Some Remarks About the Semigroup Associated to Age-Structured Diffusive Populations

0.22 MB·English

by  Christoph Walker

SOME REMARKS ABOUT THE SEMIGROUP ASSOCIATED TO
AGE-STRUCTURED DIFFUSIVE POPULATIONS
CHRISTOPHWALKER
2 Abstract. We consider linear age-structured population equations with diffusion. Supposing
1 maximal regularity of the diffusion operator, we characterize the generator and its spectral
0 propertiesoftheassociatedstronglycontinuoussemigroup. Inparticular,weprovideconditions
2 forstabilityofthezerosolutionandforasynchronous exponential growth.

n
a
J
1. Introduction
2
1
Thisworkisdedicatedtoage-structureddiffusivepopulationdynamicsgovernedbytheabstract

linear equations
]
P
∂ u + ∂ u + A(a)u =0 , t>0 , a∈(0,a ) , (1.1)
A t a m
am
h. u(t,0) = b(a)u(t,a)da , t>0 , (1.2)
t Z0
a u(0,a) = φ(a) , a∈(0,a ) . (1.3)
m
m
Here, u = u(t,a) is a function taking positive values in some ordered Banach space E . In
[ 0
applications it represents the density at time t of a population of individuals structured by age

1 a ∈ J := [0,a ), where a ∈ (0,∞] is the maximal age. Note that the age interval J may be
m m
v
unbounded. For fixed a∈J, the operator
9
8 A(a)=A (a)+µ(a) (1.4)
0
5
involvesspatialmovementofindividualsdescribedbyA (a)anddeathprocessesofindividualswith
2 0
1. mortality rate µ(a) and is assumed to be an (unbounded) linear operator A(a) : E1 ⊂ E0 → E0.
The nonlocal age-boundary condition (1.2) represents birth processes with birth rate b while φ in
0
2 (1.3) describes the initial population.
1 Equations(1.1)-(1.3)andvariantsthereof,e.g. forconstantortime-dependentoperatorsA,have
: been investigated by many authors, for example see [6, 8, 10, 11, 13, 14, 21] and the references
v
i therein though this list is far from being complete.
X
Recall,e.g. from[21],that astronglycontinuoussemigroupinL (J,E ) canbe associatedwith
1 0
r (1.1)-(1.3) if A is independent of age and generates itself a strongly continuous semigroup on E ,
a 0
see also [6, 14]. This is derived upon formally integrating (1.1) along characteristics giving the
semigroup rather explicitly. The approach has been extended to investigate the well-posedness of
modelsfeaturingnonlinearitiesinthe operatorA=A(t,u)orinthebirthratesb=b(t,u)[15,17].
A slightly different approach has been chosen in [13]. On employing methods for positive
perturbations of semigroups it has been shown that a strongly continuous semigroup for (1.1)-
(1.3)inL (J,E )isobtainedasthe derivativeofanintegratedsemigroup. Moreover,this strongly
1 0
continuoussemigroupisshowntoenjoycertaincompactnesspropertiesandtoexhibitasynchronous
exponential growth, i.e. it stabilizes as t → ∞ to a one-dimensional image of the state space of
2010 Mathematics Subject Classification. 47D06,92D25,47A10,34G10.
Key words and phrases. Agestructure,maximalregularity,semigroupsoflinearoperators,asynchronous expo-
nentialgrowth.
1
2 CHRISTOPHWALKER
initialvalues, after multiplicationby anexponentialfactor intime. This resulthasbeen recovered
as a particularcasein [11] (alsosee [10]), where time-dependent birthrates havebeen included by
means of perturbation techniques of Miyadera type. It is noteworthy that the general results of
[13] apply as well to other situations than A describing spatial diffusion.
0
Thestronglycontinuoussemigroupfrom[21,14]associatedto(1.1)-(1.3)hastheadvantagethat
certainproperties— like regularizingeffects inthe case that−A generatesananalytic semigroup,
being of utmost importance in nonlinear equations, see [15, 17] — can be read off its formula
rather easily, see Theorem 2.2 below and the subsequent remarks. The domain of the generator
of this semigroup is in general not fully identified, cf. [14, 15]. The objective of the present paper
is to characterizethe (domain ofthe) infinitesimal generatorof the stronglycontinuous semigroup
associatedto(1.1)-(1.3)andtoinvestigateitsspectralpropertiesinthecasethattheoperatorAhas
thepropertyofmaximalL -regularity. Thisassumptionissatisfiedinmanyapplications,e.g. when
p
A in (1.4) is a second order elliptic differential operator in divergence form. Maximal regularity
0
provides an adequate functional analytic setting for the characterization of the generator of the
semigroupassociatedto(1.1)-(1.3)inthe phasespaceL (J,E )anditsresolvent,seeTheorem2.7
p 0
below. Knowing the generator precisely, we shall then investigate its growth bound and derive a
stability result for the trivial solution, see Theorem 3.5 below. We also provide in Theorem 3.7 a
condition that implies asynchronous exponential growth of the semigroup.
