# On the Boundary Value Condition of an Isotropic Parabolic Equation

Authors: Qitong Ou; Huashui Zhan
Journal: Journal of Function Spaces
 (2020)

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## Abstract

The well-posedness problem of anisotropic parabolic equation with variable exponents is studied in this paper. The weak solutions and the strong solutions are introduced, respectively. By a generalized Gronwall inequality, the stability of strong solutions to this equation is established, and the uniqueness of weak solutions is proved. Compared with the related works, a new boundary value condition,∏i=1Naix,t=0,x,t∈∂Ω×0,T, is introduced the first time and has been proved that it can take place of the Dirichlet boundary value condition in some way.


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## Body

## 1. Introduction

In this paper, we mainly pay attention on the stability of solutions to the following anisotropic parabolic equation with variable exponents:(1)ut=∑i=1N∂∂xiaix,tuxipix,t−2uxi+∑i=1N∂biu,x,t∂xi,x,t∈QT,with the initial value
(2)ux,0=u0x,x∈Ωand the boundary value condition
(3)ux,t=0,x,t∈∂Ω×0,T.Here,aix,t≥0,pix,t>1, aix,t∈CQT¯, pix,t∈CQT¯, and Ω⊂ℝN is a bounded domain with the smooth boundary ∂Ω, QT=Ω×0,T.Whenpix,t=p is a constant, i=1,2,⋯,N, equation (1) arises in the mathematical modelling of various physical processes such as flows of incompressible turbulent fluids, gases in pipes, and processes of filtration in glaciology. The equation in this case has been studied widely [1–5]. When pix,t=px,t is a measurable function, i=1,2,⋯,N, equation (1) is similar with the equation with the type
(4)ut=divax,t∇upx,t∇u+fx,t,u,∇u,which arises in the phenomena of electrorheological fluids [6, 7]. The existence of solutions of the initial-boundary value problem to this equation can be found in [8–11]. Also, one can refer to [12–18] for some other related works.Ifaix,t=aix and satisfies
(5)aixx∈Ω>0,aixx∈∂Ω=0,i=1,2,⋯,N,the well-posedness problem of the following equations
(6)ut=∑i=1N∂∂xiaixuxipix−2uxi+fx,t,u,∇uhas been studied by the second author in recent years [19–21]. Instead of boundary value condition (3), only a partial boundary value condition
(7)ux,t=0,x,t∈∑p×0,Tis imposed, where ∑p∈∂Ω is a relatively open subset which has different expression according to different kinds of fx,t,u,∇u and sometimes is just an empty set [19–21].Compared with [19, 20] and [21], since the diffusion coefficient aix,t and the variable exponent pix,t both depend on the time variable t, equation (1) has a wider applications than equation (6), and in mathematical theory, there are some essential difficulties to be overcome. More than that, instead of (5), we only assume that
(8)aix,t>0,x,t∈Ω×0,T,i=1,2,⋯,N,(9)∏i=1Naix,t=0,x,t∈∂Ω×0,Tand do not require that
(10)aix,t=0,x,t∈∂Ω×0,T,i=1,2,⋯,N,which is similar as (5) in [19–21].To see the essential difference between (9) and (10), let us give a special case of equation when N=2, Ω⊂ℝ2, ∂Ω=Γ1¯∪Γ2¯, and Γ1 and Γ2 are relatively open subset of ∂Ω, Γ1∩Γ2=∅. Consider the equation
(11)ut=∂∂x1a1xux1p1x−2uxi+∂∂x2a2xux1p2x−2uxi,x,t∈QT,where
(12)a1x=0,x∈Γ1,a1x>0,x∈Γ2,(13)a2x>0,x∈Γ2,a2x=0,x∈Γ2,then (9) is true, i.e.,
(14)∏i=12aix=0,x,t∈∂Ω×0,T.More precisely, for example,(15)Ω=x=x1,x2:1<x12+x22<4,a1x=x12+x22−1,a2x=4−x12+x22.However, in (12) and (13),
(16)a1x+a2x>0,x∈∂Ω.This fact makes us feel that only under the boundary value condition (3), the uniqueness (or the stability) of weak solutions to equation (11) can be true. The following works seem to supply more evidences. One is [22] in which the equation
(17)vt=div∇vmpx,t−2∇vm+bx,t∇vm+vqx,t,x,t∈QTis studied. The others are the equations arising from the double phase obstacle problems
(18)vt=divax∇vp−2∇v+bx∇vq−2∇v+fx,t,v,∇v,x,t∈QT,where ax+bx>0, which have gained a wide attention in recent years, one can refer to [23, 24] and the references therein. In these papers, the boundary value condition (3) is imposed without exception.The main dedication of this paper is that the stability of weak solutions to equation (11)(in general, (1)) can be established independent of boundary value condition (3). Such a conclusion totally overthrows our imagination. In theory, condition (9) can take place of boundary value condition (3) is found the first time. In applications, condition (9) reflects a synthesized effect of an anisotropic diffusion process.This paper is arranged as follows. In Section1, we have given a simple introduction. In Section 2, we will introduce the definitions of weak solution and strong weak solution, respectively, quote some basic lemmas, and give the main results. In Section 3, we will study the stability of weak solutions to equation (1) with the new boundary value condition (9). In Section 4, we will study the uniqueness of weak solution to equation (1) independent of the boundary value condition (7). In Section 5, we will give the outline of the proof on the existence of strong solutions.


