# Diffusion Convection Equation with Variable Nonlinearities

Authors: Huashui Zhan
Journal: Journal of Function Spaces
 (2017)

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## Abstract

The paper studies diffusion convection equation with variable nonlinearities and degeneracy on the boundary. Unlike the usual Dirichlet boundary value, only a partial boundary value condition is imposed. If there are some restrictions in the diffusion coefficient, the stability of the weak solution based on the partial boundary value condition is obtained. In general, we may obtain a local stability of the weak solutions without any boundary value condition.


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## Body

## 1. Introduction

Consider the following diffusion convection equation with variable nonlinearities:(1)ut=divaxuαx∇upx-2∇u+∂biu∂xi,x,t∈QT=Ω×0,T,where Ω⊂RN is a bounded domain with smooth boundary ∂Ω and p(x) is a measurable function. Equation (1) comes from the so-called electrorheological fluids, comes from a motion of an ideal barotropic gas through a porous medium, and comes from the flows in fractured media, and so on (see [1, 2]). If ax≥a->0, one can impose the following initial-boundary value conditions: (2)ut=0=u0x,x∈Ω,(3)ux,t=0,x,t∈∂Ω×0,T,and there are many references recently. We would like to suggest that the first paper where parabolic equations with variable growth exponent are considered is Acerbi et al. [3]; some other interesting works are listed as [4–11] in our references. If a(x)=0,x∈∂Ω, then the equation is degenerate on the boundary; the author had shown that, in [12], besides the initial value condition (2), only a partial boundary value condition(4)ux,t=0,x,t∈Σp×0,T,is imposed, where Σp⊆∂Ω is a relatively open subset. In some cases, Σp=∅; then the solutions are determined by the initial value completely.Throughout the paper, we assume that1<p(x)∈C1(Ω¯) and denote (5)p+=maxΩ¯⁡px,1<p-=minΩ¯⁡px.The main aim of our paper is to study the stability based on the partial boundary value condition (4).Theorem 1.
Suppose thatα-≥1, a(x)>0 when x∈Ω, and a(x)=0 when x∈∂Ω. Let u,v be two solutions of (1) with the same partial boundary value condition (4) and with the different initial values u0, v0, respectively. If (6)∫Ωa-1/px-1x<∞,η1/p+-1∫Ω∖Ωη∇apxdx1/p+≤c,for small positive η, and (7)∫Ωax∇upxdx≤c,∫Ωax∇vpxdx≤c,then (8)∫Ωux,t-vx,tdx≤∫Ωu0-v0dx+csupx,t∈Σp×0,T⁡u-v.Here, Ωη={x∈Ω:dist(x,∂Ω)>η}.Ifp(x)≡p, ax=dγx,dx=dist(x,∂Ω), the two restrictions of a(x) in (6) are incompatible, and ∫Ωd-γ/px-1x<∞ implies γ<p-1, while η1/p+-1∫Ω∖Ωη∇apxdx1/p+≤c implies that γ≥p-1. So, if px≡p, it is impossible to obtain Theorem 1 (also Theorem 2). However, px is a continuous function; when p+>p-, the two restrictions of a(x) in (6) are compatible; for example, if a(x)=dγ(x), by (6), γ satisfies (9)p--1>γ>1+1p-1-2p+.One can see that only if p->3, p+ is large enough, and (9) is true.Ifα=0 in Theorem 1, without condition (7), conclusion (8) is true. In other words, we have the following important result.Theorem 2.
Suppose thata(x)>0 when x∈Ω and a(x)=0 when x∈∂Ω. Let u,v be two solutions of equation (10)ut=divax∇upx-2∇u+∂biu∂xi,x,t∈QT=Ω×0,T,with the same partial boundary value condition (4) and with the different initial values u0, v0, respectively. Then stability (8) is true only if a(x) satisfies (6).Theorem2 (also Theorem 1) has shown an essential difference between (1) and the usual evolutionary p-Laplacian equation. For the usual evolutionary p-Laplacian equation, to obtain the stability of the weak solutions, the Dirichlet boundary value condition (3) is necessary.In general, ifa(x) does not satisfy conditions (6), we have the following local stability.Theorem 3.
Suppose thatα>0, a(x)>0 when x∈Ω, and a(x)=0 when x∈∂Ω. Let u,v be two solutions of (1) with the initial values u0, v0, respectively. If (11)ax∇upx≤c,ax∇vpx≤c,then there exists a constant β≥2 such that (12)∫Ωaβux,t-vx,t2dx≤∫Ωaβu0-v02dxwhich implies that (1) with the initial value (3) is unique.


