$ (4.2)

Now, the supposition $\mathrm{Re}\,\lambda<0$ for any
$\lambda\in\sigma_{p}(\mathbb{G}_{\phi})$ ensures a negative spectral bound
$s(\mathbb{G}_{\phi})<0$ so that Corollary 3.5 implies (4.2). This proves
Theorem 4.1 (a).∎

### Preparation of the Proof of Theorem 4.1 (b): Instability

The proof of the instability result requires some preliminaries. First of all,
we infer from the supposition of Theorem 4.1 (b) and due to Theorem 3.3, that
the set

$\Sigma_{+}:=\sigma(\mathbb{G}_{\phi})\cap[\mathrm{Re}\,\lambda>0]$

is nonempty and finite; that is, $\Sigma_{+}$ is a bounded spectral set. Let

$0<\omega<\inf(\mathrm{Re}\,\Sigma_{+})\,.$ (4.3)

This yields the following spectral decomposition:

######  Lemma 4.2.

Assume (4.3). There is a projection $P\in\mathcal{L}(\mathbb{E}_{0})$ yielding
a decomposition

$\mathbb{E}_{0}=\mathbb{E}_{0}^{1}\oplus\mathbb{E}_{0}^{2}\,,\qquad\\{0\\}\not=\mathbb{E}_{0}^{1}:=P(\mathbb{E}_{0})\subset
D(\mathbb{G}_{\phi})\subset\mathbb{E}_{\alpha}\,,\qquad\mathbb{E}_{0}^{2}:=(1-P)(\mathbb{E}_{0})\,,$
(4.4a) such that
$\mathbb{G}_{\phi}|_{\mathbb{E}_{0}^{1}}\in\mathcal{L}(\mathbb{E}_{0}^{1})$.
Moreover, there are $M\geq 1$ and $\delta>0$ such that
$\|\mathbb{T}_{\phi}(t)P\|_{\mathcal{L}(\mathbb{E}_{0},\mathbb{E}_{\alpha})}\leq
Me^{(\omega+\delta)t}\,,\quad t\leq 0\,,$ (4.4b) and
$\|\mathbb{T}_{\phi}(t)(1-P)\|_{\mathcal{L}(\mathbb{E}_{0},\mathbb{E}_{\alpha})}\leq
M\,t^{-\alpha}\,e^{(\omega-\delta)t}\,,\quad t>0\,.$ (4.4c)

###### Proof.

Since $\Sigma_{+}$ is a bounded spectral set, it follows from [15, Proposition
A.1.2] (or [29, Proposition 4.15]) that there is a projection
$P\in\mathcal{L}(\mathbb{E}_{0})$ such that (4.4a) holds (noticing that
$\mathrm{dom}(\mathbb{G}_{\phi})=\mathrm{dom}(\mathbb{A}_{\phi})\subset\mathbb{E}_{\alpha}$
by Theorem 3.1, see also [26, Corollary 3.4]) and such that

$\mathbb{G}_{\phi}^{1}:=\mathbb{G}_{\phi}|_{\mathbb{E}_{0}^{1}}\in\mathcal{L}(\mathbb{E}_{0}^{1})\,.$

Moreover,

$\mathbb{G}_{\phi}^{2}:\mathrm{dom}(\mathbb{G}_{\phi})\cap\mathbb{E}_{0}^{2}\to\mathbb{E}_{0}^{2}\,,\quad\psi\mapsto\mathbb{G}_{\phi}\psi$

with

$\sigma(\mathbb{G}_{\phi}^{1})=\Sigma_{+}\,,\qquad\sigma(\mathbb{G}_{\phi}^{2})=\sigma(\mathbb{G}_{\phi})\setminus\Sigma_{+}$
(4.5a) and
$(\lambda-\mathbb{G}_{\phi}^{i})^{-1}=(\lambda-\mathbb{G}_{\phi})^{-1}|_{\mathbb{E}_{0}^{i}}\,,\qquad
i=1,2\,,\quad\lambda\in\rho(\mathbb{G}_{\phi})\,.$ (4.5b)

Choose $\delta>0$ such that

$\sup\\{\mathrm{Re}\,\lambda\,;\,\lambda\in\sigma(\mathbb{G}_{\phi}^{2})\\}<\omega-2\delta<\omega+\delta<\inf\\{\mathrm{Re}\,\lambda\,;\,\lambda\in\Sigma_{+}\\}\,.$

It follows from (4.5) that $\mathbb{G}_{\phi}^{1}$ and $\mathbb{G}_{\phi}^{2}$
generate strongly continuous semigroups on $\mathbb{E}_{0}^{1}$ respectively
$\mathbb{E}_{0}^{2}$ such that

$e^{t\mathbb{G}_{\phi}^{1}}=e^{t\mathbb{G}_{\phi}}|_{\mathbb{E}_{0}^{1}}=e^{t\mathbb{G}_{\phi}}P|_{\mathbb{E}_{0}^{1}}\,,\qquad
e^{t\mathbb{G}_{\phi}^{2}}=e^{t\mathbb{G}_{\phi}}|_{\mathbb{E}_{0}^{2}}=e^{t\mathbb{G}_{\phi}}(1-P)|_{\mathbb{E}_{0}^{2}}$
(4.6)

for $t\geq 0$. In fact, $e^{t\mathbb{G}_{\phi}^{1}}$ is extended to
$\mathbb{R}$ by (see also [15, Proposition 2.3.3])

$e^{t\mathbb{G}_{\phi}^{1}}=\frac{1}{2\pi i}\int_{\Gamma}e^{\lambda
t}\,(\lambda-\mathbb{G}_{\phi})^{-1}\,\mathrm{d}\lambda\,,\quad
t\in\mathbb{R}\,,$ (4.7)

where $\Gamma$ is a positively oriented smooth curve in
$\rho(\mathbb{G}_{\phi})$ enclosing $\Sigma_{+}$ with
$\mathrm{Re}\,\lambda\geq\omega+\delta$ for every $\lambda\in\Gamma$. Using
(4.7), we have, for $\psi\in\mathbb{E}_{0}$,

$\displaystyle\|e^{t\mathbb{G}_{\phi}^{1}}P\psi\|_{\mathbb{E}_{0}}\leq\frac{1}{2\pi}\,|\Gamma|\,\sup_{\lambda\in\Gamma}\|(\lambda-\mathbb{G}_{\phi})^{-1}\|_{\mathcal{L}(\mathbb{E}_{0})}\,e^{(\omega+\delta)t}\,\|\psi\|_{\mathbb{E}_{0}}\,,\quad
t\leq 0\,.$

Similarly, since

$\mathbb{G}_{\phi}\,e^{t\mathbb{G}_{\phi}^{1}}=\frac{1}{2\pi
i}\int_{\Gamma}\lambda\,e^{\lambda
t}\,(\lambda-\mathbb{G}_{\phi})^{-1}\,\mathrm{d}\lambda\,,\quad
t\in\mathbb{R}\,,$

we have

$\displaystyle\|\mathbb{G}_{\phi}e^{t\mathbb{G}_{\phi}^{1}}P\psi\|_{\mathbb{E}_{0}}\leq\frac{1}{2\pi}\,|\Gamma|\,\sup_{\lambda\in\Gamma}\|\lambda\,(\lambda-\mathbb{G}_{\phi})^{-1}\|_{\mathcal{L}(\mathbb{E}_{0})}\,e^{(\omega+\delta)t}\,\|\psi\|_{\mathbb{E}_{0}}\,,\quad
t\leq 0\,.$

Combining the two estimates we find $N\geq 1$ such that

$\displaystyle\|e^{t\mathbb{G}_{\phi}^{1}}P\psi\|_{D(\mathbb{G}_{\phi})}=\|e^{t\mathbb{G}_{\phi}^{1}}P\psi\|_{\mathbb{E}_{0}}+\|\mathbb{G}_{\phi}e^{t\mathbb{G}_{\phi}^{1}}P\psi\|_{\mathbb{E}_{0}}\leq
N\,e^{(\omega+\delta)t}\,\|\psi\|_{\mathbb{E}_{0}}\,,\quad t\leq 0\,.$

Consequently, since $D(\mathbb{G}_{\phi})\hookrightarrow\mathbb{E}_{\alpha}$
according to [26, Corollary 3.4], we deduce (4.4b).

Finally, since $(e^{t\mathbb{G}_{\phi}})_{t\geq 0}$ is an eventually compact
semigroup on $\mathbb{E}_{0}$ by Theorem 3.3, it follows from (4.6) that also
$(e^{t\mathbb{G}_{\phi}^{2}})_{t\geq 0}$ is an eventually compact semigroup on
$\mathbb{E}_{0}^{2}$, hence
$\omega_{0}(\mathbb{G}_{\phi}^{2})=s(\mathbb{G}_{\phi}^{2})$ due to [12,
IV.Corollary 3.11] and therefore
$\omega_{0}(\mathbb{G}_{\phi}^{2})<\omega-2\delta$ by the choice of $\delta$.
Thus, there is $N_{1}\geq 1$ such that

$\displaystyle\|e^{t\mathbb{G}_{\phi}}(1-P)\|_{\mathcal{L}(\mathbb{E}_{0})}\leq
N_{1}\,e^{(\omega-2\delta)t}\,,\quad t\geq 0\,.$ (4.8)

Noticing that also

$\displaystyle\|e^{t\mathbb{G}_{\phi}}(1-P)\|_{\mathcal{L}(\mathbb{E}_{0},\mathbb{E}_{\alpha})}\leq\|e^{t\mathbb{G}_{\phi}}\|_{\mathcal{L}(\mathbb{E}_{0},\mathbb{E}_{\alpha})}\,\|(1-P)\|_{\mathcal{L}(\mathbb{E}_{0})}\leq
N_{2}\,t^{-\alpha}\,e^{\omega_{1}t}\,,\quad t>0\,,$ (4.9)

for some $\omega_{1}>0$ and $N_{2}\geq 1$ due to (3.11), we may use
(4.8)-(4.9) and argue as in part (iv) of the proof of Corollary 3.5 to
conclude (4.4c). This proves Lemma 4.2. ∎

For the next step we introduce for a given function $h\in C([0,T],E_{0})$ and
$\gamma\in\mathbb{R}$ (sticking to the notation of [28, Definition (5.3)]) the
function $W_{0,0}^{\gamma,h}$ by setting

$W_{0,0}^{\gamma,h}(t,a)\,:=\,\left\\{\begin{aligned}
&0\,,&&(t,a)\in\mathbb{R}\times J\,,\ t\leq a\,,\\\ &e^{-\gamma
a}\,\Pi_{\phi}(a,0)\,B_{0,0}^{\gamma,h}(t-a)\,,&&(t,a)\in[0,T]\times J\,,\
t>a\,,\end{aligned}\right.$ (4.10a) where $B_{0,0}^{\gamma,h}\in
C([0,T],E_{0})$ satisfies $\displaystyle B_{0,0}^{\gamma,h}(t)\,=\,$
$\displaystyle\int_{0}^{t}b_{\phi}(a)\,e^{-\gamma
a}\,\Pi_{\phi}(a,0)\,B_{0,0}^{\gamma,h}(t-a)\,\mathrm{d}a+h(t)\,,\quad
t\in[0,T]\,,$ (4.10b)

with the understanding that $b_{\phi}(a)=0$ whenever $a\notin J$. Here,
$\Pi_{\phi}$ denotes the parabolic evolution operator associated with
$A_{\phi}$. Let $\varpi_{\phi}\in\mathbb{R}$ be such that

$\|\Pi_{\phi}(a,\sigma)\|_{\mathcal{L}(E_{\alpha})}+(a-\sigma)^{\alpha}\|\Pi_{\phi}(a,\sigma)\|_{\mathcal{L}(E_{0},E_{\alpha})}\leq
M_{*}\,e^{\varpi_{\phi}(a-\sigma)}\,,\quad 0\leq\sigma\leq a\leq a_{m}\,,$

for some $M_{*}\geq 1$ (see (B.2)). Then we have:

######  Lemma 4.3.

Let $h\in C\big{(}[0,T],E_{0}\big{)}$, $\gamma\in\mathbb{R}$, and
$\theta\in[0,1)$. Then $W_{0,0}^{\gamma,h}\in
C((-\infty,T],\mathbb{E}_{\theta})$ and there are constants $\mu=\mu(\phi)>0$
and $c_{0}=c_{0}(\phi)>0$ (both independent of $T$, $\gamma$, and $h$) such
that

$\|W_{0,0}^{\gamma,h}(t,\cdot)\|_{\mathbb{E}_{\theta}}\leq
c_{0}\int_{0}^{t}e^{(\varpi_{\phi}+\mu-\gamma)(t-a)}\,(t-a)^{-\theta}\,\|h(a)\|_{E_{0}}\,\mathrm{d}a\,,\quad
t\in[0,T]\,.$ (4.11)

###### Proof.

This is [28, Lemma 5.7]. ∎

Now, let $u_{0}\in\mathbb{E}_{\alpha}$ be arbitrary and set

$w:=u(\cdot;u_{0})-\phi\,,\qquad w_{0}:=u_{0}-\phi\,,$

where $u(\cdot;u_{0})\in C\big{(}I(u_{0}),\mathbb{E}_{\alpha}\big{)}$ is the
unique maximal solution to the nonlinear problem (1.2) provided by Proposition
2.1. Then, using the expansions (2.5c) and (2.5d) of $m$ respectively $b$ and
the notation from (4.1), we derive that $w\in C(I(u_{0}),\mathbb{E}_{\alpha})$
is the generalized solution (in the sense of (2.3), see [28, Proposition 4.2])
to

$\displaystyle\partial_{t}w+\partial_{a}w\,$
$\displaystyle=A_{\phi}(a)w+[\mathbb{B}_{\phi}w(t,\cdot)](a)+R_{m}(w(t))\,,\qquad
t\in I(u_{0})\,,\quad a\in J\,,$ (4.12a) $\displaystyle w(t,0)$
$\displaystyle=\int_{0}^{a_{m}}b_{\phi}(a)\,w(t,a)\,\mathrm{d}a+h_{w}(t)\,,\qquad
t\in I(u_{0})\,,$ (4.12b) $\displaystyle w(0,a)$
$\displaystyle=w_{0}(a)\,,\qquad a\in J\,,$ (4.12c) where $h_{w}\in
C(I(u_{0}),E_{0})$ is defined as
$h_{w}(t):=\int_{0}^{a_{m}}R_{b}(w(t))(a)\,\mathrm{d}a\,,\quad t\in
I(u_{0})\,,$ (4.12d)

and reminder terms $R_{m}$ and $R_{b}$ stemming from (2.5).

The characterization of the generator $\mathbb{A}_{\phi}$ given in Theorem 3.1
(c) gives rise to a representation of $w=u(\cdot;u_{0})-\phi$ in terms of the
semigroup $(\mathbb{T}_{\phi}(t))_{t\geq 0}$. In fact, the following result
was established in [28] (see also [16] for the non-diffusive case). It is
fundamental for the investigation of stability properties of the equilibrium
$\phi$.