Besides a precise description of the semigroup with its generator and stability of the trivial
solution, we thus obtain a similar result on asynchronous exponential growth as in [13, 11] by
another approach being inspired by the results in [19] that were dedicated to the non-diffusive
scalar case. We shall point out, however, that the results and the approachpresented herein shall
serve as a basis for a future investigation of qualitative aspects of solutions to models featuring
nonlinearitiesinthediffusionpartandintheage-boundarycondition,i.e. fordiffusionoperatorsof
theformA=A(u)andbirthratesb=b(u),bymeansoflinearizationandperturbationtechniques.
Finally, from a technical point of view it seems to be worthwhile to point out that the cases of a
finite or infinite maximal age a is treated simultaneously herein.
m
2. The Semigroup and its Generator
2.1. Notation and Assumptions. Given a closed linear operator A on a Banach space, we
let σ(A) and σ (A) denote its spectrum and point spectrum, respectively. The essentialspectrum
p
σ (A)ofAconsistsofthosespectralpointsλofAsuchthattheimageim(λ−A)isnotclosed,orλis
e
alimitpointofσ(A),orthedimensionofthekernelker(λ−A)isinfinite. Theperipheralspectrum
σ (A) is defined as σ (A) := {λ ∈ σ(A); Reλ = s(A)}, where s(A) := sup{Reλ; λ ∈ σ(A)}
0 0
denotes the spectral bound of A. The resolvent set C\σ(A) is denoted by ̺(A).
Throughout E is a real Banach lattice ordered by a closed convex cone E+. However, we do
0 0
not distinguish E from its complexification in our notation as no confusion seem likely. Recall
0
that a u∈E+ is quasi-interior if hf,ui>0 for all f in the dual space E′ with f >0 .
0 0
Let E be a densely and compactly embedded subspace of E . We fix p ∈ (1,∞), put
1 0
ς :=ς(p):=1−1/p and set
E :=(E ,E ) , E :=(E ,E )
ς 0 1 ς,p θ 0 1 θ
for θ ∈ [0,1]\{1−1/p} with (·,·) being the real interpolation functor and (·,·) being any
ς,p θ
admissible interpolation functor. We equip these interpolation spaces with the order naturally
induced by E+. Observe that E embeds compactly in E provided 0≤ϑ<θ ≤1. We put
0 θ ϑ
E :=L (J,E ) , E :=L (J,E )∩W1(J,E )
0 p 0 1 p 1 p 0
and recall that
E ֒→BUC(J,E ) (2.1)
1 ς
AGE-STRUCTURED DIFFUSIVE POPULATIONS 3
accordingto,e.g. [2,III.Thm.4.10.2],whereBUCstandsfortheboundedanduniformlycontinuous
functions. In particular, the trace γ u := u(0) is well-defined for u ∈ E and γ ∈ L(E ,E ). Let
0 1 0 1 ς
E+ denote the functions in E taking almost everywhere values in E+. Note that E is a Banach
0 0 0 0
lattice. We further assume that
A∈L (J,L(E ,E )) , σ+A∈Cρ(J,H(E ,E ;κ,ν)) (2.2)
∞ 1 0 1 0
for some ρ,ν > 0, κ ≥ 1, σ ∈ R. Here H(E ,E ;κ,ν) consists of all negative generators −A of
1 0
analytic semigroups on E with domain E such that ν+A is an isomorphismfrom E to E and
0 1 1 0
k(λ+A)xk
κ−1 ≤ E0 ≤κ , x∈E \{0} , Reλ≥ν .
1
|λ|kxk +kxk
E0 E1
Note that A generates a parabolic evolution operator Π(a,σ), 0 ≤ σ ≤ a < a , on E with
m 0
regularity subspace E according to [2, II.Cor.4.4.2] and there are M ≥1 and ̟ ∈R such that
1
kΠ(a,σ)k +(a−σ)α−β1kΠ(a,σ)k ≤Me−̟(a−σ) , 0≤σ ≤a<a , (2.3)
L(Eα) L(Eβ,Eα) m
for0≤β ≤β <α≤1withβ <β ifβ >0,see[2,II.Lem.5.1.3]. We further assumethatΠ(a,σ)
1 1
is positive for 0≤σ ≤a<a and that
m
̟ >0 if a =∞ . (2.4)
m
Moreover,we assume that
for each Re λ>−̟, the operator A :=λ+A has maximal L -regularity,
λ p
(2.5)
that is, (∂ +A ,γ ):E →E ×E is an isomorphism .
