# On the Non-Newtonian Fluid Equation with a Source Term and a Damping Term

Authors: Huashui Zhan; Yongping Li
Journal: Journal of Function Spaces
 (2018)

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## Abstract

A kind of non-Newtonian fluid equation with a damping term and a source term is considered. After giving a result of the existence, if the diffusion coefficient is degenerate on the boundary, the local stability of the weak solutions is established without any boundary condition. If the diffusion coefficient is degenerate on a part of the boundary, by imposing the homogeneous value condition on the other part of the boundary, the local stability of the weak solutions is proved. Moreover, if the equation is with a damping term, other than the finite propagation property, the results of this paper reveal the essential differences between the non-Newtonian fluid equation and the heat conduction equation in a new way.


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## Body

## 1. Introduction

Consider the parabolic equation(1)∂u∂t=divax∇up-2∇u-bx,t∇u2+fu,x,t,x,t∈Ω×0,T,related to the p-Laplacian, with the initial value(2)ux,0=u0x,x∈Ω,and the usual boundary value(3)ux,t=0,x,t∈∂Ω×0,T,where a(x)∈C(Ω¯) and a(x)≥0, b(x,t)∈C(QT¯), f(s,x,t) is a continuous function, Ω⊂RN is a bounded domain with a smooth boundary ∂Ω, p>1, q>0. The equation comes from a host of applied fields such as the theory of non-Newtonian fluid, the water infiltration through porous media, and the oil combustion process; one can refer to [1–4] and the references therein. For the evolutionary p-Laplacian equation(4)∂u∂t=div∇up-2∇u,x,t∈Ω×0,T,and with the initial-boundary value conditions (2) and (3), the weak solution is unique and has finite propagation property [3]. However, the damping term and the source term in (1) may change the situation.Bertsh et. al. [5] and Zhou et. al. [6] had discussed the existence and the properties of the viscosity solutions for the equation(5)ut=u△u-γ∇u2,and shown that the uniqueness of the weak solution is not true, where γ is a positive constant. Zhang et. al. [7] had discussed the existence and the properties of the viscosity solution for the equation(6)ut=△u-gxuq-1∇u2,and shown that the uniqueness of the weak solution is not true, where g(x)≥0 and at least there exists a point x0∈Ω such that g(x0)>0, q≥1.Meanwhile, Jirˇi´ Benedikt et. al. [8, 9] had shown that the uniqueness of the solution of the following equation is true(7)ut=div∇up-2∇u+qxuα-1u,x,t∈x,t∈Ω×0,T,is not true provided that 0<α<1, q(x)≥0 and at least there exists a point x0∈Ω such that q(x0)>0.In this paper, we first assume that(8)ax=0,x∈∂Ω,ax>0,x∈Ω,and then (1) is degenerate on the boundary. Such a degeneracy may have a substantial influence on the solutions. If one considers the well-posedness of (1), one expects that such a degeneracy may counteract the effects from the damp term and the source term. A typical example is the equation (9)ut=divdαx∇up-2∇u,x,t∈Ω×0,T,which was studied by Yin-Wang [10, 11]. Here, d(x)=dist(x,∂Ω) is the distance function from the boundary and satisfies (8). Yin-Wang showed that, if α>p-1, although the weak solution may lack the regularity to be defined and the trace on the boundary and the boundary value condition (3) cannot be imposed in the trace sense, the uniqueness of the weak solution is still true. Moreover, the author had studied the equation (10)ut=divax∇up-2∇u,x,t∈Ω×0,T,and shown that condition (8) may act as the role as the boundary value condition (3) and ensure the well-posedness of the solutions [12–15].Coming back to (1). On the one hand, based on the knowledge of (5) and (6), if a(x)=1, p=2, and b(x)≥γ>0 in (1), then the uniqueness of the solution is not true. Accordingly, in this paper, we consider the well-posedness of (1) whether p>2 or b(x)=0 on the boundary ∂Ω. On the other hand, based on the knowledge of [10–15], when p>2 and a(x) satisfies (8), we can expect that the uniqueness of the weak solution to (1) is still true, even if f(u,x,t)=q(x)uα-1u as (6).Definition 1.
