AppliedMathematics,2020,11,1219-1228
https://www.scirp.org/journal/am
ISSNOnline:2152-7393
ISSNPrint:2152-7385
StochasticViscositySolutionsforSPDEswith
DiscontinuousCoefficients
YidongZhang
CollegeofScience,NorthChinaUniversityofTechnology,Beijing,China
Howtocitethispaper:Zhang,Y.D.(2020)
StochasticViscositySolutionsforSPDEswith
DiscontinuousCoefficients.AppliedMathe-
matics,11,1219-1228.
https://doi.org/10.4236/am.2020.1111083
Received:October15,2020
Accepted:November27,2020
Published:November30,2020
Copyright
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2020byauthor(s)andScientific
ResearchPublishingInc.
ThisworkislicensedundertheCreativeCom-
monsAttributionInternationalLicense(CC
BY4.0).
http://creativecommons.org/licenses/by/4.0/
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sociisnatoquepenatibusetmagnisdispar-
turientmontes,nasceturridiculusmus.Pel-
lentesquehabitantmorbitristiquesenectuset
netusetmalesuadafamesacturpisegestas.
Phasellusidmagna.Duismalesuadainter-
dumarcu.Integermetus.Morbipulvinarpel-
lentesquemi.Suspendissesedesteumagna
molestieegestas.Quisquemilorem,pulvinar
eget,egestasquis,luctusat,ante.Proinauc-
torvehiculapurus.Fusceacnislaliquamante
hendreritpellentesque.Classaptenttacitiso-
ciosquadlitoratorquentperconubianostra,
perinceptoshymenaeos.Morbiwisi.Etiam
arcumauris,facilisissed,eleifendnon,non-
ummyut,pede.Crasutlacustempormetus
mollisplacerat.Vivamuseutortorvelmetus
interdummalesuada.
Sedeleifend,erossitametfaucibuselemen-
tum,urnasapienconsectetuermauris,quis
egestasleojustononrisus.Morbinonfelisac
liberovulputatefringilla.Maurisliberoeros,
lacinianon,sodalesquis,dapibusporttitor,
pede.Classaptenttacitisociosquadlitora
torquentperconubianostra,perinceptoshy-
menaeos.Morbidapibusmauriscondimentum
nulla.Cumsociisnatoquepenatibusetmagnis
disparturientmontes,nasceturridiculusmus.
Etiamsitameterat.Nullavarius.Etiamt-
inciduntduivitaeturpis.Donecleo.Morbi
vulputateconvallisest.Integeraliquet.Pel-
lentesquealiquetsodalesurna.
Nullameleifendjustoinnisl.Inhachabitasse
plateadictumst.Morbinonummy.Aliquam
utfelis.Invelitleo,dictumvitae,posuereid,
vulputatenec,ante.Maecenasvitaepedenec
duidignissimsuscipit.Morbimagna.Vestibu-
lumidpurusegetvelitlaoreetlaoreet.Prae-
sentsedleovelnibhconvallisblandit.Ut
rutrum.Donecnibh.Donecinterdum.Fusce
sedpedesitametelitrhoncusultrices.Nullam
atenimvitaepedevehiculaiaculis.
Classaptenttacitisociosquadlitoratorquen-
tperconubianostra,perinceptoshymenaeos.
Aeneannonummyturpisidodio.Integereu-
ismodimperdietturpis.Utnecleonecdi-
amimperdietlacinia.Etiamegetlacuseget
miultriciesposuere.Inplacerattristiquetor-
tor.Sedportavestibulummetus.Nullaia-
culissollicitudinpede.Fusceluctustellusin
dolor.Curabiturauctorvelitasem.Mor-
bisapien.Classaptenttacitisociosquadl-
itoratorquentperconubianostra,perincep-
toshymenaeos.Donecadipiscingurnavehic-
ulanunc.Sedornareleoinleo.Inrhoncus
leoutdui.Aeneandolorquam,volutpatnec,
fringillaid,consectetuervel,pede.
Nullamalesuadarisusuturna.Aenean
pretiumvelitsitametmetus.Duisiaculis.In
hachabitasseplateadictumst.Nullammo-
lestieturpisegetnisl.Duisamassaidpede
dapibusultricies.Sedeuleo.Inatmaurissit
amettortorbibendumvarius.Phasellusjusto
risus,posuerein,sagittisac,variusvel,tor-
tor.Quisqueidenim.Phasellusconsequat,
liberopretiumnonummyfringilla,tortorlacus
vestibulumnunc,utrhoncusligulanequeid
justo.Nullamaccumsaneuismodnunc.Proin
vitaeipsumacmetusdictumtempus.Namut
wisi.Quisquetortorfelis,interdumac,sodales
a,sempera,sem.Curabiturinvelitsitamet
duitristiquesodales.Vivamusmaurispede,
laciniaeget,pellentesquequis,scelerisqueeu,
est.Aliquamrisus.Quisquebibendumpede
eudolor.
Donectempusnequevitaeest.Aeneanegestas
odiosedrisusullamcorperullamcorper.Sedin
nullaatortortinciduntegestas.Namsapien
tortor,elementumsitamet,aliquamin,port-
titorfaucibus,enim.Nullamconguesuscipit
nibh.Quisqueconvallis.Praesentarcunibh,
vehiculaeget,accumsaneu,tincidunta,nibh.
Suspendissevulputate,tortorquisadipiscing
viverra,lacusnibhdignissimtellus,eususcipit
risusantefringilladiam.Quisquealiberov-
elpedeimperdietaliquet.Pellentesquenunc
nibh,eleifenda,consequatconsequat,hen-
dreritnec,diam.Sedurna.Maecenaslaoreet
eleifendneque.Vivamuspurusodio,eleifend
non,iaculisa,ultricessitamet,urna.Mau-
risfaucibusodiovitaerisus.Innisl.Praesent
purus.Integeriaculis,semeuegestaslacinia,
lacuspedescelerisqueaugue,inullamcorper
dolorerosaclacus.Nuncinlibero.
Fuscesuscipitcursussem.Vivamusrisusmi,
egestasac,imperdietvarius,faucibusquis,
leo.Aeneantincidunt.Donecsuscipit.Cras
idjustoquisnibhscelerisquedignissim.Ali-
quamsagittiselementumdolor.Aeneancon-
sectetuerjustoinpede.Curabiturullamcorper
ligulanecorci.Aliquampurusturpis,aliquam
id,ornarevitae,porttitornon,wisi.Maecenas
luctusportalorem.Donecvitaeligulaeuante
pretiumvarius.Prointortormetus,convallis
et,hendreritnon,scelerisquein,urna.Cras
quisliberoeuligulabibendumtempor.Viva-
mustellusquam,malesuadaeu,tempussed,
temporsed,velit.Doneclaciniaauctorlibero.
Praesentsednequeidpedemollisrutrum.
Vestibulumiaculisrisus.Pellentesquelacus.
Utquisnuncsedodiomalesuadaegestas.Duis
amagnasitametligulatristiquepretium.Ut
pharetra.Vestibulumimperdietmagnanec
wisi.Maurisconvallis.Sedaccumsansollic-
itudinmassa.Sedidenim.Nuncpedeenim,
laciniaut,pulvinarquis,suscipitsemper,elit.
Crasaccumsaneratvitaeenim.Crassollici-
tudin.Vestibulumrutrumblanditmassa.
Sedgravidalectusutpurus.Morbilaoreet
magna.Pellentesqueeuwisi.Prointurpis.In-
tegersollicitudinauguenecdui.Fuscelectus.
Vivamusfaucibusnullaneclacus.Integerdi-
am.Pellentesquesodales,enimfeugiatcursus
volutpat,semmaurisdignissimmauris,quis
consequatsemestfermentumligula.Nullam
justolectus,condimentumsitamet,posuerea,
fringillamollis,felis.Morbinullanibh,pellen-
tesqueat,nonummyeu,sollicitudinnec,ip-
sum.Crasneque.Nuncaugue.Nullamvitae
quamidquampulvinarblandit.Nuncsitamet
orci.Aliquameratelit,pharetranec,aliquet
a,gravidain,mi.Quisqueurnaenim,viverra
quis,suscipitquis,tinciduntut,sapien.Cras
placeratconsequatsem.Curabituracdiam.
Curabiturdiamtortor,molliset,viverraac,
tempusvel,metus.
Curabituraclorem.Vivamusnonjustoin
duimattisposuere.Etiamaccumsanligula
idpede.Maecenastinciduntdiamnecvelit.
Praesentconvallissapienacest.Aliquamul-
lamcorpereuismodnulla.Integermollisenim
veltortor.Nullasodalesplaceratnunc.Sed
tempusrutrumwisi.Duisaccumsangravi-
dapurus.Nuncnunc.Etiamfacilisisduieu
sem.Vestibulumsemper.Praesenteueros.
Vestibulumtellusnisl,dapibusid,vestibulum
sitamet,placeratac,mauris.Maecenasetelit
uteratplaceratdictum.Namfeugiat,turpis
etsodalesvolutpat,wisiquamrhoncusneque,
vitaealiquamipsumsapienvelenim.Maece-
nassuscipitcursusmi.
Quisqueconsectetuer.Insuscipitmaurisado-
lorpellentesqueconsectetuer.Maurisconvallis
nequenonerat.Inlacinia.Pellentesqueleo
eros,sagittisquis,fermentumquis,tincidun-
tut,sapien.Maecenassem.Curabitureros
odio,interdumeu,feugiateu,portaac,nis-
l.Curabiturnunc.Etiamfermentumconvallis
velit.Pellentesquelaoreetlacus.Quisquesed
elit.Namquistellus.Aliquamtellusarcu,
adipiscingnon,tincidunteleifend,adipiscing
quis,augue.Vivamuselementumplaceraten-
im.Suspendisseuttortor.Integerfaucibus
adipiscingfelis.Aeneanconsectetuermattis
lectus.Morbimalesuadafaucibusdolor.Nam
lacus.Etiamarculibero,malesuadavitae,ali-
quamvitae,blandittristique,nisl.
Maecenasaccumsandapibussapien.Duis
pretiumiaculisarcu.Curabiturutlacus.Ali-
quamvulputate.Suspendisseutpurussedsem
temporrhoncus.Utquamdui,fringillaat,dic-
tumeget,ultriciesquis,quam.Etiamsemest,
pharetranon,vulputatein,pretiumat,ipsum.
Nuncsempersagittisorci.Sedscelerisquesus-
cipitdiam.Utvolutpat,doloratullamcorp-
ertristique,erospurusmollisquam,sitamet
ornareantenuncetenim.
