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.
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1 Introduction
The dynamics of a population structured by space and age is described by a density function \(u=u(t,a,x)\ge 0\), where \(t\ge 0\) 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
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 \(m=m({\bar{u}}(t,x),a)\ge 0\) and the birth rate \(b=b({\bar{u}}(t,x),a)\ge 0\) depend on this quantity and on age, the governing equations for the density u are
where
means either Dirichlet boundary conditions \(u\vert _{\partial \Omega }=0\) 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. [25, 30] 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, 10, 11, 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
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
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
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
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 notationFootnote 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
with
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
2.1 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
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
we mean by \(D({\mathcal {A}})\) its domain \(\textrm{dom}({\mathcal {A}})\) 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
and observe \(\mathbb {E}_\theta \hookrightarrow \mathbb {E}_0\) for \(\theta \in [0,1]\). We assume that there is \(\rho >0\) such that
and
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
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).
2.2 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
where \(C_b^{1-}\) stands for locally Lipschitz continuous maps that are bounded on bounded sets. The weight function \(\varrho \) is such that
for some \(\vartheta \in (0,1)\) (if \(\alpha \in (0,1)\), then it suffices to take \(\vartheta =\alpha \)). We use the notation
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
where \(F(u):=-m({\bar{u}},\cdot )u\) and
for \(v:\mathbb {R}^+\rightarrow \mathbb {E}_\alpha \), and where \(B_u\) satisfies the Volterra equation
for \(t\ge 0\) (we set \(b({\bar{v}},a):=0\) whenever \(a\notin J\)). That is, \(u(t,0)=B_u(t)\) for \(t\ge 0\) by (2.3a), while (2.3c) implies
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
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]. \(\square \)
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 Sect. 5.
2.3 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
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 \(t\ge 0\), 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
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.
2.3.1 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
such that for \(v\in \mathbb {E}_0\) we can write (with \(\partial \) indicating Fréchet derivatives with respect to \({\bar{\phi }}\))
and
where for the reminder terms \(R_m:\mathbb {E}_\alpha \rightarrow \mathbb {E}_0\) and \(R_b:\mathbb {E}_\alpha \rightarrow \mathbb {E}_0\) 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
and
For technical reasons, we assume for the birth rate that (for some \(\vartheta \in (0,1)\), see (2.2c))
and
while for the death rate we impose that
for some \(\beta \in [0,1)\) and
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 Sect. 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
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
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 \(\textrm{Re}\,\lambda < 0\) for any eigenvalue \(\lambda \) to (2.7), then \(\phi \) is exponentially asymptotically stable in \(\mathbb {E}_\alpha \).
- (b):
-
If \(\textrm{Re}\,\lambda > 0\) 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 Sect. 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 Sect. 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).
2.4 Paper Outline
In Sect. 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 Sect. 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 Sect. 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 Sect. 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
where \(A_\ell \) satisfies
for some \(\rho >0\) and where we impose for the birth rate that there is \(\vartheta \in (0,1)\) with
We then denote by
the parabolic evolution operator on \(E_0\) with regularity subspace \(E_1\) generated by \(A_\ell \) (see Appendix B) and use the notation
3.1 Preliminaries
We first recall some of the results from [26]. Formal integrating of (3.1a) along characteristics entails that the solution
to (3.1) for given \(\psi \in \mathbb {E}_0=L_1(J,E_0)\) is of the form
with \({\textsf{B}}_\psi :=u(\cdot ,0)\) satisfying the linear Volterra equation
where \(\chi \) is the characteristic function of the interval \((0,a_m)\). Note that \({\textsf{B}}_\psi \) is such that
It follows from [26] that there is a unique solution
to (3.4b) and that \((\mathbb {S}(t))_{t\ge 0}\) 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
(a) \((\mathbb {S}(t))_{t\ge 0}\) 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 \((\mathbb {S}(t))_{t\ge 0}\) is positive.