a λ 0 1 0 ς
Let the birth rate b be such that
b∈L∞(J,L(Eθ))∩Lp′(J,L(Eθ)) , b(a)∈L+(E0) , a∈J , (2.6)
for θ ∈[0,1], where p′ is the dual exponent of p. We also assume that
b(a)Π(a,0)∈L (E ) is irreducible for a in a subset of J of positive measure . (2.7)
+ 0
Some of the assumptions above are redundant if a <∞. For instance, if a <∞ and
m m
A∈Cρ([0,a ],L(E ,E ))
m 1 0
is such that −A(a) generates an analytic semigroup on E for each a ∈ J, then (2.2) holds. We
0
shallfurthermorepointoutthatnotallassumptionswillbeneededinthisstrengthbutareimposed
for the sakeof simplicity. In particular,if µ being a real-valuednonnegativeand locally integrable
function and A(a)=A (a)+µ(a) as in the introduction, then it suffices that A satisfies (2.2) for
0 0
what follows by keeping in mind that
Π(a,σ)=e−Rσaµ(r)drU(a,σ)
with U denoting the evolution operator associated with A . Also, (2.6) is not required for the
0
whole range of θ ∈[0,1].
We remark that the assumptions above are satisfied in many applications with A describing
spatial diffusion, for example see [16, Sect.3], [18, Sect.3]. For details about parabolic evolution
operatorsandoperatorshavingmaximalregularitywereferthe reader,e.g.,to [2]. A summaryon
positive operators in ordered Banach spaces can be found e.g. in [4].
Due to (2.5), the operatorA =λ+A has maximal L -regularityonJ for eachλ∈C provided
λ p
Reλ>−̟ if a =∞ and it generates a parabolic evolution operator
m
Π (a,s):=e−λ(a−σ)Π(a,σ) , 0≤s≤a<a ,
λ m
on E . Consequently, the unique solution φ∈E to
0 1
∂ φ+A (a)φ=f(a) , a∈(0,a ) , φ(0)=φ
a λ m 0
4 CHRISTOPHWALKER
for φ ∈E and f ∈E is given by
0 ς 0
a
φ(a)=Π (a,0)φ + Π (a,s)f(s)ds , a∈J .
λ 0 λ
Z0
In particular, Π (·,0)∈L(E ,E ) for λ∈C with Reλ>−̟ if a =∞.
λ ς 1 m
2.2. The Semigroupand its Generator. Onintegrating(1.1)alongcharacteristicsweformally
derive that the solution [S(t)φ](a):=u(t,a) to (1.1)-(1.3) is given by
Π(a,a−t)φ(a−t) , 0≤t≤a<a ,
m
S(t)φ (a) := (2.8)
(Π(a,0)Bφ(t−a) , 0≤a<am, t>a ,
(cid:2) (cid:3)
with B :=u(·,0) satisfying according to (1.2) the Volterra equation
φ
t am−t
B (t) = h(a)b(a)Π(a,0)B (t−a)da+ h(a)b(a+t)Π(a+t,a)φ(a)da, t≥0, (2.9)
φ φ
Z0 Z0
with cut-off function h(a):=1 if a∈(0,a ) and h(a):=0 otherwise. Note that
m
am
B (t) = b(a) S(t)φ (a) da , t≥0 . (2.10)
φ
Z0
To make the formal integration rigorous,we fir(cid:2)st obs(cid:3)erve:
Lemma 2.1. There exists a mapping [φ 7→ B ] ∈ L E ,C(R+,E ) such that B is the unique
φ 0 0 φ
solution to (2.9). If φ∈E+, then B (t)∈E+ for t≥0. Given θ ∈[0,1], there is N :=N(θ)>0
0 φ 0 (cid:0) (cid:1)
such that
kBφ(t)kEθ ≤ Nt−θe(−̟+ζ(θ))tkφkE0 , t>0 , (2.11)
where ζ(θ):=(1+θ)Mkbk .