Phasellusfringilla,metusidfeugiatcon-
sectetuer,lacuswisiultricestellus,quislobor-
tisnibhloremquistortor.Donecegestas
ornarenulla.Maurismitellus,portafau-
cibus,dictumvel,nonummyin,est.Ali-
quameratvolutpat.Intellusmagna,port-
titorlacinia,molestievitae,pellentesqueeu,
justo.Classaptenttacitisociosquadlitora
torquentperconubianostra,perinceptoshy-
menaeos.Sedorcinibh,scelerisquesitamet,
suscipitsed,placeratvel,diam.Vestibulum
nonummyvulputateorci.Donecetvelitacar-
cuinterdumsemper.Morbipedeorci,cursus
ac,elementumnon,vehiculaut,lacus.Cras
volutpat.Namvelwisiquisliberovenenatis
placerat.Aeneansedodio.Quisqueposuere
purusacorci.Vivamusodio.Vivamusvar-
ius,nullasitametsemperviverra,odiomau-
risconsequatlacus,atvestibulumnequearcu
eutortor.Doneciaculistincidunttellus.Ali-
quameratvolutpat.Curabiturmagnalorem,
dignissimvolutpat,viverraet,adipiscingnec,
dolor.Praesentlacusmauris,dapibusvitae,
sollicitudinsitamet,nonummyeget,ligula.
Crasegestasipsumanisl.Vivamusvariusdo-
lorutdolor.Fuscevelenim.Pellentesque
accumsanligulaeteros.Crasidlacusnon
tortorfacilisisfacilisis.Etiamnislelit,cur-
sussed,fringillain,conguenec,urna.Cum
sociisnatoquepenatibusetmagnisdispar-
turientmontes,nasceturridiculusmus.In-
tegeratturpis.Cumsociisnatoquepenati-
busetmagnisdisparturientmontes,nascetur
ridiculusmus.Duisfringilla,ligulasedporta
fringilla,ligulawisicommodofelis,utadipisc-
ingfelisduiinenim.Suspendissemalesuada
ultricesante.Pellentesquescelerisqueaugue
sitameturna.Nullavolutpataliquettortor.
Crasaliquam,tellusataliquetpellentesque,
justosapiencommodoleo,idrhoncussapien
quamaterat.Nullacommodo,wisiegetsol-
licitudinpretium,orciorcialiquamorci,ut
cursusturpisjustoetlacus.Nullaveltor-
tor.Quisqueeratelit,viverrasitamet,sagittis
eget,portasitamet,lacus.
Inhachabitasseplateadictumst.Proinatest.
Curabiturtempusvulputateelit.Pellentesque
sem.Praesenteusapien.Duiselitmagna,ali-
quetat,tempussed,vehiculanon,enim.Mor-
biviverraarcunecpurus.Vivamusfringilla,
enimetcommodomalesuada,tortormetusel-
ementumligula,necaliquetestsapienutlec-
tus.Aliquammi.Utnecelit.Fusceeuismod
luctustellus.Curabiturscelerisque.Nullam
purus.Namultriciesaccumsanmagna.Morbi
pulvinarloremsitametipsum.Donecutjus-
tovitaenibhmolliscongue.Fuscequisdiam.
Praesenttempuserosutquam.
Donecinnisl.Fuscevitaeest.Vivamusante
ante,mattislaoreet,posuereeget,conguevel,
nunc.Fuscesem.Namvelorcieuerosviver-
raluctus.Pellentesquesitametaugue.Nunc
sitametipsumetlacusvariusnonummy.In-
tegerrutrumsemegetwisi.Aeneaneusapien.
Quisqueornaredignissimmi.Duisaurnavel
risuspharetraimperdiet.Suspendissepotenti.
Morbijusto.Aeneannecdolor.Inhac
habitasseplateadictumst.Proinnonummy
porttitorvelit.Sedsitametleonecmetus
rhoncusvarius.Crasante.Vestibulumcom-
modosemtinciduntmassa.Namjusto.Ae-
neanluctus,felisetcondimentumlacinia,lec-
tusenimpulvinarpurus,nonportavelitnisl
sederos.Suspendisseconsequat.Maurisadui
ettortormattispretium.Sednullametus,vo-
lutpatid,aliquameget,ullamcorperut,ipsum.
Morbieununc.Praesentpretium.Duisali-
quampulvinarligula.Utblanditegestasjusto.
Quisqueposueremetusviverrapede.
Vivamussodaleselementumneque.Viva-
musdignissimaccumsanneque.Sedatenim.
Vestibulumnonummyinterdumpurus.Mau-
risornarevelitidnibhpretiumultricies.Fusce
temporpellentesqueodio.Vivamusauguepu-
rus,laoreetin,scelerisquevel,commodoid,
wisi.Duisenim.Nullainterdum,nunceusem-
pereleifend,enimdolorpretiumelit,utcom-
modoligulanislaest.Vivamusante.Nulla
leomassa,p osuerenec,volutpatvitae,rhon-
cuseu,magna.
Quisquefacilisisauctorsapien.Pellentesque
gravidahendreritlectus.Maurisrutrumso-
dalessapien.Fuscehendreritsemvellorem.
Integerpellentesquemassavelaugue.Integer
elittortor,feugiatquis,sagittiset,ornarenon,
lacus.Vestibulump osuerepellentesqueeros.
Quisquevenenatisipsumdictumnulla.Ali-
quamquisquamnonmetuseleifendinterdum.
Namegetsapienacmaurismalesuadaadipisc-
ing.Etiameleifendnequesedquam.Nulla
facilisi.Proinaligula.Sedidduieunibh
egestastincidunt.Suspendissearcu.
Maecenasdui.Aliquamvolutpatauctorlorem.
Crasplaceratestvitaelectus.Curabitur
massalectus,rutrumeuismod,dignissimut,
dapibusa,odio.Uteroserat,vulputateut,in-
terdumnon,portaeu,erat.Crasfermentum,
felisinportacongue,velitleofacilisisodio,vi-
taeconsectetuerloremquamvitaeorci.Sed
ultrices,pedeeuplaceratauctor,anteligula
rutrumtellus,velposuerenibhlacusnecnibh.
Maecenaslaoreetdoloratenim.Donecmo-
lestiedolornecmetus.Vestibulumlibero.Sed
quiserat.Sedtristique.Duispedeleo,fer-
mentumquis,consectetuereget,vulputatesit
amet,erat.
Donecvitaevelit.Suspendisseportafermen-
tummauris.Utvelnuncnonmaurisphare-
travarius.Duisconsequatliberoquisurna.
Maecenasatante.Vivamusvarius,wisised
egestastristique,odiowisiluctusnulla,lobor-
tisdictumdolorligulainlacus.Vivamusali-
quam,urnasedinterdumporttitor,metusorci
interdumodio,sitameteuismodlectusfeliset
leo.Praesentacwisi.Namsuscipitvestibu-
lumsem.Praesenteuipsumvitaepedecursus
venenatis.Duissedodio.Vestibulumeleifend.
Nullautmassa.Proinrutrummattissapien.
Curabiturdictumgravidaante.
Phasellusplaceratvulputatequam.Maece-
nasattellus.Pellentesquenequediam,dig-
nissimac,venenatisvitae,consequatut,lacus.
Namnibh.Vestibulumfringillaarcumollisar-
cu.Sedetturpis.Donecsemtellus,volut-
patet,variuseu,commodosed,lectus.Lorem
ipsumdolorsitamet,consectetueradipiscing
elit.Quisqueenimarcu,suscipitnec,tempus
at,imperdietvel,metus.Morbivolutpatpu-
rusaterat.Donecdignissim,semidsemper
tempus,nibhmassaeleifendturpis,sedpellen-
tesquewisipurussedlibero.Nullamlobortis
tortorvelrisus.Pellentesqueconsequatnulla
eutellus.Donecvelit.Aliquamfermentum,
wisiacrhoncusiaculis,tellusnuncmalesuada
orci,quisvolutpatduimagnaidmi.Nuncvel
ante.Duisvitaelacus.Crasnecipsum.
Morbinunc.Aliquamconsectetuervariusnul-
la.Phaselluseros.Crasdapibusporttitorris-
us.Maecenasultricesmiseddiam.Praesent
gravidavelitatelitvehiculaporttitor.Phasel-
lusnislmi,sagittisac,pulvinarid,gravidasit
amet,erat.Vestibulumest.Loremipsumdo-
lorsitamet,consectetueradipiscingelit.Cur-
abituridsemelementumleorutrumhendrerit.
Utatmi.Donectinciduntfaucibusmassa.
Sedturpisquam,sollicitudina,hendreriteget,
pretiumut,nisl.Duishendreritligula.Nunc
pulvinarcongueurna.
Nuncvelit.Nullamelitsapien,eleifendeu,
commodonec,sempersitamet,elit.Nullalec-
tusrisus,condimentumut,laoreeteget,viver-
ranec,odio.Proinlobortis.Curabiturdictum
arcuvelwisi.Crasidnullavenenatistortor
congueultrices.Pellentesqueegetpede.Sed
eleifendsagittiselit.Namsedtellussitamet
lectusullamcorpertristique.Maurisenimsem,
tristiqueeu,accumsanat,scelerisquevulpu-
tate,neque.Quisquelacus.Donecetipsumsit
ametelitnonummyaliquet.Sedviverranislat
sem.Namdiam.Maurisutdolor.Curabitur
ornaretortorcursusvelit.
Morbitinciduntposuerearcu.Crasvenenatis
estvitaedolor.Vivamusscelerisquesemper
mi.Donecipsumarcu,consequatscelerisque,
viverraid,dictumat,metus.Loremipsum
dolorsitamet,consectetueradipiscingelit.Ut
pedesem,tempusut,porttitorbibendum,mo-
lestieeu,elit.Suspendissepotenti.Sedidlec-
tussitametpurusfaucibusvehicula.Praesen-
tsedsemnonduipharetrainterdum.Nam
viverraultricesmagna.
Aeneanlaoreetaliquamorci.Nuncinterdum
elementumurna.Quisqueerat.Nullamtem-
porneque.Maecenasvelitnibh,scelerisquea,
consequatut,viverrain,enim.Duismagna.
Donecodioneque,tristiqueet,tincidunteu,
rhoncusac,nunc.Maurismalesuadamalesua-
daelit.Etiamlacusmauris,pretiumvel,blan-
ditin,ultriciesid,libero.Phasellusbibendum
eratutdiam.Incongueimperdietlectus.
