Cross Diffusion Systems: Dynamics, Coexistence and Persistence

236 Pages·2022·4.096 MB·English

by  Dung Le

DungLe
CrossDiffusionSystems
De Gruyter Series in Nonlinear
Analysis and Applications
|
EditorinChief
JürgenAppell,Würzburg,Germany
Editors
CatherineBandle,Basel,Switzerland
ManueldelPino,SantiagodeChile,Chile
AvnerFriedman,Columbus,Ohio,USA
MikioKato,Tokyo,Japan
WojciechKryszewski,Torun,Poland
UmbertoMosco,Worcester,Massachusetts,USA
VicenţiuD.Rădulescu,Krakow,Poland
SimeonReich,Haifa,Israel
Volume 40
Dung Le
Cross Diffusion
Systems
|
Dynamics, Coexistence and Persistence
MathematicsSubjectClassification2020
Primary:35K45,35B65,35J47;Secondary:37C35,37C10
Author
Prof.Dr.DungLe
UniversityofTexasatSanAntonio
DepartmentofMathematics
OneUTSACircle
SanAntonio,TX78249
USA
[email protected]
ISBN978-3-11-079498-4
e-ISBN(PDF)978-3-11-079513-4
e-ISBN(EPUB)978-3-11-079517-2
ISSN0941-813X
LibraryofCongressControlNumber:2022943613
BibliographicinformationpublishedbytheDeutscheNationalbibliothek
TheDeutscheNationalbibliothekliststhispublicationintheDeutscheNationalbibliografie;
detailedbibliographicdataareavailableontheInternetathttp://dnb.dnb.de.
©2022WalterdeGruyterGmbH,Berlin/Boston
Typesetting:VTeXUAB,Lithuania
Printingandbinding:CPIbooksGmbH,Leck
www.degruyter.com
|
To my parents Thuan Le and Nghia Nguyen who have literally sacrificed their lives
forme
TomyteacherDucMinhDuongwhofirstintroducedmetothebeautyofmathematics
andconsideredmeasoneofhisbrothers

Contents
1 Introduction|1
2 Preliminaries|11
2.1 Functionalspaces|11
2.1.1 Lebesguespaces|11
2.1.2 MorreyandCampanatospaces|14
2.1.3 BMOspace|16
2.1.4 Sobolevspaces|18
2.1.5 Compactness|20
2.2 Technicallemmasandvariousinequalities|23
2.3 Fixedpointtheorems|27
2.4 WeightedGagliardo–NirenberginequalityinvolvingBMOnorms–a
simplecaseanditsimprovements|28
2.4.1 Notations|29
2.4.2 ThestrongGagliardo–Nirenberginequalityanditsproof|30
2.4.3 ProofofTheorem2.4.1|31
2.4.4 Anew(weak)Gagliardo–Nirenberginequality|34
2.4.5 Parabolicversion|38
3 Existenceresultsforcross-diffusionsystems|44
3.1 Modelsinbiologyandecology|44
3.2 Cross-diffusionmodels|45
3.3 Amoregeneralevolutioncross-diffusionmodel|46
3.4 Theconceptsofstrong,weakandstrongweaksolutions|47
3.5 Theexistenceofstrongweaksolutions|48
3.5.1 Theellipticityandspectralgapconditions|48
𝕃
3.5.2 Themap anditsfixedpoint|49
3.5.3 Mainresults|50
3.6 Uniqueness|52
3.7 Estimatesforthespatialderivatives|53
3.8 Theproofofthemainresult|56
3.8.1 Acompactnesslemma|57
𝕃: →
3.8.2 ThespaceX and X X iscompact|59
3.8.3 Proofofthemaintheorem|63
3.9 Thecorollaries|64
4 Scalartechniquesanddiagonalization|71
4.1 Onscalarequations|72
4.1.1 Globalboundednessandalocalestimate|74
4.1.2 Alocalpropertyoffunctionsinℳ(Ω,T)|77
VIII | Contents
4.1.3 Höldercontinuity|78
4.2 Applicationstosystems|83
4.2.1 Diagonalsystems|83
4.2.2 Fullsystems|85
4.3 Diagonalizationandlocaluniformboundedness|92
4.3.1 Ontheinverseofℋ(w1,w2)|99
4.4 Onconditions(iii)and(iv)ofTheorem4.3.3|102
4.4.1 AnapplicationofthestrongGNBMOinequality|103
4.4.2 AnapplicationoftheweakGNBMOinequality(parabolicversion)|110
4.5 Integrabilityoftemporalderivatives|114
4.6 Dynamicsandattractors|119
5 Existenceofsolutionstogeneralellipticsystems|121
𝕃
5.1 Themap andthespaceX |124
5.2 Estimatesforderivatives|126
5.3 Proofofthemaintheoremanditscorollaries|129
5.3.1 AnapplicationoftheweakGNBMOinequality|133
6 Existenceofsolutionstoellipticsystemsoftwoequations|137
6.1 Onscalarquasilinearellipticequations|139
6.1.1 Globalboundednessandalocalestimate|140
6.1.2 Höldercontinuity|143
6.2 Applicationstosystems|147
6.2.1 Diagonalsystems|148
6.2.2 Fullsystems|150
6.2.3 Triangularsystems|151
=
6.3 Thecasem 2|152
6.3.1 TheBMOsmallconditionandanexampleforSKTsystems|158
6.4 EigenvalueproblemsinorderedBanachspace-Indices|160
6.4.1 OBSsandpositivelinearoperators|160
6.4.2 TopologicalindicesonretractsofaBanachspace|163
6.5 Trivial,semitrivialandnontrivialsolutions|166
6.5.1 AbstracttheoryandindextheoryinorderedBanachspaces|166
6.5.2 Aconcreteexampleincross-diffusionsystems|168
6.5.3 Theindexandeigenvalueproblems|169
6.5.4 Patternformation|178
7 Persistenceinthedynamicsofevolutionprocesses|182
7.1 Persistence|183
7.2 Thedynamicsofevolutionarysolutions|186
7.2.1 Sometechnicalitiesandthegeneralcase|186
>
7.2.2 Differentboundaryconditionswhenτ 1|190
Contents | IX
7.2.3 GeneraldimensionN|194
= >
7.3 Onthenumberτ rL−1Mandtheconditionτ 1|197
λ >
7.3.1 Anapplicationoftheanti-maximumprincipletotheassumptionτ 1in
(7.2)|197
7.3.2 Aminmaxformulaforτin(7.11)|203
<
7.4 Theeffectofcross-diffusionwhenτ 1|204
>
7.5 AdiscussionontheconditionthatrL−1M 1|212
λ
Bibliography|219
Index|223