L∞(J,L(Eθ))
Proof. The proof is straightforward by standard arguments, similar statements are found in [21,
Thm.4]and[14, Lem.2.1]. We onlynote that one obtains,fort>0,onapplying (2.3) to (2.9) and
on using (2.6),
t
e̟tkBφ(t)kEθ ≤MkbkL∞(J,L(Eθ)) e̟akBφ(a)kEθ da+MkbkLp′(J,L(Eθ))kφkE0t−θ
Z0
and thus (2.11) follows from the singular Gronwall’s inequality [2, II.Cor.3.3.2]. (cid:3)
Along the lines of [21, Thm.4] (for the case p=1) and on using (2.3) and (2.11) (also see [14])
one easily proves the following:
Theorem 2.2. {S(t); t≥0} given in (2.8) is a strongly continuous positive semigroup in E with
0
sup et(̟−ζ)kS(t)kL(E0) <∞ ,
t≥0
where ζ :=ζ(0)=Mkbk .
L∞(J,L(E0))
Thoughweshallnotuseitinthefollowingletusnotethatthe semigroup{S(t); t≥0}inherits
regularizing properties from the parabolic evolution operator stated in (2.3) , e.g. there holds
kS(t)φkLp(J,Eθ) ≤c(θ)t−θet(−̟+ζ(θ))kφkE0 , t>0 , φ∈E0 , θ ∈[0,1/p) .
Let −A denote the generator of the semigroup {S(t); t ≥ 0}. Based on the assumption of
maximal regularity of the operator A in (1.1), we now fully characterize −A. First, recall that
{λ∈C; Reλ>ω(−A)} is a subset of the resolventset ̺(−A), where the growthbound ω(−A) is
given by
1
ω(−A):= lim logkS(t)k .
t→∞ t
AGE-STRUCTURED DIFFUSIVE POPULATIONS 5
Note that Theorem 2.2 entails
ω(−A)≤−̟+ζ . (2.12)
Let λ∈C be such that Reλ>−̟ if a =∞. Observe that the solution to
m
am
∂ φ+A (a)φ=0 , a∈(0,a ) , φ(0)= b(a)φ(a) da ,
a λ m
Z0
is given by
φ(a)=Π (a,0)φ(0) , a∈(0,a ) , φ(0)=Q φ(0) ,
λ m λ
where
am
Q := b(a)Π (a,0) da .
λ λ
Z0
A we shall see, the spectrum of −A and thus the asymptotic behavior of solutions to (1.1)-(1.3)
is determined by the spectral radii of the λ-dependent family Q . From (2.3), (2.4), and (2.6) we
λ
deduce the regularizing property
Q ∈L(E ,E )∩L(E ,E ) , θ ∈[0,1) , (2.13)
λ 0 θ 1−θ 1
and hence Q | ∈L(E ) is compact for θ ∈[0,1)due to the compact embedding of E in E for
λ Eθ θ α β
0≤β <α<1. Consequently, σ(Q | )\{0} consists only of eigenvalues.
λ Eθ
Lemma 2.3. Let λ∈R with λ>−̟ if a =∞. Then the spectral radius r(Q ) is positive and
m λ
a simple eigenvalue of Q ∈ L(E ) with an eigenvector in E that is quasi-interior in E+. It is
λ 0 1 0
the only eigenvalue of Q with a positive eigenvector. Moreover, σ(Q | )\{0}=σ(Q )\{0} for
λ λ Eθ λ
θ ∈[0,1).
Proof. Since Q ∈ L(E ) is compact and irreducible according to (2.7) (see the proof of [18,
λ 0
Lem.2.1]), it is a classical result that the spectral radius r(Q ) is positive and a simple eigenvalue
λ
of Q with a quasi-interior eigenvector [4, Thm.12.3]. This eigenvector belongs to E owing to
λ 1
(2.13). The regularizing property (2.3) also ensures the last statement. (cid:3)
In view of (2.13) and the observations stated in Lemma 2.3 we shall not distinguish between
Q ∈L(E ) and Q | ∈L(E ) in the sequel if θ ∈[0,1).
λ 0 λ Eθ θ
The arguments used in the proof of [18, Lem.2.2] reveal:
Lemma 2.4. Let I =R if a <∞ and I =(−̟,∞) if a =∞. Then the mapping
m m
[λ7→r(Q )]:I →(0,∞)
λ
is continuous, strictly decreasing, and lim r(Q )=0. If a <∞, then lim r(Q )=∞.
λ→∞ λ m λ→−∞ λ
Next, we characterize the resolvent of −A.
Lemma 2.5. Consider λ ∈ C such that Reλ > −̟ +ζ and suppose that 1−Q ∈ L(E ) is
λ 0
boundedly invertible. Then
a am s
(λ+A)−1φ (a)= Π (a,σ)φ(σ) dσ+Π (a,0)(1−Q )−1 b(s) Π (s,σ)φ(σ) dσds
λ λ λ λ
Z0 Z0 Z0
(cid:2) (cid:3) (2.14)
for a∈J and φ∈E .