Aeneanscelerisque.Fuscepretiumporttitor
lorem.Inhachabitasseplateadictumst.Nulla
sitametnislatsapienegestaspretium.Nunc
nontellus.Vivamusaliquet.Namadipiscing
euismoddolor.Aliquameratvolutpat.Nul-
lautipsum.Quisquetinciduntauctorau-
gue.Nuncimperdietipsumegetelit.Ali-
quamquamleo,consectetuernon,ornaresit
amet,tristiquequis,felis.Vestibulumanteip-
sumprimisinfaucibusorciluctusetultrices
posuerecubiliaCurae;Pellentesqueinterdum
quamsitametmi.Pellentesquemaurisdui,
dictuma,adipiscingac,fermentumsitamet,
lorem.
Utquiswisi.Praesentquismassa.Vivamus
egestasrisusegetlacus.Nunctincidunt,risus
quisbibendumfacilisis,lorempurusrutrum
neque,necportatortorurnaquisorci.Ae-
neanaliquet,liberosempervolutpatluctus,
pedeeratlaciniaaugue,quisrutrumsemip-
sumsitametpede.Vestibulumaliquet,nibh
sediaculissagittis,odiodolorblanditaugue,
egetmollisurnatellusidtellus.Aeneanali-
quetaliquamnunc.Nullaultriciesjustoeget
orci.Phasellustristiquefermentumleo.Sed
massametus,sagittisut,semperut,pharetra
vel,erat.Aliquamquamturpis,egestasvel,
elementumin,egestassitamet,lorem.Duis
convallis,wisisitametmollismolestie,libero
maurisportadui,vitaealiquamarcuturpisac
sem.Aliquamaliquetdapibusmetus.
Vivamuscommo doeroseleifenddui.Vestibu-
luminleoeuerattristiquemattis.Crasat
elit.Craspellentesque.Nullamidlacussit
ametliberoaliquethendrerit.Proinplacerat,
minonelementumlaoreet,eroselittincidun-
tmagna,arhoncussemarcuidodio.Nul-
laegetleoaleoegestasfacilisis.Curabitur
quisvelit.Phasellusaliquam,tortornecornare
rhoncus,purusurnap osuerevelit,etcommo-
dorisustellusquistellus.Vivamusleoturpis,
tempussitamet,tristiquevitae,laoreetquis,
odio.Proinscelerisquebibendumipsum.Eti-
amnisl.Praesentveldolor.Pellentesquevel
magna.Curabitururna.Vivamuscongueur-
nainvelit.Etiamullamcorperelementumdui.
Praesentnonurna.Sedplaceratquamnonmi.
Pellentesquediammagna,ultricieseget,ultri-
cesplacerat,adipiscingrutrum,sem.
Morbisem.Nullafacilisi.Vestibulumante
ipsumprimisinfaucibusorciluctusetultri-
cesposuerecubiliaCurae;Nullafacilisi.Mor-
bisagittisultriceslibero.Praesenteuligula
sedsapienauctorsagittis.Classaptenttaciti
sociosquadlitoratorquentperconubianos-
tra,perinceptoshymenaeos.Donecvelnunc.
Nuncfermentum,lacusidaliquamporta,dui
tortoreuismoderos,velmolestieipsumpurus
eulacus.Vivamuspedearcu,euismodac,tem-
pusid,pretiumet,lacus.Curabitursodales
dapibusurna.Nunceusapien.Doneceget
nuncapededictumpretium.Proinmauris.
Vivamusluctusliberovelnibh.
Fuscetristiquerisusidwisi.Integermolestie
massaidsem.Vestibulumveldolor.Pel-
lentesquevelurnavelrisusultricieselemen-
tum.Quisquesapienurna,blanditnec,iaculis
ac,viverrain,odio.Inhachabitasseplatea
dictumst.Morbinequelacus,convallisvitae,
commodoac,fermentumeu,velit.Sedinor-
ci.Infringillaturpisnonarcu.Donecinante.
Phasellustemporfeugiatvelit.Aeneanvar-
iusmassanonturpis.Vestibulumanteipsum
primisinfaucibusorciluctusetultricespo-
suerecubiliaCurae;
Aliquamtortor.Morbiipsummassa,imperdi-
etnon,consectetuervel,feugiatvel,lorem.
Quisqueegetloremnecelitmalesuadavestibu-
lum.Quisquesollicitudinipsumvelsem.Nul-
laenim.Proinnonummyfelisvitaefelis.Nul-
lampellentesque.Duisrutrumfeugiatfe-
lis.Maurisvelpedesedliberotinciduntmol-
lis.Phasellussedurnarhoncusdiameuismod
bibendum.Phasellussednisl.Integercondi-
mentumjustoidorciiaculisvarius.Quisqueet
lacus.Phaselluselementum,justoatdignissim
auctor,wisiodiolobortisarcu,sedsollicitudin
felisfeliseuneque.Praesentatlacus.
Vivamussitametpede.Duisinterdum,nunc
egetrutrumdignissim,nisldiamluctusleo,
ettinciduntvelitnislidtellus.Inloremtel-
lus,aliquetvitae,p ortain,aliquetsed,lectus.
Phasellussodales.Utvariusscelerisqueerat.
Invelnibheuerosimperdietrutrum.Donec
acodionecnequevulputatesuscipit.Namnec
magna.Pellentesquehabitantmorbitristique
senectusetnetusetmalesuadafamesacturpis
egestas.Nullamporta,odioetsagittisiaculis,
wisinequefringillasapien,velcommodolorem
loremidelit.Utsemlectus,scelerisqueeget,
placeratet,tinciduntscelerisque,ligula.Pel-
lentesquenonorci.
Etiamvelipsum.Morbifacilisisvestibulum
nisl.Praesentcursuslaoreetfelis.Integer
adipiscingpretiumorci.Nullafacilisi.Quisque
posuerebibendumpurus.Nullaquammauris,
cursuseget,convallisac,molestienon,enim.
Aliquamcongue.Quisquesagittisnonummy
sapien.Proinmolestiesemvitaeurna.Mae-
cenaslorem.Vivamusviverraconsequatenim.
Nuncsedpede.Praesentvitaelectus.Prae-
sentnequejusto,vehiculaeget,interdumid,
facilisiset,nibh.Phasellusatpurusetlibero
laciniadictum.Fuscealiquet.Nullaeuante
placeratleosemperdictum.Maurismetus.
Curabiturlobortis.Curabitursollicitudinhen-
dreritnunc.Donecultriceslacusidipsum.
Donecanibhutelitvestibulumtristique.In-
tegeratpede.Crasvolutpatvariusmagna.
Phaselluseuwisi.Praesentrisusjusto,lobor-
tiseget,scelerisqueac,aliquetin,dolor.Proin
idleo.Nunciaculis,mivitaeaccumsancom-
modo,nequesemlacinianulla,quisvestibulum
justosemineros.Quisquesedmassa.Morbi
lectusipsum,vulputatea,mollisut,accumsan
placerat,tellus.Nullaminwisi.Vivamuseu
ligulaanuncaccumsancongue.Suspendisse
aclibero.Aliquameratvolutpat.Donecau-
gue.Nuncvenenatisfringillanibh.Fusceac-
cumsanpulvinarjusto.Nullamsemper,duiut
dignissimauctor,orcilib erofringillamassa,b-
landitpulvinarpedetortoridmagna.Nunc
adipiscingjustosedvelittinciduntfermentum.
Integerplacerat.Pellentesquehabitantmorbi
tristiquesenectusetnetusetmalesuadafames
acturpisegestas.Sedinmassa.Classaptent
tacitisociosquadlitoratorquentperconubi-
anostra,perinceptoshymenaeos.Phasellus
tempusaliquamrisus.Aliquamrutrumpurus
atmetus.Donecposuereodioaterat.Nam
nonnibh.Phasellusligula.Quisquevene-
natislectusinaugue.Sedvestibulumdapibus
neque.
Mauristempuserosatnulla.Sedquisduidig-
nissimmaurispretiumtincidunt.Maurisac
purus.Phasellusaclibero.Etiamdapibusia-
culisnunc.Inlectuswisi,elementumeu,sollic-
itudinnec,imperdietquis,dui.Nullaviverra
nequeaclibero.Maurisurnaleo,adipiscing
eu,ultricesnon,blanditeu,dui.Maecenas
duineque,suscipitsitamet,rutruma,laoreet
in,eros.Uteunibh.Fuscenecerattempus
urnafringillatempus.Curabituridenim.Sed
ante.Crassodalesenimsitametwisi.Nunc
fermentumconsequatquam.
Utauctor,augueportadignissimvestibulum,
arcudiamlobortisvelit,velscelerisqueris-
usauguesagittisrisus.Maecenaseujusto.
Pellentesquehabitantmorbitristiquesenectus
etnetusetmalesuadafamesacturpiseges-
tas.Mauriscongueligulaegettortor.Nul-
lamlaoreeturnasedenim.Donecegeterosut
erosvolutpatconvallis.Praesentturpis.Inte-
germaurisdiam,elementumquis,egestasac,
rutrumvel,orci.Nullafacilisi.Quisqueadip-
iscing,nullavitaeelementumporta,semurna
volutpatleo,sedportaenimrisussedmassa.
Integeracenimquisdiamsodalesluctus.Ut
egeterosaligulacommodoultricies.Doneceu
urnaviverradolorhendreritfeugiat.Aliquam
acorcivelerosconguepharetra.Quisque
rhoncus,justoeuvolutpatfaucibus,augueleo
posuerelacus,arhoncuspuruspedevelest.
Proinultricesenim.
Aeneantinciduntlaoreetdui.Vestibulumante
ipsumprimisinfaucibusorciluctusetultri-
cesposuerecubiliaCurae;Integeripsumlec-
tus,fermentumac,malesuadain,eleifendut,
lorem.Vivamusipsumturpis,elementumvel,
hendreritut,semperat,metus.Vivamussapi-
entortor,eleifendid,dapibusin,egestaset,
pede.Pellentesquefaucibus.Praesentlorem
neque,dignissimin,facilisisnec,hendreritvel,
odio.Namatdiamacnequealiquetviver-
ra.Morbidapibusligulasagittismagna.In
lobortis.Donecaliquetultricieslibero.Nunc
dictumvulputatepurus.Morbivarius.Lorem
ipsumdolorsitamet,consectetueradipiscing
elit.Intempor.Phaselluscommodoporttitor
magna.Curabiturvehiculaodioveldolor.
