# On the Weak Characteristic Function Method for a Degenerate Parabolic Equation

Authors: Huashui Zhan
Journal: Journal of Function Spaces
 (2019)

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## Abstract

For a nonlinear degenerate parabolic equation, how to impose a suitable boundary value condition to ensure the well-posedness of weak solutions is a very important problem. It is well known that the classical Fichera-Oleinik theory has perfectly solved the problem for the linear case, and the optimal boundary value condition matching up with a linear degenerate parabolic equation can be depicted out by Fechira function. In this paper, a new method, which is called the weak characteristic function method, is introduced. By this new method, the partial boundary condition matching up with a nonlinear degenerate parabolic equation can be depicted out by an inequality from the diffusion function, the convection function, and the geometry of the boundary∂Ω itself. Though, by choosing different weak characteristic function, one may obtain the differential partial boundary value conditions, an optimal partial boundary value condition can be prophetic. Moreover, the new method works well in any kind of the degenerate parabolic equations.


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## Body

## 1. Introduction

For the earliest movement differential equation of a particle(1)dxdt=ft,x,the initial value (2)x0=x0is the initial position of the particle. For a second order ordinary differential equation (3)d2xdt2=ft,x,dxdt,if we regard it as a its accelerated speed differential equation, we should impose the initial value conditions as (4)x0=x0,dxdtx=0=v0,where v0 is the initial velocity. If we regard it as describing the motion of a vibrating string, we should impose the boundary value conditions (5)x0=0,x1=0,which implies that the two ends of the string are fixed at x=0 and x=1. Or even one can impose the following boundary value condition: (6)x0=x1-x0.5=0,which is called three points boundary value problem. Theoretically, all these conditions are called definite conditions. In other words, in order to solve an explicit differential equation, it is important to find a suitable definite condition. For example, considering the well-known heat conduction equation (7)ut=Δu,x,t∈Ω×0,T,besides the initial value (8)ux,0=u0x,x∈Ω,where u0 is the initial temperature, one of the following boundary value conditions should be imposed.(i) Dirichlet condition(9)ux,t=0,x,t∈∂Ω×0,T.(ii) Neumann condition(10)∂u∂n=0,x,t∈∂Ω×0,T,where n is the outer normal vector of Ω.(iii) Robin condition(11)∂u∂n+ku=0,x,t∈∂Ω×0,T,where k is a positive constant.But, if one considers the degenerate heat conduction equation(12)ut=divax,t∇uwhere a(x,t)≥0, or nonlinear heat conduction equation (13)ut=divkx,t,u∇u,where k(x,t,u)≥0, the above three boundary value conditions may be overdetermined. While, for a hyperbolic-parabolic mixed type equation (14)ut=divkx,t,u∇u+divb→u,in order to obtain the uniqueness of weak solution, besides one of the above three boundary value conditions is imposed, the entropy condition should be added additionally. In a word, for a degenerate parabolic equation, how to impose a suitable partial boundary value condition to ensure the well-posedness of weak solutions has been an interesting and important problem for a long time. Let us give a basic review of the history.First studied by Tricomi and Keldyš and later by Fichera and Oleiˇnik, the general theory of second order equation with nonnegative characteristic form, which, in particular, contains those degenerating on the boundary had been developed and perfected [1] about in 1960s. By this theory, if one wants to consider the well-posedness problem of a linear degenerate elliptic equation (15)∑r,s=1N+1arsx∂2u∂xr∂xs+∑r=1N+1brx∂u∂xr+cxu=fx,x∈Ω~⊂RN+1,only a partial boundary value condition is required. In detail, let {ns} be the unit