0
Proof. By (2.12), any λ∈C with Reλ>−̟+ζ belongs to the resolvent set of −A, so it follows
from the Laplace transform formula and (2.8) that for φ∈E and a.a. a∈J we have
0
∞ a ∞
(λ+A)−1φ (a)= e−λt S(t)φ (a) dt= Π (a,t)φ(t) dt+Π (a,0) e−λtB (t) dt .
λ λ φ
Z0 Z0 Z0
(cid:2) (cid:3) (cid:2) (cid:3)
6 CHRISTOPHWALKER
Next, from (2.11),
∞
Ψ:= e−λtB (t) dt∈E
φ 0
Z0
and, on using (2.8) and (2.10), we obtain
am ∞
Ψ= b(a) e−λt S(t)φ (a) dtda
Z0 Z0
am (cid:2) (cid:3) am a
= b(a)Π (a,0) daΨ+ b(a) Π (a,t)φ(t)dtda ,
λ λ
Z0 Z0 Z0
that is,
am a
Ψ=(1−Q )−1 b(a) Π (a,t)φ(t)dtda
λ λ
Z0 Z0
from which the claim follows. (cid:3)
Observethat Lemma 2.5 alsoholds without assumption(2.5) onmaximalregularityof−A and
forφ∈L (J,E ), i.e. for p=1. However,(2.5) allowsus tointerpretformula(2.14) inthe correct
1 0
functional setting:
Remark 2.6. Let φ∈E , let λ∈C be such that Reλ>−̟+ζ, and suppose that 1−Q ∈L(E )
0 λ 0
is boundedly invertible. Note that (1−Q )−1 ∈L(E ) by Lemma 2.3. Then, by (2.5),
λ ς
(λ+A)−1φ=v φ+w φ , (2.15)
λ λ
where maximal regularity of A implies that v φ∈E , given by
λ λ 1
a
(v φ)(a):= Π (a,σ)φ(σ) dσ , a∈J ,
λ λ
Z0
is the unique solution to the Cauchy problem
∂ v+A v =φ , a∈(0,a ) , v(0)=0 ,
a λ m
and w φ∈E , given by
λ 1
am
(w φ)(a):=Π (a,0)(1−Q )−1 b(s)(v φ)(s)ds , a∈J ,
λ λ λ λ
Z0
is the unique solution to the Cauchy problem
am
∂ w+A w =0 , a∈(0,a ) , w(0)=(1−Q )−1 b(s)(v φ)(s)ds ∈E .
a λ m λ λ ς
Z0
The characterization of the generator −A of the semigroup {S(t); t ≥ 0} from Theorem 2.2 is
now straightforward.
Theorem 2.7. φ∈E belongs to the domain dom(−A) of −A if and only if φ∈E with
0 1
am
φ(0)= b(a)φ(a) da . (2.16)
Z0
Moreover, Aφ=∂ φ+Aφ for φ∈dom(−A).
a
Proof. ByLemma2.4,wecanchooseλ>−̟+ζ suchthat1−Q ∈L(E )isboundedlyinvertible.
λ 0
Thusλbelongstotheresolventsetof−AbyTheorem2.2. Remark2.6easilygivesdom(−A)⊂E .
1
Moreover,if ψ ∈E and φ:=(λ+A)−1ψ ∈dom(−A), then φ(0)∈E by (2.1) and
0 ς
am
φ(0)=(w ψ)(0)=(1−Q )−1 b(a)(v ψ)(a) da .
λ λ λ
Z0
AGE-STRUCTURED DIFFUSIVE POPULATIONS 7
The same calculations as in the proof of Lemma 2.5 yield
am ∞ am am
b(a)φ(a)da= e−λt b(a)[S(t)ψ](a)dadt=(1−Q )−1 b(a)(v ψ)(a)da=φ(0).
λ λ
Z0 Z0 Z0 Z0
Conversely, if φ ∈E satisfies (2.16) then ψ := (∂ +A )φ ∈ E by (2.2) and, since φ(t) ∈E for
1 a λ 0 1
a.a. t∈J we have
∂
Π (a,t)φ(t) =Π (a,t)ψ(t)
λ λ
∂t
for a.a. t ∈ J and a > t due to t(cid:0)he fact that(cid:1)Π is the parabolic evolution operator for A .
λ λ
Integration with respect to t gives
(v ψ)(a)=φ(a)−Π (a,0)φ(0)
λ λ
from which
am
(w ψ)(a)=Π (a,0)(1−Q )−1 b(s) φ(s)−Π (s,0)φ(0) ds=Π (a,0)φ(0)
λ λ λ λ λ
Z0
(cid:2) (cid:3)
for a∈J, whence
φ=v ψ+w ψ =(λ+A)−1ψ ∈dom(−A) .