Praesentfacilisis,augueaadipiscingvene-
natis,liberorisusmolestieodio,pulvinarcon-
sectetuerfeliseratacmauris.Namvestibu-
lumrhoncusquam.Sedveliturna,pharetra
eu,eleifendeu,viverraat,wisi.Maecenasul-
tricesnibhatturpis.Aeneanquam.Nulla
ipsum.Aliquamposuereluctuserat.Cur-
abiturmagnafelis,laciniaet,tristiqueid,ul-
tricesut,mauris.Suspendissefeugiat.Cras
eleifendwisivitaetortor.Phasellusleopurus,
mattissitamet,auctorin,rutrumin,magna.
Inhachabitasseplateadictumst.Phasellus
imperdietmetusinsem.Vestibulumacenim
nonsemultriciessagittis.Sedveldiam.
Integervelenimsedturpisadipiscingbiben-
dum.Vestibulumpededolor,laoreetnec,po-
suerein,nonummyin,sem.Donecimperdiet
sapienplaceraterat.Donecviverra.Aliquam
eros.Nuncconsequatmassaidleo.Sedullam-
corper,loreminsodalesdapibus,risusmetus
sagittislorem,nonporttitorpurusodioneco-
dio.Sedtinciduntposuereelit.Quisqueeu
enim.Donecliberorisus,feugiatac,dapibus
eget,posuerea,felis.Quisquevellectusutme-
tustincidunteleifend.Duisutpede.Duisvelit
erat,venenatisvitae,vulputatea,pharetrasit
amet,est.Etiamfringillafaucibusaugue.
Aeneanvelitsem,viverraeu,tempusid,
rutrumid,mi.Nullamnecnibh.Proinullam-
corper,dolorincursustristique,erosaugue
tempornibh,atgravidadiamwisiatpurus.
Donecmattisullamcorpertellus.Phasellusvel
nulla.Praesentinterdum,erosinsodalessol-
licitudin,nuncnullapulvinarjusto,aeuismod
erossemnecnibh.Nullamsagittisdapibus
lectus.Nullamegetipsumeutortorlobortis
sodales.Etiampurusleo,pretiumnec,feu-
giatnon,ullamcorpervel,nibh.Sedvelelit
etquamaccumsanfacilisis.Nuncleo.Sus-
pendissefaucibuslacus.
Pellentesqueinterdumsapiensednulla.Proin
tincidunt.Aliquamvolutpatestvelmassa.
Seddolorlacus,imperdietnon,ornarenon,
commodoeu,neque.Integerpretiumsem-
perjusto.Proinrisus.Nullamidquam.
Namneque.Duisvitaewisiullamcorperdi-
amcongueultricies.Quisqueligula.Mauris
vehicula.
Curabiturnuncmagna,posuereeget,vene-
natiseu,vehiculaac,velit.Aeneanornare,
massaaaccumsanpulvinar,quamlorem
laoreetpurus,eusodalesmagnarisusmolestie
lorem.Nunceratvelit,hendreritquis,male-
suadaut,aliquamvitae,wisi.Sedposuere.
Suspendisseipsumarcu,scelerisquenec,ali-
quameu,molestietincidunt,justo.Phasellus
iaculis.Sedposuereloremnonipsum.Pel-
lentesquedapibus.Suspendissequamlibero,
laoreeta,tincidunteget,consequatat,est.
Nullamutlectusnonenimconsequatfacili-
sis.Maurisleo.Quisquepedeligula,auctor
vel,pellentesquevel,p osuereid,turpis.Cras
ipsumsem,cursuset,facilisisut,tempuseu-
ismod,quam.Suspendissetristiquedoloreu
orci.Maurismattis.Aeneansemper.Viva-
mustortormagna,facilisisid,variusmattis,
hendreritin,justo.Integerpurus.
Vivamusadipiscing.Curabiturimperdiettem-
pusturpis.Vivamussapiendolor,congueve-
nenatis,euismodeget,portarhoncus,magna.
Proincondimentumpretiumenim.Fusce
fringilla,liberoetvenenatisfacilisis,erosen-
imcursusarcu,vitaefacilisisodioauguevitae
orci.Aliquamvariusnibhutodio.Sedcondi-
mentumcondimentumnunc.Pellentesque
egetmassa.Pellentesquequismauris.Donec
utligulaacpedepulvinarlobortis.Pellen-
tesqueeuismod.Classaptenttacitisociosqu
adlitoratorquentperconubianostra,perin-
ceptoshymenaeos.Praesentelit.Utlaoreet
ornareest.Phasellusgravidavulputatenulla.
Donecsitametarcuutsemtempormalesua-
da.Praesenthendreritaugueinurna.Proin
enimante,ornarevel,consequatut,blandit
in,justo.Donecfeliselit,dignissimsed,sagit-
tisut,ullamcorpera,nulla.Aeneanpharetra
vulputateodio.
Quisqueenim.Proinvelitneque,tristiqueeu,
eleifendeget,vestibulumnec,lacus.Vivamus
odio.Duisodiourna,vehiculain,elementum
aliquam,aliquetlaoreet,tellus.Sedvelit.Sed
velmiacelitaliquetinterdum.Etiamsapi-
enneque,convalliset,aliquetvel,auctornon,
arcu.Aliquamsuscipitaliquamlectus.Proin
tinciduntmagnasedwisi.Integerblanditla-
cusutlorem.Sedluctusjustosedenim.
Morbimalesuadahendreritdui.Nuncmauris
leo,dapibussitamet,vestibulumet,commodo
id,est.Pellentesquepurus.Pellentesquetris-
tique,nuncacpulvinaradipiscing,justoeros
consequatlectus,sitametposuerelectusneque
velaugue.Crasconsectetuerliberoaceros.Ut
egetmassa.Fuscesitametenimeleifendsem
dictumauctor.Inegetrisusluctuswisiconva-
llispulvinar.Vivamussapienrisus,temporin,
viverrain,aliquetpellentesque,eros.Aliquam
euismodliberoasem.
Nuncvelitaugue,scelerisquedignissim,lobor-
tiset,aliquamin,risus.Ineueros.Vestibu-
lumanteipsumprimisinfaucibusorciluctus
etultricesposuerecubiliaCurae;Curabitur
vulputateelitviverraaugue.Maurisfringilla,
tortorsitametmalesuadamollis,sapienmi
dapibusodio,acimperdietligulaenimeget
nisl.Quisquevitaepedeapedealiquetsus-
cipit.Phasellustelluspede,viverravestibu-
lum,gravidaid,laoreetin,justo.Cumsoci-
isnatoquepenatibusetmagnisdisparturient
montes,nasceturridiculusmus.Integercom-
modoluctuslectus.Maurisjusto.Duisvar-
iuseros.Sedquam.Craslacuseros,rutrum
eget,variusquis,convallisiaculis,velit.Mau-
risimperdiet,metusattristiquevenenatis,pu-
rusnequepellentesquemauris,aultriceselit
lacusnectortor.Classaptenttacitisociosqu
adlitoratorquentperconubianostra,perin-
ceptoshymenaeos.Praesentmalesuada.Nam
lacuslectus,auctorsitamet,malesuadavel,
elementumeget,metus.Duisnequepede,fa-
cilisiseget,egestaselementum,nonummyid,
neque.
Proinnonsem.Donecnecerat.Proinlibero.
Aliquamviverraarcu.Donecvitaepurus.
Donecfelismi,semperid,scelerisqueporta,
sollicitudinsed,turpis.Nullainurna.Integer
variuswisinonelit.Etiamnecsem.Mau-
risconsequat,risusnecconguecondimentum,
ligulaligulasuscipiturna,vitaeportaodioer-
atquissapien.Proinluctusleoiderat.Etiam
massametus,accumsanpellentesque,sagittis
sitamet,venenatisnec,mauris.Praesentur-
naeros,ornarenec,vulputateeget,cursussed,
justo.Phasellusneclorem.Nullamligulaligu-
la,mollissitamet,faucibusvel,eleifendac,
dui.Aliquameratvolutpat.
Fuscevehicula,tortoretgravidaporttitor,me-
tusnibhconguelorem,uttempuspurusmau-
risapede.Integertinciduntorcisitamet
turpis.Aeneanametus.Aliquamvestibulum
lobortisfelis.Donecgravida.Sedsedurna.
Maurisetorci.Integerultricesfeugiatligu-
la.Seddignissimnibhamassa.Donecorci
dui,temporsed,tinciduntnonummy,viverra
sitamet,turpis.Quisquelobortis.Proinvene-
natistortornecwisi.Vestibulumplacerat.In
hachabitasseplateadictumst.Aliquampor-
tamiquisrisus.Donecsagittisluctusdiam.
Namipsumelit,imperdietvitae,faucibusnec,
fringillaeget,leo.Etiamquisdolorinsapien
porttitorimperdiet.
Craspretium.Nullamalesuadaipsumut
libero.Suspendissegravidahendrerittellus.
Maecenasquislacus.Morbifringilla.Vestibu-
lumodioturpis,temporvitae,scelerisquea,
dictumnon,massa.Praesenteratfelis,por-
tasitamet,condimentumsitamet,placer-
atet,turpis.Praesentplaceratlacusaen-
im.Vestibulumnoneros.Utcongue.Donec
tristiquevariustortor.Pellentesquehabitant
morbitristiquesenectusetnetusetmalesuada
famesacturpisegestas.Namdictumdictum
urna.
Phasellusvestibulumorcivelmauris.Fusce
quamleo,adipiscingac,pulvinareget,mo-
lestiesitamet,erat.Seddiam.Suspendisse
erosleo,tempuseget,dapibussitamet,tem-
puseu,arcu.Vestibulumwisimetus,dapibus
vel,luctussitamet,condimentumquis,leo.
Suspendissemolestie.Duisinante.Utso-
dalessemsitametmauris.Suspendisseornare
pretiumorci.Fuscetristiqueenimegetmi.
Vestibulumeroselit,gravidaac,pharetrased,
lobortisin,massa.Proinatdolor.Duisac-
cumsanaccumsanpede.Nullamblanditelit
inmagnalaciniahendrerit.Utnonummyluc-
tuseros.Fusceegettortor.
Utsitametmagna.Crasaligulaeuurna
dignissimviverra.Nullamtemporleoporta
ipsum.Praesentpurus.Nullamconsequat.
Maurisdictumsagittisdui.Vestibulumsol-
licitudinconsectetuerwisi.Insitametdiam.
Nullammalesuadapharetrarisus.Proinlacus
arcu,eleifendsed,vehiculaat,conguesitamet,
sem.Sedsagittispedeanisl.Sedtincidunto-
dioapede.Seddui.Nameuenim.Aliquam
sagittislacusegetlibero.Pellentesquediam
sem,sagittismolestie,tristiqueet,fermentum
ornare,nibh.Nullaettellusnonfelisimperdi-
etmattis.Aliquameratvolutpat.