inner normal vector of ∂Ω~ and denote that (16)Σ2=x∈∂Ω~:arsnrns=0,br-axsrsnr<0,Σ3=x∈∂Ω~:arsnsnr>0.Then, the partial boundary value condition is (17)uΣ2∪Σ3=0.In particular, if the matrix (ars) is positive definite, (15) is the classical elliptic equation and (17) is just the usual Dirichlet boundary condition.If the matrix(ars) is semipositive definite, the most typical is the linear degenerate parabolic equation (18)∂u∂t=∑r,s=1Naijx∂2u∂xi∂xj+∑i=1Nbix∂u∂xi+cxu-fx,x,t∈QT=Ω×0,T.To study the well-posedness problem of (18), in addition to the initial value condition (19)ux,0=u0x,x∈Ω,a partial boundary value condition should be imposed (20)ux,t=0,x,t∈Σp×0,T,where (21)Σp=x∈∂Ω:aijninj>0∪x∈∂Ω:aijninj=0,bi-axjijni<0.Now, if one considers the well-posedness problem of a nonlinear degenerate parabolic equation, it is naturally to conjecture that only a partial boundary value condition should be imposed. For example, considering the nonlinear parabolic equation(22)∂u∂t=divax,t∇upx-2∇u,x,t∈QT,with (23)ax,t=0,x,t∈∂Ω×0,T,if u1(x,t)∈BV(QT),u2(x,t)∈BV(QT) are two weak solutions of (22) with the initial values u1(x),u2(x), respectively, then it is easily to show that (24)∫Ωu1x,t-v2x,t2dx≤c∫Ωu1x-u2x2dx,t∈0,T,even without any boundary value condition. In other words, for a general nonlinear degenerate parabolic equation, (25)∂u∂t=divax,t,u,∇u∇u+fx,t,u,∇u,though we can expect that only a partial boundary value condition like (20) is enough to ensure the stability of weak solutions (or uniqueness of weak solution), since Fichera-Oleinik theory is invalid, if we insist on the partial boundary value condition (20) is still imposed in the sense of the trace, then it is difficult to assign the geometry of the partial boundary Σp appearing in (20). In this paper, we will try to find a new method to solve this problem. For the sake of convenience, we can call the new method as the weak characteristic function method. We first introduce the related definitions.Definition 1.
Ifg(x) is a nonnegative continuous function in RN, when x is near to the boundary ∂Ω, g(x) is a C2 function and satisfies (26)∂Ω=x∈RN:gx=0,Ω=x∈RN:gx>0,then we say g(x) is a weak characteristic function of Ω.Only ifΩ is with a C2 smooth boundary, the distance function d(x)=dist(x,∂Ω) is a weak characteristic function of Ω, and its square d2 is another weak characteristic function of Ω. Certainly, if u0(x) is a continuous function with u0(x)|x∈∂Ω=0, then the function a(u0(x))+d(x) also is a weak characteristic function of Ω.Definition 2.
By the weak characteristic function method it means that one can find the explicit geometric expression ofΣp in the partial boundary value condition (20) by choosing a suitable test function related to a weak characteristic function of Ω.We will choose two special nonlinear parabolic equations of (25) to verify the new method. The first one is (27)∂u∂t=ΔAu+divbu,x,t∈QT,where Ω⊂RN is an open bounded domain, b(u)={bi(u)}, and (28)Au=∫0uasds,as≥0.The second type is the evolutionary p(x)-Laplacian equation similar to (22) (see below please). We will introduce the backgrounds of these two kinds of equations, respectively.Equation (27) arises from heat flow in materials with temperature dependent on conductivity, flow in a porous medium, the conservation law, the one-dimensional Euler equation, and the boundary layer theory. It is with hyperbolic-parabolic mixed type and might have discontinuous solution. For the Cauchy problem, the well-posedness theory has been established perfectly, one can refer to [2–10] and the references therein. For the initial-boundary value problem, also there are many important papers devoting to its well-posedness problem; one can see [11–16] and the references therein. However, unlike the Cauchy problem, how to impose a suitable boundary value condition to match up with (27) has been an interesting and difficult problem for a long time. Actually, for the completely degenerate