# The Evolutionaryp(x)-Laplacian Equation with a Partial Boundary Value Condition

Authors: Huashui Zhan; Zhen Zhou
Journal: Discrete Dynamics in Nature and Society
 (2018)

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## Abstract

Consider a diffusion convection equation coming from the electrorheological fluids. If the diffusion coefficient of the equation is degenerate on the boundary, generally, we can only impose a partial boundary value condition to ensure the well-posedness of the solutions. Since the equation is nonlinear, the partial boundary value condition cannot be depicted by Fichera function. In this paper, whenα<p--1, an explicit formula of the partial boundary on which we should impose the boundary value is firstly depicted. The stability of the solutions, dependent on this partial boundary value condition, is obtained. While α>p+-1, the stability of the solutions is obtained without the boundary value condition. At the same time, only if α>0 and p->1 can the uniqueness of the solutions be proved without any boundary value condition.


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## Body

## 1. Introduction and the Main Results

The evolutionaryp(x)-Laplacian equation(1)ut=div∇upx-2∇u,x,t∈QT=Ω×0,Tcomes from a new interesting kind of fluids, the so-called electrorheological fluids (see [1, 2]), where Ω⊂RN is a bounded domain with C2 smooth boundary ∂Ω and p(x) is a measurable function. Equation (1) with the initial value(2)ux,0=u0x,x∈Ωand the homogeneous boundary value(3)ux,t=0,x,t∈∂Ω×0,Thas been researched widely; one can refer to [3–6] et al.Ifp(x)=p,(4)∂u∂t-divax∇up-2∇u-bixDiu+cx,tu=fx,t,x,t∈QTwas considered by Yin and Wang [7], where a(x) may be degenerate. Instead of the usual boundary condition (3), they classified the boundary into three parts: the nondegenerate boundary, the weakly degenerate boundary, and the strongly degenerate boundary, by means of a reasonable integral description. The boundary value condition should be supplemented definitely on the nondegenerate boundary and the weakly degenerate boundary. Even earlier, they had studied a simpler equation(5)ut=divρα∇up-2∇u,x,t∈QT,in [8]. Here ρ(x)=dist(x,∂Ω), α>0, and p>1. They showed that only if α<p-1, the usual boundary value condition (3) can be imposed; while α≥p-1, the uniqueness of the solution can be proved without any boundary value condition.In this paper, we will consider the evolutionary equation(6)ut=divρα∇upx-2∇u+gixDibu,x,t∈QT,with the initial value (2) and with a partial boundary condition(7)ux,t=0,x,t∈Σp×0,T,where gi(x)∈C1(Ω¯), Di=∂/∂xi, b(s)∈C1(R), and(8)Σp=x∈∂Ω:gix≠0.In the context,p(x) is a C1(Ω¯) function and the definitions of the function spaces with variable exponents Lp(x)(Ω),W1,p(x)(Ω),W01,p(x)(Ω) can be found in [9–16] et al. We denote that (9)p+=maxx∈Ω-⁡px,p-=minx∈Ω-⁡px.The weak solutions are defined as follows.Definition 1.
If a functionu(x,t) satisfies (10)u∈L∞QT,ut∈L2QT,ρα∇upx∈L1QT,∬QT∂u∂tφ+ρα∇upx-2∇u·∇φ+bugixφxi+bugxiiφdxdt=0,for any function φ∈L∞(0,T;W01,p(x)(Ω)), then we say that u(x,t) is the weak solution of (6). The initial value (2) is true in the sense of(11)limt→0⁡∫Ωux,t-u0xdx=0.The partial boundary condition (7) is true in the sense of trace.
If(12)u0∈L∞Ω,ρα∇u0p+∈L1Ω,we add some restrictions on b(s), by a similar method to that in [17, 18] and the existence of the weak solution of (6) with the initial value (2) can be proved. If 0<α<p--1, we can prove that the weak solution u∈L∞(0,T;W1,γ(Ω)) for some γ>1. Then the existence of the weak solutions of (6) with the initial-boundary value conditions can be obtained. The main aim of this paper is to study the stability of the weak solutions.
Firstly, we mainly pay close attention to the stability of the weak solutions based on the partial boundary value conditions.Theorem 2.
Letb(s) be a Lipschitz function and u(x,t) and v(x,t) be two weak solutions of (6) with the different initial values u0(x) and v0(x), respectively, and with the same partial homogeneous boundary value(13)uΣp×0,T=0=vΣp×0,T.If b(s) is a Lipschitz function and(14)p+>2,p--1>α≥p--1p+-2,then(15)∫Ωux,t-vx,tdx≤∫Ωu0x-v0xdx,∀t∈0,T,where Σp is the part of the boundary expressed as (8).Secondly, ifα≥p--1, the weak solutions of (6) cannot be defined as the trace on the boundary and the boundary value condition cannot be used. In order to overcome this difficulty, we will introduce a new kind of the weak solutions in Section 3 and the stability of the weak solutions can be proved when α>p+-1.Theorem 3.
Letu and v be two weak solutions of (6) with the different initial values u(x,0) and v(x,0), respectively. If α>p+-1, the constant β≥max⁡{α/p-,1}, and (16)gix≤cρβx,then for any t∈[0,T), there holds(17)∫Ωux,t-vx,t2dx≤c∫Ωux,0-vx,02dx.Last but not least, no matter whetherα<p--1 or not, the uniqueness of the weak solutions is always true. Actually, similar as [19], we can prove the following theorem.Theorem 4.
Ifbi(s) is a Lipschitz function, α>0,p->1, then the solution of (6) with the initial value (2) is unique.However, for the simplicity of the paper, we will not give the details of the proof of Theorem4 in what follows.The rest of the paper is arranged as follows. In Section2, Theorem 2 is proved. In Section 3, Theorem 3 is proved. In the last section, we give an explanation of the partial boundary value condition (8), and some conclusions similar to Theorem 2 are obtained without condition (14).


