Theory and Applications of Abstract Semilinear Cauchy Problems

558 Pages·2018·7.18 MB·English

by  Pierre Magal

Applied Mathematical Sciences
Pierre Magal
Shigui Ruan
Theory and
Applications
of Abstract
Semilinear
Cauchy Problems
Applied Mathematical Sciences
Volume 201
Editors
S.S.Antman,InstituteforPhysicalScienceandTechnology,UniversityofMaryland,CollegePark,MD,
USA
[email protected]
LeslieGreengard,CourantInstituteofMathematicalSciences,NewYorkUniversity,NewYork,NY,
USA
[email protected]
P.J.Holmes,DepartmentofMechanicalandAerospaceEngineering,PrincetonUniversity,Princeton,
NJ,USA
[email protected]
Advisors
J.Bell,LawrenceBerkeleyNationalLab,CenterforComputationalSciencesandEngineering,Berkeley,
CA,USA
P.Constantin,DepartmentofMathematics,PrincetonUniversity,Princeton,NJ,USA
R.Durrett,DepartmentofMathematics,DukeUniversity,Durham,NC,USA
R.Kohn,CourantInstituteofMathematicalSciences,NewYorkUniversity,NewYork,NY,USA
R.Pego,DepartmentofMathematicalSciences,CarnegieMellonUniversity,Pittsburgh,PA,USA
L.Ryzhik,DepartmentofMathematics,StanfordUniversity,Stanford,CA,USA
A.Singer,DepartmentofMathematics,PrincetonUniversity,Princeton,NJ,USA
A.Stevens,DepartmentofAppliedMathematics,UniversityofMu¨nster,Mu¨nster,Germany
S.Wright,ComputerSciencesDepartment,UniversityofWisconsin,Madison,WI,USA
FoundingEditors
FritzJohn,JosephP.LaSalleandLawrenceSirovich
Moreinformationaboutthisseriesathttp://www.springer.com/series/34
Pierre Magal • Shigui Ruan
Theory and Applications of
Abstract Semilinear Cauchy
Problems
123
PierreMagal ShiguiRuan
InstitutdeMathe´matiquesdeBordeaux DepartmentofMathematics
Universite´deBordeaux UniversityofMiami
Talence,France CoralGables,FL,USA
ISSN0066-5452 ISSN2196-968X (electronic)
AppliedMathematicalSciences
ISBN978-3-030-01505-3 ISBN978-3-030-01506-0 (eBook)
https://doi.org/10.1007/978-3-030-01506-0
LibraryofCongressControlNumber:2018960824
Mathematics Subject Classification: 47-XX, 47Dxx, 47D62, 34-XX, 35-XX, 58F39, 58-XX, 92-XX,
92D25
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ToEnzoandClara
and
ToYilin,Marion,andLucia
Foreword
Predictionisverydifficult,especiallyaboutthefuture.–NielsBohr
I know that in the study of material things number, order, and position are the
threefold clue to exact knowledge: and that these three, in the mathematician’s
hands, furnish the ‘first outlines for a sketch of the Universe’. – D’Arcy Thomp-
son,OnGrowthandForm(1917)
Thesubjectofdifferentialequationshasalongandstoriedhistory.Atitsfounda-
tionisthefundamentalnatureofphysicalchange.Morethantwocenturiesago,dif-
ferentialequationsdescribingphysicalchangewerestudiedandappliedwithmonu-
mentalsuccess.Thesubjecthasgrowneversincewithextraordinaryproductivityin
mathematicaltheoryandscientificapplications.Thedevelopmentofrecentmodels
ofdynamicalprocessesoffersever-increasingmathematicalchallenge.Atthecore
ofthismathematicalchallenge,therearefundamentalideas.
Oneofthemostimportantfundamentalideasinmodelsofphysicalchangeisthe
assumptionofdeterminism.Thebasicideaisthatthepresentdeterminesthefuture.
This idea is encompassed into differential equations of dynamical processes as a
knowninitialconditionataspecifiedtime0.Newton’ssecondlawstatesthattherate
ofchangeofmomentumofabodyisdirectlyproportionaltotheforceapplied:F=
mdv/dt,wheremisthemassandvisthevelocity.Iftheinitialvelocityv(0)ofthe
bodyisknown,thenthefuturevelocityisknownforalltime.Thisconceptualization
of deterministic behavior is a foundational mathematical description of scientific
phenomena.