case, i.e., A≡0, (25) becomes a first order hyperbolic equation, and it is well known that a smooth solution is constant along the maximal segment of the characteristic line in QT. When this segment intersects both {0}×Ω and ∂Ω, then the usual boundary value condition (29)ux,t=0,x,t∈∂Ω×0,T,is overdetermined if (27) is fulfilled in the traditional trace sense. Thus one needs to work within a suitable framework of entropy solutions and entropy boundary conditions. In the BV setting, the authors of [11] gave an interpretation of the boundary condition (29) as an entropy inequality on ∂Ω, which is the so-called BLN condition. However, since the trace of solutions is involved in the formulation of the BLN condition, it makes no sense if the solution is merely in L∞. The author of [12] extended the Dirichlet problem for hyperbolic equations to the L∞ setting and proved the uniqueness of the entropy solution by introducing an integral formulation of the boundary condition. This idea had been generalized to deal with the strongly degenerate parabolic equations [13–16], in which the boundary condition is not directly shown as (27) in sense of the trace but is implicitly contained in a family of entropy inequalities.If we still comprehend the boundary value condition is true in the sense of the trace, when the domainΩ=R+N is the half space of RN, in our previous work [17], we probed the initial-boundary value problem of (27) in the half space R+N×(0,T). We have proved that if bN′(0)<0, we can give the general Dirichlet boundary condition (30)ux,t=0,x,t∈∂R+N×0,T.But if bN′(0)≥0, then no boundary condition is necessary, and the solution of the equation is free from any limitation of the boundary condition.WhenΩ is a bounded smooth domain, in [18], by the parabolically regularized method, we had proved the existence of the entropy solution [18], but we could not obtain the stability based on the partial boundary value condition (20). At that time, we could not find a valid way to depict out the geometric expression of Σp in (20).The first discovery of this paper is that, by the weak characteristic new method, we find that the partial boundary value condition (20) admits the form as (31)Σp=x∈∂Ω:Δg+γ∑i=1Ngxi≥0,where the constant γ satisfies (32)biu-biv≤γu-v,and when x is near to the boundary ∂Ω, g(x) is a weak characteristic function of Ω.For example,N=2, if the domain Ω is the disc D1={(x,y):x2+y2<1}, a weak characteristic function can be chosen as g(x)=1-(x2+y2), (33)gx=-2x,gy=-2y,Δg=-4then (34)Δg+γgx+gy=-4+2γx+y,and (35)Σp=x∈∂D1:γx+y≥2,which implies that if γ≤1, then Σp=∅; if 1<γ<2, Σp is a proper subset of ∂Ω; if γ≥2, then Σp=∂Ω.It is well known and very important in applications that the boundary conditions usually stand for some physical meanings. At least from my own perspective, if we regard (27) as a nonlinear heat conduction (or heat diffusion) process, then Σp=∅ means that u(x,t)=0 occurs before x attains the boundary value ∂Ω.From mathematical theory, the partial boundary value condition (20) with the form as (27) is just as a definite condition. Since condition (31) includes g(x) and γ, we can say condition (31) is determined by the degeneracy of a, the weak characteristic function of Ω, and the first order derivative term in a special sense; this fact seems more or less likely to that (21). We will prove the stability of the entropy solutions to (27) under the partial boundary value condition (20) with expression (31).The second degenerate parabolic equation considered in this paper is(36)ut=divax∇upx-2∇u+divbu,which comes from a new kind of fluids: the so-called electrorheological fluids (see [19, 20]). If a(x)≡1, this kind of equations has been researched widely recently. One can refer to [21–29], etc. If a(x)≡1 and p(x)=p are constant, (36) is the well-known non-Newtonian fluid equation [10]. If a(x) is a C1(Ω¯) function, p(x)=p; the author of [30] considered the nonlinear equation (37)∂u∂t-divax∇up-2∇u-fixDiu+cx,tu=0,and made important progress on its study. 