A functionu(x,t) is said to be a weak solution of (1) with the initial value (2), if(11)u∈L∞QT,ut∈L2QT,ax∇up∈L1QT,and for any function φ∈C01(QT),(12)∬QTutφx+ax∇up-2∇u·∇φ+bx,t∇u2φx-fu,x,tφxdxdt=0.The initial value is satisfied in the sense(13)limt→0∫Ωux,t-u0xdx=0.Definition 2.
The functionu(x,t) is said to be the weak solution of (1) with the initial value (2) and the boundary value condition (3), if u satisfies Definition 1, and the boundary value condition (3) is satisfied in the sense of trace.Theorem 3.
Leta(x)∈C(Ω¯) satisfy (8), 0≤b(x,t)∈C(QT¯), f(s,x,t) be a Lipschitz function,(14)fs,x.t>0,ifs<0.If p>4, (15)0≤u0x∈L∞Ω,ax∇u0p∈L1Ω,and(16)∫Ωbx,t2p/p-4ax-4/p-4dx≤c,then (1) with initial value (2) has a nonnegative weak solution. Moreover, if(17)∫Ωax-1/p-1dx≤c,then the initial-boundary value problem (1), (2), (3) has a nonnegative solution in the sense of Definition 2.Sincea(x)=0 when x∈∂Ω, condition (16) implies that b(x,t)|x∈∂Ω=0; hereafter, the constants c may depend on T. We think the existence of the weak solutions can be proved only if p>2, and the condition p>4 is just a makeshift. Also condition (16) may not be necessary, but we are not ready to pay so much attentions to the existence. We will focus on the uniqueness of the weak solution.Theorem 4.
Letu(x,t),v(x,t) be two weak solutions of (1) with the initial values u0(x),v0(x), respectively. If a(x)∈C(Ω¯) satisfies (8), |b(x,t)|≤ca(x), f(s,x,t) is a continuous function, p>2, then there is a constant β≥4 such that(18)∫Ωaxβux,t-vx,tdx≤∫Ωaxβu0x-v0xdx,a.e.t∈0,T.Sincea(x) satisfies (8), Theorem 4 implies the uniqueness of the weak solution to (1) is true even without the boundary value condition. Moreover, we have the following two simple comments.(1) Theorem4 includes the case of f(u,x,t)=q(x)|u|α-1u and b(x,t)≡0; in other words, the uniqueness of the weak solution to the following equation (19)ut=divax∇up-2∇u+qxuα-1uis true, where p>2 and a(x)x∈∂Ω=0.(2) Theorem4 includes the case of f(u,x,t)≡0 and b(x,t)≡g(x)≥0 and at least there exists a point x0∈Ω such that g(x0)>0. In other words, the uniqueness of the weak solution to the following equation (20)ut=divax∇up-2∇u-gx∇u2,is true, where p>2 and a(x)|x∈∂Ω=0.Compared with (6) and (7), Theorem 4 reveals that the degeneracy of a(x) brings the new change about the property of the solutions.In order to illustrate the problem more clearly, secondly, we assume that(21)∂Ω=Σp∪Σp′,Σp¯∩Σp′¯=∅,(22)ax≥c>0,x∈Σp,(23)ax=0,x∈Σp′.In this case, we consider the uniqueness of weak solution to (1) under a partial boundary value condition. This is the following theorem.Theorem 5.
Let∂Ω satisfy (21) and u(x,t),v(x,t) be two weak solutions of (1) with the initial values u0(x),v0(x), respectively, with the same partial boundary value condition(24)ux,t=vx,t=0,x∈Σp.If p>2, b(x,t)∈C(QT¯), a(x)∈C(Ω¯) satisfies (17), (22), and (23), and f(s,x,t) is a continuous function, then there is a constant β>1 such that the local stability is true in the sense of (18).If we notice thata(x) satisfies (22), according to [5–7], the uniqueness of the solution to the equation (25)ut=Δu-gx∇u2is not true when x∈Ω is near to Σp, while Theorem 5 implies that the uniqueness of the solution to the equation (26)ut=divax∇up-2∇u-gx∇u2,is true provided that p>2. This fact shows the differences between the heat conduction equation (p=2) and the non-Newtonian fluid equation (p>2) again. It is well-known that the heat conduction equation has the infinite propagation property, while the non-Newtonian fluid equation has the finite propagation property.