## 2. Definitions and Main Results

We denote(19)pi+=maxx,t∈QT¯pix,t,pit+=maxx∈Ω¯pix,t,i=1,2,⋯,N,pi−=minx,t∈QT¯pix,t,pit−=minx∈Ω¯pix,t,i=1,2,⋯,N,p−=minp1−,p2−,⋯,pN−1−,pN−,p−>1,p+=maxp1+,p2+,⋯,pN−1+,pN+.First of all, let us introduce the definition of solutions.Definition 1.
A functionux,t is said to be a weak solution of equation (1), if
(20)u∈L∞QT,ut∈Lp+′0,T;W−1,p+′Ωaix,tuxipix,t∈L1QT,i=1,2,⋯,N,and for any function φ∈C01QT,
(21)∬QTutφdxdt+∑i=1N∬QTaix,tuxipix,t−2uxi·φxi+biu,x,t·φxidxdt=0.
This definition of weak solution is similar as that defined in [20], where aix,t=aix,pix,t=pi is a constant. Also, we can prove the existence of weak solutions similar as that defined in [20], so we do not repeat the details in this paper. As an improvement from the existing result in [20], we introduce the following definition.Definition 2.
A functionux,t is said to be a strong solution of equation (1) with the initial value (2), if
(22)u∈L∞QT,ut∈L2QT,aix,tuxipix,t∈L1QT,i=1,2,⋯,N,and for any function φ∈C01QT, u satisfies (21) and the initial value is satisfied in the sense
(23)limt→0∫Ωux,t−u0xϕxdx=0,ϕx∈C0∞Ω.
The proof of the existence of strong solution will be given at Section5 of this paper. Since aix,t is positive when x∈Ω, (22) means that ut and ∇u exist almost everywhere in QT. This is the reason that we call ux,t as a strong solution of equation (1). Moreover, from Definition 2, for all φx,t∈Lp+0,T;W01,p+Ω, we still have the integral equality (21), which implies that
(24)utLp+′0,T;W−1,p+′Ω≤c.
Thus, ifux,t is a strong solution of equation (1), then it is a weak solution.
Secondly, in order to make the paper sufficiently self-contained and present our discussions in a straightforward manner, let us briefly recall some preliminary results on properties of variable exponent Lebesgue spacesLpxΩ and variable exponent Sobolev spaces W1,pxΩ [24–26].
Set(25)C+Ω¯=h∈CΩ¯:minx∈Ω¯hx>1.
For anyh∈C+Ω¯, we define
(26)h+=supx∈Ωhx,h−=infx∈Ωhx.
For anyp∈C+Ω¯, let LpxΩ consist of all measurable real-valued functions ux satisfying
(27)∫Ωuxpxdx<∞,endowed with the Luxemburg norm
(28)uLpxΩ=infλ>0:∫Ωuxλpxdx≤1.
Define(29)W1,pxΩ=u∈LpxΩ:∇u∈LpxΩ,endowed with the norm
(30)uW1,pxΩ=uLpxΩ+∇uLpxΩ.
LetW01,pxΩ be the closure space of C0∞Ω in W1,pxΩ.From [25–27], we haveLemma 3.
The following three statements are true.(i)
The spaceLpxΩ,·LpxΩ, W1,pxΩ,∥·∥W1,pxΩ, and W01,pxΩ are reflexive Banach spaces(ii)
(px-Hölder’s inequality), let q1x and q2x be real functions satisfying 1/q1x+1/q2x=1 with q1x>1. Then, the conjugate space of Lq1xΩ is Lq2xΩ. For any u∈Lq1xΩ and v∈Lq2xΩ, we have
(31)∫Ωuvdx≤2uLq1xΩvLq2xΩ(iii)
There are the following properties:(1)
ifuLpxΩ=1, then ∫Ωupxdx=1(2)
ifuLpxΩ>1, then uLpxΩp−≤∫Ωupxdx≤uLpxΩp+(3)
ifuLpxΩ<1, then uLpxΩp+≤∫Ωupxdx≤uLpxΩp−
Basing on Lemma3, by generalizing the Gronwall inequality, we will prove the following stability theorems, in which the initial values satisfy
(32)u0∈L∞Ω,u0xi∈LpiΩ,i=1,2,⋯,N,v0∈L∞Ω,v0xi∈LpiΩ,i=1,2,⋯,N.Theorem 4.
Letux,t and vx,t be two strong solutions of (1) with the initial values u0x and v0x, respectively; p−>1 and aix,t∈CQT¯ satisfy (8) and (9) and
(33)∫Ωaix,t−1/pix,t−1dx≤cT,i=1,2,⋯,N,
bis,x,t satisfies
(34)biu,x,t−biv,x,t≤caix,t1/pix,tu−v,i=1,2,⋯,N.
If forη small enough,
(35)1η∫Ω\Ωηtaix,t∏j=1Najxxipix,tdx1/pit+≤cT,i=1,2,⋯,N,then
(36)∫Ωux,t−vx,tdx≤∫Ωu0x−v0xdx,t∈0,T,where for any t∈0,T, Ωηt=x∈Ω:∏j=1Najx,t>η.
In this paper, the constantcT represents that c depends on T. If we only want to prove the uniqueness of weak solutions, condition (34) is not necessary; we have the following result.Theorem 5.
Letp−>1,aix,t∈C1QT¯ satisfy (8) and (9) and ∂aix,t/∂t≤0,bis,x,t be a Lipschitz function on ℝ×Ω¯×0,T. If ux,t and vx,t are two strong solutions of equation (1) with the initial values u0x and v0x, respectively, then for any Ω1⊂⊂Ω,
(37)∫Ω1ux,t−vx,t2dx≤cΩ1∫Ωu0x−v0x2dx,which implies that the uniqueness of weak solution is true.
One can see that both Theorem4 and Theorem 5 imply the uniqueness of solution is true. However, in Theorem 5, the convection function bi·,x,t is independent of the diffusion coefficient aix,t, so as a uniqueness theorem, it is a better than Theorem 4.