Vestibulumsodalesipsumidaugue.Integer
ipsumpede,convallissitamet,tristiquevi-
tae,temporut,nunc.Namnonligulanon
loremconvallishendrerit.Maecenashendrerit.
Sedmagnaodio,aliquamimperdiet,portaac,
aliqueteget,mi.Cumsociisnatoquepenati-
busetmagnisdisparturientmontes,nascetur
ridiculusmus.Vestibulumnislsem,dignissim
vel,euismodquis,egestasut,orci.Nuncvi-
taerisusvelmetuseuismodlaoreet.Crassit
ametnequeaturpislobortisauctor.Sedali-
quamsemacelit.Crasvelitlectus,facilisis
id,dictumsed,portarutrum,nisl.Namhen-
dreritipsumsedaugue.Nullamscelerisque
hendreritwisi.Vivamusegestasarcusedpu-
rus.Utornarelectussederos.Suspendisse
potenti.Maurissollicitudinpedevelvelit.In
hachabitasseplateadictumst.
Suspendisseeratmauris,nonummyeget,
pretiumeget,consequatvel,justo.Pellen-
tesqueconsectetuereratsedlacus.Nullam
egestasnullaacdui.Doneccursusrhoncus
ipsum.Nuncetsemeumagnaegestasmale-
suada.Vivamusdictummassaatdolor.Mor-
biestnulla,faucibusac,posuerein,interdum
ut,sapien.Proinconsectetuerpretiumurna.
Donecsitametnibhnecpurusdignissimmat-
tis.Phasellusvehiculaelitatlacus.Nullafa-
cilisi.Crasutarcu.Sedconsectetuer.Integer
tristiqueelitquisfelisconsectetuereleifend.
Crasetlectus.
Utconguemalesuadajusto.Curabiturcongue,
felisathendreritfaucibus,maurislacusport-
titorpede,necaliquamturpisdiamfeugiatar-
cu.Nullamrhoncusipsumatrisus.Vestibu-
lumadolorseddolorfermentumvulputate.
Sednecipsumdapibusurnabibendumlobor-
tis.Vestibulumelit.Namligulaarcu,volutpat
eget,laciniaeu,lobortisac,urna.Nammol-
lisultricesnulla.Crasvulputate.Suspendisse
atrisusatmetuspulvinarmalesuada.Nullam
lacus.Aliquamtempusmagna.Aliquamut
purus.Prointellus.
Vestibulumanteipsumprimisinfaucibusorci
luctusetultricesposuerecubiliaCurae;Donec
scelerisquemetus.Maecenasnonmiutmetus
portahendrerit.Nuncsemper.Crasquiswisi
utloremp osueretristique.Nuncvestibulum
scelerisquenulla.Suspendissepharetrasollic-
itudinante.Praesentatauguesitametante
interdumporta.Nuncbibendumaugueluctus
diam.Etiamnecsem.Sederosturpis,facilisis
nec,vehiculavitae,aliquamsed,nulla.Cur-
abiturjustoleo,vestibulumeget,tristiqueut,
tempusat,nisl.
Nullavenenatisloremidarcu.Morbicur-
susurnaaipsum.Donecporttitor.Integer
eleifend,estnonmattismalesuada,minulla
convallismi,etauctorlectussapienutpu-
rus.Aliquamnullaaugue,pharetrasitamet,
faucibussemper,molestievel,nibh.Pellen-
tesquevestibulummagnaetmi.Sedfringilla
dolorveltellus.Nuncliberonunc,venenatis
eget,convallishendrerit,iaculiselementum,
mi.Nullamaliquam,felisetaccumsanvehic-
ula,magnajustovehiculadiam,eucondimen-
tumnislfelisetnunc.Quisquevolutpatmauris
avelit.Pellentesquemassa.Integeratlorem.
Nammetuserat,laciniaid,convallisut,pul-
vinarnon,wisi.Crasiaculismaurisutneque.
Crassodales,semvitaeimperdietconsequat,
pedepurussollicitudinurna,acaliquammetus
orciinleo.Utmolestieultricesmauris.Viva-
musvitaesem.Aliquameratvolutpat.Prae-
sentcommodo,nislacdapibusaliquet,tortor
orcisodaleslorem,nonornarenullaloremquis
nisl.
Sedatsemvitaepurusultricesvestibulum.
Vestibulumtinciduntlacusetligula.Pel-
lentesquevitaeelit.Vestibulumanteipsum
primisinfaucibusorciluctusetultricespo-
suerecubiliaCurae;Duisornare,erateget
laoreetvulputate,lacusipsumsuscipitturpis,
etbibendumnislorcinonlectus.Vestibu-
lumnecrisusnecliberofermentumfringilla.
Morbinonvelitinmagnagravidahendrerit.
Pellentesquequislectus.Vestibulumeleifend
lobortisleo.Vestibulumnonaugue.Vivamus
dictumtempordui.Maecenasatligulaidfe-
liscongueporttitor.Nullaleomagna,egestas
quis,vulputatesitamet,viverraid,velit.
Utlectuslectus,ultriciessitamet,semper
eget,laoreetnon,ante.Proinatmassaquis
nuncrhoncusmattis.Aliquamlorem.Cur-
abiturpharetraduiatneque.Aliquameutel-
lus.Aeneantempus,felisvitaevulputateia-
culis,estdolorfaucibusurna,inviverrawisi
nequenonrisus.Fusceveldolornecsapien
pretiumnonummy.Integerfaucibusmassaac
nullaornarevenenatis.Nullaquissapien.Sed
tortor.Phasellusegetmi.Crasnunc.Crasa
enim.
Quisquenisl.Indignissimdapibusmassa.Ae-
neansemmagna,scelerisquenec,ullamcorper
quis,porttitorut,lectus.Fuscedignissimfa-
cilisistortor.Vivamusgravidafelissitamet
nunc.Nampulvinarodiovelenim.Pellen-
tesquesitametest.Vivamuspulvinarleonon
sapien.Aliquameratvolutpat.Utelementum
auctormetus.Maurisvestibulumnequevitae
eros.Pellentesquealiquamquam.Donecve-
nenatistristiquepurus.Innisl.Nullavelit
libero,fermentumat,portaa,feugiatvitae,
urna.Etiamaliquetornareipsum.Proinnon
dolor.Aeneannuncligula,venenatissuscipit,
porttitorsitamet,mattissuscipit,magna.Vi-
vamusegestasviverraest.Morbiatrisussed
sapiensodalespretium.
Morbicongueconguemetus.Aeneansedpu-
rus.Namp edemagna,tristiquenec,porta
id,sollicitudinquis,sapien.Vestibulumblan-
dit.Suspendisseutaugueacnibhullamcorper
posuere.Integereuismod,nequeateleifend
fringilla,augueelitornaredolor,veltincidunt
purusestidlacus.Vivamusloremdui,commo-
doquis,scelerisqueeu,tinciduntnon,magna.
Crassodales.Quisquevestibulumpulvinardi-
am.Phasellustincidunt,leovitaetristiquefa-
cilisis,ipsumwisiinterdumsem,dapibussem-
pernullavelitvellectus.Crasdapibusmauris
etaugue.Quisquecursusnullainlibero.Sus-
pendisseetloremsitametmaurismalesuada
mollis.Nullamidjusto.Maecenasvenenatis.
Doneclacusarcu,egestasac,fermentumcon-
sectetuer,tempuseu,metus.Proinsodales,
seminpretiumfermentum,arcusapiencom-
modomauris,venenatisconsequataugueur-
nainwisi.Quisquesapiennunc,variuseget,
condimentumquis,laciniain,est.Fuscefacil-
isis.Praesentnecipsum.
Suspendisseadolor.Namerateros,congue
eget,sagittisa,laciniain,pede.Maecenas
inelit.Proinmolestievariusnibh.Vivamus
tristiquepurussedaugue.Proinegestassem-
pertortor.Vestibulumanteipsumprimisin
faucibusorciluctusetultricesposuerecubil-
iaCurae;Classaptenttacitisociosquadl-
itoratorquentperconubianostra,perincep-
toshymenaeos.Vestibulumorcienim,sagittis
ornare,eleifendut,mattisat,ligula.Nulla
molestieconvallisarcu.Uterostellus,condi-
mentumat,sodalesin,ultricesvel,nulla.
Duismagnaante,bibendumeget,eleifend
eget,suscipitsed,neque.Vestibuluminmi
sedmassacursuscursus.Pellentesquepulv-
inarmollisneque.Fusceutenimvitaemauris
malesuadatincidunt.Vivamusaneque.Mau-
rispulvinar,sapienidcondimentumdictum,
quamarcurhoncusdui,idtemporlacusjusto
etjusto.Proinsitametorcieudiameleifend
blandit.Nunceratmassa,luctusac,fermen-
tumlacinia,tinciduntultrices,sapien.Prae-
sentsedorcivitaedolorsollicitudinadipiscing.
Crasaneque.Utrisusdui,interdumat,plac-
eratid,tristiqueeu,enim.Vestibulumante
ipsumprimisinfaucibusorciluctusetultrices
posuerecubiliaCurae;Etiamadipiscingeros
vestibulumdolor.Pellentesquealiquam,diam
egeteleifendposuere,augueerosporttitorlec-
tus,acdignissimduimetusnecfelis.Quisque
lacinia.Vestibulumtellus.Suspendissenec
wisi.Aeneanacfelis.Aliquamultricesmetus
etnulla.
Praesentsedestnonnibhtempusvenenatis.
Praesentrhoncus.Curabitursagittisestsit
ametneque.Sedcommodomalesuadalectus.
Phasellusenimtellus,temporut,tristiqueeu,
aliquameu,quam.Aeneanquisquamquis
wisigravidavehicula.Pellentesqueamassa
aleopretiumrhoncus.Suspendisseultrices.
Doneclaciniamalesuadamassa.Classapten-
ttacitisociosquadlitoratorquentperconu-
bianostra,perinceptoshymenaeos.Donec
pretiumornaremauris.Phasellusauctorer-
ategetenim.Integerscelerisque,feliseucon-
sequatfringilla,loremwisiultriciesvelit,id
vehiculapurusnullaegetodio.Nullammat-
tis,diamarutrumfermentum,odiosapien
tristiquequam,idmollistellusquaminodi-
o.Mauriseusapien.Donecaliquamloremsit
ametlorempharetralobortis.
