Global Existence and Un iqueness of Solu tions to Evolu tion p-Lap lacian System sw ith Non linear Sou rces

WEIYingjie and GAOWen jie

InstituteofM athematics,Jilin University,Changchun 130012,China.

Global Existence and Un iqueness of Solu tions to Evolu tion p-Lap lacian System sw ith Non linear Sou rces

WEIYingjie and GAOWen jie?

InstituteofM athematics,Jilin University,Changchun 130012,China.

Received 25 February 2012;Accep ted 1Decem ber 2012Abstract.This paper p resents the global existence and uniqueness of the initial and boundary value p roblem to a system ofevolution p-Lap lacian equations coup led w ith generalnonlinear term s.Theauthorsuseskillsof inequality estim ation and them ethod of regu larization to construct a sequence of app roxim ation solu tions,hence obtain the globalexistence of solu tions to a regu larized system.Then the globalexistence of solutions to the system ofevolu tion p-Lap lacian equations isobtained w ith the app lication of a standard lim iting p rocess.The uniqueness of the solution is p roven w hen the nonlinear term sare local Lipschitz continuous.

AMSSubjectClassifications:35A 01,35A 02,35G55

ChineseLibraryClassifications:O175.29,O 175.4

Globalexistence;uniqueness;degenerate;p-Lap lacian system s.

1 In troduction

In this paper,we study theglobalexistenceand uniquenessof solutions to the initialand boundary value p roblem

w here pi＞2,i=1,2,···,m,T＞0,??Rnis an open connected bounded dom ain w ith sm ooth boundary??.

System(1.1a)m odels such as non-New tonian fluids[1,2]and non linear filtration[3], etc.In the non-New tonian fluids theory,(p1,p2,···,pm)is a characteristic quantity of thefluids.The fluidsw ith(p1,p2,···,pm)＞(2,2,···,2)are called d ilatant fluidsand thosew ith (p1,p2,···,pm)＜(2,2,···,2)are called pseudop lastics.If(p1,p2,···,pm)=(2,2,···,2),they are New tonian fluids.

For pi=2,i=1,2,m any au thorshave stud ied the p roblem above;m ostof them stud ied g lobal existence,uniqueness,boundedness,and blow up behavior of solu tions,etc(see [4–10]).Som e au thors have derived su fficient cond itions for the nonexistence of global solu tions.Such cond itions are usually related to the structu re of fi,i=1,2.And som e authors have stud ied the uniqueness of the global solu tion and blow-up of the positive solu tion,w ith non linearities in the form of

w hereα,β,γ,δare nonnegative num bers.

For pi＞2,i=1,2,in[11],the au thorsgave local existence and uniqueness theorem of solutions for the initialand boundary value p roblem on?×(0,T1),w here T1∈(0,T)(T＞0) cou ld be very sm all.

It is our goal to p rove resu lts of global existence and uniqueness for the degenerate system of m equations.Since the system is coup led w ith non linear term s,in general,the solu tionsof(1.1a)-(1.1c)w illnotexist for all tim e.Insp ired by[12],in thispaper,w e study som e specialcasesby stating constrains to non linear functions.The p roof consistsof tw o steps.First,w e p rove that the app roxim ating p roblem adm its a g lobal solu tion;then w e do som e uniform estim ates for these solu tions.Wem ain ly use skills of inequality estim ation and them ethod of regu larization to construct a sequence of app roxim ation solutions,hence obtain existence of the solu tion to a regu larized system of equations. By a standard lim iting p rocess,w e obtain the existence of solu tions to the system(1.1a)-(1.1c).

System s(1.1a)degeneratesw hen?ui=0.In general,there is no classical solu tion; therefore,w e have to study the generalized solu tions to the p roblem(1.1a)-(1.1c).The definition ofgeneralized solutions is as follow s:

forany?i∈C1(?T),s.t.?i=0,for(x,t)∈??×(0,T);and ui(x,t)=0,(x,t)∈??×(0,T),where i=1,2,···,m.

2 M ain resu lts

In order to study the p roblem(1.1a)-(1.1c),w em ake the follow ing assum p tions:

In assum p tion(H 2),w e in tend to give an exp licit form of the grow th of fi(u)for large u,fu rtherm ore to state the resu lts thatw ill follow;the nonlinear part fi(u)cou ld be allow ed to depend on x,t.In that case,in(H 2),cij,ci,w ou ld be functions of(x,t),each con tained in sam e space Lq(0,T;Lp(?)),T＞0,w here p≥1 and q≥1w ou ld be special real num bers.