## 2. The Weak Solutions Depend on the Initial Value

It is supposed thatu0 satisfies (27)0≤u0x∈L∞Ω,ax∇u0p∈L1Ω.Let uε,0(x)∈C0∞(Ω) and a(x)∇uε,0(x)p∈L1(Ω) be uniformly bounded, and let a(x)uε,0(x) converge to a(x)u0(x) in W01,p(Ω). For simplicity, we may assume that f(s,x,t) is a C1 function without loss the generality.We now consider the following regularized problem(28)uεt-divax+ε∇uε2+εp-2/2∇uε+bx,t∇uε2=fuε,x,t,x,t∈QT,(29)uεx,t=0,x,t∈∂Ω×0,T,(30)uεx,0=uε,0x,x∈Ω.Since f(s,x,t) satisfies (4), it is well-known that the above problem has a unique nonnegative classical solution [3, 16].By the maximum principle, we have(31)0≤uε⩽c.Multiplying (28) by uε and integrating it over QT, we get (32)12∫Ωuε2dx+∬QTax+ε∇uε2+εp-2/2∇uε2dxdt+∬QTbx,t∇uε2uεdxdt=12∫Ωu02dx+∬QTfuε,x,tuε.Since b(x,t)≥0, (33)∬QTbx,t∇uε2uεdxdt≥0,and by fuε,x,tuε≤c, we have(34)12∫Ωuε2dx+∬QTax+ε∇uε2+εp-2/2∇uε2dxdt⩽c,and(35)∬QTax∇uεpdxdt⩽c∬QTax+ε∇uεpdxdt⩽c.Multiplying (28) by ∂uε/∂t, integrating it over QT,(36)∬QTuεt∂uε∂tdxdt=∬QTdivax+ε∇u2+εp-2/2∇uε·∂uε∂tdxdt-∬QTbx,t∇uε2∂uε∂tdxdt+∬Qt∂uε∂tfuε,x,tdxdt.Noticing that (37)∇uε2+εp-2/2∇uε·∇∂uε∂t=12ddt∫0∇uεx,t2+εsp-2/2ds,we have(38)∬QTdivax+ε∇uε2+εp-2/2∇uε∂uε∂tdxdt=-∬QTax+ε∇uε2+εp-2/2∇uε∇∂uε∂tdxdt=-12∬QTax+εddt∫0∇uεx,t2+εsp-2/2dsdxdt=-12∫Ωax+ε∫0∇uεx,t2+εsp-2/2dsdx+12∫Ωax+ε∫0∇uεx,02+εsp-2/2dsdx.Moreover, by the Young inequality and the Hölder inequality,(39)∬QT∂uε∂tbx,t∇uε2dxdt≤34∬QTbx,t∇uε22dxdt+14∬QT∂uε∂t2dxdt≤14∬QT∂uε∂t2dxdt+c∬QTbx,t2ax-4/pp/p-4dxdtp-4/p∬QTax∇uεpdxdt4/p≤14∬QT∂uε∂t2dxdt+c.(40)∬QT∂uε∂tfuε,x,tdxdt≤c+14∬QT∂uε∂t2dxdt.By (36), (38), (39), and (40),(41)∬QT∂uε∂t2dxdt≤c.By (35), (41), we know uε can be embedded into Lloc2(QT) compactly. Then uε→u a.e. in QT. At the same time, since p>4,(42)∬QTbx,t∇uε2dxdt≤c∬QTax∇uεpdxdt2/p∬QTbx,tax-2/pp/p-2dxdtp-2/p≤c∬QTbx,tax-2/pp/p-2dxdtp-2/p=c∬QTbx,tp/p-2ax-2/p-2dxdtp-2/p≤c∬QTbx,tp/p-2ax-2/p-2p-2/24/p-4dxdtp-2/p2p-4/4p-2=c∬QTbx,t2p/p-4ax-4/p-4dxdtp-4/2p≤c.Hence, by (31), (35), (40), (41), (42) there exists a function u, n-dimensional vector function ζ→=(ζ1,…,ζn), and a function ν∈L2(QT) such that (43)u∈L∞QT,ζ→∈Lp/p-1QT,and (44)uε⇀∗u,inL∞QT,uε→u,a.e.inQT.ax∇uεp-2∇uε⇀ζ→inLp/p-1QT.bx,t∇uε2⇀ν,inMQT.Here, M(QT) is the signed Radon measures on QT. In order to prove that u satisfies (1), we notice that for any function φ∈C01(QT),(45)∬QT∂uε∂tφ+ax+ε∇uε2+εp-2/2∇uε·∇φ+bx,t∇uε2dxdt=∬QTfuε,x,tφdxdt.Now, in the first place, we can prove(46)limε→0ax+ε∇uε2+εp-2/2∇uε·∇φdxdt=limε→0∬QTax∇uεp-2∇uε·∇φdxdt=∬QTax∇up-2∇u·∇φdxdt=∬QTζ→·∇φdxdt,in a similar way to that of the usual p-Laplacian equation. Then, letting ε→0 in (45),(47)∬QT∂u∂tφ+ax∇up-2∇u·∇φ+νφdxdt=∬QTfu,x,tφdxdt.What is more, by the weak convergent theorem, for anyφ∈C01(QT),(48)limε→0∬QTax∇uεp-2∇uε·∇uφdxdt=∬QTax∇upφdxdt.For any 0≤φ∈C01(QT),(49)limε→0∬QTax∇uεp-2∇uε-∇up-2∇u·∇uεφdxdt=limε→0∬QTax∇uεp-2∇uε-∇up-2∇u∇uε-∇uφdxdt+limε→0∬QTax∇uεp-2∇uε-∇up-2∇u∇uφdxdt≥limε→0∬QTax∇uεp-2∇uε-∇up-2∇u∇uφdxdt=0.By (48),(50)limε→0∬QTax∇up-2∇u-∇uεp-2∇uε·∇uεφdxdt=limε→0∬QTax∇up-2∇u-∇uεp-2∇uε∇u-∇uεφdxdt+limε→0∬QTax∇up-2∇u-∇uεp-2∇uε∇uφdxdt≥limε→0∬QTax∇up-2∇u-∇uεp-2∇uε∇uφdxdt=0.Combining (49) with (50), we have(51)limε→0∬QTax∇uεp-2∇uε-∇up-2∇u·∇uεφdxdt=0,for any 0≤φ∈C01(QT). Clearly, for any φ∈C01(QT), (50) is still true.Then, by (48) and (51),(52)limε→0∬QTax∇uεp-∇upφdxdt=limε→0∬QTax∇uεp-2∇uε-∇up-2∇u∇uεφdxdt+limε→0∬QTax∇up-2∇u·∇uεφdxdt-∬QTax∇upφdxdt=0.By (52), by the arbitrary of 0≤φ∈C01(QT), we know that(53)limε→0∬QTax∇uεpdxdt=∬QTax∇updxdt.Thus ∇uε→∇ua.e. in QT.Sincep>4, by (53), for any function φ∈C01(QT)(54)limε→0∬QTbx,t∇uε2-∇u2φdxdt=0is clearly. Then(55)∬QTbx,t∇u2φdxdt=∬QTνφdxdt,for any function φ∈C01(QT). Combining (46) with (55), u satisfies (7).At last, we are able to prove (13) as in [17]; thus we have Theorem 3.