(b) In fact, given \(\alpha \in [0,1)\), the restriction \((\mathbb {S}(t)\vert _{\mathbb {E}_\alpha })_{t\ge 0}\) defines a strongly continuous semigroup on \(\mathbb {E}_\alpha \), and there are \(M_\alpha \ge 1\) and \(\varkappa _\alpha \in \mathbb {R}\) such that
(c) Denote by \(\mathbb {A}\) the infinitesimal generator of the semigroup \((\mathbb {S}(t))_{t\ge 0}\) on \(\mathbb {E}_0\). Then \(\psi \in \textrm{dom}(\mathbb {A})\) if and only if there exists \(\zeta \in \mathbb {E}_0\) such that \(\psi \in C(J,E_0)\) is the mild solution to
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]Footnote 2. \(\square \)
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
The spectrum of the generator \(\mathbb {A}\) has been investigated in [26] when the generated semigroup \((\mathbb {S}(t))_{t\ge 0}\) is positive. In the following, let
be the spectral bound of the generator \(\mathbb {A}\) and
be the growth bound of the corresponding semigroup \(\mathbb {S}(t)=e^{t\mathbb {A}}\), \(t\ge 0\).
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 positiveFootnote 3 for a in a subset of J of positive measure. Then
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
Proof
This is [26, Corollary 4.3]. \(\square \)
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
with \(q(a,\sigma )=q(a)(\sigma )\) satisfying
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
For the birth rate we impose additionally that
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 \((\mathbb {S}(t))_{t\ge 0}\) defined in (3.4).Then the semigroup \((\mathbb {T}(t))_{t\ge 0}\) on \(\mathbb {E}_0\) generated by \(\mathbb {G}:=\mathbb {A}+\mathbb {B}\) is eventually compact. In particular,
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 \(\{\lambda \in \sigma (\mathbb {G})\,;\,\textrm{Re}\, \lambda \ge r\}\) 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 \((\mathbb {T}(t))_{t\ge 0}\) 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
Then \({\mathcal {V}}\mathbb {S}:(0,\infty )\rightarrow {\mathcal {K}}(\mathbb {E}_0)\), 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 \((\mathbb {S}(t))_{t\ge 0}\) 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.
3.3 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 \((\mathbb {S}(t))_{t\ge 0}\) 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 \((\mathbb {T}(t))_{t\ge 0}\) 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. \(\square \)
We can derive now also properties of the semigroup \((\mathbb {T}(t))_{t\ge 0}\) 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 \((\mathbb {S}(t))_{t\ge 0}\) defined in (3.4). Let \((\mathbb {T}(t))_{t\ge 0}\) 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 \(\left( \mathbb {T}(t)\vert _{\mathbb {E}_\alpha }\right) _{t\ge 0}\) 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 \(N_\alpha \ge 1\) such that
Proof
(i) It was shown in [26, Theorem 1.2] that the semigroup \((\mathbb {T}(t))_{t\ge 0}\) on \(\mathbb {E}_0\) generated by \(\mathbb {G}=\mathbb {A}+\mathbb {B}\) is given by
and that there are \(C_\alpha \ge 1\) and \(\varsigma _\alpha \ge 0\) such that
We then infer from (3.7) and (3.12) that
for \(\phi \in \mathbb {E}_\alpha \) so that Gronwall’s inequality implies
for some \(c_\alpha \ge 1\) and \(\omega _1>0\). Moreover, since
for \(t\ge 0\) and \(\phi \in \mathbb {E}_\alpha \), the strong continuity of \(\left( \mathbb {T}(t)\vert _{\mathbb {E}_\alpha }\right) _{t\ge 0}\) on \(\mathbb {E}_\alpha \) follows from the strong continuity of \((\mathbb {S}(t)\vert _{\mathbb {E}_\alpha })_{t\ge 0}\) on \(\mathbb {E}_\alpha \) guaranteed by Theorem 3.1 and from (3.7) and (3.14). Therefore, \(\left( \mathbb {T}(t)\vert _{\mathbb {E}_\alpha }\right) _{t\ge 0}\) is a strongly continuous semigroup on \(\mathbb {E}_\alpha \). Writing
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 \(\left( \mathbb {T}(t)\vert _{\mathbb {E}_\alpha }\right) _{t\ge 0}\) 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 \(\mathbb {T}(t)\vert _{\mathbb {E}_\alpha }=e^{t\mathbb {G}_\alpha }\) for \(t\ge 0\). 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 \(\zeta \in \textrm{dom} (\mathbb {G}_\alpha )\). Then
hence \(\zeta \in \textrm{dom} (\mathbb {G})\) and
Consequently, \(\zeta \in \textrm{dom} (\mathbb {G}_{\mathbb {E}_\alpha })\) and \(\mathbb {G}_{\mathbb {E}_\alpha }\zeta =\mathbb {G}_\alpha \zeta \).