## 3. The Stability of Strong Solutions Independent of the Boundary Value Condition

For smallη>0, let
(38)hηs=2η1−∣s∣η+,Sηs=∫0shητdτ.Obviously,hηs∈Cℝ, and
(39)Sηs≤1;limη→0Sηs=sgns,limη→0sSη′s=0.In addition, if we denoteHηs=∫0sSητdτ, then
(40)limη→0Hηs=s,s∈−∞,+∞.At first, we give a generalization of the Gronwall inequality.Lemma 6.
Ifax,tux,tpx,t,ax,tvx,tpx,t∈L1QT and
(41)∫Ωax,sux,s−vx,spx,sdx≤∫Ωax,τux,τ−vx,τpx,τdx+c∫τs∫Ωax,tu−vpx,tdxdtl,where 0<l≤1, then,
(42)∫Ωax,tux,s−vx,spx,tdx≤c∫Ωax,τux,τ−vx,τpx,tdx.Proof.
Define thatκt=∫Ωax,tux,t−vx,tpx,tdx. Without loss of the generality, we may assume that there exists τ∈0,T, κτ>0. Then, for any s>τ, ∫τsktdt>0. Denoting
(43)τ0=maxt∈τ,s,κt>0,∫ττ0ktdt=c1,then, τ<τ0≤s, and
(44)∫τsktdt≥∫ττ0ktdt=c1.
Sinceax,tux,tpx,t,ax,tvx,tpx,t∈L1QT, there exists a constant C>0 such that
(45)c∫τsktdtl∫τsktdt≤c∫τsktdtlc1≤C=Cc,c1,T.
Combing (41) with (45), we obtain
(46)κs−κτ≤C+c∫τsktdt.
Using the Gronwall inequality, we have(47)∫Ωax,sux,s−vx,spx,sdx≤c∫Ωax,τux,τ−vx,τpx,τdx.
Ifux,t,vx,t∈L∞QT, this lemma has been proved in [19] recently.
Secondly, we give the proof of Theorem4.Proof of Theorem 4.
Letux,t and vx,t be two strong solutions of equation (1) with the initial values u0x and v0x, respectively.
For anyt∈0,T, set Ωηt=x∈Ω:∏i=1Naix,t>η, and
(48)ϕηtx=1,ifx∈Ωηt,1η∏i=1Naix,t,ifx∈Ω\Ωηt.
By a process of limit, we can chooseχτ,sϕηtxSηu−v as the test function. Here, χτ,s is the characteristic function of τ,s⊂0,T. Then,
(49)∫τs∫ΩϕηtSηu−v∂u−v∂tdxdt+∑i=1N∫τs∫Ωaix,tuxipix,t−2uxi−vxipix,t−2∇vuxi−vxihηu−vϕηtxdxdt+∑i=1N∫τs∫Ωaix,tuxipix,t−2uxi−vxipix,t−2vxiu−vSηu−v∂ϕηt∂xidxdt+∑i=1N∫τs∫Ωbiu,x,t−biv,x,t∂ϕηt∂xiSηu−vdxdt+∑i=1N∫τs∫Ωbiu,x,t−biv,x,tu−vxiϕηthηu−vdxdt=0.
In the first place, using the dominated convergence theorem, we have(50)limη→0∫τs∫ΩϕηtxSηu−v∂u−v∂tdxdt=∫τs∫Ωsgnu−v∂u−v∂tdxdt=limη→0∫τs∫Ω∂Hηu−v∂tdxdt=limη→0∫ΩHηu−vx,s−Hηu−vx,τdx=∫Ωu−vx,sdx−∫Ωu−vx,τdx,and for any 1≤i≤N,
(51)∫Ωaix,tuxipix,t−2uxi−vxipix,t−2vxiuxi−vxihηu−vϕηtxdx≥0.