Donecacvelit.Sedconvallisvestibulumsapi-
en.Vivamustemporlacussedlacus.Nunc
utlorem.Utettortor.Nullamvariuswisiat
diam.Etiamultricies,dolorsitametfermen-
tumvulputate,nequeliberovestibulumorci,
vitaefringillanequearcualiquetante.Lorem
ipsumdolorsitamet,consectetueradipiscing
elit.Quisquevenenatislobortisaugue.Sed
tempor,tellusiaculispellentesquepharetra,
pededuimalesuadamauris,velultricesurna
maurisacnibh.Etiamnibhodio,ultriciesve-
hicula,vestibulumvitae,feugiateleifend,fe-
lis.Vivamuspulvinar.Aliquameratvolutpat.
Nullaegestasvenenatismetus.Namfeugiat
nuncquiselitegestassagittis.Sedvitaefe-
lis.Inliberoarcu,rhoncusin,commodoeget,
auctorin,enim.Vivamussuscipitest.Nul-
ladapibus,magnavelaliquetegestas,massa
massahendreritlacus,acrutrumtellustellus
sitametfelis.Crasviverra.
Suspendisseeununc.Aliquamdignissimurna
sitametmauris.Crascommodo,urnautport-
titorvenenatis,arcumetussodalesrisus,vitae
gravidasapienligulainest.Donecvulputate
sollicitudinwisi.Donecvehicula,estidinter-
dumornare,nibhtellusconsectetuerjusto,a
ultricesfeliseratatlectus.Inestmassa,male-
suadanon,suscipitat,ullamcorpereu,elit.
Namnullalacus,bibendumsitamet,sagit-
tissed,temporeget,libero.Praesentligula.
Suspendissenulla.Etiamdiam.Nullaante
diam,vestibulumet,aliquetac,imperdietvi-
tae,urna.Fuscetinciduntlacusvelelit.Mae-
cenasdictum,tortornoneuismodbibendum,
pedenibhpretiumtellus,atdignissimleoeros
egetpede.Nullavenenatiseleifenderos.Ae-
neanutodiodignissimauguerutrumfaucibus.
Fusceposuere,tellusegetviverramattis,er-
attellusportami,atfacilisissemnibhnon
urna.Phasellusquisturpisquismaurissus-
cipitvulputate.Sedinterdumlacusnonvelit.
Vestibulumanteipsumprimisinfaucibusorci
luctusetultricesposuerecubiliaCurae;
Vivamusvehiculaleoajusto.Quisquenecau-
gue.Morbimauriswisi,aliquetvitae,dignis-
simeget,sollicitudinmolestie,ligula.Indic-
tumenimsitametrisus.Curabiturvitaevelit
eudiamrhoncushendrerit.Vivamusutelit.
Praesentmattisipsumquisturpis.Curabitur
rhoncusnequeeudui.Etiamvitaemagna.
Namullamcorper.Praesentinterdumbiben-
dummagna.Quisqueauctoraliquamdolor.
Morbieuloremetestporttitorfermentum.
Nuncegestasarcuattortorvariusviverra.
Fusceeunullautnullainterdumconsectetuer.
Vestibulumgravida.Morbimattisliberosed
est.
Abstract
Inthispaper,aclassofnonlinearstochasticpartialdifferentiale-
quationswithdiscontinuouscoefficientsisinvestigated.Thisstudy
ismotivatedbysomeresearchonstochasticviscositysolutionsun-
dernon-Lipschitzconditionsrecently.Bystudyingthesolutionsof
backwarddoublystochasticdifferentialequationswithdiscontinu-
ouscoefficientsandconstructinganewapproximationfunctionf
n
tothecoefficientf,wegettheexistenceofstochasticviscositysub-
solutions(orsuper-solutions).Theresultsofthispapercanbeseen
astheextensionandapplicationofrelatedarticles.
Keywords
StochasticPartialDifferentialEquation,StochasticViscosity
Solution,BackwardDoublyStochasticDifferentialEquation
1.Introduction
Thenotionof(deterministic)viscositysolutionwasfirstintroduced
byCrandallandLions[1]asanextensionoftheclassicalsolutions
ofpartialdifferentialequations(PDEsinshort).Itiswidelyusedin
optimizationcontrol,differentialgame,font-endevolutionproblems
andsoon.Theexistenceanduniquenessofviscositysolutionshave
beenextensivelystudiedinthepastfewdecades(seee.g.[2]and[3]).
Basedonthefactthatalmostallthedeterministicproblemsinthe
aboveappliedfieldshavecorrespondingstochasticproblems,Lionsand
Souganidis[4]introducedthenotionofstochasticviscositysolutions
forthefirsttime.Theyused”stochasticcharacteristics”toeliminate
thestochasticintegralsinthestochasticpartialdifferentialequations
(SPDEsinshort),soastosolvethedifficultyofdefiningthestochas-
ticviscositysolutionscausedbytheexistenceofthemartingaleterm.
Withthesamepurpose,BuckdahnandMa[5]transformedSPDEsinto
ordinaryPDEswithrandomcoefficientsbyusinganonlinearversion
ofDoss-Sussmanntransformation,sothatstochasticviscositysolu-
tionsofSPDEscanbeconnectedwithsolutionsofbackwarddoubly
stochasticdifferentialequations(BDSDEsinshort).
DOI:10.4236/am.2020.1111083Nov.30,20201219AppliedMathematics
Y.D.Zhang
Backwardstochasticdifferentialequations(BSDEsinshort)were
introducedbyPardouxandPeng[6],whichhavewideconnections
withmathematicalfinanceandPDEs,andthentheyintroducedanew
classofBDSDEsunderLipschitzconditionstoproduceaprobabilistic
representationofcertainquasilinearSPDEsin[7].Manyliteratures
havemadeeffortstoobtaintheexistenceanduniquenessresultsof
solutionstoBDSDEunderrelaxedLipschitzconditions.Shi,Guand
Liu[8]provedtheexistenceofminimumsolutionofBDSDEandob-
tainedthecorrespondingcomparisontheoremundertheconditionthat
fiscontinuouswithrespectto(y,z)andsatisfiesthelineargrowth
condition.Lin[9]gotthesameresultsundertheconditionthatfis
left-continuouswithrespecttoy,non-decreasingandbounded.Re-
cently,Zhang,XiaoandFan[10]consideredtheconditionthatfis
left-continuousandleft-Lipschitzwithrespecttoy,lineargrowthon
(y,z),and{f(t,0,0)}
t[0,T]
issquareintegrable.Today,BDSDEsare
oneofthemostusefultoolsinthestudyofSPDEs.Boufoussi,Van
CasterenandMrhardy[11]gavetheexistenceofstochasticviscosi-
tysolutionsforsemi-linearSPDEswithaNeunannboundarycon-
ditionbyestablishingtherelationshipbetweengeneralizedBDSDEs
andSPDEsas[5]did,whenthecoefficientsareLipschitzcontinuous.
AfterthatDjehiche,N’ziandOwo[12]relaxedtheLipschitzcontinu-
ityconditionsandconsideredtheexistenceresulttononlinearSPDEs
withcontinuouscoefficients,butwithDirichletboundaryconditions.
Inthispaper,weconsidertheexistenceofstochasticviscositysolu-
tionsfornonlinearSPDEswithdiscontinuouscoefficients.Thenon-
linearSPDEswedealwithinthispaperareoftheform:
u(t,x)=l(x)+
T
t
{Lu(s,x)+f(s,x,u(s,x)
(x)Du(s,x))}ds
+
T
t
g(s,x,u(s,x))
←−
dB
s
,t[0,T],
(1)
whereDudenotesthegradientofu;
←−
dB
t
integralisabackwards-
tochasticKunita-Itˆointegral;l,f,garesomemeasurablefunctions;L
isasecond-orderdifferentialoperatordefinedby
L=
1
2
d
i,j=1
(σσ
)
i,j
2
∂x
i
∂x
j
+
d
i=1
β
i
∂x
i
,
From[5],wecanconnecttheSPDE(1)withthefollowingBDSDE
(2)byusingaDoss-Sussmanntransformation.
Y
t
s
=l(X
t
T
)+
T
s
f(r,X
t
r
,Y
t
r
,Z
t
r
)dr
+
T
s
g(r,X
t
r
,Y
t
r
)
←−
dB
r
T
s
Z
t
r
dW
r
,
tsT,
(2)
wheredW
t
integralisaforwardItˆointegraland{X
t
s
;tsT}is
thesolutionofSDE:
X
t
s
(x)=x+
t
s
b(X
t
r
(x))dr+
t
s
σ(X
t
r
(x))dW
r
,tsT.
WeprovethattheminimalsolutionofBDSDE(2)isexactlyas-
tochasticviscositysub-solutionofSPDE(1)whenthecoefficientf
DOI:10.4236/am.2020.11110831220AppliedMathematics
Y.D.Zhang
isleftcontinuous,non-decreasingandlineargrowthwithrespectto
(x,y,z).Weproposeanewapproximationfunctionf
n
ofthecoef-
ficientfsuchthatf
n
isLipschitzcontinuouswithrespecttoy,thus
solvingtheproblemscausedbydiscontinuity.Ourworkgeneralizesthe
resultof[11]andof[12],whichcanalsobeseenassomeapplications
oftheresultsin[10].
Thepresentpaperisorganizedasfollows.InSection2,somepre-
liminaries,notationsandthedefinitionofstochasticviscositysolutions
areshown.Theexistenceresultsofstochasticviscositysub-solution
forSPDE(1)isgiveninSection3,andtheconclusionisgivenin
Section4.
2.PreliminariesandDefinitions
2.1.NotationsandAssumptions
Let(Ω
1
,F
1
,P
1
)and(Ω
2
,F
2
,P
2
)aretwocompleteprobabilityspaces
wheretwomutuallyindependentBrownianmotions{W
t
:0tT}
and{B
t
:0tT}aredefined,respectively.T>0isafixedfinal
time.LetF
η
s,t
=σ{η
r
η
s
:srt}∨Nforanyprocess{η
t
},where
NdenotestheclassofP
i
-nullsetsofF
i
.DefineF
t
,F
W
t
F
B
t,T
.Note
thatF
t
isnotafiltration.SetF
B
,{F
B
t,T
}
0tT
,F
W
,{F
W
t,T
}
0tT
.
