Table Of ContentMathematics for Industry 31
Nikos I. Kavallaris
Takashi Suzuki
Non-Local Partial
Differential
Equations for
Engineering and
Biology
Mathematical Modeling and Analysis
Mathematics for Industry
Volume 31
Editor-in-Chief
MasatoWakayama(KyushuUniversity,Japan)
ScientificBoardMembers
RobertS.Anderssen(CommonwealthScientificandIndustrialResearchOrganisation,Australia)
HeinzH.Bauschke(TheUniversityofBritishColumbia,Canada)
PhilipBroadbridge(LaTrobeUniversity,Australia)
JinCheng(FudanUniversity,China)
MoniqueChyba(UniversityofHawaiiatMānoa,USA)
Georges-HenriCottet(JosephFourierUniversity,France)
JoséAlbertoCuminato(UniversityofSãoPaulo,Brazil)
Shin-ichiroEi(HokkaidoUniversity,Japan)
YasuhideFukumoto(KyushuUniversity,Japan)
JonathanR.M.Hosking(IBMT.J.WatsonResearchCenter,USA)
AlejandroJofré(UniversityofChile,Chile)
KerryLandman(TheUniversityofMelbourne,Australia)
RobertMcKibbin(MasseyUniversity,NewZealand)
AndreaParmeggiani(UniversityofMontpellier2,France)
JillPipher(BrownUniversity,USA)
KonradPolthier(FreeUniversityofBerlin,Germany)
OsamuSaeki(KyushuUniversity,Japan)
WilSchilders(EindhovenUniversityofTechnology,TheNetherlands)
ZuoweiShen(NationalUniversityofSingapore,Singapore)
Kim-ChuanToh(NationalUniversityofSingapore,Singapore)
EvgenyVerbitskiy(LeidenUniversity,TheNetherlands)
NakahiroYoshida(TheUniversityofTokyo,Japan)
Aims&Scope
Themeaningof“MathematicsforIndustry”(sometimesabbreviatedasMIorMfI)isdifferentfrom
thatof“MathematicsinIndustry”(orof“IndustrialMathematics”).Thelatterisrestrictive:ittends
tobeidentifiedwiththeactualmathematicsthatspecificallyarisesinthedailymanagementand
operationofmanufacturing.Theformer,however,denotesanewresearchfieldinmathematicsthat
may serve as a foundation for creating future technologies. This concept was born from the
integrationandreorganizationofpureandappliedmathematicsinthepresentdayintoafluidand
versatileformcapableofstimulatingawarenessoftheimportanceofmathematicsinindustry,as
well as responding to the needs of industrial technologies. The history of this integration and
reorganizationindicatesthatthisbasicideawillsomedayfindincreasingutility.Mathematicscanbe
akeytechnologyinmodernsociety.
Theseriesaimstopromotethistrendby(1)providingcomprehensivecontentonapplicationsof
mathematics, especially to industry technologies via various types of scientific research,
(2)introducingbasic,useful,necessaryandcrucialknowledgeforseveralapplicationsthroughcon-
crete subjects, and (3) introducing new research results and developments for applications of
mathematicsintherealworld.Thesepointsmayprovidethebasisforopeninganewmathematics
orientedtechnologicalworldandevennewresearchfieldsofmathematics.
More information about this series at http://www.springer.com/series/13254
Nikos I. Kavallaris Takashi Suzuki
(cid:129)
Non-Local Partial
Differential Equations
for Engineering and Biology
Mathematical Modeling and Analysis
123
NikosI. Kavallaris Takashi Suzuki
Department ofMathematics Department ofMathematics
University of Chester Osaka University
Chester Osaka
UK Japan
ISSN 2198-350X ISSN 2198-3518 (electronic)
Mathematics for Industry
ISBN978-3-319-67942-6 ISBN978-3-319-67944-0 (eBook)
https://doi.org/10.1007/978-3-319-67944-0
LibraryofCongressControlNumber:2017957187
©SpringerInternationalPublishingAG2018
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I dedicate this monograph to my beloved wife
Yota and to my dear parents.