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. On the strongly degenerate boundary, they formulated a new approach to prescribe the boundary value condition rather than defining the Fichera function as treating the linear case. Moreover, they formulated the boundary value condition on this strongly degenerate boundary in a much weak sense since the regularity of the solutions much weaker near this boundary.In this paper, we assume thata(x)∈C1(Ω¯) satisfies condition (38)ax>0,x∈Ω;ax=0,x∈∂Ω,and bi(s) is a C1 function on R. The second discovery of this paper is that, by choosing a(x) as the weak characteristic function of Ω, we deduce that Σp can be depicted out by (39)Σp=x∈∂Ω:∑i=1Naxix≠0.By (39), we can prove the stability of the entropy solutions of (36) under the partial boundary value condition (20) with the expression (38).Let us give a simple summary. For a nonlinear degenerate parabolic equation, to the best knowledge of the author, there are three ways to deal with the boundary value condition. The traditional way is to comprehend (29) (also (20)) in the sense of the trace as in [2, 4, 10, 17, 18, 31]. The second way, the boundary value condition (29) is understood in weaker sense than the trace and is elegantly implicitly contained in family entropy inequalities [11–16]. In this way, if the equation is completely degenerate, then the boundary value condition is replaced by BLN condition. Moreover, in [12–16], the entropy solutions are in L∞ space, the existence of the traditional trace on the boundary is not guaranteed, and it is impossible to depict out Σp in a geometric way. The third way, the boundary value condition (29) is decomposed into two parts; on one part (the nondegenerate part and the weak degenerate part in [30]) the boundary value condition is true in the sense of trace, while on the other part (the strongly degenerate part in [30]), the boundary value condition is true in a much weaker sense than the trace. In this paper, we still use the traditional way to deal with the boundary value condition. The most innovation lies in the fact that if one chooses the different weak characteristic function ϕ(x) of Ω, then one obtains the different partial boundary value condition (40)ux,t=0,x,t∈Σϕ×0,T,where Σϕ⊆∂Ω depends on ϕ(x). Thus, we can predict that the optimal partial boundary value condition matching up with a nonlinear degenerate parabolic equation should have the form (41)ux,t=0,x,t∈Σϕ×0,T,with that (42)Σ=⋂ϕΣϕ.But we can not prove this conjecture for the time being.


## 2. Main Results

For smallη>0, let (43)Sηs=∫0shητdτ,hηs=2η1-sη+.Obviously hη(s)∈C(R), and (44)hηs≥0,shηs≤1,Sηs≤1;limη→0Sηs=sgns,limη→0sSη′s=0.Definition 3.
A functionu is said to be the entropy solution of (27) with the initial value condition (19), if
(1)u satisfies (45)u∈BVQT∩L∞QT,∂∂xi∫0uasds∈L2QT.
(2) For anyφ∈C02(QT), φ≥0, for any k∈R, for any small η>0, u satisfies (46)∬QTIηu-kφt-Bηiu,kφxi+Aηu,kΔφ-Sη′u-k∇∫0uasds2φdxdt≥0.
(3) The initial value is true in the sense of(47)limt→0∫Ωux,t-u0xdx=0.
(4) If the partial boundary value condition (20) is true in the sense of the trace, then we say u is the solution of (27) with the initial-boundary value conditions (19) and (20).Here the pairs of equal indices imply a summation from1 up to N, and (48)Bηiu,k=∫kubi′sSηs-kds,Aηu,k=∫kuasSηs-kds,Iηu-k=∫0u-kSηsds.On one hand, if (27) has a classical solution u, multiplying (27) by φSη(u-k) and integrating over QT, we are able to show that u satisfies Definition 3.On the other hand, letη→0 in (46). We have (49)∬QTu-kφt-sgnu-kbiu-bikφxidxdt+∬QTsgnu-kAu-AkΔφdxdt≥0.Thus if u is the entropy solution in Definition 3, then u is a entropy solution defined in [2, 10], etc.The existence of the entropy solution in the sense of Definition3 can be proved similar to Theorem 2.3 in [18]; we omit the details here.Theorem 4.