Let(Ω,F,P)denotetheproductspace,whereΩ=Ω
1
×
2
,
F=F
1
F
2
,P=P
1
P
2
.Anyrandomvariableξ(ω
1
)
1
1
(resp.η(ω
2
)
2
2
)isconsideredasarandomvariableonΩif
ξ(ω)=ξ(ω
1
)(resp.η(ω)=η(ω
2
)),ω,(ω
1
2
).Throughoutthis
article,wedenoteEandE
1
begenericEuclideanspacesand|·|bethe
norminE.M
B
0,T
denotesthesetofF
B
-stoppingtimesτsuchthat
0τT,P
2
-a.e.
WedenoteC
k,n
([0,T]×E;E
1
)tobethespaceofallE
1
-valued
functionsdefinedon[0,T]×Ewhicharek-timescontinuouslyd-
ifferentiableintandn-timescontinuouslydifferentiableinxE;
C
k,n
b
([0,T]×E;E
1
)isthesubsetofC
k,n
([0,T]×E;E
1
)whereallE
1
-
valuedfunctionshaveuniformlyboundedpartialderivatives.
C
k,n
(F
B
,[0,T]×E;E
1
)(resp.C
k,n
b
(F
B
,[0,T]×E;E
1
))denotesthe
spaceofrandomfieldsφC
k,n
(F
B
,[0,T]×E;E
1
)
(resp.C
k,n
b
(F
B
,[0,T]×E;E
1
))suchthat(t,ω
2
)7→φ(t,x,ω
2
)isF
B
-
progressivelymeasurableforfixedxE.
LSC([0,T]×E;E
1
)(resp.USC([0,T]×E;E
1
))isthespaceofall
E-valuedfunctionsdefinedon[0,T]×Ewhicharelower(resp.upp er)
semi-continuous.Similarly,wecandefineLSC(F
B
,[0,T]×E;E
1
)and
USC(F
B
,[0,T]×E;E
1
).
Forsub- σ-fieldGF
B
T
andconstantp0,letL
P
(G;E)denote
G-measurablerandomvariablesξwhicharevaluedinEandsatisfy
E|ξ|
p
<.
Henceforth,foreaseofnotations,wedenoteD=D
x
=
(∂/∂x
1
,...,∂/∂x
q
),D
y
=∂/∂y,D
t
=∂/∂tandD
xx
=
(
2
x
i
x
j
)
i,j=1,...,q
.WehavetopointoutthatKissomeconstant,
allowingdifferenceshereandthere.
Inthispaper,weassumethatgalwayssatisfiesthefollowingas-
sumptions:
(H1)gC
0,2,3
b
([0,T]×R
q
×R;R
l
);
Thefunctionf :Ω
2
×[0,T]×R
q
×R×R
d
7→Risarandom
fieldwhichisdiscontinuousinyandcontinuousinz,suchthatfor
all(ω
2
,t,x,y,z)
2
×[0,T]×R
q
×R×R
d
,f(ω
2
,t,x,y,z)isF
t
-
DOI:10.4236/am.2020.11110831221AppliedMathematics
Y.D.Zhang
measurable.Moreover,f(ω
2
,·,·,0,0)issquare-integrable.Wegive
thefollowingassumptions:
(H2)Thefunctionf(ω
2
,t,x,·,z)isleftcontinuousandnon-
decreasing;
(H3)Thereexistsapositive-valuedandF
t
-measurableprocessh(·)
satisfyingE(
T
0
|h(t)|
2
dt)<andapositive-valuedconstantK,such
that
|f(ω
2
,t,x,y,z)|≤h(t)+K(|x|+|y|+|z|)
foreach(ω
2
,t,x,y,z)
2
×[0,T]×R
q
×R×R
d
;
(H4)ThereexistsaconstantK>0suchthatforall(ω
2
,y)
2
×R
and(t
i
,x
i
,z
i
)[0,T]×R
q
×R
d
,i=1,2,
|f(ω
2
,t
1
,x
1
,y,z
1
)f(ω
2
,t
2
,x
2
,y,z
2
)|
K(|t
1
t
2
|+|x
1
x
2
|+|z
1
z
2
|).
(H5)Thefunctionsσ:R
q
7→R
q×q
andβ:R
q
7→Rareuniformly
LipschitzcontinuouswithLipschitzconstantK>0.
(H6)l(x)isacontinuousfunctionsuchthatforsomep>0,it
satisfies
|l(x)|≤K(1+|x|
p
),xR
q
.
2.2.DefinitionofStochasticViscositySolutions
Inthelightofthedefinitionofstochasticviscositysolutionsin[5],the
followingstochasticflowηC
0,0,0
(F
B
,[0,T]×R
q
×R;R),whichis
theuniquesolutionofthefollowingBSDEintheStratonovichsense,
playsanimportantrole.
η(t,x,y)=y+
T
t
g(s,x,η(s,x,y))
←−
dB
s
.(3)
WecanrewriteBSDE(3)toitsequivalentItˆoform:
η(t,x,y)=y+
1
2
T
t
g(s,x,η(s,x,y))D
y
g(s,x,η(s,x,y))ds
+
T
t
g(s,x,η(s,x,y))
←−
dB
s
,0tT,
wherethemappingy7→η(t,x,y)isadiffeomorphismunderassump-
tion(H1).Therefore,letE(t,x,y)denotethey-inverseofη(t,x,y).
WecangeteasilythatE(t,x,y)isthesolutionofthefollowingBSDE:
E(t,x,y)=y
T
t
D
y
E(s,x,y)g(s,x,y)
←−
dB
s
.
Foranyrandomfieldψ:[0,T]×R
q
7→R,wecanintroducethe
transformation
φ(t,x)=E(t,x,ψ(t,x)),
namelyψ(t,x)=η(t,x,φ(t,x)),thenψC
0,p
(F
B
,[0,T]×R)ifand
onlyifψC
0,p
(F
B
,[0,T]×R
q
;R),forp=0,1,2.
ApplyingItˆo’sformulatoψ(t,x),itfollowsthat
ψ(t,x)=η(t,x,φ(t,x))
=φ(T,x)+
1
2
T
t
g(s,x,η(s,x,φ(s,x))D
y
g(s,x,η(s,x,φ(s,x))ds
+
T
t
g(s,x,η(s,x,y))
←−
dB
s
T
t
D
y
η(s,x,φ(s,x))D
t
φ(s,x)ds.
DOI:10.4236/am.2020.11110831222AppliedMathematics
Y.D.Zhang
Accordingtothediscussiononconstructingstochasticviscosityso-
lutionin[5],itisnaturaltocompare
(t,x)+f(t,x,ψ(t,x)
(x)(t,x))
and
1
2
g(t,x,η(t,x,φ(t,x))D
y
g(t,x,η(t,x,φ(t,x))D
y
η(t,x,φ(t,x))D
t
φ(t,x).
Thus,wegivethefollowingdefinition:
Definition.Arandomfield
u∈LSC(F
B
,[0,T]×R
q
;R)(resp.USC(F
B
,[0,T]×R
q
;R))iscalleda
stochasticviscositysub-solution(resp.super-solution)ofSPDE(1)if
u(T,x)l(x)(resp.u(T,x)l(x)),forallxR
q
andifforany
τ∈M
B
0,T
L
0
(F
B
τ
;R
q
)andφC
1,2
(F
B
τ
,[0,T]×R
q
;R)satisfying
u(ω
2
,t,x)η(ω
2
,t,x,φ(t,x))(resp.)0
=u(τ(ω
2
)( ω
2
))η(τ(ω
2
)(ω
2
)(τ(ω
2
)(ω
2
))
forall(t,x)inaneighborhoodof(τ,ξ),P
2
-a.e.Then,on{0<τ<T},
(τ,ξ)+f(τ,ξ,ψ(τ,ξ)
(ξ)(τ,ξ))(resp.)
1
2
g(τ,ξ,ψ(τ,ξ))D
y
g(τ,ξ,ψ(τ,ξ))D
y
η(τ,ξ,φ(τ,ξ))D
t
φ(τ,ξ),
whereψ(t,x),η(t,x,φ(t,x)).
3.ExistenceofStochasticViscosity
Sub-Solutions(orSuper-Solutions)
Wearenowinpositiontodiscusstheexistenceofstochasticviscos-
itysub-solutionsofSPDE(1).Tobeginwith,itisindispensableto
introducetwosetsthatarewidelyusedinBDSDEstudy,suchas[7].
ForanynN,M
2
(0,T;R
n
)denotesthesetof(dP×dta.e.equal)
ndimensionaljointlymeasurablerandomprocessed{φ
t
:0tT}
satisfying
E
T
t
|φ
t
|
2
dt<;
φ
t
isF
t
-measurablefora.e.t[0,T].
LetS
2
([0,T];R
n
)denotethesetofcontinuousndimensionalran-
domprocessessatisfying:
E(sup
0tT
|φ
t
|
2
)<;
φ
t
ismeasurableforanyt[0,T].
WeconsiderthefollowingtwoBDSDEs:
Y
t,1
s
=l(X
t
T
)+
T
s
f(r,X
t
r
,Y
t,1
r
,Z
t,1
r
)dr+
T
s
g(r,X
t
r
,Y
t,1
r
)
←−
dB
r
T
s
Z
t,1
r
dW
r
,tsT,
(4)
DOI:10.4236/am.2020.11110831223AppliedMathematics
Y.D.Zhang
Y
t,2
s
=l(X
t
T
)+
T
s
[h(r)+K(
X
t
r
+|Y
t,2
r
|+|Z
t,2
r
|)]dr
+
T
s
g(r,X
t
r
,Y
t,2
r
)
←−
dB
r
T
s
Z
t,2
r
dW
r
,tsT,
(5)
wherethefunctionfin(4)satisfiesassumptions(H2)-(H4),andgin
both(4)and(5)satisfyassumption(H1).
ByTheorem3.1in[10],BDSDE(4)hasatleastonesolution.
Moreover,thereexistsaminimalsolution(Y
t,1
s
,Z
t,1
s
)ofBDSDE(4)
inthesensethatforanyothersolution(Y
t,1
s
,Z
t,1
s
)of(4),wehave
Y
t,1
s
Y
t,1
s
,tsT.
ItiseasytocheckthatBDSDE(4)andBDSDE(5)meetthecondi-
tionsofthecomparisontheoremin[10]underassumptions(H1)-(H5),
thenifwedenotetheminimalsolutionsofBDSDE(4)and(5)by
(Y
t,1
s
,Z
t,1
s
)and(Y
t,2
s
,Z
t,2
s
),respectively,wehaveY
t,1
s
Y
t,2
s
.Thus
wecanturntostudytheboundednessofY
t,2
s
ratherthanY
t,1
s
directly.