In the second place, we notice∂ϕηt/∂xi=1/η∏j=1Najx,txi when x∈Ω\Ωηt; in the other places, it is identical to zero. Since we assume that
(52)∏j=1Najx,t=0,x,t∈∂Ω×0,T,by Lemma 3, we have
(53)∫Ωaix,tuxipix,t−2uxi−vxipix,t−2vxi∂ϕηt∂xiSηu−vdx=∫Ω\Ωηtaix,tuxipix,t−2uxi−vxipix,t−2vxi∂ϕηt∂xiSηu−vdx≤1η∫Ω\Ωηtaix,tuxipix,t−1+vxipix,t−1∏j=1Najx,txiSηu−vdx≤c1η∫Ω\Ωηtaix,tuxipix,t+vxipix,tdx1/p′it+·∫Ω\Ωηtaix,t∏j=1Najx,txipix,tdx1/pit+≤c∫Ω\Ωηtaix,tuxipix,tdxpi−1/pi+∫Ω\Ωηtaixvxipix,tdx1/p′it+·1η∫Ω\Ωηtaix,t∏j=1Najx,txipitdx1/pit+≤c∫Ω\Ωηtaix,tuxipix,tdx1/p′it++∫Ω\Ωηtaix,tvxipitdx1/p′it+,where pit+=maxx∈Ω¯px,t and pi′x,t=px,t/px,t−1.
(53) implies
(54)limη→0∫τs∫Ωaix,tuxipix,t−2uxi−vxipix,t−2vxi∂ϕηt∂xiSηu−vdxdt≤climη→0∫τs∫Ω\Ωηtaix,tuxipix,tdx1/p′it++∫Ω\Ωηtaix,tvxipix,tdx1/p′it+dt=0.
In the third place, sinceaix,t satisfies condition (34), we can deduce that
(55)limη→0∑i=1N∫τs∫Ωbiu,x,t−biv,x,tu−vxiϕηhηu−vdxdt=0.
In details,(56)∫Ωbiu,x,t−biv,x,tu−vxihηu−vdx=∫Ω:∣u−v∣<ηbiu,x,t−biv,x,tu−vxihηu−vdx≤∫Ω:∣u−v∣<ηaix,t1/pix,tu−vxiaix,t−1/pix,tu−vhηu−vdx.
If the setΩ:u−v=0 has a positive measure, by condition (34), we have
(57)limη→0∫Ω:∣u−v∣<ηbiu,x,t−biv,x,tu−vxihηu−vdx≤c∫Ω:∣u−v∣=0aix,t1/pix,tuxi−vxipix,tdx1/pit+∫Ωaix,t−1/pix,t−1dx1/p′1it=0,where p′1it=p′it+ or p′it− according to ∫Ωaix,t−1/pix,t−1dx≤1 or ∫Ωaix,t−1/pix,t−1dx>1 from Lemma 3.
If the setΩ:u−v=0 only has zero measure, since aix,t−1/pix,t−1∈L1Ω,
(58)limη→0∫Ω:∣u−v∣<ηbiu,x,t−biv,x,tu−vxihηu−vdx≤c∫Ωai1/piuxi−vxipidx1/pi∫Ω:∣u−v∣=0aix,t−1/pi−1dxpi−1/pi≤c∫Ωaixuxipi+vxipidx1/pi∫Ω:∣u−v∣=0aix,t−1/pi−1dxpi−1/pi=0.
Finally, by condition (34), we have
(59)limη→0∫Ωbiu,x,t−biv,x,t∂ϕηt∂xiSηu−vdx=limη→0∫Ω\Ωηtbiu,x,t−biv,x,t∂ϕηt∂xiSηu−vdx≤limη→01η∫Ω\Ωηtai1/pix,t∏j=1Najx,txiSηu−vu−vdx≤limη→01η∫Ω\Ωηtaix,t∏j=1Najx,txipix,tdx1/pit+∫Ω\ΩηtSηu−vu−vpix,t/pix,t−1dx1/p′it+≤climη→0∫Ω\ΩηtSηu−vu−vpix,t/pix,t−1dx1/p′it+≤c∫Ωu−vdx1/p′it+,since ux,t,vx,t∈L∞QT.
Now, letη→0 in (42). We easily obtain that
(60)∫Ωux,s−vx,sdx≤∫Ωux,τ−vx,τdx+c∫0t∫Ωu−vdxdtl.where l<1.
Using Lemma6, we have
(61)∫Ωux,s−vx,sdx≤c∫Ωux,τ−vx,τdx.Then by the arbitrary of τ,
(62)∫Ωux,s−vx,sdx≤c∫Ωu0x−v0xdx.