## 2. The Stability of Solutions Whenα<p--1

Lemma 5 (see [9]).
(i) Letp(x) and q(x) be real functions with 1/p(x)+1/q(x)=1 and p(x)>1. Then, for any u∈Lp(x)(Ω) and v∈Lq(x)(Ω), we have(18)∫Ωuvdx≤2uLpxΩvLqxΩ.(ii)(19)IfuLpxΩ=1,then∫Ωupxdx=1,IfuLpxΩ>1,thenuLpxp-≤∫Ωupxdx≤uLpxp+,IfuLpxΩ<1,thenuLpxp+≤∫Ωupxdx≤uLpxp-.Proof of Theorem2.
For a small positive constantλ>0, let(20)ϕx=1,ifx∈Ωλ,ρλ,ifx∈Ω∖Ωλ,where(21)Ωλ=x∈Ω:ρx=distx,∂Ω>λ.Then(22)∇ϕ=1λ∇ρ,x∈Ω∖Ωλ.
For smallη>0, let(23)Sηs=∫0shητdτ,hηs=2η1-sη+.Obviously, hη(s)∈C(R) and(24)hηs≥0,shηs≤1,Sηs≤1,limη→0⁡Sηs=sgns,limη→0⁡sSη′s=0.
Ifu and v are two weak solutions of (6) with the same partial homogeneous boundary value (13) and Sη(ϕ(u-v)) is chosen to be the test function, then(25)∫ΩSηϕu-v∂u-v∂tdx+∫Ωρα∇upx-2∇u-∇vpx-2∇v·ϕ∇u-vSη′ϕu-vdx+∫Ωρα∇upx-2∇u-∇vpx-2∇v·∇ϕu-vSη′ϕu-vdx+∫Ωgxiixbu-bvSηϕu-vdx+∫Ωgixbu-bvu-vxiSη′ϕu-vϕdx+∫Ωgixbu-bvϕxiu-vSη′ϕu-vdx=0.Thus(26)limη→0⁡limλ→0⁡∫ΩSηϕu-v∂u-v∂tdx=ddt∫Ωu-vdx,∫Ωρα∇upx-2∇u-∇vpx-2∇v·ϕ∇u-vSη′ϕu-vdx≥0.Obviously, we have(27)u-vSη′ϕu-v=ϕu-vSη′ϕu-v1ϕ≤cϕ,∇ϕϕ≤cλ.
Using the Young inequality, we have(28)∫Ωρα∇upx-2∇u-∇vpx-2∇v·∇ϕu-vSη′ϕu-vdx≤∫Ω∖Ωλρα∇upx-1+∇vpx-1∇ϕϕϕu-vSη′ϕu-vdx≤c∫Ω∖Ωλ1λρα∇upx-1+∇vpx-1ϕu-vSη′ϕu-vdx≤cλ∫Ω∖Ωλρα-α/px-1ρα/px-1∇upx-1+∇vpx-1dx≤c∫Ω∖Ωλρα∇upx+∇vpx+1λpxρpxα-α/px-1dx,which goes to 0 as λ→0, due to the assumption that p--1>α≥p--1/p+-2 implies that(29)1λpxρpxα-α/px-1≤λα-1-α/px-1px⟶0,while (30)∫Ωgixbu-bvSη′ϕu-vu-vϕxixdx≤c∫Ω∖Ωλbu-bvgixλdx.Since b(s) is a Lipschitz function, |b(u)-b(v)|≤c|u-v|. According to the definition of the trace, by the partial boundary value condition (7),(31)ux,t=vx,t=0,x∈Σ1=x∈∂Ω:gix≠0,gix=0,x∈Σ2=x∈∂Ω:gix=0,we have(32)limλ→0⁡∫Ωgixbu-bvSη′ϕu-vu-vϕxixdx≤c∫∂Ωgixu-vdΣ=c∫Σ1∪Σ2gixu-vdΣ=0.Moreover, as in [17], we can prove that(33)limη→0⁡limλ→0⁡∫Ωgixbu-bvSη′ϕu-vu-vxiϕxdx=0.The details of the proof of (33) are omitted here.
Once again,(34)limη→0⁡limλ→0⁡∫Ωgxiixbu-bvSηϕu-vdx≤c∫Ωu-vdx.
Now, after lettingλ→0, let η→0 in (25). Then, by (26), (28), (32), (33), and (34), we have(35)ddt∫Ωu-vdx⩽c∫Ωu-vdx,and by the Gronwall inequality, we have(36)∫Ωux,t-vx,tdx⩽c∫Ωu0x-v0xdx,∀t∈0,T.Theorem 2 is proved.