## 3. The Proof of Theorem4

For smallη>0, let (56)Sηs=∫0shητdτ,hηs=2η1-sη+.Obviously hη(s)∈C(R), and (57)hηs≥0,shηs≤1,Sηs≤1;limη→0Sηs=sgn⁡s,limη→0sSη′s=0.Theorem 6.
Letu(x,t),v(x,t) be two weak solutions of (1) with the initial values u0(x),v0(x), respectively. If a(x)∈C(Ω¯) satisfies (8), b(x,t)∈C(QT¯), there is a constant β>1 such that one of the following conditions is true:
( i )   p > 4;(58)∫Ωax2pβ-4/p-4bx,t2p/p-4dx≤c,(ii)2<p≤4,(59)ax2β-1/p-2bx,tp/p-2≤c,(ii)1<p≤2, and(60)axβ/2bx,t∇u2dx≤c,axβ/2bx,t∇v2dx≤c;then(61)∫Ωaxβux,t-vx,t2dx≤∫Ωaxβu0x-v0x2dx,a.e.t∈0,T.Proof.
Letu(x,t), v(x,t) be two solutions of (1) with the initial values u0(x),v0(x). We can choose a(x)β(u-v) as the test function. Then(62)∫Ωaxβu-v∂u-v∂tdx+∫Ωaxβ+1∇up-2∇u-∇vp-2∇v·∇u-vdx+∫Ωax∇up-2∇u-∇vp-2∇v·∇aβu-vdx+∫Ωbx,t∇u2-∇v2axβu-vdx=∫Ωfu,x,t-fv,x,taxβu-vdx.Thus(63)∫Ωaxβu-v∂u-v∂tdx=12ddt∫Ωaxβux,t-vx,t2dx,(64)∫Ωaxβ+1∇up-2∇u-∇vp-2∇v·∇u-vdx≥0.
By that∇ax≤c in Ω, we have(65)∫Ωax∇up-2∇u-∇vp-2∇v·∇aβu-vdx≤β∫Ωax1/p+β-1∇au-vpdx1/p∫Ωax∇up+∇vpdxp-1/p≤c∫Ωaxpβ-1+1u-vpdx1/p.
Ifp>2, since β>1, p(β-1)+1-β>0, we have(66)∫Ωaxpβ-1+1u-vpdx1/p=∫Ωaxpβ-1+1-βu-vp-2axβu-v2dx1/p≤c∫Ωaxβu-v2dx1/p.
Ifp=2, since β>1, p(β-1)+1-β>0, we have(67)∫Ωaxpβ-1+1u-vpdx1/p≤c∫Ωaxβu-v2dx1/2.
If1<p<2(68)∫Ωaxpβ-1+1u-vpdx1/p≤∫Ωaxpβ-1+1-βp/22/2-pdx2-p/2p∫Ωaxβu-v2dx1/2≤c∫Ωaxβu-v2dx1/2,only if (69)∫Ωaxpβ-1+1-βp/22/2-pdx<∞.However, this inequality is natural since β>1.