## 4. The Uniqueness of Weak Solution

Lemma 7.
Letut∈Lp′0,T;W−1,p′Ω. For any continuous function hs, Hs=∫0shσdσ, a.e.t1,t2∈0,T, it holds
(63)∫t1t2∫Ωhuutdxdt=∫ΩHux,t2−Hux,t1dx.
This lemma can be generalized from of Lemma 2.2 in [28] simply; we do not give the details here.Theorem 8.
Letp−>1,aix,t∈C1QT¯ satisfy (8)(9) and ∂aix,t/∂t≤0, bis,x,t be a Lipschitz function on ℝ×Ω¯×0,T. If ux,t and vx,t are two weak solutions of equation (1) with the initial values u0x and v0x, respectively, then, there exists a positive constant βj≥2 such that
(64)∫Ω∏j=1Najβjx,tux,t−vx,t2dx≤c∫Ω∏j=1Najβjx,0u0x−v0x2dx.Proof.
Letux,t and vx,t be two weak solutions of equation (1) with the initial values u0x and v0x, respectively. By a process of limitation, we may choose φ=χτ,s∏j=1Najβju−v as a test function. Denoting that Qτs=Ω×τ,s, then,
(65)∬Qτsu−v∏j=1Najβj∂u−v∂tdxdt=−∑i=1N∬Qτsaixuxipi−2uxi−vxipi−2vxiu−v∏j=1Najβjxidxdt−∑i=1N∬Qτsbiu,x,t−biv,x,tu−v∏j=1Najβjxidxdt.
At first, we have(66)∬Qτsaix,tuxipix,t−2uxi−vxipix,t−2vxiu−vxi∏j=1Najβjdxdt≥0,(67)∬Qτsu−vaix,tuxipix,t−2uxi−vxipix,t−2vxi∏j=1Najβjxidxdt≤∬Qτsu−vaix,tuxipix,t−1+vxipix,t−1∏j=1Najβjxidxdt≤c∫τs∫Ωaix,tuxipix,t+vxipix,tdxdt1/p′1i∫τs∫Ωaix,t∏j=1Najβjxipix,tu−vpix,tdxdt1/p1i≤c∫τs∫Ωaixuxipi+vxipidxdtpi−1/pi·∫τs∫Ωaix∏j=1Najβj−1ajxipiu−vpidxdt1/pi≤c∫τs∫Ωaix∏j=1Najβj−1ajxipiu−vpidxdt1/pi.
Here, we have used the fact thataxi≤c and p1i=pi+ or pi− according to
(68)∫τs∫Ωaix,t∏j=1Najβjxipix,tu−vpix,tdxdt≤1,or
(69)∫τs∫Ωaix,t∏j=1Najβjxipix,tu−vpix,tdxdt>1.
p′1i has a similar meaning. Now, if we denote that Q1=x,t∈QT:1<px,t<2 and Q2=x,t∈QT:px,t≥2, by that βi≥2, we have
(70)∬Qτs⋂Q2aix,t∏j=1Najβj−1ajxipix,tu−vpix,tdxdt1/p1i≤c∫τs∫Ω∏j=1Najβju−v2dxdt1/p1i.and by the Hölder inequality,
(71)∬Qτs⋂Q1aix,t∏j=1Najβj−1ajxipix,tu−vpix,tdxdt1/p1i≤c∫τs∫Ω∏j=1Najβju−v2dxdt1/k.where k<1.
Combing (67)-(71), we obtain
(72)∬Qτsu−vaix,tuxipix,t−2uxi−vxipix,t−2vxi∏j=1Najβjxidxdt≤c∫τs∫Ω∏j=1Najβju−v2dxdtl,where l<1.