Nikos I. Kavallaris
Preface
Manyclassicalpartialdifferentialequations,arisinginmodelingofvariousphysical
and biological phenomena, are only taking into consideration local spatial or time
variables, ignoring any possible spatial dependence on the neighboring points as
well as neglecting any feasible memory effects. For instance, the classical (local)
nonlinear heat equation provides the rate of change of temperature at every (local-
ized)pointandforanytimewherethesourcetermoverlookinganypossibleeffects
fromanynearbypointsandignoringthehistoryoftheheatingprocess.Incontrast,
non-localmodelsconsideranypossiblerelianceoftheinvolvedphysicalquantities
on the evolution of the inspected process all over the nearby spatial points or any
dependenceonprecedingtimes.Suchanon-localdependenceusuallystemsfroma
distance interaction or from several conservation laws. Some of the first non-local
equations appeared in the literature are encountered in the field of phase transition
and are related to theories due to van der Waals, Ginzburg & Landau and Cahn &
Hilliard[5].Lately,awidevarietyofnon-localequationsemergedintheliterature
withsignificantapplicationsinengineering,astrophysics,andbiology.Forinstance
such models with non-local spatial terms are encountered in the Ohmic heating
production[29,30],intheshearbandingformationinmetalsbeingdeformedunder
highstrainrates[1,2],inthetheoryofgravitationalequilibriumofpolytropicstars
[27],intheinvestigationofthefullyturbulentbehaviorofrealflows,usinginvariant
measures for the Euler equation [3], in population dynamics [8], in modeling
aggregationofcellsviainteractionwithachemicalsubstance(chemotaxis)[32],just
to mention a few of them. Some more applications of non-local equations with
memory (integral) terms can be found in [31]. Therefore, non-locality is not a
technicalobstructiontoscientificresearchbutitactuallyprovidesanessenceofwhat
happens in reality, and for that purpose, its mathematical study can provide very
useful predictions in many areas ofapplications. In general, non-local models pro-
vide more accurate predictions compared to their local counterparts since they
actually use all the available information regarding the evolution of the inspected
process.Ontheotherhand,thepresenceofthenon-localtermsmightberesponsible
for the lack of some fundamental features, that share the local analogue problems,
vii
viii Preface
likethemaximumprinciple[30,31].Additionally,mostofthenon-localproblems
exhibit quite rich dynamics, which is usually more complicated to the dynamics
of their local counterparts. In particular, the long-time behavior of non-local para-
bolicequationsmightbemorecomplexthantheoneoftheirlocalcomplements,see
forexample[4,6,7,8].Anotherveryintriguingphenomenonwhichisbasicallydue
tothepresenceofthenonlinearity,butwhoseimpactgrowswhenanon-localtermis
alsopresent,istheoccurrenceoffinite-timeblow-uporfinite-timequenching,when
solutions of nonlinear equations cannot be extended after a finite time. Notably,
blow-up and quenching fight against the well-posedness of nonlinear evolution
equations, since both, under some circumstances, they rule out the possibility of
existence of global in time solutions. Detailed profiles of the blowing up and
quenching solutions, however, are heavily associated with the form of the nonlin-
earity,andthewholemechanismsofthesephenomenahavenotyetbeenclarifiedin
many equations as it happens with the blow-up of the solutions for the famous
exampleoftheNavier–Stokesequation.Stillalotofeffortshavebeenpaidsincethe
pioneeringworkofFujita[9,10]ontheblow-upofsemilinearparabolicequations,
andmanyimportantoutcomeshavebeenproduced.Attheearlieststage,studieson
blow-up and quenching were thought to be related only within the field of pure
mathematics where several toy models were investigated. However, recently it has
beenrecognizedthatmanyrealisticmodelsarereducedtosemilinearequationswith
non-local terms and many of them exhibit the phenomena of finite blow-up and
quenching. This monograph isdevoted to this type of nonlinear partial differential
equations, investigating both their mathematical modeling and their mathematical
analysis.
PartIofthecurrentmonographisdevotedtotheinvestigationofsomenon-local
models linked with applications from engineering. Chapter 1 focuses on the study
of the following non-local model associated with electrostatic MEMS control
@u k
¼Duþ (cid:2) R (cid:3) in X(cid:3)ð0;TÞ ð1Þ
@t ð1(cid:2)uÞ2 1þa 1 2
X1(cid:2)u
uðx;tÞ¼0 on @X(cid:3)ð0;TÞ ð2Þ
uðx;0Þ¼u ðxÞ in X; ð3Þ
0
where uðx;tÞ denotes the deformation of an elastic membrane which is part of the
MEMS device. Here and henceforth, k stands for a positive parameter. In the first
place,theconstructionoftheabovemodelispresented.Tothisend,wedescribethe
two main physical problems which build up the operation of an idealized MEMS
device: the elastic and the electric problem . In the second place, we proceed with
its mathematical analysis. First, the structure of the set of radially symmetric
steady-state solutions is investigated together with their stability. Then, the cir-
cumstances under which finite-time quenching, or otherwise called touching down
Preface ix
intheMEMScontext,occursareinvestigatedfollowingtheapproachdevelopedin
[11, 16]. FinallyChap. 1 closes up withthe investigation ofa hyperbolic variation
of Eqs. (1)–(3) along the lines of the approach introduced in [14].