######  Proposition 4.4.

Given $u_{0}\in\mathbb{E}_{\alpha}$ let $u(\cdot;u_{0})\in
C\big{(}I(u_{0}),\mathbb{E}_{\alpha}\big{)}$ with
$I(u_{0})=[0,T_{max}(u_{0}))$ be the unique maximal solution to the nonlinear
problem (1.2) provided by Proposition 2.1. Set $w=u(\cdot;u_{0})-\phi$ and
$w_{0}=u_{0}-\phi$. Then $w\in C(I(u_{0}),\mathbb{E}_{\alpha})$ can be written
as

$\begin{split}w(t)&=\mathbb{T}_{\phi}(t)w_{0}+\int_{0}^{t}\mathbb{T}_{\phi}(t-s)\left(\big{(}\gamma+\mathbb{B}_{\phi}\big{)}W_{0,0}^{\gamma,h_{w}}(s,\cdot)+R_{m}(w(s))\right)\,\mathrm{d}s+W_{0,0}^{\gamma,h_{w}}(t,\cdot)\end{split}$
(4.13)

for $t\in I(u_{0})$ and every $\gamma\in\mathbb{R}$, where $h_{w}\in
C(I(u_{0}),E_{0})$ stems from (4.12d) and

$W_{0,0}^{\gamma,h_{w}}\in
C\big{(}(-\infty,T_{max}(u_{0})),\mathbb{E}_{\alpha}\big{)}$

from (4.10).

###### Proof.

This is [28, Proposition 6.1]. ∎

In the following, assume (4.3) and let the projection
$P\in\mathcal{L}(\mathbb{E}_{0})$ and the constants $M$ and $\delta$ be as in
Lemma 4.2. Further, let $\varpi_{\phi}$, $c_{0}$, and $\mu$ be as in Lemma
4.3. Choose then $\gamma>0$ such that

$\kappa:=\omega-\varpi_{\phi}-\mu+\gamma>0\,$

and set

$R_{0}:=\frac{c_{0}\Gamma(1-\alpha)}{\kappa^{1-\alpha}}+\frac{M}{\delta}\,\big{(}1+\delta^{\alpha}\Gamma(1-\alpha)\big{)}\Big{(}\|\gamma+\mathbb{B}_{\phi}\|_{\mathcal{L}(\mathbb{E}_{0})}\frac{c_{0}}{\kappa}+1\Big{)}\,,$
(4.14)

where $\Gamma$ denotes the Gamma function. Recalling the function $d_{o}$ from
(2.5e) and (2.5f), we may choose $r>0$ and $w_{0}\in\mathbb{E}_{0}^{1}$ is
such that

$R_{0}\,d_{o}(r)<\frac{\|w_{0}\|_{\mathbb{E}_{\alpha}}}{r}<\frac{1}{2M}\leq\frac{1}{2}\,.$
(4.15)

In order to prove the instability of $\phi$, we now show the existence of a
sequence $(u_{0}^{k})_{k\geq 1}$ such that $u_{0}^{k}\to\phi$ in
$\mathbb{E}_{\alpha}$ and

$\inf_{k\geq 1}\|u(k;u_{0}^{k})-\phi\|_{\mathbb{E}_{\alpha}}>0\,.$

To this end, we first derive backwards solutions to problem (1.2):

######  Lemma 4.5.

Let $r>0$ and $w_{0}\in\mathbb{E}_{0}^{1}$ be as in (4.15). Then, for each
integer $k\geq 1$, there exists a unique function $v_{k}\in
C\big{(}(-\infty,k],\mathbb{E}_{\alpha}\big{)}$ such that

$\displaystyle v_{k}(t)$
$\displaystyle=\mathbb{T}_{\phi}(t-k)w_{0}+W_{0,0}^{\gamma,h_{v_{k}}}(t,\cdot)$
$\displaystyle\quad-\int_{t}^{k}\mathbb{T}_{\phi}(t-s)P\left(\big{(}\gamma+\mathbb{B}_{\phi}\big{)}W_{0,0}^{\gamma,h_{v_{k}}}(s,\cdot)+R_{m}(v_{k}(s))\right)\,\mathrm{d}s$
(4.16)
$\displaystyle\quad+\int_{-\infty}^{t}\mathbb{T}_{\phi}(t-s)(1-P)\left(\big{(}\gamma+\mathbb{B}_{\phi}\big{)}W_{0,0}^{\gamma,h_{v_{k}}}(s,\cdot)+R_{m}(v_{k}(s))\right)\,\mathrm{d}s$

for $t\leq k$ and satisfying

$\|v_{k}(t)\|_{\mathbb{E}_{\alpha}}\leq re^{\omega(t-k)}\,,\quad t\leq k\,.$
(4.17)

###### Proof.

Let $k\geq 1$ be fixed. We introduce the complete metric space $Z=(Z,d_{Z})$
by

$Z:=\big{\\{}v\in
C\big{(}(-\infty,k],\mathbb{E}_{\alpha}\big{)}\,;\,\|v(t)\|_{\mathbb{E}_{\alpha}}\leq
re^{\omega(t-k)}\,,\,t\leq k\big{\\}}$

equipped with the metric

$d_{Z}(v,\bar{v}):=\sup_{t\leq
k}\Big{(}e^{-\omega(t-k)}\|v(t)-\bar{v}(t)\|_{\mathbb{E}_{\alpha}}\Big{)}\,,\quad
v,\bar{v}\in Z\,,$

and claim that

$\displaystyle H(v)(t)$
$\displaystyle:=\mathbb{T}_{\phi}(t-k)w_{0}+W_{0,0}^{\gamma,h_{v}}(t,\cdot)$
$\displaystyle\qquad-\int_{t}^{k}\mathbb{T}_{\phi}(t-s)\,P\,\left(\big{(}\gamma+\mathbb{B}_{\phi}\big{)}W_{0,0}^{\gamma,h_{v}}(s,\cdot)+R_{m}(v(s))\right)\,\mathrm{d}s$
$\displaystyle\qquad+\int_{-\infty}^{t}\mathbb{T}_{\phi}(t-s)\,(1-P)\,\left(\big{(}\gamma+\mathbb{B}_{\phi}\big{)}W_{0,0}^{\gamma,h_{v}}(s,\cdot)+R_{m}(v(s))\right)\,\mathrm{d}s$

defines a contraction $H:Z\to Z$. Indeed, for $v\in Z$ we have
$\|v(t)\|_{\mathbb{E}_{\alpha}}\leq r$ for $t\leq k$ and thus, invoking
(2.5e),

$\|R_{m}(v(t))\|_{\mathbb{E}_{0}}\leq
d_{o}(r)\,\|v(t)\|_{\mathbb{E}_{\alpha}}\leq
d_{o}(r)\,r\,e^{\omega(t-k)}\,,\quad t\leq k\,,$ (4.18a) and, together with
(4.12d), $\|h_{v}(t)\|_{E_{0}}\leq\|R_{b}(v(t))\|_{\mathbb{E}_{0}}\leq
d_{o}(r)\,\|v(t)\|_{\mathbb{E}_{\alpha}}\leq
d_{o}(r)\,re^{\omega(t-k)}\,,\quad t\leq k\,.$ Therefore, (4.11) implies for
$v\in Z,$ $t\leq k$, and $\theta\in\\{0,\alpha\\}$ that
$\displaystyle\|W_{0,0}^{\gamma,h_{v}}(t,\cdot)\|_{\mathbb{E}_{\theta}}$
$\displaystyle\leq\chi_{\mathbb{R}^{+}}(t)c_{0}d_{o}(r)r\int_{0}^{t}e^{(\varpi_{\phi}+\mu-\gamma)(t-a)}\,(t-a)^{-\theta}\,e^{\omega(a-k)}\,\mathrm{d}a$
$\displaystyle\leq\frac{c_{0}\Gamma(1-\theta)}{\kappa^{1-\theta}}d_{o}(r)re^{\omega(t-k)}\,.$
(4.18b)

We then use (4.4), (4.18), and (4.14) to derive

$\displaystyle\|H(v)(t)\|_{\mathbb{E}_{\alpha}}$
$\displaystyle\leq\|\mathbb{T}_{\phi}(t-k)P\|_{\mathcal{L}(\mathbb{E}_{\alpha})}\,\|w_{0}\|_{\mathbb{E}_{\alpha}}+\|W_{0,0}^{\gamma,h_{v}}(t,\cdot)\|_{\mathbb{E}_{\alpha}}$
$\displaystyle\quad+\int_{t}^{k}\|\mathbb{T}_{\phi}(t-s)P\|_{\mathcal{L}(\mathbb{E}_{0},\mathbb{E}_{\alpha})}\Big{(}\|\gamma+\mathbb{B}_{\phi}\|_{\mathcal{L}(\mathbb{E}_{0})}\,\|W_{0,0}^{\gamma,h_{v}}(s,\cdot)\|_{\mathbb{E}_{0}}+\|R_{m}(v(s))\|_{\mathbb{E}_{0}}\Big{)}\,\mathrm{d}s$
$\displaystyle\quad+\int_{-\infty}^{t}\|\mathbb{T}_{\phi}(t-s)(1-P)\|_{\mathcal{L}(\mathbb{E}_{0},\mathbb{E}_{\alpha})}\Big{(}\|\gamma+\mathbb{B}_{\phi}\|_{\mathcal{L}(\mathbb{E}_{0})}\,\|W_{0,0}^{\gamma,h_{v}}(s,\cdot)\|_{\mathbb{E}_{0}}+\|R_{m}(v(s))\|_{\mathbb{E}_{0}}\Big{)}\,\mathrm{d}s$
$\displaystyle\leq
Me^{(\omega+\delta)(t-k)}\,\|w_{0}\|_{\mathbb{E}_{\alpha}}+\frac{c_{0}\Gamma(1-\alpha)}{\kappa^{1-\alpha}}\,d_{o}(r)\,r\,e^{\omega(t-k)}$
$\displaystyle\quad+M\,r\,d_{o}(r)\Big{(}\|\gamma+\mathbb{B}_{\phi}\|_{\mathcal{L}(\mathbb{E}_{0})}\,\frac{c_{0}}{\kappa}+1\Big{)}\int_{t}^{k}e^{(\omega+\delta)(t-s)}\,e^{\omega(s-k)}\,\mathrm{d}s$
$\displaystyle\quad+M\,r\,d_{o}(r)\Big{(}\|\gamma+\mathbb{B}_{\phi}\|_{\mathcal{L}(\mathbb{E}_{0})}\,\frac{c_{0}}{\kappa}+1\Big{)}\int_{-\infty}^{t}e^{(\omega-\delta)(t-s)}\,e^{\omega(s-k)}\,(t-s)^{-\alpha}\,\mathrm{d}s$
$\displaystyle\leq
Me^{\omega(t-k)}\,\|w_{0}\|_{\mathbb{E}_{\alpha}}+\frac{c_{0}\Gamma(1-\alpha)}{\kappa^{1-\alpha}}\,d_{o}(r)\,r\,e^{\omega(t-k)}$
$\displaystyle\quad+\frac{M}{\delta}\,r\,d_{o}(r)\,\big{(}1+\delta^{\alpha}\Gamma(1-\alpha)\big{)}\Big{(}\|\gamma+\mathbb{B}_{\phi}\|_{\mathcal{L}(\mathbb{E}_{0})}\,\frac{c_{0}}{\kappa}+1\Big{)}e^{\omega(t-k)}$
$\displaystyle=\left(\frac{M}{r}\|w_{0}\|_{\mathbb{E}_{0}^{1}}+R_{0}d_{o}(r)\right)\,r\,e^{\omega(t-k)}$

so that (4.15) implies

$\displaystyle\|H(v)(t)\|_{\mathbb{E}_{\alpha}}\leq\,r\,e^{\omega(t-k)}\,,\qquad
v\in Z\,,\quad t\leq k\,.$

That is, $H:Z\to Z$. Next, notice from (2.5f) that, for $v_{1},v_{2}\in Z$,

$\|R_{m}(v_{1}(t))-R_{m}(v_{2}(t))\|_{\mathbb{E}_{0}}\leq
d_{o}(r)\,d_{Z}(v_{1},v_{2})\,e^{\omega(t-k)}\,,\quad t\leq k\,,$ (4.19a)
while from (4.12d) and (2.5f),
$\|h_{v_{1}}(t)-h_{v_{2}}(t)\|_{E_{0}}\leq\|R_{b}(v_{1}(t))-R_{b}(v_{2}(t))\|_{\mathbb{E}_{0}}\leq
d_{o}(r)\,d_{Z}(v_{1},v_{2})\,e^{\omega(t-k)}\,,\quad t\leq k\,.$ Therefore,
since the mapping $[h\to W_{0,0}^{\gamma,h}]$ is linear, it follows from
(4.11) that
$\displaystyle\|W_{0,0}^{\gamma,h_{v_{1}}}(t,\cdot)-W_{0,0}^{\gamma,h_{v_{2}}}(t,\cdot)\|_{\mathbb{E}_{\alpha}}\leq\frac{c_{0}\Gamma(1-\alpha)}{\kappa^{1-\alpha}}d_{o}(r)d_{Z}(v_{1},v_{2})e^{\omega(t-k)}\,,\quad
t\leq k\,.$ (4.19b)

Using then (4.19) we derive similarly as above that, for $v_{1},v_{2}\in Z$,

$\displaystyle\|H(v_{1})(t)-H(v_{2})(t)\|_{\mathbb{E}_{\alpha}}$
$\displaystyle\leq
R_{0}\,d_{o}(r)\,d_{Z}(v_{1},v_{2})\,\,e^{\omega(t-k)}\,,\quad t\leq k\,,$

so that (4.15) implies

$\displaystyle
d_{Z}\big{(}H(v_{1}),H(v_{2})\big{)}\leq\,\frac{1}{2}\,d_{Z}(v_{1},v_{2})\,,\quad
v_{1},v_{2}\in Z\,.$

Consequently, $H:Z\to Z$ is indeed a contraction, and Lemma 4.5 follows from
Banach’s fixed point theorem. ∎

In fact, for positive times we have a simpler representation of $v_{k}$:

######  Corollary 4.6.

Let $r>0$ and $w_{0}\in\mathbb{E}_{0}^{1}$ be as in (4.15). Then $v_{k}\in
C\big{(}(-\infty,k],\mathbb{E}_{\alpha}\big{)}$ from Lemma 4.5 satisfies

$\displaystyle v_{k}(t)$
$\displaystyle=W_{0,0}^{\gamma,h_{v_{k}}}(t,\cdot)+\mathbb{T}_{\phi}(t)v_{k}(0)$
$\displaystyle\quad+\int_{0}^{t}\mathbb{T}_{\phi}(t-s)\left(\big{(}\gamma+\mathbb{B}_{\phi}\big{)}W_{0,0}^{\gamma,h_{v_{k}}}(s,\cdot)+R_{m}(v_{k}(s))\right)\,\mathrm{d}s$

for $0\leq t\leq k$.