Conversely, let \(\psi \in \textrm{dom} (\mathbb {G}_{\mathbb {E}_\alpha })\) 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
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 \(\zeta \in \textrm{dom} (\mathbb {G}_{\alpha })\) 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 \(\psi =\zeta \in \textrm{dom} (\mathbb {G}_\alpha )\). 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 \(\psi \in \textrm{dom} (\mathbb {G})\setminus \{0\}\) with
by Theorem 3.1. From (ii) we deduce that \(\psi \in \textrm{dom} (\mathbb {G}_\alpha )\) 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 \(\zeta \in \textrm{dom} (\mathbb {G}_\alpha )\setminus \{0\}\) 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
Thus, for \(\omega -\varepsilon >s(\mathbb {G})\) with \(\varepsilon >0\) there is \(c_\alpha \ge 1\) such that
On the one hand, it follows from (3.13) and (3.16) for \(t> 1\) that
On the other hand, due to (3.13) we have, for \(0<t\le 1\),
Consequently, there is \(N_\alpha \ge 1\) such that
and the assertion follows. \(\square \)
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 Sect. 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
In the following, we demonstrate that this linear problem fits into the framework of Sect. 3. More precisely, the solution v is given by a semigroup \((\mathbb {T}_\phi (t))_{t\ge 0}\) 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 \((\mathbb {T}_\phi (t))_{t\ge 0}\), 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
where we set
so that
according to (2.5g), (2.5h), and (2.2c). We also introduce
and infer from (2.1), (2.5i), and [3, I.Theorem 1.3.1] that
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
and infer from (2.5a), (2.5j), and (2.2c) that
for \(\zeta \in \mathbb {E}_0\), where \(q\in C\big (J,C(J,\mathcal {L}(E_0))\big )\) is given by
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
Theorem 3.3 implies that \((\mathbb {T}_\phi (t))_{t\ge 0}\) 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 \(\textrm{Re}\,\lambda < 0\) for any \(\lambda \in \sigma _p(\mathbb {G}_\phi )\), then \(\phi \) is exponentially asymptotically stable in \(\mathbb {E}_\alpha \).
- (b):
-
If \(\textrm{Re}\,\lambda > 0\) for some \(\lambda \in \sigma _p(\mathbb {G}_\phi )\), then \(\phi \) is unstable in \(\mathbb {E}_\alpha \).
4.1 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
Now, the supposition \(\textrm{Re}\,\lambda < 0\) 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). \(\square \)
4.2 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
is nonempty and finite; that is, \(\Sigma _+\) is a bounded spectral set. Let
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
such that \(\mathbb {G}_\phi \vert _{\mathbb {E}_0^1}\in \mathcal {L}(\mathbb {E}_0^1)\). Moreover, there are \(M\ge 1\) and \(\delta >0\) such that
and
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 \(\textrm{dom}(\mathbb {G}_\phi )=\textrm{dom}(\mathbb {A}_\phi )\subset \mathbb {E}_\alpha \) by Theorem 3.1, see also [26, Corollary 3.4]) and such that
Moreover,
with
and
Choose \(\delta >0\) such that
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
for \(t\ge 0\). In fact, \(e^{t\mathbb {G}_\phi ^1}\) is extended to \(\mathbb {R}\) by (see also [15, Proposition 2.3.3])
where \(\Gamma \) is a positively oriented smooth curve in \(\rho (\mathbb {G}_\phi )\) enclosing \(\Sigma _+\) with \(\textrm{Re}\,\lambda \ge \omega +\delta \) for every \(\lambda \in \Gamma \). Using (4.7), we have, for \(\psi \in \mathbb {E}_0\),
Similarly, since
we have
Combining the two estimates we find \(N\ge 1\) such that
Consequently, since \(D(\mathbb {G}_\phi )\hookrightarrow \mathbb {E}_\alpha \) according to [26, Corollary 3.4], we deduce (4.4b).