Secondly,(73)∬Qτsbiu,x,t−biv,x,tu−v∏j=1Najβjxidxdt=∬Qτsbiu,x,t−biv,x,tu−v∏j=1Najβjxidxdt+∬Qsbiu,x,t−biv,x,tu−vxi∏j=1Najβjdxdt.
For the first term on the right hand side of (73), by that βj≥2, ∣ajxi∣≤c, using the Hölder inequality, we have
(74)∬Qτsbiu,x,t−biv,x,tu−v∏j=1Najβjxidxdt=∫τs∫Ωbiu,x,t−biv,x,tu−v∑k=1Nβkakβk−1akxi∏j=1,j≠kNajβjdxdt≤c∫τs∫Ωu−v∑k=1Nβkakβk−1akxi∏j=1,j≠kNajβjdxdt≤c∫τs∫Ω∏j=1Najβju−v2dxdt1/2.
For the second term on the right hand side of (73), since βi≥2, denoting pi′x,t=pix,t/pix,t−1 as usual, we have
(75)βi−1pix,tpi′x,t≥βi.
By this inequality, we deduce(76)∬Qτsbiu,x,t−biv,x,tu−vxi∏j=1Najβjdxdt≤c∫τs∫Ωaiβi−1/pix,tpi′x,t∏j=1,j≠iNajβjbiu,x,t−biv,x,tpi′x,tdxdt1/p1i′∫τs∫Ωaix,tuxipix,t+vxipix,tdxdt1/p1i≤c∫τs∫Ωaiβi−1/pix,tpi′x,t∏j=1,j≠iNajβjbiu,x,t−biv,x,tpi′x,tdxdt1/p1i′≤c∫τs∫Ω∏j=1Najβju−vpi′x,tdxdt1/p1i′.
DefiningQ1,Q2 as above, if pix,t≥2, then 1<pi′x,t≤2. By the Hölder inequality,
(77)∬Qτs⋂Q2∏j=1Najβju−vpi′x,tdxdt1/p1i′≤c∫τs∫Ω∏j=1Najβju−v2dxdt1/k,where k<1.
If1<pix,t<2, then pi′x,t>2,
(78)∬Qτs⋂Q1∫Ω∏j=1Najβju−vpi′x,tdxdt1/p1i′≤c∫τs∫Ω∏j=1Najβju−v2dxdt1/p1i′.
From (76)-(78), we obtain
(79)∬Qτsbiu,x,t−biv,x,tu−vxi∏j=1Najβjdxdt≤c∫τs∫Ω∏j=1Najβju−v2dxdtl,where l<1.
Once more, by Lemma7, we have
(80)∬Qτsu−v∏j=1Najβj∂u−v∂tdxdt=∬Qτsu−v∏j=1Najβj∂∏j=1Najβju−v∂tdxdt−∬Qτsu−v2∏j=1Najβj∂∏j=1Najβj∂tdxdt=∫Ω∏j=1Najβjux,s−vx,s2dx−∫Ω∏j=1Najβjux,τ−vx,τ2dx−∬Qτsu−v2∏j=1Najβj∂∏j=1Najβj∂tdxdt.
According to (65), (66), (72), (74), (79), and (80), since ∂aix,t/∂t≤0 and βj≥2, we have
(81)∫Ω∏j=1Najβjux,s−vx,s2dx−∫Ω∏j=1Najβjux,τ−vx,τ2dx≤c∫τs∫Ω∏j=1Najβjux,t−vx,t2dxdtl+∬Qτsu−v2∏j=1Najβj∂∏j=1Najβj∂tdxdt≤c∫τs∫Ω∏j=1Najβjux,t−vx,t2dxdtl,where l<1. By (81), using the generalized Gronwall inequality, we deduce
(82)∫Ω∏j=1Najx,sβjux,s−vx,s2dx≤∫Ω∏j=1Najx,τβjux,τ−vx,τ2dx.
Thus, by the arbitrary ofτ, we have
(83)∫Ω∏j=1Najx,sβjux,s−vx,s2dx≤∫Ω∏j=1Najx,0βju0x−v0x2dx.
By (83), we have the conclusion. The proof is complete.By this theorem, Theorem5 is true clearly.