Suppose thatA(s) is a C2(R) function and bi(s) is a C1(R) function; u and v are two solutions of (27) with the different initial values u0(x), v0(x)∈L∞(Ω), respectively. If u and v are with the same homogeneous partial boundary value condition (20), then (50)∫Ωux,t-vx,tdx≤∫Ωu0x-v0xdx.Definition 5.
A functionu(x,t) is said to be a weak solution of (36) with the initial value (18), provided that (51)u∈L∞QT,ut∈L2QT,ax∇upx∈L1QT,and for any function φ1∈C01(QT) and φ2∈L∞0,T;Wloc1,p(x)(Ω) there holds (52)∬QT∂u∂tφ1φ2+ax∇upx-2∇u∇φ1φ2+biuφ1φ2xidxdt=0.The initial value (18) is satisfied in the sense of (47). If the partial boundary value condition (20) is true in the sense of the trace, then we say u is the solution of (36) with the initial-boundary value conditions (19) and (20).Here,W1,p(x)(Ω) is the variable exponent Sobolev space [23]. Suppose that p-=minx∈Ω¯>1, a(x) satisfies (38), and bi(s) is a C1 function on R. If (53)u0x∈L∞Ω,axu0x∈W1,pxΩ,and there are some other restrictions in a(x) and bi(x), in a similar way as that of Theorem 2.5 of [32], we can prove the existence of a weak solution of (36) with the initial value (19) in the sense of Definition 5. We omit the details here. We mainly pay attentions to the stability of the weak solutions.According to Lemma 3.2 of [32], if (54)∫Ωax-1/px-1dx<∞,i=1,2,…,N,then (55)∫Ω∇udx<∞,and u can be defined the trace on the boundary ∂Ω. If the homogeneous boundary value condition (29) is imposed, the stability can be established in a way analogous to the one of the evolutionary p-Laplacian equation [10]. In this paper, we will use the weak characteristic function method to prove the following stability theorems based on the partial boundary value condition (20).Theorem 6.
Letu(x,t) and v(x,t) be two weak solutions of (36) with the initial values u0(x) and v0(x) respectively, with the same partial boundary value condition (56)ux,t=vx,t=0,x,t∈Σp×0,T.If a(x) satisfies (38) and (54), bi(s) is a Lipschitz function, and (57)1ηp+-1/p-∫Ω∖Dη∇apxdx1/p-≤c,then the stability (50) is true, where Σp has the form as (39), (58)Dη=x∈Ω:ax>ηfor the sufficiently small η.The last but not least, we would like to suggest that the weak characteristic function method introduced in this paper can be widely used to study the boundary value problem of any kind of the degenerate parabolic or hyperbolic equations.


## 3. The Proof of Theorem4

LetΓu be the set of all jump points of u∈BV(QT),v be the normal of Γu at X=(x,t), and u+(X) and u-(X) be the approximate limits of u at X∈Γu with respect to (v,Y-X)>0 and (v,Y-X)<0, respectively. For the continuous function p(u,x,t) and u∈BV(QT), we define (59)p^u,x,t=∫01pτu++1-τu-,x,tdτ,which is called the composite mean value of p. For a given t, we denote Γut, Ht, (v1t,…,vNt) and u±t as all jump points of u(·,t), Housdorff measure of Γut, the unit normal vector of Γut, and the asymptotic limit of u(·,t), respectively. Moreover, if f(s)∈C1(R), u∈BV(QT), then f(u)∈BV(QT) and (60)∂fu∂xi=f′^u∂u∂xi,i=1,2,…,N,N+1,where xN+1=t.Lemma 7.
Letu be a solution of (27). Then in the sense of Hausdorff measure HN(Γu), we have (61)as=0,s∈Iu+x,t,u-x,ta.e.onΓu,where I(α,β) denotes the closed interval with endpoints α and β.This lemma can be proved in a similar way as described in [9]; we omit the details here.Proof of Theorem4.
Letu,v be two entropy solutions of (27) with initial values (62)ux,0=u0x,vx,0=v0x.