###### Proof.

Define for $t\leq k$

$q(t):=\big{(}\gamma+\mathbb{B}_{\phi}\big{)}W_{0,0}^{\gamma,h_{v_{k}}}(t,\cdot)+R_{m}(v_{k}(t))\,,\qquad
p(t):=W_{0,0}^{\gamma,h_{v_{k}}}(t,\cdot)$

so that

$\displaystyle v_{k}(t)-p(t)$
$\displaystyle=\mathbb{T}_{\phi}(t-k)w_{0}-\int_{t}^{k}\mathbb{T}_{\phi}(t-s)\,P\,q(s)\,\mathrm{d}s+\int_{-\infty}^{t}\mathbb{T}_{\phi}(t-s)\,(1-P)\,q(s)\,\mathrm{d}s\,.$

Then $q\in C((-\infty,k],\mathbb{E}_{0})$ and $p\in
C((-\infty,k],\mathbb{E}_{\alpha})$ by Lemma 4.3. However, we may approximate
$q,p$ uniformly on compact intervals by continuously differentiable functions
with compact support and $w_{0}$ by a sequence in
$\mathrm{dom}(\mathbb{G}_{\phi})$ to justify the formal computation

$\displaystyle\frac{\mathrm{d}}{\mathrm{d}t}\big{(}v_{k}(t)-p(t)\big{)}$
$\displaystyle=\mathbb{G}_{\phi}\,\mathbb{T}_{\phi}(t-k)w_{0}+Pq(t)$
$\displaystyle\quad-\int_{t}^{k}\mathbb{G}_{\phi}\mathbb{T}_{\phi}(t-s)P\,q(s)\,\mathrm{d}s$
$\displaystyle\quad+\int_{-\infty}^{t}\mathbb{G}_{\phi}\mathbb{T}_{\phi}(t-s)(1-P)q(s)\,\mathrm{d}s+(1-P)q(t)$
$\displaystyle=\mathbb{G}_{\phi}\big{(}v_{k}(t)-p(t)\big{)}+q(t)\,.$

Thus, for $0\leq t\leq k$,

$\displaystyle v_{k}(t)-p(t)$
$\displaystyle=\mathbb{T}_{\phi}(t)\big{(}v_{k}(0)-p(0)\big{)}+\int_{0}^{t}\mathbb{T}_{\phi}(t-s)q(s)\,\mathrm{d}s$

and since $p(0)=W_{0,0}^{\gamma,h_{v_{k}}}(0,\cdot)=0$, the assertion follows.
∎

We are now in a position to provide the proof of Theorem 4.1 (b).

### Proof of Instability: Theorem 4.1 (b)

In order to prove instability, we may assume without loss of generality that
all solutions $u(\cdot;u_{0})$ to (1.2) provided by Proposition 2.1 exist
globally – that is, $T_{max}(u_{0})=\infty$ – whenever the initial values
$u_{0}$ are close to the equilibrium $\phi$. We set

$u_{0}^{k}:=v_{k}(0)+\phi\,,\quad k\geq 1\,,$

and note from (4.17) that

$\|u_{0}^{k}-\phi\|_{\mathbb{E}_{\alpha}}=\|v_{k}(0)\|_{\mathbb{E}_{\alpha}}\leq
re^{-\omega k}\rightarrow 0\,,\quad m\to\infty\,.$

Hence $T_{max}(u_{0}^{k})=\infty$ as just agreed. Let
$w_{k}:=u(\cdot;u_{0}^{k})-\phi\in C(\mathbb{R}^{+},\mathbb{E}_{\alpha})$.
Then Lemma 4.6 and Proposition 4.4 entail that both $w_{k},v_{k}\in
C([0,k],\mathbb{E}_{\alpha})$ satisfy the fixed point equation

$\begin{split}z(t)&=\mathbb{T}_{\phi}(t)\big{(}u_{0}^{k}-\phi\big{)}+\int_{0}^{t}\mathbb{T}_{\phi}(t-s)\left(\big{(}\gamma+\mathbb{B}_{\phi}\big{)}W_{0,0}^{\gamma,h_{z}}(s,\cdot)+R_{m}(z(s))\right)\,\mathrm{d}s+W_{0,0}^{\gamma,h_{z}}(t,\cdot)\end{split}$

for $t\in[0,k]$. It is easily seen that Gronwall’s inequality ensures
uniqueness in $C([0,k],\mathbb{E}_{\alpha})$ of this fixed point equation,
hence $v_{k}=w_{k}$ on $[0,k]$. Thus, we deduce from (4.5), (4.4), and (4.18)
that

$\displaystyle\|u(k;u_{0}^{k})$
$\displaystyle-\phi\|_{\mathbb{E}_{\alpha}}=\|w_{k}(k)\|_{\mathbb{E}_{\alpha}}=\|v_{k}(k)\|_{\mathbb{E}_{\alpha}}$
$\displaystyle\geq\|w_{0}\|_{\mathbb{E}_{\alpha}}-\|W_{0,0}^{\gamma,h_{v_{k}}}(k,\cdot)\|_{\mathbb{E}_{\alpha}}$
$\displaystyle\quad-\int_{-\infty}^{k}\|\mathbb{T}_{\phi}(k-s)(1-P)\|_{\mathcal{L}(\mathbb{E}_{0},\mathbb{E}_{\alpha})}$
$\displaystyle\qquad\qquad\times\left(\|\gamma+\mathbb{B}_{\phi}\|_{\mathcal{L}(\mathbb{E}_{0})}\,\|W_{0,0}^{\gamma,h_{v_{k}}}(s,\cdot)\|_{\mathbb{E}_{0}}+\|R_{m}(v_{k}(s))\|_{\mathbb{E}_{0}}\right)\,\mathrm{d}s$
$\displaystyle\geq\|w_{0}\|_{\mathbb{E}_{\alpha}}-\frac{c_{0}\Gamma(1-\alpha)}{\kappa^{1-\alpha}}\,d_{o}(r)r$
$\displaystyle\quad-M\left(\|\gamma+\mathbb{B}_{\phi}\|_{\mathcal{L}(\mathbb{E}_{0})}\,\frac{c_{0}}{\kappa}+1\right)d_{o}(r)r\int_{-\infty}^{k}e^{(\omega-\delta)(k-s)}e^{\omega(s-k)}\,(k-s)^{-\alpha}\mathrm{d}s$
$\displaystyle=\|w_{0}\|_{\mathbb{E}_{\alpha}}-\frac{c_{0}\Gamma(1-\alpha)}{\kappa^{1-\alpha}}d_{o}(r)r$
$\displaystyle\quad-\frac{M}{\delta^{1-\alpha}}\Gamma(1-\alpha)\left(\|\gamma+\mathbb{B}_{\phi}\|_{\mathcal{L}(\mathbb{E}_{0})}\frac{c_{0}}{\kappa}+1\right)d_{o}(r)r=:\xi_{0}\,,$

where, due to (4.15) and (4.14),

$\xi_{0}>\left(R_{0}-\frac{c_{0}\Gamma(1-\alpha)}{\kappa^{1-\alpha}}-\frac{M}{\delta^{1-\alpha}}\Gamma(1-\alpha)\left(\|\gamma+\mathbb{B}_{\phi}\|_{\mathcal{L}(\mathbb{E}_{0})}\,\frac{c_{0}}{\kappa}+1\right)\right)\,d_{o}(r)\,r>0\,.$

Consequently, we have shown that there exists a sequence $(u_{0}^{k})_{k\geq
1}$ such that $u_{0}^{k}\to\phi$ in $\mathbb{E}_{\alpha}$ as $k\to\infty$
while $\|u(k;u_{0}^{k})-\phi\|_{\mathbb{E}_{\alpha}}\geq\xi_{0}$ for $k\geq
1$. This proves that $\phi$ is unstable in $\mathbb{E}_{\alpha}$ and thus
Theorem 4.1 (b).∎

### Rephrasing the Eigenvalue Problem

According to Theorem 4.1, the stability of an equilibrium $\phi$ is determined
from the eigenvalues of the operator
$\mathbb{G}_{\phi}=\mathbb{A}_{\phi}+\mathbb{B}_{\phi}$. Clearly,
$\lambda\in\mathbb{C}$ is an eigenvalue of
$\mathbb{G}_{\phi}=\mathbb{A}_{\phi}+\mathbb{B}_{\phi}$ if and only if there
is some $\psi\in\mathrm{dom}(\mathbb{A}_{\phi})$ such that
$(\lambda-\mathbb{A}_{\phi}-\mathbb{B}_{\phi})\psi=0$. Now, due to Theorem 3.1
(c) and (4.1), this is equivalent to $\psi\in C(J,E_{0})$ solving (in a mild
sense)

$\displaystyle\partial_{a}\psi$
$\displaystyle=\big{(}-\lambda+A_{\phi}(a)\big{)}\psi+(\mathbb{B}_{\phi}\psi)(a)\,,\quad
a\in J\,,$ (4.20a) $\displaystyle\psi(0)$
$\displaystyle=\int_{0}^{a_{m}}b_{\phi}(a)\,\psi(a)\,\mathrm{d}a\,.$ (4.20b)

Note that (4.20a) entails

$\displaystyle\psi(a)=e^{-\lambda
a}\Pi_{\phi}(a,0)\psi(0)+\int_{0}^{a}e^{-\lambda(a-\sigma)}\,\Pi_{\phi}(a,\sigma)(\mathbb{B}_{\phi}\psi)(\sigma)\,\mathrm{d}\sigma\,,\quad
a\in J\,,$ (4.21)

which, when plugged into (4.20b), yields

$\displaystyle\left(1-\int_{0}^{a_{m}}b_{\phi}(a)\,e^{-\lambda
a}\,\Pi_{\phi}(a,0)\,\mathrm{d}a\right)\psi(0)$
$\displaystyle=\int_{0}^{a_{m}}b_{\phi}(a)\int_{0}^{a}e^{-\lambda(a-\sigma)}\,\Pi_{\phi}(a,\sigma)(\mathbb{B}_{\phi}\psi)(\sigma)\,\mathrm{d}\sigma\mathrm{d}a\,.$
(4.22)

Recall from (4.1c) that

$[\mathbb{B}_{\phi}\psi](a)=-\partial
m(\bar{\phi},a)[\bar{\psi}]\phi(a)\,,\quad a\in J\,.$

We thus introduce

$\displaystyle
K_{\phi,\lambda}(a)\bar{\psi}:=\int_{0}^{a}e^{-\lambda(a-\sigma)}\,\Pi_{\phi}(a,\sigma)\,\partial
m(\bar{\phi},\sigma)[\bar{\psi}]\phi(\sigma)\,\mathrm{d}\sigma\,,\quad a\in
J\,,$ (4.23)

and then obtain from (4.22) that

$\displaystyle\left(1-\int_{0}^{a_{m}}b_{\phi}(a)\,e^{-\lambda
a}\,\Pi_{\phi}(a,0)\,\mathrm{d}a\right)\psi(0)$
$\displaystyle=-\int_{0}^{a_{m}}b_{\phi}(a)K_{\phi,\lambda}(a)\,\mathrm{d}a\,\bar{\psi}\,.$

Moreover, (4.21) implies that $\bar{\psi}$ satisfies

$\displaystyle\left(1+\int_{0}^{a_{m}}\varrho(a)K_{\phi,\lambda}(a)\,\mathrm{d}a\right)\bar{\psi}=\int_{0}^{a_{m}}e^{-\lambda
a}\,\varrho(a)\Pi_{\phi}(a,0)\,\mathrm{d}a\,\psi(0)\,.$

Therefore, $\lambda\in\mathbb{C}$ is an eigenvalue of
$\mathbb{G}_{\phi}=\mathbb{A}_{\phi}+\mathbb{B}_{\phi}$ if and only if there
is a nontrivial eigenvector $(\psi(0),\bar{\psi})\in E_{0}\times E_{0}$ in the
sense that

$\begin{pmatrix}1-\displaystyle\int_{0}^{a_{m}}e^{-\lambda
a}\,b_{\phi}(a)\,\Pi_{\phi}(a,0)\,\mathrm{d}a&\displaystyle\int_{0}^{a_{m}}b_{\phi}(a)K_{\phi,\lambda}(a)\,\mathrm{d}a\\\\[10.0pt]
-\displaystyle\int_{0}^{a_{m}}e^{-\lambda
a}\,\varrho(a)\Pi_{\phi}(a,0)\,\mathrm{d}a&1+\displaystyle\int_{0}^{a_{m}}\varrho(a)K_{\phi,\lambda}(a)\,\mathrm{d}a\end{pmatrix}\begin{pmatrix}\psi(0)\\\\[3.0pt]
\bar{\psi}\end{pmatrix}=0\,,$ (4.24)

where $K_{\phi,\lambda}$ is defined in (4.23) with $\Pi_{\phi}$ denoting the
evolution operator associated with $A_{\phi}$ given by

$A_{\phi}(a)v=A(a)v-m(\bar{\phi},a)v\,,\quad v\in E_{1}\,,\quad a\in J\,,$

and

$b_{\phi}(a)v=b(\bar{\phi},a)v+\int_{0}^{a_{m}}\partial
b(\bar{\phi},\sigma)[\varrho(a)v]\,\phi(\sigma)\,\mathrm{d}\sigma\,,\quad a\in
J\,,\quad v\in E_{0}\,.$

Consequently, we obtain from Theorem 4.1:

######  Proposition 4.7.

Let $\alpha\in[0,1)$. Assume (2.1), (2.2), (2.5) and let
$\phi\in\mathbb{E}_{1}\cap C(J,E_{\alpha})$ be an equilibrium to (1.2). The
following hold:

  * (a)

If $\mathrm{Re}\,\lambda<0$ for every $\lambda\in\mathbb{C}$ for which there
is a nontrivial $(\psi(0),\bar{\psi})\in E_{0}\times E_{0}$ satisfying (4.24),
then the equilibrium $\phi$ is exponentially asymptotically stable in
$\mathbb{E}_{\alpha}$.

  * (b)

If there are $\lambda\in\mathbb{C}$ with $\mathrm{Re}\,\lambda>0$ and a
nontrivial $(\psi(0),\bar{\psi})\in E_{0}\times E_{0}$ satisfying (4.24), then
the equilibrium $\phi$ is unstable in $\mathbb{E}_{\alpha}$.

It is worth emphasizing that the (spectral radius of the) compact operator

$\displaystyle Q_{\lambda}(\phi):=\int_{0}^{a_{m}}e^{-\lambda
a}\,b_{\phi}(a)\,\Pi_{\phi}(a,0)\,\mathrm{d}a\in\mathcal{K}(E_{0})$ (4.25)

occurring in the linear eigenvalue problem (4.24) plays a particular role in
the analysis. This becomes also apparent in the next section where we will
focus on applications. For $\lambda=0$ one may interpret its spectral radius
$r(Q_{0}(\phi))$ as the expected number of offspring per individual during its
life span at equilibrium.