λ λ
Finally, (λ+A)φ=ψ =(∂ +A )φ and the proof is complete. (cid:3)
a λ
Remark 2.8. Theorem 2.2 and Theorem 2.7 show that for any initial value φ ∈ E satisfying
1
(2.16), the unique solution u ∈ C(R+,E )∩C1(R+,E ) to (1.1)-(1.3) is given by u(t) = S(t)φ,
1 0
t ≥ 0, with S(t) defined in (2.8). If φ is only in E , then u(t) = S(t)φ, t ≥ 0, defines a mild
0
solution in C(R+,E ). Moreover, u(t)∈E+ for t≥0 if φ∈E+.
0 0 0
3. Stability of the Trivial Solution and Asynchronous Exponential Growth
We now shall characterize the growth bound ω(−A) of −A. We first characterize the point
spectrum of −A and extend formula (2.14) to a larger class of λ values.
Lemma 3.1. (i) Let λ ∈ C with Reλ > −̟ if a = ∞ and let m ∈ N\{0}. Then λ ∈ σ (−A)
m p
with geometric multiplicity m if and only if 1∈σ (Q ) with geometric multiplicity m.
p λ
(ii) Formula (2.14) holds for any λ∈̺(−A) provided Reλ>−̟ if a =∞.
m
Proof. (i) Let λ∈C with Reλ>−̟ if a =∞. Suppose λ∈σ (−A) has geometric multiplicity
m p
msothattherearelinearlyindependentφ ,...,φ ∈dom(−A)with(λ+A)φ =0forj =1,...,m.
1 m j
From Theorem 2.7 we deduce
φ (a)=Π (a,0)φ (0) with φ (0)=Q φ (0) .
j λ j j λ j
Hence, φ (0),...φ (0)arenecessarilylinearlyindependenteigenvectorsofQ correspondingtothe
1 m λ
eigenvalue 1. Now, suppose 1 ∈ σ (Q ) has geometric multiplicity m so that there are linearly
p λ
independent Φ ,...Φ ∈E with Q Φ =Φ for j =1,...,m. Put φ :=Π (·,0)Φ ∈E and note
1 m ς λ j j j λ j 1
that, for j =1,...,m,
am
∂ φ +A φ =0 , b(a)φ (a) da=Q Φ =Φ =φ (0) .
a j λ j j λ j j j
Z0
Thus φ ∈ dom(−A) and (λ+A)φ = 0 by Theorem 2.7, i.e. λ ∈ σ (−A). If α ,...,α are any
j j p 1 m
scalars, the unique solvability of the Cauchy problem
∂ φ+A φ=0 , a∈(0,a ) , φ(0)= α Φ
a λ m j j
j
X
8 CHRISTOPHWALKER
ensures that φ ,...,φ are linearly independent. This proves (i).
1 m
(ii)Letλ∈̺(−A)withReλ>−̟ ifa =∞. ThenA hasmaximalregularitydue to(2.5)and,
m λ
by Theorem 2.7, (λ+A)ψ =φ with φ∈E and ψ ∈E if and only if (∂ +A )ψ =φ with
0 1 a λ
am
ψ(0)= b(a)ψ(a) da ,
Z0
that is,
a
ψ(a)=Π (a,0)ψ(0)+ Π (a,σ)φ(σ) dσ ,
λ λ
Z0
am a
(1−Q )ψ(0)= b(a) Π (a,σ)φ(σ) dσda .
λ λ
Z0 Z0
Since λ ∈ ̺(−A) and since Q is compact, (i) ensures that 1 ∈ ̺(Q ), hence 1−Q is invertible
λ λ λ
and so
am a
ψ(0)=(1−Q )−1 b(a) Π (a,σ)φ(σ) dσda .
λ λ
Z0 Z0
As ψ =(λ+A)−1φ, this gives formula (2.14). (cid:3)
Recall that the α-growth bound ω (−A) of −A is defined by
1
1
ω (−A):= lim log α(S(t)) ,
1
t→∞ t
(cid:0) (cid:1)
where α denotes Kuratowski’s measure of non-compactness. That is, if B is a subset of a normed
vector space X, then α(B) is defined as the infimum over all δ > 0 such that B can be covered
with finitely many sets of diameter less than δ, and if T is a bounded operatoron X,then α(T)is
the infimum over all ε > 0 such that α(T(B)) ≤ εα(B) for any bounded set B ⊂ X. Recall that
ω (−A)≤ω(−A).
1
We next provide bounds on ω (−A).