By (65)-(68), we have(70)∫Ωax∇up-2∇u-∇vp-2∇v·∇aβu-vdx≤c∫Ωaxβu-v2dxq,where q<1.
If4≥p>2, then p/p-2≥2. Since (71)ax2β-1/p-2bx,tp/p-2=axpβ-2/p-2-βbx,tp/p-2≤c, we have(72)∫Ωbx,t∇u2-∇v2axβu-vdx≤∫Ωax∇u2+∇v2p/2dx2/p∫Ωaxβ-2/pbx,tu-vp/p-2dxp-2/p≤c∫Ωaxpβ-2/p-2bx,tp/p-2u-vp/p-2dxp-2/p≤c∫Ωaxpβ-2/p-2-βbx,tp/p-2aβu-vp/p-2dxp-2/p≤c∫Ωaxβu-v2dxp-2/p.If p>4, then p/p-2<2. Since (73)∫Ωax2pβ-4/p-4bx,t2p/p-4dx≤c,and we have(74)∫Ωbx,t∇u2-∇v2axβu-vdx≤∫Ωax∇u2+∇v2p/2dx2/p∫Ωaxβ-2/pbx,tu-vp/p-2dxp-2/p≤c∫Ωaxβ-2p/p-2bx,tp/p-2u-vp/p-2dxp-2/p≤c∫Ωaxβ-2p/p-2-β/2p/p-2bx,tp/p-2axβ/2p/p-2u-vp/p-2dxp-2/p≤∫Ωaxβ-2p/p-2-β/2p/p-2bx,tp/p-22p-2/p-4dxp-4/2p-2·∫Ωaxβu-v2dx1/2≤∫Ωaxpβ-4/p-2bx,tp/p-22p-2/p-4dxp-4/2p-2·∫Ωaxβu-v2dx1/2≤∫Ωax2pβ-4/p-4bx,t2p/p-4dxp-4/2p-2·∫Ωaxβu-v2dx1/2≤c∫Ωaxβu-v2dx1/2.
If1<p≤2, by (60), we have(75)∫Ωbx,t∇u2-∇v2axβu-vdx≤c∫Ωaxβ/2u-vdx≤c∫Ωaxβu-v2dx1/2.
Moreover, sinceux,tL∞(QT)≤c, ux,tL∞(QT)≤c, f(s,x,t) is a continuous function,(76)∫Ωfu,x,t-fv,x,taxβux,t-vx,tdxdt≤c∫Ωaxβux,t-vx,tdxdt.
Now, letη→0 in (62). Then(77)ddt∫Ωaxβux,t-vx,t2dx⩽∫Ωaxβux,t-vx,t2dxq,where q≤1. This inequality implies that (78)∫Ωaxβux,t-vx,t2dx⩽c∫Ωaxβu0x-v0x2dx.Proof of Theorem4.
Sincebx,t≤ca(x), β≥4, and p>2, conditions (58) and (59) are true naturally, by Theorem 6, we have the conclusion.