##  5\. Examples

In order to shed some light on the previous results, we consider concrete
examples. We impose for simplicity stronger assumptions than actually
required444In particular, the results require less regularity assumptions than
imposed in (5.2) and also apply to any other uniformly elliptic second-order
differential operator.. In the following, $\Omega\subset\mathbb{R}^{n}$ is a
bounded domain with smooth boundary and outer unit normal $\nu$. We consider

$\displaystyle\partial_{t}u+\partial_{a}u$
$\displaystyle=\mathrm{div}_{x}\big{(}d(a,x)\nabla_{x}u\big{)}-m\big{(}\bar{u}(t,x),a,x\big{)}u\
,\quad t>0\,,\quad(a,x)\in(0,a_{m})\times\Omega\,,$ (5.1a) $\displaystyle
u(t,0,x)$
$\displaystyle=\int_{0}^{a_{m}}b\big{(}\bar{u}(t,x),a,x\big{)}u(t,a,x)\,\mathrm{d}a\,,\quad
t>0\,,\quad x\in\Omega\,,$ (5.1b) $\displaystyle\mathcal{B}u(t,a,x)$
$\displaystyle=0\ ,\quad t>0\,,\quad(a,x)\in(0,a_{m})\times\partial\Omega\,,$
(5.1c) $\displaystyle u(0,a,x)$ $\displaystyle=u_{0}(a,x)\
,\quad(a,x)\in(0,a_{m})\times\Omega\,,$ (5.1d)

where

$\mathcal{B}u:=(1-\delta)u+\delta\partial_{\nu}u\,,\quad\delta\in\\{0,1\\}\,,$

either refers to Dirichlet boundary conditions $u|_{\partial\Omega}=0$ if
$\delta=0$ or Neumann boundary conditions $\partial_{\nu}u=0$ if $\delta=1$
and

$\bar{u}(t,x)=\int_{0}^{a_{m}}\varrho(a,x)\,u(t,a,x)\,\mathrm{d}a\,,\qquad
t\geq 0\,,\quad x\in\Omega\,.$

We set $J=[0,a_{m}]$ and assume for the data that

$\displaystyle q>n\,,\quad
2\alpha\in(n/q,2)\setminus\\{\delta+1/q\\}\,,\quad\rho>0\,,$ (5.2a)
$\displaystyle d\in C^{\rho,1}\big{(}J\times\bar{\Omega},(0,\infty)\big{)}\,,$
(5.2b) $\displaystyle m\in C^{4,\rho,2}\big{(}\mathbb{R}\times
J\times\bar{\Omega},\mathbb{R}^{+}\big{)}\,,$ (5.2c) $\displaystyle b\in
C^{4,0,2}\big{(}\mathbb{R}\times J\times\bar{\Omega},(0,\infty)\big{)}\,,$
(5.2d) $\displaystyle\varrho\in
C^{0,2}\big{(}J\times\bar{\Omega},\mathbb{R}^{+}\big{)}\,.$ (5.2e)

Note that one may choose $2\alpha=1$ in the following. We introduce
$E_{0}:=L_{q}(\Omega)$ and

$E_{1}:=W_{q,\mathcal{B}}^{2}(\Omega):=\\{v\in
W_{q}^{2}(\Omega)\,;\;\mathcal{B}w=0\text{ on }\partial\Omega\\}\,.$

Then $E_{1}$ is compactly embedded in the Banach lattice $E_{0}$ and, for real
interpolation with $\theta\in(0,1)\setminus\\{1/2\\}$,

$\displaystyle E_{\theta}$
$\displaystyle:=\big{(}L_{q}(\Omega),W_{q,\mathcal{B}}^{2}(\Omega)\big{)}_{\theta,q}\doteq
W_{q,\mathcal{B}}^{2\theta}(\Omega)$
$\displaystyle:=\left\\{\begin{array}[]{ll}\left\\{v\in
W_{q}^{2\theta}(\Omega)\,;\;\mathcal{B}w=0\text{ on
}\partial\Omega\right\\}\,,&\delta+1/q<2\theta\leq 2\,,\\\\[3.0pt]
W_{q}^{2\theta}(\Omega)\,,&0\leq 2\theta<\delta+1/q\,,\end{array}\right.$

while, for complex interpolation with $\theta=1/2$,

$E_{1/2}:=\big{[}L_{q}(\Omega),W_{q,\mathcal{B}}^{2}(\Omega)\big{]}_{1/2}\doteq
W_{q,\mathcal{B}}^{1}(\Omega)\,.$

Setting

$A(a,x)w:=\mathrm{div}_{x}\big{(}d(a,x)\nabla_{x}w\big{)}\,,\quad w\in
W_{q,\mathcal{B}}^{2}(\Omega)\,,\quad a\in J\,,\quad x\in\Omega\,,$

it follows from (5.2b) and e.g. [1] that $A\in
C^{\rho}\big{(}J,\mathcal{H}\big{(}W_{q,\mathcal{B}}^{2}(\Omega),L_{q}(\Omega)\big{)}\big{)}$
while the maximum principle ensures that $A(a)$ is resolvent positive for each
$a\in J$. Therefore, (2.1) holds. It follows from (5.2a), (5.2c), (5.2d), and
[23, Proposition 4.1] that

$[v\mapsto b(v,\cdot,\cdot)]\,,\,[v\mapsto m(v,\cdot,\cdot)]\in
C^{1}\big{(}W_{q,\mathcal{B}}^{2\alpha}(\Omega),L_{\infty}(J,W_{q,\mathcal{B}}^{2\eta}(\Omega))\big{)}\,,\quad
2\eta\in(0,2\alpha)\setminus\\{\delta+1/q\\}\,,$

with

$\displaystyle\big{(}\partial
b(v,\cdot)[h]\big{)}(a)(x)=\partial_{1}b(v(x),a,x)h(x)\,,\qquad(a,x)\in
J\times\Omega\,,\quad v,h\in W_{q,\mathcal{B}}^{2\alpha}(\Omega)\,,$

and correspondingly for $m$. In particular, using the continuity of pointwise
multiplication

$W_{q,\mathcal{B}}^{2\eta}(\Omega)\times
W_{q,\mathcal{B}}^{2\alpha}(\Omega)\to L_{q}(\Omega)\,,$

we deduce that (2.5a)-(2.5f) are valid and hence also (2.2a) and (2.2b).
Clearly, (5.2e) implies (2.2c). Moreover, if
$\phi\in\mathbb{E}_{1}=L_{1}\big{(}J,W_{q,\mathcal{B}}^{2}(\Omega)\big{)}$ is
an equilibrium to (5.1), then

$\bar{\phi}=\int_{0}^{a_{m}}\varrho(a,\cdot)\,\phi(a,\cdot)\,\mathrm{d}a\in
W_{q,\mathcal{B}}^{2}(\Omega)$

due to (5.2e), hence $b(\bar{\phi},\cdot,\cdot)\in
L_{\infty}\big{(}J,W_{q,\mathcal{B}}^{2}(\Omega)\big{)}$. Since pointwise
multiplication

$W_{q,\mathcal{B}}^{2}(\Omega)\times W_{q,\mathcal{B}}^{2\alpha}(\Omega)\to
W_{q,\mathcal{B}}^{2\alpha}(\Omega)$

is continuous [2], we deduce (2.5g). Moreover, since
$\partial_{1}b(\bar{\phi},\cdot,\cdot)\in
L_{\infty}\big{(}J,W_{q,N}^{2-\varepsilon}(\Omega)\big{)}$ for every
$\varepsilon>0$ small and since pointwise multiplication

$W_{q,N}^{2-\varepsilon}(\Omega)\times W_{q,N}^{2\theta}(\Omega)\to
W_{q,N}^{2\theta}(\Omega)$

is continuous for $\theta=0,\alpha$, we obtain (2.5h) and similarly (2.5j).
Clearly, (5.2c) implies (2.5i). Consequently, assumptions (2.1), (2.2), and
(2.5) are all satisfied owing to (5.2).

Recall for $(a,x)\in J\times\Omega$ that

$A_{\phi}(a,x)w=\mathrm{div}_{x}\big{(}d(a,x)\nabla_{x}w\big{)}-m(\bar{\phi}(x),a,x)w\,,\quad
w\in W_{q,\mathcal{B}}^{2}(\Omega)\,,$

and

$b_{\phi}(a,x)=b(\bar{\phi}(x),a,x)+\int_{0}^{a_{m}}\partial_{1}b(\bar{\phi}(x),\sigma,x)\,\phi(\sigma,x)\,\mathrm{d}\sigma\,\varrho(a,x)\,.$

Moreover,

$(\mathbb{B}_{\phi}\psi)(a,x)=-\partial_{1}m(\bar{\phi}(x),a,x)\,\phi(a,x)\int_{0}^{a_{m}}\varrho(\sigma,x)\,\psi(\sigma,x)\,\mathrm{d}\sigma\,,\quad(a,x)\in
J\times\Omega\,.$

### The Trivial Equilibrium

For the particular case of the trivial equilibrium $\phi=0$, we observe that
(with dot referring to the suppressed $x$-variable)

$b_{0}(a)=b(0,a,\cdot)\,,\qquad A_{0}(a)=A(a)-m(0,a,\cdot)\,.$

Then $v(a,\cdot)=\Pi_{0}(a,0)v_{0}$, $a\in J$, is for each $v_{0}\in
L_{q}(\Omega)$ the unique strong solution to the heat equation

$\partial_{a}v=\mathrm{div}_{x}\big{(}d(a,x)\nabla_{x}v\big{)}-m(0,a,x)v\,,\quad(a,x)\in
J\times\Omega\,,\qquad v(0,x)=v_{0}(x)\,,\quad x\in\Omega\,,$

subject to Dirichlet boundary conditions if $\delta=0$ or Neumann boundary
conditions if $\delta=1$. Also note that $\mathbb{B}_{0}=0$ and
$K_{0,\lambda}=0$ in (4.23). The eigenvalue equation (4.24) then reduces to

$\displaystyle(1-Q_{\lambda}(0))\psi(0)$ $\displaystyle=0\,,$

where

$\displaystyle Q_{\lambda}(0)=\int_{0}^{a_{m}}e^{-\lambda
a}\,b(0,a,\cdot)\,\Pi_{0}(a,0)\,\mathrm{d}a$

is a strongly positive compact operator on $L_{q}(\Omega)$ for
$\lambda\in\mathbb{R}$ due to [8, Corollary 13.6] and the strict positivity of
$b(0,\cdot)$ assumed in (5.2d). As for its spectral radius $r(Q_{\lambda}(0))$
we note that the mapping

$\mathbb{R}\to(0,\infty)\,,\quad\lambda\mapsto r(Q_{\lambda}(0))$

is continuous and strictly decreasing with

$\lim_{\lambda\to-\infty}r(Q_{\lambda}(0))=\infty\,,\qquad\lim_{\lambda\to\infty}r(Q_{\lambda}(0))=0$

according to [26, Lemma 3.1]. Thus, there is a unique
$\lambda_{0}\in\mathbb{R}$ such that $r(Q_{\lambda_{0}}(0))=1$. In fact, it
follows from [26, Proposition 4.2] that $\lambda_{0}=s(\mathbb{G}_{0})$; that
is, $\lambda_{0}$ coincides with the spectral bound of the generator
$\mathbb{G}_{0}=\mathbb{A}_{0}$ (see Proposition 3.2).

Consequently, we can state the stability of the trivial equilibrium according
to Proposition 4.7 as follows:

######  Proposition 5.1.

Assume (5.2). Then:

  * (a)

If $r(Q_{0}(0))<1$, then the trivial equilibrium $\phi=0$ to (5.1) is
exponentially asymptotically stable in the phase space
$L_{1}\big{(}J,W_{q,\mathcal{B}}^{2\alpha}(\Omega)\big{)}$.

  * (b)

If $r(Q_{0}(0))>1$, then the trivial equilibrium $\phi=0$ to (5.1) is unstable
in the phase space $L_{1}\big{(}J,W_{q,\mathcal{B}}^{2\alpha}(\Omega)\big{)}$.

Proposition 1.1 is the special case of Proposition 5.1 with $\alpha=1/2$ and
$x$-independent vital rates $m\in C^{4,\rho}\big{(}\mathbb{R}\times
J,(0,\infty)\big{)}$ and $b\in C^{4,0}\big{(}\mathbb{R}\times
J,(0,\infty)\big{)}$, noticing that in this case

$\Pi_{0}(a,0)=\exp\left(-\int_{0}^{a}m(0,\sigma)\,\mathrm{d}\sigma\right)\,\Pi_{*}(a,0)\,,$

where $v(a,\cdot)=\Pi_{*}(a,0)v_{0}$, $a\in J$, is for given $v_{0}\in
L_{q}(\Omega)$ the unique strong solution to the heat equation

$\partial_{a}v=\mathrm{div}_{x}\big{(}d(a,x)\nabla_{x}u\big{)}\,,\quad(a,x)\in
J\times\Omega\,,\qquad v(0,x)=v_{0}(x)\,,\quad x\in\Omega\,,$

subject to Dirichlet boundary conditions if $\delta=0$ or Neumann boundary
conditions if $\delta=1$.

It is worth pointing out for the case of $x$-independent vital rates $m$ and
$b$ and Neumann boundary conditions $\mathcal{B}=\partial_{\nu}$ (i.e.
$\delta=1$) that the constant function $\mathbf{1}:=[x\mapsto 1]$ belongs to
$W_{q,N}^{2}(\Omega)$ (with subscript $N$ referring to the Neumann boundary
conditions) and satisfies $\Pi_{*}(a,0)\mathbf{1}=\mathbf{1}$. Therefore,

$Q_{0}(0)\mathbf{1}=\int_{0}^{a_{m}}b(0,a)\,e^{-\int_{0}^{a}m(0,s)\mathrm{d}s}\,\mathrm{d}a\,\mathbf{1}$

so that $\mathbf{1}$ is a positive eigenfunction of the strongly positive
compact operator $Q_{0}(0)$. Krein-Rutman’s theorem (e.g., see [8, Theorem
12.3]) implies

$r(Q_{0}(0))=\int_{0}^{a_{m}}b(0,a)\,e^{-\int_{0}^{a}m(0,s)\mathrm{d}s}\,\mathrm{d}a\,.$

Consequently, we obtain from Proposition 5.1:

######  Corollary 5.2.

Assume (5.2) with $x$-independent vital rates $m=m(\bar{u},a)$ and
$b=b(\bar{u},a)$ and $\delta=1$ (case of Neumann boundary conditions). Set

$r_{0}:=\int_{0}^{a_{m}}b(0,a)\,e^{-\int_{0}^{a}m(0,s)\mathrm{d}s}\,\mathrm{d}a\,.$

If $r_{0}<1$, then the trivial equilibrium $\phi=0$ to (5.1) is exponentially
asymptotically stable in $L_{1}\big{(}J,W_{q,N}^{2\alpha}(\Omega)\big{)}$.

If $r_{0}>1$, then the trivial equilibrium $\phi=0$ to (5.1) is unstable in
$L_{1}\big{(}J,W_{q,N}^{2\alpha}(\Omega)\big{)}$.

In particular, if the death rate dominates the birth rate in the sense that

$b(0,a)\leq m(0,a)\,,\quad a\in J\,,$

then

$r_{0}=\int_{0}^{a_{m}}b(0,a)\,e^{-\int_{0}^{a}m(0,s)\mathrm{d}s}\,\mathrm{d}a\leq\int_{0}^{a_{m}}m(0,a)\,e^{-\int_{0}^{a}m(0,s)\mathrm{d}s}\,\mathrm{d}a=1-e^{-\int_{0}^{a_{m}}m(0,s)\mathrm{d}s}<1\,.$

Hence, the trivial equilibrium is stable.

We also remark the following result on global stability of the trivial
solution in the special case of Corollary 5.2 (a). It is the analogue to the
non-diffusive case from [16, Theorem 4].