By Definition3, for φ∈C02(QT), we have (63)∬QTIηu-kφt-Bηiu,kφxi+Aηu,kΔφdxdt-∬QTSη′u-k∇∫0uasds2φdxdt≥0,(64)∬QTIηv-lφτ-Bηiv,lφyi+Aηv,lΔφdydτ-∬QTSη′v-l∇∫0vasds2φdydτ≥0.
Letφ=ψ(x,t,y,τ)=ϕ(x,t)jh(x-y,t-τ), where ϕ(x,t)≥0, ϕ(x,t)∈C0∞(QT), and (65)jhx-y,t-τ=ωht-τΠi=1Nωhxi-yi,(66)ωhs=1hωsh,ωs∈C0∞R,ωs≥0,ωs=0,ifs>1,∫-∞∞ωsds=1.
We choosek=v(y,τ),l=u(x,t), and φ=ψ(x,t,y,τ) in (63) and (64) and integrate it over QT. It yields (67)∬QT∬QTIηu-vψt+ψτ-Bηiu,vψxi+Bηiv,uψyidxdtdydτ+∬QT∬QTAηu,vΔxψ+Aηv,uΔyψdxdtdydτ-∬QT∬QTSη′u-v∇x∫0uasds2+∇y∫0vasds2ψdxdtdydτ≥0.Here Δx is the usual Laplacian operator corresponding to the variable x, and ∇x is the gradient operator corresponding to the variable x.
By the basic relations(68)∂jh∂t+∂jh∂τ=0,∂jh∂xi+∂jh∂yi=0,i=1,⋯,N;∂ψ∂t+∂ψ∂τ=∂ϕ∂tjh,∂ψ∂xi+∂ψ∂yi=∂ϕ∂xijh,using Lemma 7, just by the same calculations as in the proof of Theorem 2.4 in [18], letting η→0,h→0 in (67), we can deduce that (69)∬QTux,t-vx,tϕt-sgnu-vbiu-bivϕxidxdt+∬QTAu-AvΔϕdxdt≥0.If we let (70)ϕx,t=ηtξx,where η(t)∈C0∞(0,T) and ξ(x)∈C0∞(Ω), then (71)∬QTux,t-vx,tηtξx-sgnu-vbiu-bivηtξxidxdt+∬QTAu-AvηtΔξdxdt≥0.For 0<τ<s<T, we choose (72)ηt=∫τ-ts-tαϵσdσ,ϵ<minτ,T-s,where αϵ(t) is the kernel of mollifier with αϵ(t)=0 for t∉(-ϵ,ϵ).
By (71), since |bi(u)-bi(v)|≤γ|u-v|, we have (73)∫Ωux,s-vx,sξxdx≤∫Ωux,τ-vx,τξxdx+∫sτ∫Ωu-vΔξ+γ∑i=1Nξxidxdt.For any small λ>0, we choose (74)ξx=1,gx>λ,1-gx-λ2λ2,0≤gx⩽λ,where g(x) is a weak characteristic function of Ω. Then (75)ξxi=-2gx-λλ2gxi,Δξ=-2λ2∇g2-2g-λλ2Δg.By (73), we have (76)∫Ωux,s-vx,sξxdx≤∫Ωux,τ-vx,τξxdx+∫sτ∫Ωλu-vΔξ+∑i=1Nγξxidxdt,where Ωλ={x∈Ω:g(x)<λ}.
Then, since-2(g-λ)/λ2>0 in Ωλ, we have (77)∫Ωux,s-vx,sξxdx≤∫Ωux,τ-vx,τξxdx+∫sτ∫Ωλu-v-2g-λλ2Δg+γ∑i=1N2gx-λλ2gxidxdt=∫Ωux,τ-vx,τξxdx+∫sτ∫Ωλu-v-2g-λλ2Δg+γ∑i=1Ngxidxdt≤∫Ωux,τ-vx,τξxdx+cλ∫sτ∫Ωλu-vΔg+γ∑i=1Ngxidxdt.