OntheboundednessofY
t,2
s
,wehavethefollowinglemma:
Lemma1.ThereexistsaconstantB>0dependingonK,Tand
l(X
t
T
)suchthatE(sup
0sT
Y
t,2
s
2
)B.
Proof.Thepro ofofthislemmaisparalleltothatofLemma4.2in[8].
Soweomitit.
Inaddition,thefollowinglemmaisnecessarytoprovestochastic
viscositysub-solution.
Lemma2.{f
n
}
n0
isasequenceoffunctionsdefinedby
f
n
=inf
uR
{n
u
u
1
n
f(t,x,r,z)dr+nK|yu|}.
Thenitfollowsthat
(i)|f
n
(t,x,y,z)|≤h(t)+K(|x|+|y|+|z|);
(ii)|f
n
(t,x,y
1
,z)f
n
(t,x,y
2
,z)|≤nK|y
1
y
2
|;
(iii)|f
n
(t
1
,x
1
,y,z
1
)f
n
(t
2
,x
2
,y,z
2
)K(|t
1
t
2
|+|x
1
x
2
|+|z
1
z
2
|);
(iv)ifsequencey
n
increasestoy,thenlim
n→∞
f
n
(ω
2
,t,x,y
n
,z)=
f(ω
2
,t,x,y,z),a.s..
Proof.Thepro ofofthislemmaismainlyinspiredby[13]and[14].
Indeed,f
n
fh(t)+K(|x|+|y|+|z|)isobvious,andontheother
hand,weobtain
f
n
inf
u
{n
u
u
1
n
(h(t)K(|x|+|r|+|z|))dr+nK|yu|)}
inf
u
{−h(t)K(|x|+|z|)nK
u
u
1
n
|r|dr+nK|yu|}
inf
u
{−h(t)K(|x|+|u|+|z|)+K|yu|}
≥−h(t)K(|x|+|y|+|z|).
Thus,(i)holds.Fromthedefinitionoff
n
andinequality
|inf
xD
f
1
(x)inf
xD
f
2
(x)|≤sup
xD
|f
1
(x)f
2
(x)|,wecanreadily
DOI:10.4236/am.2020.11110831224AppliedMathematics
Y.D.Zhang
get(ii).(iii)isanimmediateresultof(H4).Nowwestarttoprove
(iv).Inthelightofthedefinitionoff
n
,wecandeducethatthere
existsasequence{u
n
}whichconvergestoysuchthatdP×dt-a.e.
n
u
n
u
n
1
n
f(t,x,r,z)dr+nK(|y
n
u
n
|)
1
n
f
n
(t,x,y
n
,z)f(t,x,y
n
,z).(6)
ByLemma3.1in[14],weobtain
lim
y
n
y
n
y
n
y
n
1
n
f(t,x,r,z)dr=f(t,x,y,z)
foranyfixedyR.Moreover,becauseoftheleftcontinuityoff,
weobtainthatlim
y
n
y
f(t,x,y
n
,z)=f(t,x,y,z).Takingthelimit
onbothsidesof(6),wegetlim
n→∞
f
n
(t,x,y
n
,z)=f(t,x,y,z),for
which(iv)holds.
Now,wearegoingtoverifythatY
t,1
t
isastochasticviscositysub-
solutionofSPDE(1).
Lemma3.Letu
:Ω
2
×[0 ,T]×R
q
7→Rbearandomfielddefined
by
u
(t,x),Y
t,1
t
forall(t,x)[0,T]×R
q
.Thenu
∈LSC(F
B
,[0,T]×R
q
;R).
Proof.Let(f
n
)
n1
bethenon-decreasingsequenceofLipschitzcon-
tinuousfunctionsasdefinedinLemma2toapproximatef.
Let(Y
n,t
s
,Z
n,t
s
,tsT)bethesolutionofthefollowingBDSDE:
Y
n,t
s
=l(X
t
T
)+
T
s
f
n
(r,X
t
r
,Y
n,t
r
,Z
n,t
r
)dr+
T
s
g(r,X
t
r
,Y
n,t
r
)
←−
dB
r
T
s
Z
n,t
r
dW
r
.
(7)
SimilartotheproofofTheorem3.1in[10],wecanverifythat
(Y
n,t
s
,Z
n,t
s
)convergesto(Y
t,1
s
,Z
t,1
s
)whichistheminimalsolutionof
BDSDE(4).
Letu
n
:[0,T]×R
q
7→Rbearandomfieldsdefinedby
u
n
(t,x)=Y
n,t
t
,(t,x)[0,T]×R
q
.
Itisobviousthat(u
n
)
n1
isanon-decreasingsequenceanditcon-
vergestou
.Thus,u
∈LSC(F
B
,[0,T]×R
q
;R).
Remark.BDSDE(4)hasamaximalsolutionY
t,1
s
whenthecoeffi-
cientfisrightcontinuousandnon-decreasing.Usingthesimilarproof
ofLemma3wheref
n
isreplacedbyf
n
,sup
uR
{n
u+
1
n
u
f(t,x,r,z)
nK|yu|},wecanprovethatu
(t,x),Y
t,1
t
satisfiesu
USC(F
B
,[0,T]×R
q
;R).
Next,wewillstateourmainresult.ThetechniqueofTheorem3.2
in[12]isusedintheproofofthefollowingtheorem.Butforsakeof
thereaders’convenience,westillgiveashortproof.
DOI:10.4236/am.2020.11110831225AppliedMathematics
Y.D.Zhang
Theorem4.Underassumptions(H1)-(H6),therandomfieldu
LSC(F
B
,[0,T]×R
q
;R)definedbyu
,Y
t,1
t
isastochasticviscosity
sub-solutionforSPDE(1).Further,foreachclassicalsolutionv
C
0,2
(F
B
,[0,T]×R
q
;R)ofSPDE(1),wehaveu
(t,x)v(t,x).
Proof.ForallxR
q
,wereadilyhavethatu(T,x)l(x).Now,we
considerundertheconditionthatifforanyτM
B
0,T
L
0
(F
B
T
;R
q
)
andφC
1,2
(F
B
τ
,[0,T]×R
q
;R)suchthat
u
(t,x)η(t,x,φ(t,x))0=u
(τ,ξ)η(τ,ξ,φ(τ,ξ))
forall(t,x)inaneighborhoodof(τ,ξ),P
2
-a.e.Itisnothardtocheck
thatBDSDE(7)meetstheconditionsin[11]underassumptions(H1)
-(H6).Thus,u
n
isastochasticviscositysolutionforSPDE:
u(t,x)=l(x)+
T
t
{Lu(s,x)+f
n
(s,x,u(s,x)
(x)Du(s,x)}ds
+
T
t
g(s,x,u(s,x))
←−
dB
s
,t[0,T].
Therefore,ifτ
n
M
B
0,T
n
L
0
(F
B
τ
,R
q
)andφ
n
C
1,2
(F
B
T
,[0,T]×
R
q
;R)satisfying
u
n
(t,x)η(t,x,φ
n
(t,x))0=u
n
(τ
n
n
)η(τ
n
n
n
(τ
n
n
)),
foreach(t,x)inaneighborhoodof(τ
n
n
),P
2
-a.s.,suchthat
(τ
n
n
n
)(τ,ξ,φ).ItfollowsthatP
2
-a.s.
n
(τ
n
n
)+f
n
(τ
n
n
n
(τ
n
n
)
(ξ
n
)
n
(τ
n
n
))
1
2
g(τ
n
n
n
(τ
n
n
))D
y
g(τ
n
n
n
(τ
n
n
))
D
y
η(τ
n
n
n
(τ
n
n
))D
t
φ
n
(τ
n
n
),
(8)
whereψ
n
(t,x)=η(t,x,φ
n
(t,x))convergestoψ(t,x)=η(t,x,φ(t,x))
in[0,T]×R
q
.Sendingn→∞in(8),weget
(τ,ξ)+f(τ,ξ,ψ(τ,ξ)
(ξ)(τ,ξ))
1
2
g(τ,ξ,ψ(τ,ξ))D
y
g(τ,ξ,ψ(τ,ξ))D
y
η(τ,ξ,φ(τ,ξ))D
t
φ(τ,ξ).
Consequently,u
∈LSC(F
B
,[0,T]×R
q
;R)isastochasticviscosity
sub-solutionofSPDE(1).From[7],{v(s,X
t
s
),(σ
Dv)(s,X
t
s
),0s
tT}isasolutionofBDSDE(4),sov(s,X
t
s
)Y
t,1
s
.Specially,we
havev(t,x)u
(t,x).
Remark.Analogously,ifthecoefficientfisrightcontinuousand
non-decreasing,thenu
(t,x),Y
t,1
t
isastochasticviscositysuper-
solutionforSPDE(1).Moreover,foreachclassicalsolutionv
C
0,2
(F
B
,[0,T]×R
q
;R)ofSPDE(1),wehaveu
(t,x)v
(t,x).
Finally,wegiveacorollaryabouttheboundednessofu
(t,x)which
isanimmediateresultofLemma1.
Corollary1.Letu
(t,x)∈LSC(F
B
,[0,T]×R
q
;R)beastochastic
viscositysub-solutionofSPDE(1),thenthereexistsaconstantA>0
suchthatE(u
(t,x))A.
DOI:10.4236/am.2020.11110831226AppliedMathematics
Y.D.Zhang
4.Conclusion
Thispaperisdevotedtothestudyofstochasticviscositysolutions
toSPDE(1)whosecoefficientfisleft(resp.right)continuous,non-
decreasingwithrespecttoyandlineargrowthwithrespectto(x,y,z).
Wepresentanewfunctionf
n
=inf
uR
{n
u
u
1
n
f(t,x,r,z)dr+
nK|yu|}(resp.f
n
,sup
uR
{n
u+
1
n
u
f(t,x,r,z)nK|yu|})
whichsatisfiesLipschitzconditionstoapproximatef,thussolving
thetroublecausedbythediscontinuouscoefficient.Wegetthatthe
minimal(resp.maximal)solutionofBDSDE(2)isastochasticvis-
cositysub-solution(resp.super-solution)ofSPDE(1)undergiven
conditions.Theresultsobtainedinthispaperextendtheexistence
conditionsofstochasticviscositysolutionstomorerelaxedconditions
andgeneralizesomeresultsinrelatedliterature.
Acknowledgements
ThisworkissupportedbyYuyouProjectofNorthUniversityofTech-
nology(No.207051360020XN140/007)andScientificResearchFoun-
dationofNorthUniversityofTechnology(No.110051360002).
ConflictsofInterest
Theauthordeclaresnoconflictsofinterestregardingthepublication
ofthispaper.
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