######  Corollary 5.3.

Assume (5.2) with $\delta=1$ (case of Neumann boundary conditions) and assume
that there are $b_{*}\in C(J,(0,\infty))$ and $m_{*}\in
C^{\rho}(J,\mathbb{R}^{+})$ such that

$\displaystyle b(r,a,x)\leq b_{*}(a)\,,\quad m(r,a)\geq
m_{*}(a)\,,\quad(r,a,x)\in\mathbb{R}\times J\times\bar{\Omega}\,,$ (5.3a) and
$\displaystyle\int_{0}^{a_{m}}b_{*}(a)\,e^{-\int_{0}^{a}m_{*}(s)\mathrm{d}s}\,\mathrm{d}a<1\,.$
(5.3b) Moreover, assume that there is $C_{*}>0$ such that
$\displaystyle|\partial_{1}b(r,a,x)|+|m(r,a,x)|+|\partial_{1}m(r,a,x)|\leq
C_{*}\,,\quad(r,a)\in\mathbb{R}\times J\times\bar{\Omega}\,.$ (5.3c)

Then the maximal solution $u(\cdot;u_{0})$ to (5.1) exists globally for each
$u_{0}\in L_{1}(J,W_{q}^{1}(\Omega))$ with $u_{0}\geq 0$ and $u(t;u_{0})\to 0$
in $L_{1}(J,L_{q}(\Omega))$ as $t\to\infty$.

###### Proof.

Note that

$A_{\ell}(a,x)w:=\mathrm{div}_{x}\big{(}d(a,x)\nabla_{x}w\big{)}-m_{*}(a)w\,,\quad
w\in W_{q,N}^{2}(\Omega)\,,\quad(a,x)\in J\times\Omega\,,$

and $b_{\ell}:=b_{*}$ satisfy (3.2) and (3.3). We then denote by
$(\mathbb{S}(t))_{t\geq 0}$ the corresponding positive semigroup on
$\mathbb{E}_{0}=L_{1}(J,L_{q}(\Omega))$ defined in (3.4) for these $A_{\ell}$,
$b_{\ell}$ (see Theorem 3.1). As in the proof of Corollary 5.2, supposition
(5.3b) implies that the semigroup $(\mathbb{S}(t))_{t\geq 0}$ has a negative
growth bound $\omega_{0}<0$ (see Proposition 3.2).

Let $u_{0}\in\mathbb{E}_{1/2}=L_{1}(J,W_{q}^{1}(\Omega))$ with $u_{0}\geq 0$
be arbitrary and $u(\cdot;u_{0})\in C(I(u_{0}),\mathbb{E}_{1/2})$ be the
maximal, positive solution to problem (5.1) (guaranteed by Proposition 2.1).
We simply write $u=u(\cdot;u_{0})$ and note that $u$ satisfies (in a mild
sense)

$\displaystyle\partial_{t}u+\partial_{a}u$
$\displaystyle=A_{\ell}(a)+f(t,a,x)\ ,$ $\displaystyle t\in I(u_{0})\,,$
$\displaystyle a\in(0,a_{m})\,,$ $\displaystyle x\in\Omega\,,$ $\displaystyle
u(t,0,x)$
$\displaystyle=\int_{0}^{a_{m}}b_{\ell}(a)u(t,a,x)\,\mathrm{d}a+h(t,x)\,,$
$\displaystyle t\in I(u_{0})\,,$ $\displaystyle x\in\Omega\,,$
$\displaystyle\mathcal{B}u(t,a,x)$ $\displaystyle=0\ ,$ $\displaystyle t\in
I(u_{0})\,,$ $\displaystyle a\in(0,a_{m})\,,$ $\displaystyle
x\in\partial\Omega\,,$

where we have introduced the negative functions $f\in
C(I(u_{0}),\mathbb{E}_{0})$ and $h\in C(I(u_{0}),E_{0})$ by

$\displaystyle f(t,a,x)$
$\displaystyle:=\big{(}m_{*}(a)-m(\bar{u}(t,x),a,x)\big{)}u(t,a,x)\leq 0\,,$
$\displaystyle h(t,x)$
$\displaystyle:=\int_{0}^{a_{m}}\big{(}b(\bar{u}(t,x),a,x)-b_{\ell}(a)\big{)}\,u(t,a,x)\,\mathrm{d}a\leq
0\,.$

Here, in a mild sense means that $u$ satisfies

$u(t)=\mathbb{S}(t)u_{0}+\int_{0}^{t}\mathbb{S}(t-s)f(s)\,\mathrm{d}s+W_{0,0}^{0,h}(t,\cdot)\,,\quad
t\in I(u_{0})\,,$ (5.4)

according to [28, Corollary 5.8], where $W_{0,0}^{0,h}\in
C(I(u_{0}),\mathbb{E}_{0})$ stems from Lemma 4.3. Since $f(t,\cdot,\cdot)\leq
0$ and $W_{0,0}^{0,h}(t,\cdot)\leq 0$ for $t\in I(u_{0})$ due to (5.3a) and
(4.10), it follows from (5.4) and the positivity of $(\mathbb{S}(t))_{t\geq
0}$ that

$0\leq u(t)\leq\mathbb{S}(t)u_{0}\,,\quad t\in I(u_{0})\,,$

in the Banach lattice $\mathbb{E}_{0}=L_{1}(J,L_{q}(\Omega))$. Therefore,

$\|u(t)\|_{\mathbb{E}_{0}}\leq\|\mathbb{S}(t)\|_{\mathcal{L}(\mathbb{E}_{0})}\,\|u_{0}\|_{\mathbb{E}_{0}}\leq
c_{0}\,e^{\omega_{0}t}\,\|u_{0}\|_{\mathbb{E}_{0}}\,,\quad t\in I(u_{0})\,.$
(5.5)

Assumption (5.3c) ensures that there is a constant $C_{1}>0$ such that

$\|f(t)\|_{L_{1}(J,W_{q}^{1}(\Omega))}+\|h(t)\|_{W_{q}^{1}(\Omega)}\leq
C_{1}\big{(}1+\|u(t)\|_{L_{1}(J,W_{q}^{1}(\Omega))}\big{)}\,,\quad t\in
I(u_{0})\,,$ (5.6)

while (3.7) yields for every $T>0$ a constant $c(T)>0$ such that

$\|\mathbb{S}(t)\|_{\mathcal{L}(L_{1}(J,W_{q}^{1}(\Omega)))}\leq c(T)\,,\quad
t\in[0,T]\,.$ (5.7)

It then readily follows from (5.4), (5.6), (5.7), Lemma 4.3, and Gronwall’s
inequality that

$\|u(t)\|_{L_{1}(J,W_{q}^{1}(\Omega))}\leq c_{1}(T)\,,\quad t\in
I(u_{0})\cap[0,T]\,,$

for every $T>0$. Proposition 2.1 now implies that the solution $u$ exists
globally, i.e. $I(u_{0})=[0,\infty)$. Consequently, we may let $t\to\infty$ in
(5.5) and use $\omega_{0}<0$ to conclude that $u(t;u_{0})\to 0$ in the phase
space $\mathbb{E}_{0}=L_{1}(J,L_{q}(\Omega))$. ∎

Corollary 5.2 is not restricted to the particular case of Neumann boundary
conditions (just replace the left-hand side of (5.3b) by the corresponding
spectral radius).

### An Instability Result

We provide the analogue to [16, Theorem 6]:

######  Proposition 5.4.

Assume (5.2). Consider a positive equilibrium

$\phi\in L_{1}\big{(}J,W_{q,\mathcal{B}}^{2}(\Omega)\big{)}\cap
C(J,W_{q,\mathcal{B}}^{2\alpha}(\Omega))\,,\quad\phi\geq 0\,,$

to (5.1) for which

$\partial_{1}b(\bar{\phi}(x),a,x)\geq
0\,,\quad\partial_{1}m(\bar{\phi}(x),a,x)\leq 0\,,\qquad(a,x)\in
J\times\Omega\,.$ (5.8)

If $s(\mathbb{G}_{\phi})\not=0$, then $\phi$ is unstable in
$L_{1}\big{(}J,W_{q,\mathcal{B}}^{2\alpha}(\Omega)\big{)}$.

###### Proof.

(i) Observe that (5.2d), (5.2e), (5.8), and the positivity of $\phi$ entail
the strict positivity $b_{\phi}>0$ and that $\mathbb{B}_{\phi}$ is a positive
operator on $\mathbb{E}_{0}=L_{1}(J,L_{q}(\Omega))$. Moreover, the maximum
principle ensures that $A_{\phi}(a)$ is resolvent positive for $a\in J$.
Theorem 3.3 now implies that
$\mathbb{G}_{\phi}=\mathbb{A}_{\phi}+\mathbb{B}_{\phi}$ is resolvent positive.
We then claim that for its spectral bound we have
$s(\mathbb{G}_{\phi})>-\infty$. Indeed, since $\mathbb{B}_{\phi}\geq 0$, it
follows from [4, Proposition 12.11] that $s(\mathbb{G}_{\phi})\geq
s(\mathbb{A}_{\phi})$. Next, the strict positivity $b_{\phi}>0$ and [8,
Corollary 13.6] imply that $b_{\phi}(a)\Pi_{\phi}(a,0)$ is strongly positive
on $L_{q}(\Omega)$ for $a\in J$. Thus, for $\lambda\in\mathbb{R}$, the
operator

$\displaystyle Q_{\lambda}(\phi)=\int_{0}^{a_{m}}e^{-\lambda
a}\,b_{\phi}(a)\,\Pi_{\phi}(a,0)\,\mathrm{d}a$

is compact and strongly positive on $L_{q}(\Omega)$. As in the previous
section we infer from [26, Lemma 3.1] that the mapping $[\lambda\mapsto
r(Q_{\lambda}(\phi))]$ is continuous and strictly decreasing on $\mathbb{R}$
with

$\lim_{\lambda\to-\infty}r(Q_{\lambda}(\phi))=\infty\,,\qquad\lim_{\lambda\to\infty}r(Q_{\lambda}(\phi))=0\,,$

and then from [26, Proposition 3.2] that $s(\mathbb{A}_{\phi})=\lambda_{0}$
with $\lambda_{0}\in\mathbb{R}$ being the unique real number such that
$r(Q_{\lambda_{0}}(\phi))=1$. Thus $s(\mathbb{G}_{\phi})\geq\lambda_{0}$ and
$s(\mathbb{G}_{\phi})$ is an eigenvalue of $\mathbb{G}_{\phi}$.

(ii) Let now $\lambda>0$ be large enough, i.e.
$\lambda>\max\\{s(\mathbb{G}_{\phi}),0\\}$ with $r(Q_{\lambda}(\phi))<1$. Set

$v:=(\lambda-\mathbb{G}_{\phi})^{-1}\phi$

and note that $v\geq 0$ since $\mathbb{G}_{\phi}$ is resolvent positive and
$\lambda>s(\mathbb{G}_{\phi})$, see [4, Remark 12.12 (b)]. Then Theorem 3.1
(c) entails that $v$ satisfies (in the sense of mild solutions)

$\partial_{a}v=(-\lambda+A_{\phi}(a))v+\mathbb{B}_{\phi}v+\phi\,,\quad a\in
J\,,\qquad v(0)=\int_{0}^{a_{m}}b_{\phi}(a)v(a)\,\mathrm{d}a\,.$

Thus, since $\mathbb{B}_{\phi}v\geq 0$, we deduce

$\displaystyle v(a)\geq e^{-\lambda
a}\,\Pi_{\phi}(a,0)\,v(0)+\int_{0}^{a}e^{-\lambda(a-\sigma)}\,\Pi_{\phi}(a,\sigma)\,\phi(\sigma)\,\mathrm{d}\sigma\,,\quad
a\in J\,,$ (5.9)

and, when plugging this into the initial condition,

$\displaystyle v(0)$
$\displaystyle\geq\int_{0}^{a_{m}}b_{\phi}(a)\,e^{-\lambda
a}\,\Pi_{\phi}(a,0)\,\mathrm{d}a\,v(0)+\int_{0}^{a_{m}}b_{\phi}(a)\int_{0}^{a}e^{-\lambda(a-\sigma)}\,\Pi_{\phi}(a,\sigma)\,\phi(\sigma)\,\mathrm{d}\sigma\,\mathrm{d}a$
$\displaystyle\geq\int_{0}^{a_{m}}b(\bar{\phi},a)\,e^{-\lambda
a}\,\Pi_{\phi}(a,0)\,\mathrm{d}a\,v(0)+\int_{0}^{a_{m}}b(\bar{\phi},a)\int_{0}^{a}e^{-\lambda(a-\sigma)}\,\Pi_{\phi}(a,\sigma)\,\phi(\sigma)\,\mathrm{d}\sigma\,\mathrm{d}a\,,$
(5.10)

where we used that $b_{\phi}(a)\geq b(\bar{\phi},a)$ due to (5.8). Now, the
equilibrium $\phi$ satisfies

$\phi(a)=\Pi_{\phi}(a,0)\phi(0)\,,\quad a\in
J\,,\qquad\phi(0)=Q_{0}(\phi)\phi(0)\,.$

Therefore, using the evolution property

$\Pi_{\phi}(a,\sigma)\Pi_{\phi}(\sigma,0)=\Pi_{\phi}(a,0)\,,\quad
0\leq\sigma\leq a\leq a_{m}\,,$

we infer that

$\displaystyle\int_{0}^{a}e^{-\lambda(a-\sigma)}\,\Pi_{\phi}(a,\sigma)\,\phi(\sigma)\,\mathrm{d}\sigma=\frac{1}{\lambda}\big{(}1-e^{-\lambda
a}\big{)}\,\Pi_{\phi}(a,0)\,\phi(0)$ (5.11)

and thus

$\displaystyle\int_{0}^{a_{m}}b(\phi,a)\int_{0}^{a}e^{-\lambda(a-\sigma)}\,\Pi_{\phi}(a,\sigma)\,\phi(\sigma)\,\mathrm{d}\sigma\,\mathrm{d}a=\frac{1}{\lambda}\big{(}1-Q_{\lambda}(\phi)\big{)}\phi(0)\,.$
(5.12)

From (5.10) and (5.12) it then follows that

$\big{(}1-Q_{\lambda}(\phi)\big{)}v(0)\geq\frac{1}{\lambda}\big{(}1-Q_{\lambda}(\phi)\big{)}\phi(0)$

and thus, since $\big{(}1-Q_{\lambda}(\phi)\big{)}^{-1}\geq 0$ as
$r(Q_{\lambda}(\phi))<1$, we conclude that $\lambda v(0)\geq\phi(0)$. Using
this along with (5.11) in (5.9) we derive $\lambda v(a)\geq\phi(a)$ for $a\in
J$. By definition of $v$, this means that

$\lambda\big{(}\lambda-\mathbb{G}_{\phi}\big{)}^{-1}\phi\geq\phi\,,\quad\lambda\gg
0\,.$

Invoking the exponential representation of the semigroup we obtain

$e^{t\mathbb{G}_{\phi}}\phi=\lim_{n\to\infty}\left(1-\frac{t}{n}\mathbb{G}_{\phi}\right)^{-n}\phi\geq\phi\,,\quad
t\geq 0\,,$

from which $\|e^{t\mathbb{G}_{\phi}}\|_{\mathcal{L}(\mathbb{E}_{0})}\geq 1$
for $t\geq 0$, since $\mathbb{E}_{0}=L_{1}(J,L_{q}(\Omega))$ is a Banach
lattice. This implies that
$s(\mathbb{G}_{\phi})=\omega_{0}(\mathbb{G}_{\phi})\geq 0$. By supposition, we
then even have $s(\mathbb{G}_{\phi})>0$. Since $s(\mathbb{G}_{\phi})$ is an
eigenvalue of $\mathbb{G}_{\phi}$, we conclude from Theorem 4.1 that $\phi$ is
unstable. ∎

A simple consequence is:

######  Corollary 5.5.

Assume (5.2). Consider a positive equilibrium

$\phi\in L_{1}\big{(}J,W_{q,\mathcal{B}}^{2}(\Omega)\big{)}\cap
C(J,W_{q,N}^{2\alpha}(\Omega))\,,\quad\phi\geq 0\,,$

to (5.1) for which (5.8) holds. If $r(Q_{0}(\phi))>1$, then $\phi$ is unstable
in $L_{1}\big{(}J,W_{q,N}^{2\alpha}(\Omega)\big{)}$.