1
Lemma 3.2. There holds
sup{Reλ; λ∈σ (−A)} ≤ ω (−A) ≤ −̟ .
e 1
Moreover, if a <∞, then ω (−A)=−∞ and the semigroup {S(t); t≥0} is eventually compact.
m 1
Proof. Thefirstinequalityoftheassertionisgenerallytrueforstronglycontinuoussemigroups[21,
Prop.4.13]. We thus merely have to show that ω (−A)≤−̟ what can be done along the lines of
1
the scalar case [21, Thm.4.6]: Let t>0 and write S(t)=U(t)+W(t), where U(t),W(t)∈L(E )
0
are defined as
0 , a∈(0,t) , S(t)φ (a) , a∈(0,t) ,
U(t)φ (a) := W(t)φ (a) :=
( S(t)φ (a) , a∈(t,am) , ((cid:2)0 , (cid:3) a∈(t,am) ,
(cid:2) (cid:3) (cid:2) (cid:3)
for a∈J, φ∈E . O(cid:2)bservi(cid:3)ng that α(S(t))≤α(U(t))+α(W(t)) and, by (2.3) and (2.8),
0
α(U(t))≤kU(t)kL(E0) ≤Me−̟t , t<am , (3.1)
the assertion follows from the definition of ω (−A) provided we can show that α(W(t)) = 0. For
1
this it suffices to show that if B is any bounded subset of E , then W(t)B is relatively compactin
0
E . We use Kolmogorov’scompactness criterion [7, Thm.A.1]. Without loss of generality we may
0
AGE-STRUCTURED DIFFUSIVE POPULATIONS 9
assume that a =∞. Clearly, Theorem 2.2 ensures that W(t)B is bounded in E . If φ∈E and
m 0 0
h>0, then
∞
k[W(t)φ](a+h)−[W(t)φ](a)kp da
E0
Z0
t−h
≤ kΠ(a+h,0)−Π(a,0)kp kB (t−a−h)kp da
L(Eξ,E0) φ Eξ
Z0
t−h
+ kΠ(a,0)kp kB (t−a−h)−B (t−a)kp da
L(E0) φ φ E0
Z0
t
+ kΠ(a,0)kp kB (t−a)kp da ,
L(E0) φ E0
Zt−h
where ξ ∈ (0,1/p). On using (2.3) and Lemma 2.1 it is readily seen that the second and third
integral on the right side tend to zero as h → 0, uniformly with respect to φ ∈ B. For the first
integral we use the fact (see [2, II.Eq.(5.3.8)]) that
kΠ(a+h,0)−Π(a,0)k ≤c(t)hξ , a≤t ,
L(Eξ,E0)
to obtain from Lemma 2.1 the estimate
t−h t−h
kΠ(a+h,0)−Π(a,0)kp kB (t−a−h)kp da≤c(t)phξp (t−a−h)−ξp da kφkp
L(Eξ,E0) φ Eξ E0
Z0 Z0
with right hand side tending to zero as h → 0, uniformly with respect to φ ∈ B. Next, by (2.3),
(2.8), and Lemma 2.1,
k[W(t)φ](a)k ≤c(B,t)(t−a)−ξ , a<t ,
Eξ
for some constant c(B,t). Given ε∈(0,t) let R be the E -closure of the ball in E centered at 0
ε 0 ξ
of radius c(B,t)ε−ξ. Then R is compact in E since E embeds compactly in E and
ε 0 ξ 0
[W(t)φ](a)∈R , a∈R+\[t−ε,t] , φ∈B .
ε
Therefore, [7, Thm.A.1] implies that W(t)B is relatively compact in E , hence ω (−A)≤−̟.
0 1
Finally, if a <∞ and t>a , then U(t)=0 and so S(t)=W(t). This proves the lemma. (cid:3)
m m
Lemma 3.3. Let λ ∈ σ(−A) with Reλ > −̟ if a = ∞. Then λ ∈ σ (−A)\σ (−A) and
m p e
1 ∈ σ (Q ). Moreover, λ is isolated in σ(−A) and a pole of the resolvent [τ 7→ (τ +A)−1]. The
p λ
residue of the resolvent at λ,
1
P := (τ +A)−1 dτ ,
λ
2πi
ZΓ
is a projection on E and E = im(P )⊕im(1−P ) with im(P ) = ker(λ+A)m, where Γ is a
0 0 λ λ λ
positively oriented closed curve in the complex plane such that no point in σ(−A) lies in or on Γ
and m∈N is the order of the pole λ.