## 5. The Strong Solutions Dependent on the Initial Value

For the completeness of the paper, we will give a basic theorem about the existence of strong solutions.Theorem 9.
Ifp−>2, aix,t∈CΩ¯ satisfies (8) and (9), bis,x,t is a C1 function on ℝ×Ω¯×0,T,
(84)u0∈L∞Ω,u0xi∈LpiΩ,i=1,2,⋯,N,(85)∫Ωaix,t−1/pix,t−1dx<∞,i=1,2,⋯,N,(86)bis,x,t≤caix,t1/pix,t,biss,x,t≤caix,t1/pix,t,i=1,2,⋯,N,then equation (1) with initial value (2) has a weak solution.
Here,biss,x,t=∂bis,x,t/∂s. Before we give the proof of Theorem 9, we would like to point out that condition (86) is just a sufficient condition; we also can use other conditions to replace them. For example, when aix,t≡ax,pix,t=pi, by the conditions
(87)∫Ωaix−2/pi−2dx<c,biss,x,t≤cax1/pi.
Theorem9 had been obtained in [19].Proof of Theorem 9.
Consider the following regularized problem(88)uεt−∑i=1N∂∂xiaix,t+εuεxipix,t−2uεxi−∑i=1N∂biuε,x,t∂xi=0,x,t∈QT,(89)uεx,t=0,x,t∈∂Ω×0,T,(90)uεx,0=uε0x,x∈Ω.
Here,uε0∈C0∞Ω, uε0L∞Ω≤u0L∞Ω, and ∇uε0 converges to ∇u0x in Lp+Ω. It is well-known that the above problem has a unique weak solution uε∈L∞QT and aix,tuxipix,t∈L1QT [8], and
(91)uεL∞QT≤cM,where M=u0xL∞Ω.
Multiplying (88) by uε and integrating it over QT, then
(92)12∫Ωuε2dx+∑i=1N∬QTaix,t+εuεxipix,tdxdt+∑i=1N∬QT∂biuε,x,t∂xiuεdxdt=12∫Ωuε02dx.
Ifbiuε,x,t≤caix,t1/pix,t, by that ∫Ωaix,t−1/pix,t−1dx<∞, then
(93)∫Ω∂biuε,x,t∂xiuεdx=−∫Ω∂uε∂xibiuε,x,tdx≤c∫Ωaix,t1/pix,t∂uε∂xidx≤c∫Ωaix,tuxipix,tdx+c.
Accordingly, by (92), we have
(94)∫Ωuε2dx+∑i=1N∬QTaix,t+εuεxipix,tdxdt≤c.
Multiplying (88) by uεt, integrating it over QT, then it yields
(95)∬QTuεt2dxdt=∑i=1N∬QT∂∂xiaix,t+εuεxipix,t−2uεxiuεtdxdt+∑i=1N∬QTuεt∂biuε,x,t∂xidxdt.
Note thatp−>2,
(96)−uεxi2+εpix,t−2/2uεxi·∂∂xiuεt=−12∂∂t∫0uεxi2+εspix,t−2/2ds+12∫0uεxi2+ε∂∂tspix,t−2/2ds=−12∂∂t∫0uεxi2+εspix,t−2/2ds+14∫0uεxi2+εspix,t−2/2lns∂pix,t∂tds=−12∂∂t∫0uεxi2+εspix,t−2/2ds+14∫0uεxi2+εspix,t−2/2lns∂pix,t∂tds,and by the Young inequality,
(97)∬QTaix,t+ε∫0uεxi2+εspix,t−2/2lns∂pix,t∂tdsdxdt≤c∬QTaix,t+εuεxi2+εdxdt≤c∬QTaix,t+εuεxipix,t+1dxdt≤c.We have
(98)∑i=1N∬QT∂∂xiaix,t+εuεxi2+εpix,t−2/2uεxiuεtdxdt=−∑i=1N∬QTaix,t+εuεxi2+εpix,t−2/2uεxi·∂∂xiuεtdxdt=−∑i=1N12∬QTaix,t+εddt∫0uεxi2+εspix,t−2/2dsdxdt+14∑i=1N∬QTaix,t+ε∫0uεxix,t2+εspix,t−2/2lns∂pix,t∂tdsdxdt.
If∣biuuε,x,t∣≤caix,t1/pix,t,
(99)∬QTuεt∂biuε,x,t∂xidxdt≤∬QTbiuuε,x,tuεxiuεtdxdt+∬QTbixiuε,x,tuεtdxdt≤12∬QTuεt2dxdt+c∬QTaix,tuεxipix,tdxdt+c.
Combining (95), (98), and (99), we have
(100)∬QTuεt2dxdt+∑i=1N∬QTaix,t+εddt∫0uεxi2spix,t−2/2dsdxdt≤c.
By the above inequality, we have(101)∬QTuεt2dxdt≤c+c∑i=1N∫Ωaix,0+εuε0xipidx≤c.