###### Proof.

It has been observed in the proof of Proposition 5.4 that
$s(\mathbb{G}_{\phi})\geq\lambda_{0}$, where $\lambda_{0}\in\mathbb{R}$ is the
uniquely determined from the condition $r(Q_{\lambda_{0}}(\phi))=1$. Since
$[\lambda\mapsto r(Q_{\lambda}(\phi))]$ is strictly decreasing on $\mathbb{R}$
and $r(Q_{0}(\phi))>1$, we necessarily have $\lambda_{0}>0$. ∎

##  Appendix A Proof of Proposition 3.4

We provide here the proof of Proposition 3.4 which is fundamental for Theorem
3.3. We thus impose (3.2), (3.3), (3.9), and recall that we consider nonlocal
perturbations

$\displaystyle[\mathbb{B}\zeta](a):=\int_{0}^{a_{m}}q(a,\sigma)\,\zeta(\sigma)\,\mathrm{d}\sigma\,,\quad
a\in J\,,\quad\zeta\in\mathbb{E}_{0}\,,$

for some $q(a,\sigma)=q(a)(\sigma)$ satisfying

$\displaystyle q\in C\big{(}J,L_{\infty}(J,\mathcal{L}(E_{0}))\big{)}\,.$

Then $\mathbb{B}\in\mathcal{L}(\mathbb{E}_{0})$ with

$\|\mathbb{B}\|_{\mathcal{L}(\mathbb{E}_{0})}\leq
a_{m}\,\|q\|_{\infty}\,,\qquad\|q\|_{\infty}:=\|q\|_{C(J,L_{\infty}(J,\mathcal{L}(E_{0})))}\,.$

For the birth rate we recall that $b_{\ell}\in
C\big{(}J,\mathcal{L}(E_{0})\big{)}$. We begin with an auxiliary result:

######  Lemma A.1.

Suppose (3.2), (3.3), and (3.9). Given $\psi\in\mathbb{E}_{0}$, let
$\mathsf{B}_{\psi}\in C(\mathbb{R}^{+},E_{0})$ be defined as in (3.4b). Then,
given $T>0$, there is $c_{\mathsf{B}}(T)>0$ such that

$\|\mathsf{B}_{\psi}(\tau)\|_{E_{\theta}}\leq
c_{\mathsf{B}}(T)\,\tau^{-\theta}\,\|\psi\|_{\mathbb{E}_{0}}\,,\quad\tau\in(0,T]\,,\quad\theta\in\\{0,\vartheta\\}\,.$
(A.1)

Moreover, given $\varepsilon>0$ and $\kappa\in(0,a_{m}/2)$, there is
$\delta:=\delta(T,\kappa,\varepsilon)>0$ such that

$\|\mathsf{B}_{\mathbb{B}\zeta}(\tau_{1})-\mathsf{B}_{\mathbb{B}\zeta}(\tau_{2})\|_{E_{0}}\leq\varepsilon\|\zeta\|_{\mathbb{E}_{0}}\,,\qquad\tau_{1},\tau_{2}\in[\kappa,T]\
\text{ with }\ |\tau_{1}-\tau_{2}|\leq\delta\,,$

whenever $\zeta\in\mathbb{E}_{0}$.

###### Proof.

Estimate (A.1) follows for $\theta=0$ from the fact that
$[\psi\mapsto\mathsf{B}_{\psi}]\in\mathcal{L}\big{(}\mathbb{E}_{0},C(\mathbb{R}^{+},E_{0})\big{)}$,
see (3.6). It is derived from Gronwall’s inequality as shown for both cases
$\theta=0,\vartheta$ in [26, Formula (2.2)].

In order to prove the continuity property of $\mathsf{B}_{\mathbb{B}\zeta}$
for $\zeta\in\mathbb{E}_{0}$, set $\psi:=\mathbb{B}\zeta\in\mathbb{E}_{0}$. We
write

$\displaystyle\mathsf{B}_{\psi}(\tau)$ $\displaystyle=\int_{(\tau-
a_{m})_{+}}^{\tau}b_{\ell}(\tau-a)\,\Pi_{\ell}(\tau-a,0)\,\mathsf{B}_{\psi}(a)\,\mathrm{d}a$
$\displaystyle\qquad+\int_{0}^{(a_{m}-\tau)_{+}}b_{\ell}(\tau+a)\,\Pi_{\ell}(\tau+a,a)\,\psi(a)\,\mathrm{d}a$
$\displaystyle=:\mathsf{B}_{\psi}^{1}(\tau)+\mathsf{B}_{\psi}^{2}(\tau)$ (A.2)

for $\tau\geq 0$ and note from (B.2) that

$\displaystyle c_{1}:=\max_{0\leq\sigma\leq a\leq
a_{m}}\|\Pi_{\ell}(a,\sigma)\|_{\mathcal{L}(E_{0})}<\infty\,.$ (A.3)

Choose
$\delta_{1}:=\delta_{1}(T,\varepsilon,\kappa)\in(0,\min\\{a_{m}/2,\kappa\\})$
such that

$2\,c_{1}\,c_{\mathsf{B}}(T)\,\|b\|_{L_{\infty}(J,\mathcal{L}(E_{0}))}\,\max\\{T,1\\}\,\delta_{1}\leq\frac{\varepsilon}{4\|\mathbb{B}\|_{\mathcal{L}(\mathbb{E}_{0})}}\,.$
(A.4)

Due to Lemma B.1 we may choose
$\delta_{2}:=\delta_{2}(T,\varepsilon,\kappa)>0$ such that

$T\,c_{\mathsf{B}}(T)\,\|b\|_{L_{\infty}(J,\mathcal{L}(E_{0}))}\,\|\Pi_{\ell}(s_{1},0)-\Pi_{\ell}(s_{2},0)\|_{\mathcal{L}(E_{0})}\leq\frac{\varepsilon}{4\|\mathbb{B}\|_{\mathcal{L}(\mathbb{E}_{0})}}$
(A.5)

for $s_{1},s_{2}\in[\delta_{1},T\wedge a_{m}]$ with
$|s_{1}-s_{2}|\leq\delta_{2}$ and

$\|b_{\ell}(\bar{s}_{1})-b_{\ell}(\bar{s}_{2})\|_{\mathcal{L}(E_{0})}\,c_{1}\,\max\\{c_{\mathsf{B}}(T)T,1\\}\leq\frac{\varepsilon}{4\|\mathbb{B}\|_{\mathcal{L}(\mathbb{E}_{0})}}$
(A.6)

for $\bar{s}_{1},\bar{s}_{2}\in[0,a_{m}]$ with
$|\bar{s}_{1}-\bar{s}_{2}|\leq\delta_{2}$. Set

$\delta_{0}:=\delta_{0}(T,\varepsilon,\kappa):=\min\\{\delta_{1},\delta_{2}\\}>0\,.$

Then, for $\kappa\leq\tau_{2}\leq\tau_{1}\leq T$ with
$|\tau_{1}-\tau_{2}|\leq\delta_{0}$ we note that
$(\tau_{1}-a_{m})_{+}<\tau_{2}-\delta_{1}$ and obtain from (A.1)-(A.6)

$\displaystyle\|\mathsf{B}_{\psi}^{1}$
$\displaystyle(\tau_{1})-\mathsf{B}_{\psi}^{1}(\tau_{2})\|_{E_{0}}$
$\displaystyle\leq\int_{(\tau_{2}-a_{m})_{+}}^{(\tau_{1}-a_{m})_{+}}\|b_{\ell}(\tau_{2}-a)\|_{\mathcal{L}(E_{0})}\,\|\Pi_{\ell}(\tau_{2}-a,0)\|_{\mathcal{L}(E_{0})}\,\|\mathsf{B}_{\psi}(a)\|_{E_{0}}\,\mathrm{d}a$
$\displaystyle\quad+\int_{(\tau_{1}-a_{m})_{+}}^{\tau_{2}}\|b_{\ell}(\tau_{1}-a)-b_{\ell}(\tau_{2}-a)\|_{\mathcal{L}(E_{0})}\,\|\Pi_{\ell}(\tau_{1}-a,0)\|_{\mathcal{L}(E_{0})}\,\|\mathsf{B}_{\psi}(a)\|_{E_{0}}\,\mathrm{d}a$
$\displaystyle\quad+\int_{(\tau_{1}-a_{m})_{+}}^{\tau_{2}-\delta_{1}}\|b_{\ell}(\tau_{2}-a)\|_{\mathcal{L}(E_{0})}\,\|\Pi_{\ell}(\tau_{1}-a,0)-\Pi_{\ell}(\tau_{2}-a,0)\|_{\mathcal{L}(E_{0})}\,\|\mathsf{B}_{\psi}(a)\|_{E_{0}}\,\mathrm{d}a$
$\displaystyle\quad+\int_{\tau_{2}-\delta_{1}}^{\tau_{2}}\|b_{\ell}(\tau_{2}-a)\|_{\mathcal{L}(E_{0})}\,\|\Pi_{\ell}(\tau_{1}-a,0)-\Pi_{\ell}(\tau_{2}-a,0)\|_{\mathcal{L}(E_{0})}\,\|\mathsf{B}_{\psi}(a)\|_{E_{0}}\,\mathrm{d}a$
$\displaystyle\quad+\int_{\tau_{2}}^{\tau_{1}}\|b_{\ell}(\tau_{1}-a)\|_{\mathcal{L}(E_{0})}\,\|\Pi_{\ell}(\tau_{1}-a,0)\|_{\mathcal{L}(E_{0})}\,\|\mathsf{B}_{\psi}(a)\|_{E_{0}}\,\mathrm{d}a$
$\displaystyle\leq\|b_{\ell}\|_{L_{\infty}(J,\mathcal{L}(E_{0}))}\,c_{1}\,c_{\mathsf{B}}(T)\,\|\psi\|_{\mathbb{E}_{0}}\,|(\tau_{1}-a_{m})_{+}-(\tau_{2}-a_{m})_{+}|$
$\displaystyle\quad+\sup_{\begin{subarray}{c}s_{1},s_{2}\in[0,a_{m}]\\\
|s_{1}-s_{2}|\leq\delta_{0}\end{subarray}}\|b_{\ell}(s_{1})-b_{\ell}(s_{2})\|_{\mathcal{L}(E_{0})}\,c_{1}\,c_{\mathsf{B}}(T)\,T\,\|\psi\|_{\mathbb{E}_{0}}$
$\displaystyle\quad+\|b_{\ell}\|_{L_{\infty}(J,\mathcal{L}(E_{0}))}\,\sup_{\begin{subarray}{c}s_{1},s_{2}\in[\delta_{1},a_{m}]\\\
|s_{1}-s_{2}|\leq\delta_{0}\end{subarray}}\|\Pi_{\ell}(s_{1},0)-\Pi_{\ell}(s_{2},0)\|_{\mathcal{L}(E_{0})}\,c_{\mathsf{B}}(T)\,T\,\|\psi\|_{\mathbb{E}_{0}}$
$\displaystyle\quad+2\,\|b_{\ell}\|_{L_{\infty}(J,\mathcal{L}(E_{0}))}\,c_{1}\,\delta_{1}\,c_{\mathsf{B}}(T)\,T\,\|\psi\|_{\mathbb{E}_{0}}+\|b_{\ell}\|_{L_{\infty}(J,\mathcal{L}(E_{0}))}\,c_{1}\,c_{\mathsf{B}}(T)\,\|\psi\|_{\mathbb{E}_{0}}\,|\tau_{1}-\tau_{2}|$
$\displaystyle\leq\frac{\varepsilon}{\|\mathbb{B}\|_{\mathcal{L}(\mathbb{E}_{0})}}\,\|\psi\|_{\mathbb{E}_{0}}\,.$

Consequently, using $\psi=\mathbb{B}\zeta$ with
$\mathbb{B}\in\mathcal{L}(\mathbb{E}_{0})$ we derive

$\displaystyle\|\mathsf{B}_{\psi}^{1}$
$\displaystyle(\tau_{1})-\mathsf{B}_{\psi}^{1}(\tau_{2})\|_{E_{0}}\leq\varepsilon\,\|\zeta\|_{\mathbb{E}_{0}}\,,\qquad\kappa\leq\tau_{2}\leq\tau_{1}\leq
T\,,\quad|\tau_{1}-\tau_{2}|\leq\delta_{0}\,.$ (A.7)

For $\mathsf{B}_{\psi}^{2}$ we use Lemma B.1 to find
$\eta:=\eta(\varepsilon,\kappa)>0$ such that

$\|b_{\ell}\|_{L_{\infty}(J,\mathcal{L}(E_{0}))}\,\|\Pi_{\ell}(a+\tau_{1},a)-\Pi_{\ell}(a+\tau_{2},a)\|_{\mathcal{L}(E_{0})}\leq\frac{\varepsilon}{4\|\mathbb{B}\|_{\mathcal{L}(\mathbb{E}_{0})}}\,,$

whenever $\kappa\leq\tau_{2}\leq\tau_{1}\leq T$,
$a\in[0,(a_{m}-\tau_{1})_{+}]$, $|\tau_{1}-\tau_{2}|\leq\eta$. Let
$\delta_{3}>0$ with

$\|q\|_{\infty}\,\delta_{3}\leq\frac{\varepsilon}{4}$

and set

$\delta:=\delta(T,\varepsilon,\kappa):=\min\\{\delta_{0},\eta,\delta_{3}\\}\,.$

Then we obtain for $\kappa\leq\tau_{2}\leq\tau_{1}\leq T$ with
$|\tau_{1}-\tau_{2}|\leq\delta$ that