Finally, since \((e^{t\mathbb {G}_\phi })_{t\ge 0}\) is an eventually compact semigroup on \(\mathbb {E}_0\) by Theorem 3.3, it follows from (4.6) that also \((e^{t\mathbb {G}_\phi ^2})_{t\ge 0}\) 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 \(N_1\ge 1\) such that
Noticing that also
for some \(\omega _1>0\) and \(N_2\ge 1\) 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. \(\square \)
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
where \(B_{0,0}^{\gamma ,h}\in C([0,T],E_0)\) satisfies
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
for some \(M_*\ge 1\) (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
Proof
This is [28, Lemma 5.7]. \(\square \)
Now, let \(u_0\in \mathbb {E}_\alpha \) be arbitrary and set
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
where \(h_w\in C(I(u_0),E_0)\) is defined as
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 \((\mathbb {T}_\phi (t))_{t\ge 0}\). 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
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
from (4.10).
Proof
This is [28, Proposition 6.1]. \(\square \)
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
and set
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
In order to prove the instability of \(\phi \), we now show the existence of a sequence \((u_0^k)_{k\ge 1}\) such that \(u_0^k\rightarrow \phi \) in \(\mathbb {E}_\alpha \) and
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 \(k\ge 1\), there exists a unique function \(v_k\in C\big ((-\infty ,k],\mathbb {E}_\alpha \big )\) such that
for \(t\le k\) and satisfying
Proof
Let \(k\ge 1\) be fixed. We introduce the complete metric space \(Z=(Z,d_Z)\) by
equipped with the metric
and claim that
defines a contraction \(H:Z\rightarrow Z\). Indeed, for \(v\in Z\) we have \(\Vert v(t)\Vert _{\mathbb {E}_\alpha }\le r\) for \(t\le k\) and thus, invoking (2.5e),
and, together with (4.12d),
Therefore, (4.11) implies for \(v\in Z,\) \(t\le k\), and \(\theta \in \{0,\alpha \}\) that
We then use (4.4), (4.18), and (4.14) to derive
so that (4.15) implies
That is, \(H:Z\rightarrow Z\). Next, notice from (2.5f) that, for \(v_1, v_2\in Z\),
while from (4.12d) and (2.5f),
Therefore, since the mapping \([h\rightarrow W_{0,0}^{\gamma ,h}]\) is linear, it follows from (4.11) that
Using then (4.19) we derive similarly as above that, for \(v_1, v_2\in Z\),
so that (4.15) implies
Consequently, \(H:Z\rightarrow Z\) is indeed a contraction, and Lemma 4.5 follows from Banach’s fixed point theorem. \(\square \)
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
for \(0\le t\le k\).
Proof
Define for \(t\le k\)
so that
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 \(\textrm{dom}(\mathbb {G}_\phi )\) to justify the formal computation
Thus, for \(0\le t\le k\),
and since \(p(0)=W_{0,0}^{\gamma ,h_{v_k}}(0,\cdot )=0\), the assertion follows. \(\square \)
We are now in a position to provide the proof of Theorem 4.1 (b).