Thus, by (91), (94), and (101), there exist a function u and a n−dimensional vector function ζ←=ζ1,⋯,ζn satisfying that uε→u a.e. in QT, and
(102)u∈L∞QT,ζi∈L10,T;Lpix,t/pix,t−1Ω,uε⇀∗u,inL∞QT,biuε,x,t→biu,x,t,a.e.inQT,aix,tuεxipix,t−2uεxi⇀ζi,inL10,T;Lpix,t/pix,t−1Ω.
Similar as with Theorem 2.1 of Chapter 2 in [1] (also, one can refer to [19] in which pix,t=pi is just a constant), we are able to show that
(103)limε→0∑i=1N∬QTaix,t+εuεxipix,t−2uεxiφxidxdt=limε→0∑i=1N∬QTaix,tuεxipix,t−2uεxiφxidxdt=∬QTζ←·∇φdxdt=∑i=1N∬QTaix,tuxipix,t−2uxiφxidxdt,for any φ∈C01QT. At last, by a process of limit, we can choose the test function φx,t=χt1,t2ϕx in which ϕx∈C0∞Ω and χt1,t2 is the characteristic function of t1,t2⊂0,T. Then,
(104)∑i=1N∫t1t2∫Ωaix,tuxipix,t−2uxiϕxi+biu,x,tϕxidxdt=∫Ωux,t2−ux,t1ϕxdx,
Lett=t2 and t1→0. Then, we have (23) and u is a strong solution of equation (1) with the initial value (2) in the sense of Definition 2.
At last, we give a simple comment. The conditionaiuxi∈L1Ω can not assure the boundary value condition
(105)ux,t=0,x,t∈Σp×0,T⊆∂Ω×0,Tis imposed in the sense of the trace. In fact, we have the following proposition.Proposition 10.
Ifaix,t satisfies (85), i.e.,
(106)∫Ωaix,t−1/pix,t−1dx<∞,i=1,2,⋯,N.
Then,(107)∫Ωuxidx≤cT<∞,i=1,2,⋯,N.
Thus, ifaix,t satisfies (85), the partial boundary value condition (105) can be imposed in the sense of trace. However, in this paper, we pay our attention on the studying the stability (or the uniqueness) of solutions independent of the boundary value condition (105), so Proposition 10 is not important.


## 6. Conclusion

It is clear of that Lemma6 has a wider applications than the classical Gronwall inequality. Moreover, compared with reference [18], there is at least the essential difference in two aspects. The first one is that condition (9), i.e.,
(108)∏i=1Naix,t=0,x,t∈∂Ω×0,T,is much weaker than condition (5) appearing in [19], i.e.
(109)aix=0,x∈∂Ω,i=1,2,⋯,N.Such a degeneracy is the special nature of the anisotropic equation. The second one is that we have not used boundary value condition (3) throughout this paper; in other words, condition (9) may replace boundary value condition (3) in some way. Moreover, using some techniques developed by the second author in his work [10], in which the well-posedness of weak solutions to equation,
(110)vβ−1vt=divbx,t∇vpx,t−2∇v+∑i=1Ngix,t∂γiv∂xi,x,t∈QThas been discussed; the method used in this paper can be applied to study a more general equation
(111)uβ−1ut=∑i=1N∂∂xiaix,tuxipix,t−2uxi+fx,t,u,∇u,x,t∈QT,in the future.



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*Source: 2180830-2020-11-27.xml*