$\displaystyle\|\mathsf{B}_{\psi}^{2}$
$\displaystyle(\tau_{1})-\mathsf{B}_{\psi}^{2}(\tau_{2})\|_{E_{0}}$
$\displaystyle\leq\int_{0}^{(a_{m}-\tau_{1})_{+}}\|b_{\ell}(a+\tau_{1})-b_{\ell}(a+\tau_{2})\|_{\mathcal{L}(E_{0})}\,\|\Pi_{\ell}(a+\tau_{1},a)\|_{\mathcal{L}(E_{0})}\,\|\psi(a)\|_{E_{0}}\,\mathrm{d}a$
$\displaystyle\quad+\int_{0}^{(a_{m}-\tau_{1})_{+}}\|b_{\ell}(a+\tau_{2})\|_{\mathcal{L}(E_{0})}\,\|\Pi_{\ell}(a+\tau_{1},a)-\Pi_{\ell}(a+\tau_{2},a)\|_{\mathcal{L}(E_{0})}\,\|\psi(a)\|_{E_{0}}\,\mathrm{d}a$
$\displaystyle\quad+\int_{(a_{m}-\tau_{1})_{+}}^{(a_{m}-\tau_{2})_{+}}\|b_{\ell}(a+\tau_{2})\|_{\mathcal{L}(E_{0})}\,\|\Pi_{\ell}(a+\tau_{2},a)\|_{\mathcal{L}(E_{0})}\,\|\psi(a)\|_{E_{0}}\,\mathrm{d}a$
$\displaystyle\leq\sup_{\begin{subarray}{c}s_{1},s_{2}\in[0,a_{m}]\\\
|s_{1}-s_{2}|\leq\delta\end{subarray}}\|b_{\ell}(s_{1})-b_{\ell}(s_{2})\|_{\mathcal{L}(E_{0})}\,c_{1}\,\|\psi\|_{\mathbb{E}_{0}}$
$\displaystyle\quad+\|b_{\ell}\|_{L_{\infty}(J,\mathcal{L}(E_{0}))}\,\sup_{a\in[0,(a_{m}-\tau_{1})_{+}]}\|\Pi_{\ell}(a+\tau_{1},a)-\Pi_{\ell}(a+\tau_{2},a)\|_{\mathcal{L}(E_{0})}\,\|\psi\|_{\mathbb{E}_{0}}$
$\displaystyle\quad+\|b_{\ell}\|_{L_{\infty}(J,\mathcal{L}(E_{0}))}\,c_{1}\int_{(a_{m}-\tau_{1})_{+}}^{(a_{m}-\tau_{2})_{+}}\|\psi(a)\|_{E_{0}}\,\mathrm{d}a\,.$

Since $\psi=\mathbb{B}\zeta$ we get from (3.9) that

$\displaystyle\int_{(a_{m}-\tau_{1})_{+}}^{(a_{m}-\tau_{2})_{+}}\|\psi(a)\|_{E_{0}}\,\mathrm{d}a$
$\displaystyle\leq\int_{(a_{m}-\tau_{1})_{+}}^{(a_{m}-\tau_{2})_{+}}\int_{0}^{a_{m}}\|q(a,\sigma)\zeta(\sigma)\|_{E_{0}}\,\mathrm{d}\sigma\mathrm{d}a$
$\displaystyle\leq\|\zeta\|_{\mathbb{E}_{0}}\,\|q\|_{\infty}\,|(a_{m}-\tau_{2})_{+}-(a_{m}-\tau_{1})_{+}|\,.$

Gathering the previous computations and using (A.6) and $\psi=\mathbb{B}\zeta$
with $\mathbb{B}\in\mathcal{L}(\mathbb{E}_{0})$, we deduce that

$\displaystyle\|\mathsf{B}_{\psi}^{2}$
$\displaystyle(\tau_{1})-\mathsf{B}_{\psi}^{2}(\tau_{2})\|_{E_{0}}\leq\varepsilon\,\|\zeta\|_{\mathbb{E}_{0}}\,.$
(A.8)

for $\kappa\leq\tau_{2}\leq\tau_{1}\leq T$ with
$|\tau_{1}-\tau_{2}|\leq\delta$. Consequently, Lemma A.1 follows from (A.2),
(A.7), and (A.8). ∎

### Proof of Proposition 3.4

For the proof of Proposition 3.4 we suppose (3.2), (3.3), and (3.9). We have
to show that $\mathcal{V}\mathbb{S}(t)\in\mathcal{K}(\mathbb{E}_{0})$ for each
$t>0$, where

$\displaystyle\mathcal{V}\mathbb{S}(t)\zeta:=\int_{0}^{t}\mathbb{S}(t-s)\,\mathbb{B}\,\mathbb{S}(s)\zeta\,\mathrm{d}s\,,\quad\zeta\in\mathbb{E}_{0}\,.$

To this end we use Simon’s criterion for compactness in
$\mathbb{E}_{0}=L_{1}(J,E_{0})$ (see [19, Theorem 1]). Note first from (3.7)
that, for $t>0$ and $\zeta\in\mathbb{E}_{0}$,

$\displaystyle\ \|\mathcal{V}\mathbb{S}(t)\zeta\|_{\mathbb{E}_{0}}$
$\displaystyle\leq\int_{0}^{t}\|\mathbb{S}(t-s)\|_{\mathcal{L}(\mathbb{E}_{0})}\,\|\mathbb{B}\|_{\mathcal{L}(\mathbb{E}_{0})}\,\|\mathbb{S}(s)\|_{\mathcal{L}(\mathbb{E}_{0})}\,\|\zeta\|_{\mathbb{E}_{0}}\,\mathrm{d}s$
$\displaystyle\leq
M_{0}^{2}\,\|\mathbb{B}\|_{\mathcal{L}(\mathbb{E}_{0})}\,e^{\varkappa_{0}t}\,t\,\|\zeta\|_{\mathbb{E}_{0}}\,,$

so that $\mathcal{V}\mathbb{S}:(0,\infty)\to\mathcal{L}(\mathbb{E}_{0})$.

Let $t\in(0,T)$ be fixed and consider a sequence $(\phi_{j})_{j\in\mathbb{N}}$
with $\|\phi_{j}\|_{\mathbb{E}_{0}}\leq k_{0}$ for $j\in\mathbb{N}$. Then
$(\mathcal{V}\mathbb{S}(t)\phi_{j})_{j\in\mathbb{N}}$ is bounded in
$\mathbb{E}_{0}$ as just shown.

(a) We introduce

$\psi_{s}^{j}:=\mathbb{B}\,\mathbb{S}(s)\phi_{j}\,,\quad s>0\,,\quad
j\in\mathbb{N}\,.$

Note that $[s\mapsto\psi_{s}^{j}]\in C\big{(}(0,\infty),\mathbb{E}_{0}\big{)}$
with, recalling (3.9),

$\begin{split}\|\psi_{s}^{j}(a)\|_{E_{0}}&\leq\|q(a,\cdot)\|_{L_{\infty}(J,\mathcal{L}(E_{0}))}\,\|\mathbb{S}(s)\phi_{j}\|_{\mathbb{E}_{0}}\,,\quad
a\in J\,,\quad s\in(0,T]\,,\quad j\in\mathbb{N}\,,\end{split}$

Hence, invoking (3.7),

$\begin{split}\|\psi_{s}^{j}(a)\|_{E_{0}}&\leq c_{0}(T)\,k_{0}\,,\quad a\in
J\,,\quad s\in(0,T]\,,\quad j\in\mathbb{N}\,,\end{split}$ (A.9)

where $c_{0}(T):=\|q\|_{\infty}M_{0}\,e^{|\varkappa_{0}|T}$, and then

$\|\psi_{s}^{j}\|_{\mathbb{E}_{0}}\leq a_{m}\,c_{0}(T)\,k_{0}\,,\quad
s\in(0,T]\,,\quad j\in\mathbb{N}\,.$ (A.10)

Together with (A.1) this yields

$\|\mathsf{B}_{\psi_{s}^{j}}(\tau)\|_{E_{\theta}}\leq
c_{\mathsf{B}}(T)\,a_{m}\,c_{0}(T)\,k_{0}\,\tau^{-\theta}\,,\quad
s,\tau\in(0,T]\,,\quad j\in\mathbb{N}\,,\quad\theta\in\\{0,\vartheta\\}\,.$
(A.11)

Let $0<h<\min\\{a_{m}/2,t\\}$. Then, due to (3.4) we have

$\displaystyle\int_{0}^{a_{m}-h}\left\|{[\mathcal{V}\mathbb{S}(t)\phi_{j}]}(a+h)-[\mathcal{V}\mathbb{S}(t)\phi_{j}](a)\right\|_{E_{0}}\,\mathrm{d}a$
$\displaystyle\leq\int_{0}^{a_{m}-h}\int_{0}^{t}\left\|[\mathbb{S}(t-s)\psi_{s}^{j}](a+h)-[\mathbb{S}(t-s)\psi_{s}^{j}](a)\right\|_{E_{0}}\,\mathrm{d}s\,\mathrm{d}a$
$\displaystyle\leq\int_{0}^{a_{m}-h}\int_{(t-a)_{+}}^{t}\|\Pi_{\ell}(a+h,a+h-t+s)-\Pi_{\ell}(a,a-t+s)\|_{\mathcal{L}(E_{0})}\,$
$\displaystyle\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\times\|\psi_{s}^{j}(a+h-t+s)\|_{E_{0}}\,\mathrm{d}s\,\mathrm{d}a$
$\displaystyle\quad+\int_{0}^{a_{m}-h}\int_{(t-a)_{+}}^{t}\|\Pi_{\ell}(a,a-t+s)\|_{\mathcal{L}(E_{0})}\,\|\psi_{s}^{j}(a+h-t+s)-\psi_{s}^{j}(a-t+s)\|_{E_{0}}\,\mathrm{d}s\,\mathrm{d}a$
$\displaystyle\quad+\int_{0}^{a_{m}-h}\int_{(t-a-h)_{+}}^{(t-a)_{+}}\big{\|}\Pi_{\ell}(a+h,a+h-t+s)\psi_{s}^{j}(a+h-t+s)$
$\displaystyle\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad-\Pi_{\ell}(a,0)\mathsf{B}_{\psi_{s}^{j}}(t-s-a)\big{\|}_{E_{0}}\,\mathrm{d}s\,\mathrm{d}a$
$\displaystyle\quad+\int_{0}^{a_{m}-h}\int_{0}^{(t-a-h)_{+}}\big{\|}\Pi_{\ell}(a+h,0)\mathsf{B}_{\psi_{s}^{j}}(t-s-
a-h)$
$\displaystyle\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad-\Pi_{\ell}(a,0)\mathsf{B}_{\psi_{s}^{j}}(t-s-a)\big{\|}_{E_{0}}\,\mathrm{d}s\,\mathrm{d}a$
$\displaystyle=:I+II+III+IV\,.$ (A.12)

We then treat each integral separately. Let $\varepsilon>0$ be arbitrary in
the following.

(i) Choose $\kappa:=\kappa(\varepsilon,T,t)\in(0,\min\\{a_{m}/2,t\\})$ such
that

$2c_{1}\,c_{0}(T)\,k_{0}\,(T+a_{m})\,\kappa\leq\frac{\varepsilon}{6}$

and $\eta_{1}:=\eta_{1}(T,\varepsilon,\kappa)\in(0,a_{m})$ such that (see
Lemma B.1)

$\sup_{\begin{subarray}{c}(a_{1},\sigma_{1}),(a_{2},\sigma_{2})\in
S_{\kappa}\\\
|(a_{1},\sigma_{1})-(a_{2},\sigma_{2})|\leq\eta_{1}\end{subarray}}\|\Pi_{\ell}(a_{1},\sigma_{1})-\Pi_{\ell}(a_{2},\sigma_{2})\|_{\mathcal{L}(E_{0})}\,c_{0}(T)\,k_{0}\,a_{m}\,T\leq\frac{\varepsilon}{6}\,.$

Then, from (A.9), (A.3) we have, for $2h<\eta_{1}$,

$\displaystyle I$ $\displaystyle\leq
c_{0}(T)k_{0}\int_{0}^{\kappa}\int_{(t-a)_{+}}^{t}\|\Pi_{\ell}(a+h,a+h-t+s)-\Pi_{\ell}(a,a-t+s)\|_{\mathcal{L}(E_{0})}\,\mathrm{d}s\,\mathrm{d}a$
$\displaystyle\
+c_{0}(T)k_{0}\int_{\kappa}^{a_{m}-h}\int_{(t-a)_{+}}^{t-\kappa}\|\Pi_{\ell}(a+h,a+h-t+s)-\Pi_{\ell}(a,a-t+s)\|_{\mathcal{L}(E_{0})}\,\mathrm{d}s\,\mathrm{d}a$
$\displaystyle\
+c_{0}(T)k_{0}\int_{\kappa}^{a_{m}-h}\int_{t-\kappa}^{t}\|\Pi_{\ell}(a+h,a+h-t+s)-\Pi_{\ell}(a,a-t+s)\|_{\mathcal{L}(E_{0})}\,\mathrm{d}s\,\mathrm{d}a$
$\displaystyle\leq 2c_{1}c_{0}(T)k_{0}(T+a_{m})\kappa$ $\displaystyle\
+c_{0}(T)k_{0}\int_{\kappa}^{a_{m}-h}\int_{(t-a)_{+}}^{t-\kappa}\|\Pi_{\ell}(a+h,a+h-t+s)-\Pi_{\ell}(a,a-t+s)\|_{\mathcal{L}(E_{0})}\,\mathrm{d}s\,\mathrm{d}a$
$\displaystyle\leq 2c_{1}c_{0}(T)k_{0}(T+a_{m})\,\kappa$ $\displaystyle\
+c_{0}(T)\,k_{0}\,a_{m}\,T\,\sup_{\begin{subarray}{c}(a_{1},\sigma_{1}),(a_{2},\sigma_{2})\in
S_{\kappa}\\\
|(a_{1},\sigma_{1})-(a_{2},\sigma_{2})|\leq\eta_{1}\end{subarray}}\|\Pi_{\ell}(a_{1},\sigma_{1})-\Pi_{\ell}(a_{2},\sigma_{2})\|_{\mathcal{L}(E_{0})}$

and therefore

$\displaystyle I$ $\displaystyle\leq\frac{\varepsilon}{3}\,,\quad
2h<\eta_{1}\,.$ (A.13)

(ii) We choose $\eta_{2}:=\eta_{2}(T,\varepsilon)>0$ according to (3.9) such
that

$c_{1}\sup_{\begin{subarray}{c}\tau_{1},\tau_{2}\in J\\\
|\tau_{1}-\tau_{1}|\leq\eta_{2}\end{subarray}}\left\|q(\tau_{1},\cdot)-q(\tau_{2},\cdot)\right\|_{L_{\infty}(J,\mathcal{L}(E_{0}))}a_{m}\,T\,M_{0}\,e^{|\varkappa_{0}|T}\,k_{0}\leq\frac{\varepsilon}{3}\,.$

We then use (A.3), recall $\psi_{s}^{j}=\mathbb{B}\,\mathbb{S}(s)\phi_{j}$,
and invoke (3.9) and (3.7) to get, for $h<\eta_{2}$,