Since by (20) and (31), (78)limλ→01λ∫Ωλu-vΔg+γ∑i=1Ngxidx=∫∂Ωu-vΔg+γ∑i=1NgxidΣ=∫Σp∪Σp′u-vΔg+γ∑i=1NgxidΣ≤∫Σpu-vΔg+γ∑i=1NgxidΣ=0.Accordingly, letting λ→0, we have (79)∫Ωux,s-vx,sdx≤∫Ωux,τ-vx,τdx.Let τ→0. Then (80)∫Ωux,s-vx,sdx≤∫Ωux,0-vx,0dx.
Theorem4 is proved.


## 4. The Proof of Theorem6

LetW1,p(x)(Ω) be the variable exponent Sobolev space. One can refer to [22–24] for the following lemma.Lemma 8.
(i) The spaceLp(x)(Ω), W1,p(x)(Ω), and W01,p(x)(Ω) are reflexive Banach spaces.
(ii)p(x)-Hölder’s inequality. Let q1(x) and q2(x) be real functions with 1/q1(x)+1/q2(x)=1 and q1(x)>1. Then, the conjugate space of Lq1(x)(Ω) is Lq2(x)(Ω). For any u∈Lq1(x)(Ω) and v∈Lq2(x)(Ω), (81)∫Ωuvdx≤2uLq1xΩvLq2xΩ.
(iii)(82)IfuLpxΩ=1,then∫Ωupxdx=1.IfuLpxΩ>1,thenupLpxΩ-≤∫Ωupxdx≤upLpxΩ+.IfuLpxΩ<1,thenupLpxΩ+≤∫Ωupxdx≤upLpxΩ-.
(iv) Ifp1(x)≤p2(x), then (83)Lp1xΩ⊃Lp2xΩ.
(v) Ifp1(x)≤p2(x), then (84)W1,p2xΩ↪W1,p1xΩ.
(vi)p(x)-Poincarés inequality. If p(x)∈C(Ω), then there is a constant C>0, such that (85)uLpxΩ≤C∇uLpxΩ,∀u∈W01,pxΩ.This implies that ∇uLp(x)(Ω) and uW1,p(x)(Ω) are equivalent norms of W01,p(x)(Ω).In order to prove Theorem6, we let g(x) be a weak characteristic function of Ω and define Sη(s), hη(s), and Iη(s) as in Section 2.Theorem 9.
Letu(x,t) and v(x,t) be two weak solutions of (36) with the initial values u0(x) and v0(x), respectively, and with the same partial boundary value condition (86)Σp=x∈∂Ω:∑i=1Ngxix≠0.If bi(s) is a Lipschitz function, a(x) satisfies (38), (87)η-p+/p-∫Ω∖Dηax∂g∂xipxdx1/p-≤c,i=1,2,…,N,then there holds (88)∫Ωux,t-vx,tdx≤c∫Ωu0x-v0xdx,t∈0,T,where Dη=x∈Ω:g(x)>η and g(x) is a weak characteristic function of Ω.Proof.
For any given weak characteristic functiong(x), we define (89)φηx=1ηgx,gx<η,1,gx≥η,where η is a positive constant small enough.
In view of the definition of weak solution, by a process of limit, lettingφ1=χs,tφn(x) and φ2=gn(u-v), we can choose χs,tφηSη(u-v) as the test function, where [s,t]⊆(0,T), and χs,t is its characteristic function. Then we have (90)∫st∫ΩφηxSηu-v∂u-v∂tdxdt+∫st∫Ωax∇upx-2∇u-∇vpx-2∇v∇u-vhηu-vφηxdxdt+∫st∫Ωax∇upx-2∇v-∇vpx-2∇vSηu-v∇φηdxdt+∑i=1N∫st∫Ωbiu-biv·u-vxihηu-vφηxdxdt+∑i=1N∫st∫Ωbiu-biv·Sηu-vφηxixdxdt=0.As n→∞, we have (91)limη→0∫τs∫ΩφηxSηu-v∂u-v∂tdxdt=limη→0∫τs∫Ω∂φηxIηu-v∂tdxdt=limη→0∫ΩφηxIηu-vx,s-Iηu-vx,τdx=∫Ωu-vx,sdx-∫Ωu-vx,τdx.