We begin by regu larizing p roblem(1.1a)-(1.1c).

Since the nonlinear term fi(u)cou ld be super-linear for large u,w ew illapp roxim ate it by a sequence of linear m aps for large u.Let{Rq}q∈Nbe an increasing sequence of positive realnum berss.t.limq→+∞Rq=+∞and fiqbe sm ooth functions that linearize for the functions fifor|u|＞Rq(actually they shou ld also satisfy thequasi-positive cond ition), and fiq≤fi,for ui≥0,q∈N.

We consider the follow ing regu larizing p roblem for every q≥1:

We p rove the follow ing lemm a by using a sim ilarm ethod as in[12].

Lemma2.1.Forevery q≥1,problem(2.1a)-(2.1c)existsa classicalglobal solution

and

Proof.We consider the system

w ith

This is a quasilinear nondegenerate parabolic system.The system(2.3)w ith initial and boundary cond itions(2.1b)-(2.1c)adm its a unique classicalsolution

Let

Wew illshow that the functions viq(x,t)are greater than zero.It is clear viq(x,0)≥0 in?and viq(x,t)≥0 in??×(0,T).Now suppose that for som e j∈{1,2,···,m},vjq(x,t)take negative values,then itm ust have a negativem inim um at a point(x0,t0);therefore,the inequality

is true at(x0,t0).On the other hand,due to(2.3),

at(x0,t0).Ifw e takeassum p tions(H 0),(H 1)into account,w e have

at(x0,t0).

Hence the righ t-hand side of equality(2.5)is positive at(x0,t0).This con trad icts to

Wenow p rove som ea p rioriestim ates for the solution uqof(2.1a)-(2.1c).Webegin by p roving that uiqare equilim ited in?T,T≥0.

Lemma2.2.Assume that cij＞0.If

(1)αij＜pi-1,i,j=1,2,···,m,

or

(2)αij≤pi-1,i,j=1,2,···,m,and d iam(?)issufficiently small, then thefollow ing a prioriestimate

is valid for uq=(u1q,u2q,···,umq)which is a classical solution of(2.1)-(2.3),where cijandαijcomefrom(H 2),and C1denotesa constant independent ofq.

T＞0,w e have

Therefore

M oreover

Ifw e take assum p tion(H 2)(fiq≤fi)into accoun t,w e have

App lying Young’s inequality,w e have

w hereαij＜s＜r w ill be suitably chosen.App lying the Sobolev em bedd ing theorem,w e have

w here C denotes various constants independent of r and q.In d ifferent form u lae these constantsw ill in generalhave d ifferentvalues.Chooseαij＜s＜r,s.t.

Then

Accord ing to assum p tionαij＜pi-1,w e know that s＜r.i.e.w e can choose such s. From Young’s inequality,w e obtain

and

By(2.9)-(2.13),w e get

Therefore

Using Gronw all’s lemm a(see e.g.[14])and that(2.15)is true for every T＞0,for every t＜T,w e have

Therefore

Let r→+∞,w e have

from w hich(2.6)follow s.

(2)The p roof is sim ilar to case(1)w hen m ax{αij}＜pi-1.Ifm ax{αij}=pi-1,then the firstpartof righthand of(2.8)isas follow s.

App lying Young’s inequality,w e have

App lying Poincar′e inequality,w e have

and

Sim ilar to case(1),w e can get(2.14)-(2.16)p rovided that d iam(?)is su fficiently sm all. Then(2.6)follow s.

Lemma2.3.Under theassumptions ofLemma2.2,wehave

where Cj(j=2,3)are constants independent ofq,q≥1.

Proof.M u ltip lying(2.1a)by uiqand in tegrating over?T,w e have

Fu rtherm ore

By(2.6)and the p roperty of fiq,w e have

M u ltip lying(2.1a)by uiqtand integrating over?T,w e have

By H¨older inequality and integrating by parts,w e obtain

Therefore

The p roof is com p lete.

Now w e are able to p rove an existence theorem for(1.1a)-(1.1c).

for every T＞0,then there exists a generalized solution u=(u1,···,um)to problem(1.1a)-(1.1c) in?T.Furthermore,

and

w here?stands forw eak convergence,and

We can p rove that wixl=|?ui|pi-2uixlusing sim ilarm ethod as in[11].