An alternative view of determinism is that the past determines the future. This
ideaencompassesintothedifferentialequationsofdynamicalprocessesarequire-
mentthatthefutureforwardfromaninitialtime0isdependentonthehistoryofthe
processuptotime0.Theinitialconditionofsuchaprocessmustincorporatemore
thanthecurrentstate,butinaddition,apasthistoryofthecurrentstate.
Themathematicaltheoryofthedifferentialequationsofhistory-determinedpro-
cesses has a much more recent development. The subject is known as functional
vii
viii Foreword
differentialequations.Akeyroleinthedevelopmentoffunctionaldifferentialequa-
tionswasplayedbyJackHale.Inhis1977monograph(TheoryofFunctionalDif-
ferential Equations, Springer-Verlag), the theory of ordinary differential equations
in finite dimensional spaces was extended to functional differential equations in
a comprehensive treatment. Theoretical results about existence, uniqueness, initial
conditions, stability, periodicity, asymptotic behavior, and other basic ideas were
developed.Oneofthekeyideaswastheformulationoffunctionaldifferentialequa-
tionsasabstractordinarydifferentialequationsininfinitedimensionalspaces.This
idea is accomplished by utilizing the theory of semigroups of linear operators in
infinitedimensionalspaces.
Thetheoryoflinearsemigroupsofoperatorshasbeendevelopedextensively,and
keyroleswereplayedbythemonographsofE.HilleandR.S.Phillips(Functional
Analysis and Semi-Groups, Amer. Math. Soc.,1948; 1957), K. Yosida (Functional
Analysis, Springer-Verlag, 1965), T. Kato (Perturbation Theory of Linear Opera-
tors,Springer-Verlag,1966),andA.Pazy(SemigroupsofLinearOperatorandAp-
plicationstoPartialDifferentialEquations,Springer-Verlag,1983).Thebasicidea
of a semigroup of operators is the idea of an exponential process. The solution of
theabstractdifferentialequationdx(t)/dt=Ax(t),inaninfinitedimensionalspace
X,withinitialconditionx(0)=x ,isx(t)=etAx ,wherex ∈X andetA istheex-
0 0 0
ponentialoftA.IfAisaboundedoperator(matrix),thenetAis∑∞ tnAn/n!.IfAis
n=0
anunboundedlinearoperator,thenetA=limn→∞(I−t/nA)−n,where(I−λA)−1is
theresolventofA.TheoperatorAiscalledtheinfinitesimalgeneratorofthesemi-
groupoflinearoperatorsT(t)=etA,t≥0.Inclassicallinearsemigrouptheory,Ais
denselydefinedinthestatespaceX.
Alinearsemigroupofoperatorscanbeviewedasageneralizedversionoftheex-
ponentialoftheinfinitesimalgenerator.Linearoperatorsemigrouptheoryiscalled
abstractCauchytheory.Thetheoryoffirst-ordernonlinearperturbationsofunderly-
inglinearabstractCauchyproblemsiscalledabstractsemilinearCauchytheory.A
history-dependent deterministic dynamical process can be viewed, in an appropri-
atesettingandanappropriateformulation,asanexponentialprocessoranonlinear
versionofanexponentialprocess.TherearemanyapplicationsofabstractCauchy
problems,bothlinearandnonlinear.
InthismonographPierreMagalandShiguiRuandevelopanextensionoflinear
operator semigroup theory to the case that the semigroup has an integrated form.
Thiscaseariseswhentheinfinitesimalgeneratorisnotdenselydefinedinthestate
space of the operators. In this case the theoretical results for the classical case of
denselydefinedinfinitesimalgeneratorsmustbeextended,andsometimeswithvery
elaborate theoretical extensions. The theory of integrated semigroups of operators
with non-densely defined infinitesimal generators reveals the power of the funda-
mentalconceptofexponentialprocesses.PierreMagalandShiguiRuanhavebeen
at the forefront of this development, inboth its theoretical aspects and its applica-
tionstoscientificproblems.