## 3. The Stability of Solutions without the Boundary Value Condition

As we have said in the introduction, whenα≥p--1, since the weak solutions of (6) generally lack the regularity, we cannot define the trace on the boundary. Thus, we cannot use the boundary value condition to research the stability or the uniqueness of the weak solution. In order to overcome this difficulty, we introduce another kind of the weak solutions as follows.Definition 6.
A functionu(x,t) is said to be a weak solution of (6) with the initial value (2), if u satisfies(37)u∈L∞QT,∂u∂t∈L2QT,ρα∇upx∈L1QT,and for any function φ1∈L1(0,T;C01(Ω)),  φ2∈L∞(QT) such that for any given t∈[0,T),  φ2(x,·)∈Wloc1,p(x)(Ω),(38)∬QT∂u∂tφ1φ2+ρα∇upx-2∇u·∇φ1φ2+gixbuφ1φ2xi+bugxiiφ1φ2dxdt=0,and the initial value (2) is satisfied in the sense of(39)limt→0⁡∫Ωux,t-u0xdx=0.
We first introduced this kind of the weak solutions in our previous paper [19], in which the following equation was studied:(40)∂u∂t-divax∇up-2∇u-bixDiu+cx,tu=fx,t,x,t∈Ω×0,T,where 0≤a(x)∈C1(Ω¯) with a(x)|x∈∂Ω=0. It is not difficult to prove the existence of the weak solution in the sense of Definition 6.Proof of Theorem3.
For any fixeds,τ∈[0,T), after an approximate procedure, we may choose χ[τ,s](u-v)ϕ as a test function in equality (38), where χ[τ,s] is the characteristic function on [τ,s] and ϕ(x) is defined as (20). Thus we have (41)∬Qτsu-vϕ∂u-v∂tdxdt=-∬Qsρα∇upx-2∇u-∇vpx-2∇v·∇u-vϕdxdt-∬Qτsgixbu-bvu-vϕxidxdt-∬Qτsbu-bvgxiiϕu-vdxdt,where Qτs=Ω×[τ,s].
We can rewrite (41) as follows:(42)∫Ωux,s-vx,s2ϕdx=∫Ωux,0-vx,02ϕdx-2∬Qτsϕρα∇upx-2∇u-∇vpx-2∇v·∇u-vdxdt-2∬Qτsu-vρα∇upx-2∇u-∇vpx-2∇v∇ϕdxdt-2∬Qτsgixbu-bvu-vϕxidxdt-2∬Qτsbu-bvgxiiϕu-vdxdt≤∫Ωux,0-vx,02ϕdx-2∬Qτsu-vρα∇upx-2∇u-∇vpx-2∇v∇ϕdxdt-2∬Qτsgixbu-bvu-vϕxidxdt-2∬Qτsbu-bvgxiiϕu-vdxdt.
In the first place, since(43)-2∬Qτsu-vρα∇upx-2∇u-∇vpx-2∇v·∇ϕdxdt≤2∬Qτsu-vρα∇upx-1+∇vpx-1∇ϕdxdt≤c∫τs∫Ω∖Ωλpx-1pxρα∇upx+∇vpx+1pxρα∇ϕpxdxdt≤c∫τs∫Ω∖Ωλpx-1pxρα∇upx+∇vpx+1pxλα-pxdxdt,by α>p+-1 leading to α-p(x)>-1, (43) yields(44)limλ→0⁡-2∬Qτsu-vρα∇upx-2∇u-∇vpx-2∇v∇ϕdxdt=0.