M u ltip lying(2.1a)by(uiq-ui)φiand integrating over?T,φi∈C1(?T),φi≥0,w e get

Hence

On the other hand,since?ui∈Lpi(?T),w e have

Note that

By(2.30)and(2.31),w e have

Since

and

w e have

i.e.

Hence wixl=|?ui|pi-2uixl,i=1,2,···,m.

Follow ing a standard lim iting p rocess,w e obtain that u=(u1,···,um)satisfies the initialand boundary value cond itions and the integrating exp ression.Thus u is a generalized solu tion to(1.1a)-(1.1c).

Theorem2.2.Assume f=(f1,f2,···,fm)is local Lipschitz continuous in u,then the solution is unique.

Proof.Assum e that u=(u1,···,um)and v=(v1,···,vm)are tw o solu tions to(1.1a)-(1.1c), then u,v are bounded.Considering that f is local Lipschitz continuous in u,w e get that f is Lipschitz continuouson[0,m ax{‖u‖L∞(QT),‖v‖L∞(QT)}].

Let?i=ui-vi,then by(1.2), Z

Subtracting the tw o equations,w e get

Note that

Using the p revious inequality and the Lipschitz cond ition,a sim p le calcu lation show s that

Set

then the above inequality can bew ritten as

A standard argum entshow s that F(T)≡0 since F(0)≡0,ui≡vi.The p roof is com p lete.

Acknow ledgm en ts

The p roject is supported by NSFC(10771085),by Key Lab of Sym bolic Com pu tation and Know ledge Engineering of M inistry of Education and by the 985 p rogram of Jilin University.

Theau thorsw ou ld like to thank the refereesand ed itors for their valuable suggestions and comm entson this paper.

[1]Astrita G.,M arrucciG.,Princip les of Non-New tonian Fluid M echanics,M cGraw-H ill,1974.

[2]M artinson L.K.,Pav lov K.B.,Unsteady shear flow s of a conducting fluid w ith a rheological pow er law.M agnitnayaGidrodinamika,2(1971),50-58.

[3]Esteban J.R.,Vazquez J.L.,On the equation of turbu lent filteration in one-d im ensional porousm ed ia.Nonlinear Analysis,10(1982),1303-1325.

[4]Chen H.,Global existence and blow-up for a non linear reaction-d iffusion system.J.M ath. Anal.Appl.,12(1997),481-492.

[5]Escobedo M.,Herrero M.A.,Boundedness and blow up for a sem ilinear reaction-diffusion system.J.Differential Equation,89(1991),176-202.

[6]Escobedo M.,LevineM.A.,Fu jita type exponents for reaction-diffusion system.Arch.RationalM ed.Anal.,129(1995),47-100.

[7]Zhang J.,Boundedness and blow-up behavior for reaction-d iffusion system s in a bounded dom ain.Nonlinear Analysis,35(1999),833-844.

[8]Zheng S.,Globalexistence and globalnon-existence of solutions to a reaction-diffusion system.Nonlinear Analysis,39(2000),327-340.

[9]Levine H.A.,A Fu jita type global existence-global noexistence theorem for a w eak ly coup led system of reaction-d iffusion equations.ZAMP,42(1992),408-430.

[10]Dickstein F.,Escobedo M.,A m axim um p rincip le for sem ilinear parabolic system s and app lications.NonlinearAnalysis,45(2001),825-837.

[11]WeiY.,Gao W.,Existence and uniqueness of local solutions to a class of quasilinear degenerate parabolic system s.Appl.M ath.Comp.,190(2007),1250-1257.

[12]M addalena L.,Existence of global solu tion for reaction-d iffusion system w ith density dependentd iffusion.Nonlinear Anal.,TMA,8(11)(1984),1383-1394.

[13]LadyzenskajaO.A.,Solonnikov V.A.and Ural’cevaN.N.,Linearand Quasilinear Equations of Parabolic Type.Am er.M ath.Soc.,Providence,RI,1968.

[14]Rao M.Ram a M ohana,Ordinary DifferentialEquations,A rnold,1980.

10.4208/jpde.v26.n1.1 M arch 2013

?Correspond ing au thor.Emailaddresses:weiyj@j lu.edu.cn(Y.Wei),wjgao@j lu.edu.cn(W.Gao)