4.3 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
and note from (4.17) that
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
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 (4.16), (4.4), and (4.18) that
where, due to (4.15) and (4.14),
Consequently, we have shown that there exists a sequence \((u_0^k)_{k\ge 1}\) such that \(u_0^k\rightarrow \phi \) in \(\mathbb {E}_\alpha \) as \(k\rightarrow \infty \) while \(\Vert u(k;u_0^k)-\phi \Vert _{\mathbb {E}_\alpha }\ge \xi _0\) for \(k\ge 1\). This proves that \(\phi \) is unstable in \(\mathbb {E}_\alpha \) and thus Theorem 4.1 (b).\(\square \)
4.4 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 \(\psi \in \textrm{dom}(\mathbb {A}_\phi )\) 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)
Note that (4.20a) entails
which, when plugged into (4.20b), yields
Recall from (4.1c) that
We thus introduce
and then obtain from (4.22) that
Moreover, (4.21) implies that \({\bar{\psi }}\) satisfies
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
where \(K_{\phi ,\lambda }\) is defined in (4.23) with \(\Pi _{\phi }\) denoting the evolution operator associated with \(A_\phi \) given by
and
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 \(\textrm{Re}\,\lambda < 0\) 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 \(\textrm{Re}\,\lambda > 0\) 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
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 requiredFootnote 4. In the following, \(\Omega \subset \mathbb {R}^n\) is a bounded domain with smooth boundary and outer unit normal \(\nu \). We consider
where
either refers to Dirichlet boundary conditions \(u\vert _{\partial \Omega }=0\) if \(\delta =0\) or Neumann boundary conditions \(\partial _\nu u=0\) if \(\delta =1\) and
We set \(J=[0,a_m]\) and assume for the data that
Note that one may choose \(2\alpha =1\) in the following. We introduce \(E_0:=L_q(\Omega )\) and
Then \(E_1\) is compactly embedded in the Banach lattice \(E_0\) and, for real interpolation with \(\theta \in (0,1)\setminus \{1/2\}\),
while, for complex interpolation with \(\theta =1/2\),
Setting
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
with
and correspondingly for m. In particular, using the continuity of pointwise multiplication
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
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
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
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
and
Moreover,
5.1 The Trivial Equilibrium
For the particular case of the trivial equilibrium \(\phi =0\), we observe that (with dot referring to the suppressed x-variable)
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
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
where
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
is continuous and strictly decreasing with
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
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
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 \({\textbf{1}}:=[x\mapsto 1]\) belongs to \(W_{q,N}^2(\Omega )\) (with subscript N referring to the Neumann boundary conditions) and satisfies \(\Pi _*(a,0){\textbf{1}}={\textbf{1}}\). Therefore,
so that \({\textbf{1}}\) 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
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
- (a):
-
If \(r_0<1\), then the trivial equilibrium \(\phi =0\) to (5.1) is exponentially asymptotically stable in \(L_1\big (J,W_{q,N}^{2\alpha }(\Omega )\big )\).
- (b):
-
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
then
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
and
Moreover, assume that there is \(C_*>0\) such that
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 \(u_0\ge 0\) and \(u(t;u_0)\rightarrow 0\) in \(L_1(J,L_q(\Omega ))\) as \(t\rightarrow \infty \).
Proof
Note that
and \(b_\ell :=b_*\) satisfy (3.2) and (3.3). We then denote by \((\mathbb {S}(t))_{t\ge 0}\) 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 \((\mathbb {S}(t))_{t\ge 0}\) 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 \(u_0\ge 0\) 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)
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
Here, in a mild sense means that u satisfies
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 \(f(t,\cdot ,\cdot )\le 0\) and \(W_{0,0}^{0,h}(t,\cdot )\le 0\) for \(t\in I(u_0)\) due to (5.3a) and (4.10), it follows from (5.4) and the positivity of \((\mathbb {S}(t))_{t\ge 0}\) that
in the Banach lattice \(\mathbb {E}_0=L_1(J,L_q(\Omega ))\). Therefore,
Assumption (5.3c) ensures that there is a constant \(C_1>0\) such that
while (3.7) yields for every \(T>0\) a constant \(c(T)>0\) such that
It then readily follows from (5.4), (5.6), (5.7), Lemma 4.3, and Gronwall’s inequality that
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 \(t\!\rightarrow \! \infty \) in (5.5) and use \(\omega _0<0\) to conclude that \(u(t;u_0)\!\rightarrow \! 0\) in the phase space \(\mathbb {E}_0=L_1(J,L_q(\Omega ))\). \(\square \)
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).