Chapter2discussessomenon-localmodelsdescribingOhmicheatproductionin
various industrial processes. In the first part of the chapter, the process of food
sterilization by using Ohmic heating is considered and the following
one-dimensional non-local model is formulated
@u @u @2u kfðuÞ
@t þ @x ¼ @x2 þ (cid:4)R1 fðuÞdx(cid:5)2 in ð0;1Þ(cid:3)ð0;TÞ; ð4Þ
(cid:2)1
uð0;tÞ¼u ð1;tÞ¼0 in ð0;TÞ; ð5Þ
x
uðx;0Þ¼u ðxÞ in ð0;1Þ; ð6Þ
0
along thelinesof [24, 29].Here,uðx;tÞstands forthe temperatureof thesterilized
food, while the nonlinearity fðuÞ represents either electrical conductivity or resis-
tivity and is taken to be positive. Next, and under different circumstances, a
hyperbolicvariationofEqs.(4)–(6)withanon-localconvectionvelocityisbuiltup
following the approach introduced in [25]. Both of these non-local models are
inspectedintermsoftheirstabilityandtheoccurrenceoffinite-timeblow-up,where
thelatterinthecurrentcontextmeansfoodburning.Differentapproachesshouldbe
followed though depending on the monotonicity of the nonlinearity f; since no
maximum principle is available for the non-local parabolic problem (Eqs. (4)–(6))
whenf isincreasing.Inthecasewheref isdecreasing,someusefulestimatesofthe
blow-up(burning)timearegivenviathemethoddevelopedin[19,28].Finally,the
hyperbolic problem istreated via themethod ofcharacteristics. The second part of
Chap. 2 deals with another application of Ohmic heating process in the thermistor
device which is modeled by the following
@u kfðuÞ
¼rðkðuÞruÞþ (cid:2)R (cid:3) in X(cid:3)ð0;TÞ; ð7Þ
@t fðuÞ 2
X
@u
þbu¼0 on @X(cid:3)ð0;TÞ; ð8Þ
@m
uðx;0Þ¼u ðxÞ in X; ð9Þ
0
whereuðx;tÞisthetemperatureacrossthethermistordevice,while @ standsforthe
@m
normal outward derivative to @X and 0(cid:4)b(cid:4)1; kðuÞ[0: The finite-time
blow-up, which suggests either the destruction of the thermistor device or the
failure of the model, is investigated via the methods developed in [18].
x Preface
Chapter3debatesanapplicationarisingintheprocessoflinearfrictionwelding
applied in metallurgy. Initially, the following one-dimensional non-local model is
constructed
eu
u ¼u þ (cid:2)R (cid:3) ; in ð0;1Þ(cid:3)ð0;TÞ; a[0; ð10Þ
t xx 1eudx 1þa
0
u ð0;tÞ;¼0; lim u ðx;tÞ¼(cid:2)1; in ð0;TÞ; ð11Þ
x x
x!1
uðx;0Þ¼u ðxÞ; in ð0;1Þ; ð12Þ
0
forthesoft-materialcase,whereuðx;tÞservesasthetemperatureacrossthewelding
region.Next,asimilarnon-localmodelisderivedforthehard-materialcasewhere
the exponential nonlinearity is replaced by fðuÞ¼ð(cid:2)uÞ(cid:2)p for p¼1: The stability
a
of Eqs. (10)–(12) is investigated using the analytical approach developed in [13],
which actually proves convergence to the unique steady state. On the other hand,
thestabilityinthecaseofthepower-lawnonlinearityisinvestigatedbymeansofa
numerical approach [13].
The following degenerate non-local model
kfðbðuÞÞ
u ¼DbðuÞþ R ; in X(cid:3)ð0;TÞ; T[0; ð13Þ
t ð fðbðuÞÞdxÞ2
X
@bðuÞ
bðuÞþkðxÞ ¼0; on @X(cid:3)ð0;TÞ; ð14Þ
@m
uðx;0Þ¼u ðxÞ; in X; ð15Þ
0
isproducedinChap.4,where0(cid:4)kðxÞ(cid:4)1andbðuÞ(cid:5)0arecontinuousfunctions
with bð0Þ¼0: Non-local model (Eqs. (13)–(15)) is associated with the industrial
process of resistance spot welding and uðx;tÞ serves as the temperature in the
weldingarea.Byusinganumericalschemedevelopedin[26],theoccurrenceofan
emerging interface (free boundary), stemming from the degeneracy due to the
condition bð0Þ¼0; is revealed.
PartIIishandlingsomeapplicationsofnon-localmodelscomingforthefieldof
biology. In Chap. 5, the following non-local problem is derived
up
ut ¼Du(cid:2)uþ (cid:6)Z (cid:7)c; in X(cid:3)ð0;TÞ; ð16Þ
(cid:2) urdx
X
@u
¼0; on @X(cid:3)ð0;TÞ; ð17Þ
@m