In the second place, since|ρxi|≤|∇ρ|=1, by (16), |gi(x)|≤cρβ(x), then(45)limλ→0⁡∬Qτsgixbu-bvu-vϕxidxdt=limλ→0⁡∫τs∫Ω∖Ωλgixbu-bvu-vϕxidxdt≤climλ→0⁡∫τs∫Ω∖Ωλu-v2dxdt=0.By u,v∈L∞(QT) and the condition (16), we have(46)∬Qτsgixbu-bvu-vxiϕdxdt≤∑i=1N∫τs∫Ωgixpx/px-1ρ-α/px-1bu-bvpx/px-1dxdt1/q1∫τs∫Ωρα∇upx+∇vpxdxdt1/p1≤c∑i=1N∫τs∫Ωρβpx/px-1-α/px-1u-vpx/px-1dxdt1/q1≤c∑i=1N∫τs∫Ωu-vpx/px-1dxdt1/q1,by which β≥α/p-. Here p1=p+ or p- and  q1=max⁡{p(x)/p(x)-1} or min{p(x)/p(x)-1} according to (iii) of Lemma 5.
We denote that(47)Ω1=x∈Ω:pxpx-1≥2,Ω2=x∈Ω:pxpx-1<2,and then (48)∫Ω1u-vpx/px-1dx≤c∫Ω1u-v2dx,∫Ω2u-vpx/px-1dx≤c∫Ω1u-v2dx1/q2,where q2=max{2(p(x)-1)/p(x)} or min{2(p(x)-1)/p(x)} according to (iii) of Lemma 5.
Combining (48) with (46), we have(49)∬Qτsgixbu-bvu-vxiϕdxdt≤c∫τs∫Ωu-v2dxdtl,where l<1.
Once more,(50)∬Qτsbu-bvϕu-vgxiidxdt≤c∫τs∫Ωu-v2dxdt.
Letλ→0 in (42). By (44), (45), (49), and (50), we have(51)∫Ωux,s-vx,s2dx≤∫Ωux,0-vx,02dx+c∫τs∫Ωu-v2dxdt+c∫τs∫Ωu-v2dxdtl.Let κ(s)=∫Ω‍|u(x,s)-v(x,s)|2dx. Without loss of the generality, we may assume that there exist τ∈[0,T) and κ(τ)>0. Then for any s>τ, ∫τs‍k(t)dt>0. If we denote that(52)τ0=sup⁡t∈τ,s,κt>0,∫ττ0ktdt=c1,then τ<τ0≤s and(53)∫τsktdt≥∫ττ0ktdt=c1.By u,v∈L∞(QT), there exists a constant C>0 such that(54)c∫τsktdtl∫τsktdt≤c∫τsktdtlc1≤C=Cc,c1,T,q.By (51) and (54), we have(55)κs-κτ≤C+c∫τsktdt,and using the Gronwall inequality, we have(56)∫Ωux,s-vx,s2dx≤c∫Ωux,τ-vx,τ2dx,where c depends on C. Thus, we have (57)∫Ωux,s-vx,s2dx≤c∫Ωu0x-v0xdx.The proof is complete.


## 4. The Partial Boundary Value Condition

Consider the linear degenerate equation(58)∂u∂t-divax∇u-∑i=1NfixDiu=0,which is a particular case of (6) (where p=2, gi=fi,b(s)=s). Rewrite it as(59)∂u∂t-axΔu-∑i=1Naxix+fixDiu=0.According to the Fichera-Oleinik theory [20, 21], besides the initial value condition (2), since ax∂Ω=0, the partial boundary, where we should impose the boundary value condition, is(60)Σ1=x∈∂Ω:∑i=1Nfixnix<0,where n→={ni} is the inner normal vector of ∂Ω.By reviewing the formula of (8), One can see that (61)Σ1⊆Σp⊆∂Ω.By this token, condition (13) in Theorem 2 is reasonable.



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