5.2 An Instability Result
We provide the analogue to [16, Theorem 6]:
Proposition 5.4
Assume (5.2). Consider a positive equilibrium
to (5.1) for which
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 \(\mathbb {B}_\phi \ge 0\), it follows from [4, Proposition 12.11] that \(s(\mathbb {G}_\phi )\ge s(\mathbb {A}_\phi )\). 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
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
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 \(s(\mathbb {G}_\phi )\ge \lambda _0\) 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
and note that \(v\ge 0\) 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)
Thus, since \(\mathbb {B}_\phi v\ge 0\), we deduce
and, when plugging this into the initial condition,
where we used that \(b_\phi (a)\ge b({\bar{\phi }},a)\) due to (5.8). Now, the equilibrium \(\phi \) satisfies
Therefore, using the evolution property
we infer that
and thus
From (5.10) and (5.12) it then follows that
and thus, since \(\big (1-Q_\lambda (\phi )\big )^{-1}\ge 0\) as \(r(Q_{\lambda }(\phi ))<1\), we conclude that \(\lambda v(0)\ge \phi (0)\). Using this along with (5.11) in (5.9) we derive \(\lambda v(a)\ge \phi (a)\) for \( a\in J\). By definition of v, this means that
Invoking the exponential representation of the semigroup we obtain
from which \(\Vert e^{t\mathbb {G}_\phi }\Vert _{\mathcal {L}(\mathbb {E}_0)}\ge 1\) for \(t\ge 0\), since \(\mathbb {E}_0=L_1(J,L_q(\Omega ))\) is a Banach lattice. This implies that \(s(\mathbb {G}_\phi )=\omega _0(\mathbb {G}_\phi )\ge 0\). 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. \(\square \)
A simple consequence is:
Corollary 5.5
Assume (5.2). Consider a positive equilibrium
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 \(s(\mathbb {G}_\phi )\ge \lambda _0\), 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 \(\lambda _0>0\). \(\square \)
Notes
We 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 \(u(t,a)\in E_0\).
Recall 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 \(\langle z',Tz\rangle _{E} >0\) for every \(z'\in (E')^+\setminus \{0\}\).
In particular, the results require less regularity assumptions than imposed in (5.2) and also apply to any other uniformly elliptic second-order differential operator.
An operator \(A\in {\mathcal {A}}(E_0)\) is resolvent positive, if there is \(\lambda _0\ge 0\) such that \((\lambda _0,\infty )\subset \rho (A)\) and
$$\begin{aligned} (\lambda -A)^{-1}\in \mathcal {L}_+(E_0)\,,\quad \lambda >\lambda _0\,. \end{aligned}$$
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Appendices
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
for some \(q(a,\sigma )=q(a)(\sigma )\) satisfying
Then \(\mathbb {B}\in \mathcal {L}(\mathbb {E}_0)\) with
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 \({\textsf{B}}_\psi \in C(\mathbb {R}^+,E_0)\) be defined as in (3.4b). Then, given \(T>0\), there is \(c_{{\textsf{B}}}(T)>0\) such that
Moreover, given \(\varepsilon >0\) and \(\kappa \in (0,a_m/2)\), there is \(\delta :=\delta (T,\kappa ,\varepsilon )>0\) such that
whenever \(\zeta \in \mathbb {E}_0\).
Proof
Estimate (A.1) follows for \(\theta =0\) from the fact that \([\psi \mapsto {\textsf{B}}_\psi ]\in \mathcal {L}\big (\mathbb {E}_0, C(\mathbb {R}^+,E_0)\big )\), 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 \({\textsf{B}}_{\mathbb {B}\zeta }\) for \(\zeta \in \mathbb {E}_0\), set \(\psi :=\mathbb {B}\zeta \in \mathbb {E}_0\). We write
for \(\tau \ge 0\) and note from (B.2) that
Choose \(\delta _1:=\delta _1(T,\varepsilon ,\kappa )\in (0,\min \{a_m/2,\kappa \})\) such that
Due to Lemma B.1 we may choose \(\delta _2:=\delta _2(T,\varepsilon ,\kappa )>0\) such that
for \(s_1, s_2\in [\delta _1,T\wedge a_m]\) with \(\vert s_1-s_2\vert \le \delta _2\) and
for \({\bar{s}}_1, {\bar{s}}_2\in [0,a_m]\) with \(\vert {\bar{s}}_1-{\bar{s}}_2\vert \le \delta _2\). Set
Then, for \(\kappa \le \tau _2\le \tau _1\le T\) with \(\vert \tau _1-\tau _2\vert \le \delta _0\) we note that \((\tau _1-a_m)_+<\tau _2-\delta _1\) and obtain from (A.1)–(A.6)
Consequently, using \(\psi =\mathbb {B}\zeta \) with \(\mathbb {B}\in \mathcal {L}(\mathbb {E}_0)\) we derive
For \({\textsf{B}}_\psi ^2\) we use Lemma B.1 to find \(\eta :=\eta (\varepsilon ,\kappa )>0\) such that
whenever \(\kappa \le \tau _2\le \tau _1\le T\), \( a\in [0,(a_m-\tau _1)_+]\), \(\vert \tau _1-\tau _2\vert \le \eta \). Let \(\delta _3>0\) with
and set
Then we obtain for \(\kappa \le \tau _2\le \tau _1\le T\) with \(\vert \tau _1-\tau _2\vert \le \delta \) that
Since \(\psi =\mathbb {B}\zeta \) we get from (3.9) that
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
for \(\kappa \le \tau _2\le \tau _1\le T\) with \(\vert \tau _1-\tau _2\vert \le \delta \). Consequently, Lemma A.1 follows from (A.2), (A.7), and (A.8). \(\square \)
1.1 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
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 \(\zeta \in \mathbb {E}_0\),
so that \({\mathcal {V}}\mathbb {S}:(0,\infty )\rightarrow \mathcal {L}(\mathbb {E}_0)\).