Proof. Let λ ∈ σ(−A) with Reλ > −̟ if a = ∞. Since −A is closed and λ ∈ σ(−A)\σ (−A)
m e
by Lemma 3.2, itfollowsfrom[3]thatλ∈σ (−A)is isolatedinσ(−A) andapole of the resolvent
p
[τ 7→ (τ +A)−1]. In particular, λ ∈ σ (−A) implies 1 ∈ σ (Q ) by Lemma 3.1(i). Since λ is
p p λ
isolated in σ(−A), the remaining assertions follow from Laurent series theory described in [9, 22],
for details we refer to [19, Prop.4.8,Prop.4.11]. (cid:3)
Thefollowingcharacterizationoftheperipheralspectrumσ (−A)intermsofthespectralbound
0
s(−A)inthe Banachlattice E isa usefultoolforourpurpose. Its proofis basedon[5]andfound
0
in [20, Prop.2.5]:
Proposition 3.4. If s(−A)>ω (−A), then σ (−A)={s(−A)}.
1 0
10 CHRISTOPHWALKER
We now give a criterionfor the spectralbound to be negative. Note that this result implies the
global asymptotic stability of the trivial solution to (1.1)-(1.2).
Theorem 3.5. If r(Q )<1, then ω(−A)<0.
0
Proof. Let λˆ :=s(−A). Since r(Q )<1 there is δ >0 such that r(Q )<1 by Lemma 2.4. Fix
0 0 −δ
ε∈(0,δ)withε<̟ifa =∞,see(2.4). Supposethereisλ∈CwithReλ≥−εand1∈σ (Q ).
m p λ
Then λ ∈ σ(−A) due to Lemma 3.1, whence λˆ ≥ −ε. Since ε < ̟ if a = ∞, Lemma 3.2 and
0 m
Proposition 3.4 entail σ (−A)={λˆ }. Invoking Lemma 3.1 again we see that 1∈σ (Q ) and so
0 0 p λˆ0
r(Q )≥1. Lemma 2.4 then gives λˆ <−δ contradicting λˆ ≥−ε and ε<δ. Consequently,
λˆ0 0 0
sup Reλ; 1∈σ (Q ) ≤−ε<0 . (3.2)
p λ
Recall e.g. from [19, Prop.4.13]tha(cid:8)t (cid:9)
ω(−A)=max ω (−A), sup Reλ .
1
λ∈σ(−A)\σe(−A)
(cid:8) (cid:9)
Since ̟ > 0 if a = ∞, the assertion is an immediate consequence of (3.2), Lemma 3.2, and
m
Lemma 3.3. (cid:3)
Next,weprovideacriterionforasynchronousexponentialgrowthofthesemigroup{S(t);t≥0}
which is analogous to the scalar case A≡µ in [19]. For similar results in the spatially inhomoge-
neous setting we refer to [13, 11]. We first need an auxiliary result.
Lemma 3.6. Let λ ∈R with λ >−̟ if a =∞ be such that r(Q )=1. Then λ is a simple
0 0 m λ0 0
eigenvalue of −A.
Proof. ReferringtoLemma2.3thereisaquasi-interioreigenvectorΦ ∈E ofQ correspondingto
0 1 λ0
thesimpleeigenvaluer(Q )=1. ByTheorem2.7andLemma3.1,ker(λ +A)isone-dimensional
λ0 0
and spanned by ϕ:=Π (·,0)Φ . It thus remains to show that ker(λ +A)2 ⊂ ker(λ +A). Let
λ0 0 0 0
ψ ∈ker(λ +A)2 and set
0
φ:=(λ +A)ψ ∈ker(λ +A) .
0 0
Then φ = ξϕ for some ξ ∈ R. Suppose ξ 6= 0, so without loss of generality ξ > 0. Let τ > 0 be
suchthat τΦ +ψ(0)∈E+\{0}and put q :=τϕ+ψ ∈dom(−A). Then (λ +A)q =φ and, from
0 ς 0
Theorem 2.7,
a
q(a)=Π (a,0)q(0)+ξ Π (a,σ)Π (σ,0)Φ dσ =Π (a,0)q(0)+aξΠ (a,0)Φ
λ0 λ0 λ0 0 λ0 λ0 0
Z0
and
am
q(0)= b(a)q(a) da .
Z0
Plugging the former into the second formula yields
am
(1−Q )q(0)=ξ b(a)aΠ (a,0)Φ da .
λ0 λ0 0
Z0
As q(0) and the right hand side are both positive and nonzero, we derive from [4, Cor.12.4] a
contradiction to r(Q )=1. Consequently, ξ =0 and the claim follows because now φ=0. (cid:3)
λ0
Exponential asynchronous growth of the semigroup {S(t);t ≥ 0} given in (2.8) is now an easy
consequenceof [20]. RecallfromLemma 3.3 thatP :E →ker(λ +A) is a projectionwith rank
λ0 0 0
one.