$\displaystyle II$ $\displaystyle\leq
c_{1}\int_{0}^{a_{m}-h}\int_{(t-a)_{+}}^{t}\int_{0}^{a_{m}}\left\|\big{(}q(a+h-t+s,\sigma)-q(a-t+s,\sigma)\big{)}[\mathbb{S}(s)\phi_{j}](\sigma)\right\|_{E_{0}}\,\mathrm{d}\sigma\mathrm{d}s\mathrm{d}a$
$\displaystyle\leq
c_{1}\int_{0}^{a_{m}-h}\int_{(t-a)_{+}}^{t}\left\|q(a+h-t+s,\cdot)-q(a-t+s,\cdot)\right\|_{L_{\infty}(J,\mathcal{L}(E_{0}))}\,\|\mathbb{S}(s)\phi_{j}\|_{\mathbb{E}_{0}}\,\mathrm{d}s\,\mathrm{d}a$
$\displaystyle\leq c_{1}\sup_{\begin{subarray}{c}\tau_{1},\tau_{2}\in J\\\
|\tau_{1}-\tau_{1}|\leq\eta_{2}\end{subarray}}\left\|q(\tau_{1},\cdot)-q(\tau_{2},\cdot)\right\|_{L_{\infty}(J,\mathcal{L}(E_{0}))}a_{m}\,T\,M_{0}\,e^{|\varkappa_{0}|T}\,k_{0}$

and therefore

$\displaystyle II$ $\displaystyle\leq\frac{\varepsilon}{3}\,,\quad
h<\eta_{2}\,.$ (A.14)

(iii) It follows from (A.3), (A.9), and (A.11) that

$\displaystyle III$ $\displaystyle\leq
c_{1}\,c_{0}(T)\,k_{0}\,a_{m}\big{(}1+c_{\mathsf{B}}(T)\,a_{m}\big{)}\,h\,.$
(A.15)

(iv) Finally, we choose
$\bar{\kappa}:=\bar{\kappa}(\varepsilon,T)\in(0,a_{m}/2)$ such that

$2c_{1}\,a_{m}^{2}\,c_{\mathsf{B}}(T)\,c_{0}(T)\,k_{0}\,\bar{\kappa}\leq\frac{\varepsilon}{6}$

and invoke then (A.9) and Lemma A.1 to find
$\eta_{3}:=\eta_{3}(T,\varepsilon)>0$ such that

$\displaystyle\|\mathsf{B}_{\psi_{s}^{j}}(\tau_{1})-\mathsf{B}_{\psi_{s}^{j}}(\tau_{2})\|_{E_{0}}\leq\frac{\varepsilon}{6c_{1}a_{m}T}\,,\qquad\tau_{1},\tau_{2}\in[\bar{\kappa},T]\
\text{ with }\ |\tau_{1}-\tau_{2}|\leq\eta_{3}\,.$

Let $C(\vartheta)>0$ be the constant from (B.4) (for $\Pi$ replaced by
$\Pi_{\ell}$). Using the previous estimate along with (A.3) and (A.11) we then
derive, for $h<\eta_{3}$,

$\displaystyle IV$
$\displaystyle\leq\int_{0}^{a_{m}-h}\int_{0}^{(t-a-h-\bar{\kappa})_{+}}\left\|\Pi_{\ell}(a+h,0)-\Pi_{\ell}(a,0)\right\|_{\mathcal{L}(E_{\vartheta}E_{0})}\|\mathsf{B}_{\psi_{s}^{j}}(t-s-
a-h)\|_{E_{\vartheta}}\,\mathrm{d}s\,\mathrm{d}a$
$\displaystyle\quad+\int_{0}^{a_{m}-h}\int_{0}^{(t-a-h-\bar{\kappa})_{+}}\|\Pi_{\ell}(a,0)\|_{\mathcal{L}(E_{0})}\,\|\mathsf{B}_{\psi_{s}^{j}}(t-s-
a-h)-\mathsf{B}_{\psi_{s}^{j}}(t-s-a)\|_{E_{0}}\,\mathrm{d}s\,\mathrm{d}a$
$\displaystyle\quad+\int_{0}^{a_{m}-h}\int_{(t-a-h-\bar{\kappa})_{+}}^{(t-a-h)_{+}}\Big{\\{}\|\Pi_{\ell}(a+h,0)\|_{\mathcal{L}(E_{0})}\,\|\mathsf{B}_{\psi_{s}^{j}}(t-s-
a-h)\|_{E_{0}}$
$\displaystyle\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad+\|\Pi_{\ell}(a,0)\|_{\mathcal{L}(E_{0})}\,\|\mathsf{B}_{\psi_{s}^{j}}(t-s-a)\|_{E_{0}}\Big{\\}}\,\mathrm{d}s\,\mathrm{d}a$
$\displaystyle\leq
C(\vartheta)\,h^{\vartheta}\,c_{\mathsf{B}}(T)\,a_{m}\,c_{0}(T)\,k_{0}\,\int_{0}^{a_{m}-h}\int_{0}^{(t-a-h-\bar{\kappa})_{+}}(t-s-
a-h)^{-\vartheta}\,\mathrm{d}s\,\mathrm{d}a$
$\displaystyle\quad+c_{1}\,a_{m}\,T\,\frac{\varepsilon}{6c_{1}a_{m}T}+2c_{1}\,a_{m}^{2}\,c_{\mathsf{B}}(T)\,c_{0}(T)\,k_{0}\,\bar{\kappa}$

and therefore

$\displaystyle IV$
$\displaystyle\leq\frac{\varepsilon}{3}+C(\vartheta)\,c_{\mathsf{B}}(T)\,a_{m}\,c_{0}(T)\,k_{0}\,a_{m}\,T^{1-\vartheta}\,h^{\vartheta}\,,\quad
h<\eta_{3}\,.$ (A.16)

(v) Consequently, we conclude from (A.12)-(A.16) that

$\varlimsup_{h\to
0}\,\sup_{j\in\mathbb{N}}\int_{0}^{a_{m}-h}\left\|{[\mathcal{V}\mathbb{S}(t)\phi_{j}]}(a+h)-[\mathcal{V}\mathbb{S}(t)\phi_{j}](a)\right\|_{E_{0}}\,\mathrm{d}a\leq\varepsilon$

and thus, since $\varepsilon>0$ was arbitrary,

$\displaystyle\lim_{h\to
0}\,\sup_{j\in\mathbb{N}}\int_{0}^{a_{m}-h}\left\|{[\mathcal{V}\mathbb{S}(t)\phi_{j}]}(a+h)-[\mathcal{V}\mathbb{S}(t)\phi_{j}](a)\right\|_{E_{0}}\,\mathrm{d}a=0\,.$
(A.17)

(b) Finally, for $j\in\mathbb{N}$ we have from (B.2), (A.11), and (A.9) that

$\displaystyle\int_{0}^{a_{m}}\|[\mathcal{V}\mathbb{S}(t)\phi_{j}](a)\|_{E_{\vartheta}}\,\mathrm{d}a$
$\displaystyle\leq\int_{0}^{a_{m}}\int_{0}^{(t-a)_{+}}\|\Pi(a,0)\|_{\mathcal{L}(E_{0},E_{\vartheta})}\,\|\mathsf{B}_{\psi_{s}^{j}}(t-s-a)\|_{E_{0}}\,\mathrm{d}s\,\mathrm{d}a$
$\displaystyle\
+\int_{0}^{a_{m}}\int_{(t-a)_{+}}^{t}\|\Pi(a,a-t+s)\|_{\mathcal{L}(E_{0},E_{\vartheta})}\,\|\psi_{s}^{j}(a-t+s)\|_{E_{0}}\,\mathrm{d}s\,\mathrm{d}a$
$\displaystyle\leq
M_{\vartheta}\big{(}c_{\mathsf{B}}(T)\,a_{m}+1\big{)}c_{0}(T)\,k_{0}\int_{0}^{a_{m}}e^{\varpi
a}\,a^{-\vartheta}\,\mathrm{d}a\,.$

Since the right-hand side is finite and due to the compact embedding of
$E_{\vartheta}$ in $E_{0}$ we conclude that

$\displaystyle\left\\{\int_{a_{1}}^{a_{2}}[\mathcal{V}\mathbb{S}(t)\phi_{j}](a)\,\mathrm{d}a\,;\,j\in\mathbb{N}\right\\}\
\text{ is relatively compact in $E_{0}$ for $0<a_{1}<a_{2}<a_{m}$}\,.$ (A.18)

We now infer from (A.17), (A.18), and [19, Theorem 1] that the sequence
$(\mathcal{V}\mathbb{S}(t)\phi_{j})_{j\in\mathbb{N}}$ is relatively compact in
$\mathbb{E}_{0}=L_{1}(J,E_{0})$. Since $t>0$ was arbitrary, this yields
Proposition 3.4. ∎

##  Appendix B Parabolic Evolution Operators

Parabolic evolution operators are thoroughly treated in [3] to which we refer.
We only provide here their most important properties that we have used in the
previous sections.

### Basic Definition

Let $E_{1}\hookrightarrow E_{0}$ be a densely injected Banach couple,
$J=[0,a_{m}]$, and

$J_{\Delta}:=\\{(a,\sigma)\in J\times J\,;\,0\leq\sigma\leq a\\}\,,\quad
J_{\Delta}^{*}:=\\{(a,\sigma)\in J\times J\,;\,0\leq\sigma<a\\}\,.$

We consider

$A:J\to\mathcal{A}(E_{0})$

with $\mathrm{dom}(A(a))=E_{1}$ for each $a\in J$, where $\mathcal{A}(E_{0})$
means the closed linear operators in $E_{0}$.

Following [3, Section II.2.1] we say that $A$ generates a parabolic evolution
operator $\Pi$ on $E_{0}$ with regularity subspace $E_{1}$, provided that
$\Pi:J_{\Delta}\to\mathcal{L}(E_{0})$ is such that

$\Pi\in C\big{(}J_{\Delta},\mathcal{L}_{s}(E_{0})\big{)}\cap
C\big{(}J_{\Delta}^{*},\mathcal{L}(E_{0},E_{1})\big{)}$ (B.1a) satisfying
$\Pi(a,a)=1_{E_{0}}\,,\qquad\Pi(a,\sigma)=\Pi(a,s)\Pi(s,\sigma)\,,\quad(a,s),(s,\sigma)\in
J_{\Delta}\,,$ (B.1b) and, for $a\in J$, $\Pi(\cdot,a)\in
C^{1}\big{(}(a,a_{m}),\mathcal{L}(E_{0})\big{)}\,,\quad\Pi(a,\cdot)\in
C^{1}\big{(}[0,a),\mathcal{L}_{s}(E_{1},E_{0})\big{)}$ (B.1c) with
$\partial_{1}\Pi(a,\sigma)=A(a)\Pi(a,\sigma)\,,\quad\partial_{2}\Pi(a,\sigma)=-\Pi(a,\sigma)A(\sigma)\,,\quad(a,\sigma)\in
J_{\Delta}^{*}\,.$ (B.1d)

In the following, let

$A\in C^{\rho}\big{(}J,\mathcal{H}(E_{1},E_{0})\big{)}$

be fixed with $\rho>0$. Then [3, II. Corollary 4.4.2] ensures that $A$
generates a unique parabolic evolution operator $\Pi$ on $E_{0}$ with
regularity subspace $E_{1}$ in the above sense.

### Basic Estimates

Given an interpolation space $E_{\theta}=(E_{0},E_{1})_{\theta}$ with
$\theta\in[0,1]$, there are $\varpi\in\mathbb{R}$ and $M_{\theta}\geq 1$ such
that

$\|\Pi(a,\sigma)\|_{\mathcal{L}(E_{\theta})}+(a-\sigma)^{\theta}\,\|\Pi(a,\sigma)\|_{\mathcal{L}(E_{0},E_{\theta})}\leq
M_{\theta}e^{\varpi(a-\sigma)}\,,\qquad(a,\sigma)\in J_{\Delta}\,,$ (B.2)

according to [3, II. Lemma 5.1.3].

### Solvability of Cauchy Problems

For $x\in E_{0}$ and $f\in L_{1}(J,E_{0})$, the mild solution $v\in
C(J,E_{0})$ to the Cauchy problem

$\partial_{a}v=A(a)v+f(a)\,,\quad a\in\dot{J}:=(0,a_{m}]\,,\qquad v(0)=x\,,$
(B.3a) is given by
$v(a)=\Pi(a,0)x+\int_{0}^{a}\Pi(a,\sigma)\,f(\sigma)\,\mathrm{d}\sigma\,,\quad
a\in J\,.$ (B.3b)

If $x\in E_{\vartheta}$ for some $\vartheta\in[0,1]$ and $f\in
C^{\theta}(J,E_{0})+C(J,E_{\theta})$ with $\theta\in(0,1]$ (with admissible
interpolation functors), then

$v\in C(J,E_{\vartheta})\cap C^{1}(\dot{J},E_{0})\cap C(\dot{J},E_{1})$

is a strong solution to (B.3a). Actually, if, in addition, $x\in E_{1}$, then

$v\in C^{1}(J,E_{0})\cap C(J,E_{1})\,.$

See [3, II. Theorem 1.2.1, Theorem 1.2.2].

### Continuity Properties

Given $\vartheta\in(0,1)$, there is $C(\vartheta)>0$ such that

$\|\Pi(a+h,0)-\Pi(a,0)\|_{\mathcal{L}(E_{\vartheta},E_{0})}\leq
C(\vartheta)h^{\vartheta}\,,\quad 0\leq a\leq a+h\leq a_{m}\,,$ (B.4)

according to [3, II. Equation (5.3.8)]. Moreover:

######  Lemma B.1.

For $\varepsilon>0$ and $\kappa\in(0,a_{m})$ given, there is
$\eta:=\eta(\varepsilon,\kappa)>0$ such that

$\|\Pi(a_{1},\sigma_{1})-\Pi(a_{2},\sigma_{2})\|_{\mathcal{L}(E_{0})}\leq\varepsilon\,,\quad(a_{1},\sigma_{1}),(a_{2},\sigma_{2})\in
S_{\kappa}\,,\quad|(a_{1},\sigma_{1})-(a_{2},\sigma_{2})|\leq\eta\,,$

where $S_{\kappa}:=\\{(a,\sigma)\in\Delta_{J}^{*}\,;\,\kappa\leq a-\sigma\\}.$

###### Proof.

This follows from the fact that $\Pi\in
C\big{(}\Delta_{J}^{*},\mathcal{L}(E_{0})\big{)}$ is uniformly continuous on
the compact subset $S_{\kappa}$ of $\Delta_{J}^{*}$. ∎

### Positivity

If $E_{0}$ is an ordered Banach space and $A(a)$ is resolvent positive555An
operator $A\in\mathcal{A}(E_{0})$ is resolvent positive, if there is
$\lambda_{0}\geq 0$ such that $(\lambda_{0},\infty)\subset\rho(A)$ and
$(\lambda-A)^{-1}\in\mathcal{L}_{+}(E_{0})\,,\quad\lambda>\lambda_{0}\,.$ for
each $a\in J$, then [3, II. Theorem 6.4.1, Theorem 6.4.2] imply that the
evolution operator $\Pi$ is positive, that is,
$\Pi(a,\sigma)\in\mathcal{L}_{+}(E_{0})$ for each $(a,\sigma)\in J_{\Delta}$.

## Declarations

### Ethical Approval

The submitted work is original and has not been published elsewhere in any
form or language (partially or in full).

### Competing interests

Not applicable.

### Funding

Not applicable.

### Availability of data and materials

Not applicable.

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