Let \(t\in (0,T)\) be fixed and consider a sequence \((\phi _j)_{j\in \mathbb {N}}\) with \(\Vert \phi _j\Vert _{\mathbb {E}_0}\le k_0\) 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
Note that \([s\mapsto \psi _s^j]\in C\big ((0,\infty ),\mathbb {E}_0\big )\) with, recalling (3.9),
Hence, invoking (3.7),
where \(c_0(T):=\Vert q\Vert _{\infty }M_0\,e^{\vert \varkappa _0\vert T}\), and then
Together with (A.1) this yields
Let \(0<h<\min \{a_m/2,t\}\). Then, due to (3.4) we have
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
and \(\eta _1:=\eta _1(T,\varepsilon ,\kappa )\in (0,a_m)\) such that (see Lemma B.1)
Then, from (A.9), (A.3) we have, for \(2h<\eta _1\),
and therefore
(ii) We choose \(\eta _2:=\eta _2(T,\varepsilon )>0\) according to (3.9) such that
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 \(h<\eta _2\),
and therefore
(iii) It follows from (A.3), (A.9), and (A.11) that
(iv) Finally, we choose \({\bar{\kappa }}:={\bar{\kappa }}(\varepsilon ,T)\in (0,a_m/2)\) such that
and invoke then (A.9) and Lemma A.1 to find \(\eta _3:=\eta _3(T,\varepsilon )>0\) such that
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 \(h<\eta _3\),
and therefore
(v) Consequently, we conclude from (A.12)–(A.16) that
and thus, since \(\varepsilon >0\) was arbitrary,
(b) Finally, for \(j\in \mathbb {N}\) we have from (B.2), (A.11), and (A.9) that
Since the right-hand side is finite and due to the compact embedding of \(E_\vartheta \) in \(E_0\) we conclude that
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. \(\square \)
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.
1.1 Basic Definition
Let \(E_1\hookrightarrow E_0\) be a densely injected Banach couple, \(J=[0,a_m]\), and
We consider
with \(\textrm{dom}(A(a))=E_1\) 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 \(\Pi :J_\Delta \rightarrow \mathcal {L}(E_0)\) is such that
satisfying
and, for \(a\in J\),
with
In the following, let
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.
1.2 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 \(M_\theta \ge 1\) such that
according to [3, II. Lemma 5.1.3].
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
is given by
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
is a strong solution to (B.3a). Actually, if, in addition, \( x\in E_1\), then
See [3, II. Theorem 1.2.1, Theorem 1.2.2].
1.4 Continuity Properties
Given \(\vartheta \in (0,1)\), there is \(C(\vartheta )>0\) such that
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
where \( S_\kappa :=\{(a,\sigma )\in \Delta _J^*\,;\, \kappa \le a-\sigma \}. \)
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 \(\Delta _J^*\). \(\square \)
1.5 Positivity
If \(E_0\) is an ordered Banach space and A(a) is resolvent positiveFootnote 5 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 \((a,\sigma )\in J_\Delta \).
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Walker, C. Stability and Instability of Equilibria in Age-Structured Diffusive Populations. J Dyn Diff Equat (2024). https://doi.org/10.1007/s10884-023-10340-9
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DOI: https://doi.org/10.1007/s10884-023-10340-9