## 2. The Definition of the Weak Solutions

Here, the basic function spaces with variable exponents are quoted; for more details, see [13–16] et al. Set (13)C+Ω¯=h∈CΩ¯:minx∈Ω¯⁡hx>1.For any h∈C+(Ω¯) we define (14)h+=supx∈Ω⁡hx,h-=infx∈Ω⁡hx.For any p∈C+Ω¯, we introduce the variable exponent Lebesgue spaces and the variable exponent Sobolev space.(1)
L
p
x
Ω
Space
(15)
L
p
x
Ω
=
u
:
u
is a measurable real-valued function
,
∫
Ω
u
x
p
x
d
x
<
∞
,and it is equipped with the following Luxemburg’s norm: (16)uLpxΩ=inf⁡λ>0:∫Ωuxλpxdx≤1.The space LpxΩ,·LpxΩ is a separable, uniformly convex Banach space.(2)
W
1
,
p
x
Ω
Space
(17)
W
1
,
p
x
Ω
=
u
∈
L
p
x
Ω
:
∇
u
∈
L
p
x
Ω
,and it is endowed with the following norm: (18)uW1,px=uLpxΩ+∇uLpxΩ,∀u∈W1,pxΩ.We use W01,pxΩ to denote the closure of C0∞Ω in W1,px.Some properties of the function spacesW1,pxΩ are quoted in the following lemma.Lemma 4.
(i) The spaceLpxΩ,·Lpx(Ω),  W1,pxΩ,·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). And, for any u∈Lq1(x)(Ω) and v∈Lq2(x)(Ω), we have (19)∫Ωuvdx≤2uLq1xΩvLq2xΩ.
(iii)(20)IfuLpxΩ=1,then∫Ωupxdx=1,IfuLpxΩ>1,thenuLpxp-≤∫Ωupxdx≤uLpxp+,IfuLpxΩ<1,thenuLpxp+≤∫Ωupxdx≤uLpxp-.
(iv) Ifp1x≤p2x, then (21)Lp1xΩ⊃Lp2xΩ.
(v) Ifp1x≤p2x, then (22)W1,p2xΩ↪W1,p1xΩ.
(vi)p(x)-Poincarés inequality: if p(x)∈C(Ω), then there is a constant C>0, such that (23)uLpxΩ≤C∇uLpxΩ,∀u∈W01,pxΩ.This implies that ∇uLp(x)(Ω) and uW1,pxΩ are equivalent norms of W01,p(x).
In [14], Zhikov showed that (24)W01,pxΩ≠v∈W01,pxΩv∂Ω=0=W˚1,pxΩ.Hence, the property of the space is different from the case when p is a constant. This fact can make the general methods used in studying the well-posedness of the solutions to the evolutionary p-Laplacian equation not be used directly.
If the exponentp(x) is required to satisfy logarithmic Hölder continuity condition, (25)px-py≤ωx-y,∀x,y∈Ω,x-y<12,with (26)lims→0+-⁡ωsln⁡1s=C<∞,then (27)W01,pxΩ=W˚1,pxΩ.Lemma 5.
LetΩ⊂RN be an open, bounded set with Lipschitz boundary and p∈C+(Ω¯) with 1<p-≤p+<N satisfy the log-Hölder continuity (25). If r∈L∞(Ω) with r->1 satisfies (28)rx⩽p∗x≔NpxN-px∀x∈Ω,then we have (29)W1,pxΩ↪LrxΩand the embedding is compact if infx∈Ω(p∗(x)-r(x))>0.Remark 6.
Furthermore, under the same assumptions as in the above lemma, if we remove the log-Hölder continuity condition (25), then there is also a continuous and compact embedding (30)W1,pxΩ↪LrxΩ∀x∈Ω,where p,r∈C+(Ω¯) and r(x)<p∗(x).Definition 7.
A functionu(x,t) is said to be a weak solution of (1) with the initial value (2) and the partial boundary value condition (4), if (31)u∈L∞QT,∂u∂t∈L1QT,axuαx∇upx∈L1QT,and for any function φ∈C01(QT), (32)∬QT∂u∂tφ+axuαx∇upx,t-2∇u·∇φ+biuφxidxdt=0.The initial value (2) is satisfied in the sense of (33)limt→0⁡∫Ωux,t-ux,0dx=0.The partial boundary value condition (4) is satisfied in the sense of the trace.
Ifu0 satisfies (34)u0∈L∞QT,axu0αx∇upx∈L1QT,it is not difficult to prove there exists a weak solution in the sense of Definition 7.Definition 8.
A functionu(x,t) is said to be a solution of (1) with the initial value (2), if u satisfies (31) and(35)∬QTutφ1φ2+axuαx∇upx-2∇u·∇φ1φ2+biuφ1φ2xidxdt=0,where φ2∈C01(QT), and if we denote that (36)Wα1,px=u:usatisfies31,then, for any given t, φ1(x,t)∈Wα1,p(x) and, for any given x, φ1x,t≤c. The initial value (2) is satisfied in the sense of (33).Based on the existence of the weak solution in the sense of Definition7, one also can be able to prove the existence of the weak solution in the sense of Definition 8. Since we mainly are concerned with the stability of the weak solutions, we are not ready to give the proof of the existence of the weak solutions in what follows.