One of these applications is to functional differential equations with partial
derivativeterms.Thesemodelshaveapplicationstoproblemsinvolvingspatialbe-
havior, for example in models in which spatial diffusion plays a role. Another ap-
Foreword ix
plicationistostructuredpopulationmodels.Thesemodelstracktheevolutionofa
populationintime,butalsointheorganizationoftheirstructurewithrespecttoage,
size,orother individual variation.Agestructureisveryusefulindescribing many
biological species, such as humans in demographic contexts. The continuum ver-
sionofsuchmodelsleadstoanabstractCauchyprobleminaspaceofpossibleage
densitiesofthegivenpopulation.Size-structuredpopulationsareanotherversionof
ausefulwaytoorganizepopulationinvestigations.Sizestructureissometimesmore
appropriateforanalyzingpopulationbehavior,forexampleinmicrospeciessuchas
cell populations. The evolution of a size-structured population can be modeled as
anabstractCauchyprobleminanappropriateinfinitedimensionalspaceofpossible
sizedensities.Alltheissuesofpopulationbehavior,suchasexistence,uniqueness,
asymptotic behavior, stability, and periodicity, can be investigated using abstract
CauchytheoryandsemilinearabstractCauchytheory.
There is a connection between structured population equations and functional
differential equations. For example, the evolution of age structure in a population
canbeviewedasdeterminedbytheinitialagestructureofthepopulationinanin-
finite dimensional space of age densities at initial time 0. It can also be viewed as
determinedbyahistory-dependentagestructureofthepopulationbeforetheinitial
time0.Iftheageofallindividualsinapopulationisknown,thentheirbirthdatesare
known. Conversely, if the birth dates and the history of all individuals are known,
thentheirageisknownatthepresenttime.Structuredpopulationmodelsandfunc-
tionaldifferentialequationsmodelshavegreatutilityinscientificapplications,and
theirtheoreticalanalysisisgroundedinthedevelopmentfoundinthismonograph.
The subject of abstract Cauchy theory has developed rapidly in recent years,
withanexpandingcommunityofresearchers.Thereisanimportantneedforacom-
prehensive treatment of this expanding subject. This monograph provides such a
comprehensivetreatmentandhasgreatvaluetoresearchersinthisfield,boththeo-
reticiansandappliedscientists.
Nashville,TN,USA GlennWebb
July2017
Preface
Although mathematics ranks last in the Six Arts (rites, music, archery, chariot
racing, calligraphy and mathematics), it is used in the most practical issues and
affairs.Maximally,itenablesunderstandingoftheunderlyingmythsofthingsand
comprehensionoftheirnatureanddevelopmentalregularities.Minimally,itcanbe
usedindealingwithsmallaffairsandsolvingmultipletrivialissues.–QINJiushao,
Prefaceto“MathematicalTreatiseinNineSections”(1247)
Mathematicshasathreefoldpurpose.Itmustprovideaninstrumentforthestudy
of nature. But this is not all: it has a philosophical purpose, and, I daresay, an
aestheticpurpose.–HenriPoincare´
We first met in Nashville, Tennessee, in the fall of 2001, when one of us (SR)
was on sabbatical at Vanderbilt University while the other one (PM) was visiting
the school. Both of us were working with Glenn Webb on various problems in
mathematical biology, in particular age-structured biological models described by
first-orderhyperbolicpartialdifferentialequations.
There are different approaches to study age-structured population models. One
approach,usingthetheoryofsemigroupsofoperatorssincethelate1970s,became
very powerful and important, mainly due to the work by Glenn Webb. His mono-
graph, Theory of Nonlinear Age-Dependent Dynamics (Marcel Dekker, 1985), re-
mainstheclassicalreferenceintreatingage-structuredmodelsusingfunctionalan-
alytic techniques of nonlinear semigroups and evolution operators. The principle
of linearized stability, established in Webb’s monograph, says that a steady state
is exponentially stable if the spectrum of the infinitesimal generator of the lin-
earized semigroup lies entirely in the open left half-plane, whereas it is unstable
if there is at least one spectral value lying in the open right half-plane (i.e., with
positive real part). This not only provides a fundamental tool to study the stabil-
ity of age-structured models but also indicates that periodic solutions may exist
in age-structured models via Hopf bifurcation when spectral values leave the left
half-plane, cross the purely imaginary axis, and enter the right half-plane as some
xi