Denote(92)Dη=x∈Ω:gx>ηandqx=pxpx-1.Note that |φηxi|=1/η|gxi| and x∈Ω∖Dη. We may assume that (93)1ηax1/pxSηu-vgxiLpxΩ∖Dη>1,without loss the generality. Using (ii) of Lemma 8, we have (94)1ηax1/pxSηu-vgxiLpxΩ∖Dη≤1ηax1/pxgxiLpxΩ∖Dη≤∫Ω∖Dηax1ηpx∂g∂xipxdx1/p-≤1ηp+/p-∫Ω∖Dηax∂g∂xipxdx1/p-≤c.We further have (95)∫Ωax∇upx-2∇u-∇vpx-2∇v∇φηSηu-vdx=∫Ω∖Dηax∇upx-2∇u-∇vpx-2∇v∇φηSηu-vdx≤axpx-1/px∇upx-1+∇vpx-1LqxΩ∖Dη·1ηax1/pxSηu-v∇gLpxΩ∖Dη≤c∫Ω∖Dηax∇upxdx1/q1+∫Ω∖Dηax∇upxdx1/q1,which goes to zero as n→0, where q1 is taken to be q- (or q+) if (96)∇upx-1LqxΩ∖Dη>1or≤1.
Consider the convection term(97)∫x∈Ω:u-v<ηax1/1-pxbiu-bivu-vpx/px-1dx⩽c∫Ωax1/1-pxdx⩽c,since bi is a Lipschitz function.
If{x∈Ω:|u-v|=0} is a set with measure zero, it has (98)limη→0∫x∈Ω:u-v<ηax1/1-pxdx=∫x∈Ω:u-v=0ax1/1-pxdx=0.
If the set{x∈Ω:|u-v|=0} has a positive measure, due to the fact that a(x)|∇|p(x), a(x)|∇v|p(x)∈L1(QT), we have (99)limη→0∫x∈Ω:u-v<ηax∇u-vpxdx=∫x∈Ω:u-v=0ax∇u-vpxdx=0.
According to (81), when s⩽η, it has (100)sSη′s⩽c.By Lemma 8, we have (101)∫x∈Ω:u-v<ηφηbiu-bivhηu-vu-vxidx⩽c∫x∈Ω:u-v<ηbiu-bivu-vu-vxidx≤cax-1/pxbiu-bivu-vLqxΩηax1/pxu-vxiLpxΩη⩽c∫x∈Ω:u-v<ηax1/1-pxbiu-bivu-vpx/px-1dx1/q1·∫x∈Ω:u-v<ηaxu-vxipxdx1/p1,where Ωη=x∈Ω:|u(x,t)-v(x,t)|<η, q1 is taken to be q- (or q+) according to (iii) of Lemma 8, and p1 is p- (or p+) similarly.
By (97)-(99) and (101), we have (102)limη→0∫Ωbiu-bivφηxhηu-vu-vxidx=0.
Meanwhile,(103)limη→0∫Ωbiu-bivSηu-vφηxixdx≤climη→01η∫Ω∖Dηu-v∑i=1N∂g∂xidx≤c∫Σpu-vdx=0.
Letη→0 in (90). Using (91) and (95) in combination with (102)-(103), we obtain (104)ddtu-vL1Ω⩽0and thus arrive at (105)∫Ωux,t-vx,tdx⩽∫Ωu0x-v0xdx,∀t∈0,T.Proof of Theorem6.
If we choose the weak characteristic functiong(x)=a(x), then the part of the boundary Σp of (39) is just the same as (86) and condition (57) is just the same as (87). Theorem 6 follows from Theorem 9 directly.



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*Source: 9040284-2019-08-26.xml*