## 4. The Proof of Theorem5

Lemma 7.
If∫Ωa(x)-1/p-1dx<∞, u is a weak solution of (7) with the initial condition (2). Then the trace of u on the boundary ∂Ω can be defined in the traditional way.This lemma can be found in [14]. Recall that we have assumed (21)-(23), i.e., (79)∂Ω=Σp∪Σp′,Σp¯∩Σp′¯=∅,ax≥c>0,x∈Σp,ax=0,x∈Σp′.Let φ(x) be a C1(Ω¯) function satisfying that(80)φxx∈Σp′=0,φxx∈Ω¯∖Σp′>0,and (81)Ωη=x∈Ω:φx>η.Let(82)φηx=1,ifx∈Ωη,1ηφx,ifx∈Ω∖Ωη.Then φη(x)|x∈Σp′=0.Theorem 8.
Letu(x,t),v(x,t) be two weak solutions of (1) with the initial values u0(x),v0(x), respectively, with the same partial boundary value condition (83)ux,t=vx,t,x∈Σp.If b(x,t)∈C(QT¯), a(x)∈C(Ω¯) satisfies (17), (22)-(23), and there is a constant β>1 such that one of the following conditions is true,(84)∫Ωax-1/p-1dx≤c,axpβ∇φpηp-1≤c,
( i )   p > 2;
( i i )   1 < p ≤ 2, and (60) is true;
then the local stability (18) is true.Proof.
Since∫Ωa(x)-1/p-1dx≤c, then ∫Ω∇udx<∞; accordingly, we can choose φη(x)a(x)β(u-v) as the test function. Then(85)∫Ωφηxaxβu-v∂u-v∂tdx+∫Ωaxβ+1∇up-2∇u-∇vp-2∇v·∇u-vφηxdx+∫Ωaxβ+1φηx∇up-2∇u-∇vp-2∇v·∇φηxu-vdx+∫Ωax∇up-2∇u-∇vp-2∇v·∇aβu-vφηxdx+∫Ωbx,t∇u2-∇v2axβu-vφηxdx=∫Ωfu,x,t-fv,x,taxβu-vφηxdx.At first(86)limη→0∫Ωφηxaxβu-v∂u-v∂tdx=12ddt∫Ωaxux,t-vx,t2dx,(87)∫Ωaxβ+1∇up-2∇u-∇vp-2∇v·∇u-vφηxdx≥0.In the second place, by the fact that ∇ax≤c in Ω, we have(88)limη→0∫Ωφηxax∇up-2∇u-∇vp-2∇v·∇aβu-vdx∫Ωax∇up-2∇u-∇vp-2∇v·∇aβu-vdx≤β∫Ωax1/paxβ-1∇au-vpdx1/p∫Ωax∇up+∇vpdxp-1/p≤c∫Ωaxpβ-1+1u-vpdx1/p≤c∫Ωaxβu-v2dxq.The last inequality is obtained similar to (66)-(68), where q<1. Thirdly,(89)limη→0∫Ωaxβ+1∇up-2∇u-∇vp-2∇v·∇φηxu-vdx=limη→01η∫Ω∖Ωηaxβ+1∇up-2∇u-∇vp-2∇v·∇φxu-vdx≤limη→0⁡1η∫Ω∖Ωηaxpβ+1∇φxu-vpdx1/p·∫Ω∖Ωηax∇up+∇vpdxp-1/p.Since u=v=0 when x∈Σp, and a(x) satisfies (90)axpβ∇φpηp-1≤c,axx∈Σp′=0, we have(91)limη→01η∫Ω∖Ωηaxpβ+1∇φxu-vpdx1/p≤climη→01η∫Ω∖Ωηaxpβ+1∇φpηp-1u-vpdx1/p≤climη→01η∫Ω∖Ωηaxu-vpdx1/p=c∫Σpaxu-vpdx+∫Σp′axu-vdx1/p=0.Fourthly, if p>2,(92)limη→0∫Ωbx,tφηx∇u2-∇v2axβu-vdx≤limη→0∫Ωbx,t∇u2+∇v22/pdx2/p∫Ωbx,tφηxaxβu-vp/p-2dxp-2/p≤climη→0∫Ωbx,tφηxaβu-vp/p-2dxp-2/p≤c∫Ωaxβu-v2dxq,where q<1.
If1<p≤2, by (60), we have(93)limη→0∫Ωbx,tφηx∇u2-∇v2axβu-vdx=∫Ωbx,t∇u2-∇v2axβu-vdx≤c∫Ωaxβ/2u-vdx≤c∫Ωaxβu-v2dx1/2.Last but not least,(94)limη→0∫Ωφηxfu,x,t-fv,x,taxβux,t-vx,tdx=∫Ωfu,x,t-fv,x,taxβux,t-vx,tdx≤c∫Ωaxβux,t-vx,tdx.
Now, letη→0 in (85). Then(95)ddt∫Ωaxβux,t-vx,t2dx⩽∫Ωaxβux,t-vx,t2dxq,where q≤1. This inequality implies that (96)∫Ωaxβux,t-vx,t2dx⩽c∫Ωaxβu0x-v0x2dx.



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*Source: 9689476-2018-11-25.xml*