## 3. The Proofs of Theorems

Proof of Theorem1.
Letu,v be two solutions of (1) with the partial homogeneous boundary values (4) and with the initial values u0, v0, respectively. From the definition of the weak solution in the sense of Definition 7, for all φ∈C01(QT), (37)∫Ωφ∂u-v∂tdx=-∫Ωaxuαx∇upx-2∇u-vα∇vpx-2∇v·∇φdx.For small η>0, let (38)Sηs=∫0shητdτ,hηs=2η1-sη+.Obviously hη(s)∈C(R), and (39)hηs≥0,shηs≤1,Sηs≤1;limη→0⁡Sηs=sgns,limη→0⁡sSη′s=0.Let (40)aηx=axη,ax<η,1,ax≥η.By a process of limit, we can choose aηSη(u-v) as the test function; then (41)∫ΩaηSηu-v∂u-v∂tdx+∫Ωaxaηuα∇upx-2∇u-∇vpx-2∇v·∇u-vSη′u-vdx+∫Ωaxaηuα-vα∇vpx-2∇v·∇u-vSη′u-vdx+∫Ωaxuα∇upx-2∇u-∇vpx-2∇v·∇aηSηu-vdx+∫Ωaxuα-vα∇vpx-2∇v·∇aηSηu-vdx+∫Ωbiu-biv·u-vxiSη′u-vaηxdx+∫Ωbiu-biv·Sηu-vaηxixdx=0.Thus (42)limη→0⁡∫ΩaηSηu-v∂u-v∂tdx=ddtu-v1,∫Ωaηaxuα∇upx-2∇u-∇vpx-2∇v·∇u-vSη′u-vdx⩾0.
Thus, by (ii) and (iii) in Lemma4, (43)∫Ωaηaxuα-vα∇vpx-2∇v·∇u-vSη′u-vdx≤∫Ωaxuα-vα∇vpx-1∇u+∇vSη′u-vdx≤c∫Ωaxuα-vα∇vpx-1∇uSη′u-vdx+∫Ωaxuα-vα∇vpxSη′u-vdx≤c∫Ωaxuα-vα∇vpxSη′u-vdx1/q1∫Ωaxuα-vα∇upxSη′u-vdx1/p1+∫Ωax∇vpxuα-vαSη′u-vdx.Here, p1=p+ or p- according to (iii) of Lemma 4; also q1 has the same meaning if we denote that q(x)=p(x)/p(x)-1.
Using the mean theorem, by (7) and α-≥1, (44)∫Ωaxaηuα-vα∇vpxSη′u-vdx=∫Ωaxaηαxξα-1u-v∇vpxSη′u-vdx≤c∫Ωax∇vpxu-vSη′u-vdxwhich goes to zero as η→0 by the Lebesgue dominated convergence theorem. Then, letting η→0 in (43), we have (45)limη→0⁡∫Ωaηaxuα-vα∇vpx-2∇v·∇u-vSη′u-vdx=0.
LetΩη={x∈Ω:a(x)>η}. (46)∫Ωaxuα∇upx-2∇u-∇vpx-2∇v·∇aηSηu-vdx=∫Ω∖Ωηaxuα∇upx-2∇u-∇vpx-2∇v·∇aηSηu-vdx≤capx-1/px∇upx-1+∇vpx-1LqxΩ∖Ωη1ηa1/px∇aLpxΩ∖Ωη≤c∫Ω∖Ωηax∇upx1/q++∫Ω∖Ωηax∇vpx1/q+which goes to 0 as η→0 by (47)∫Ωax∇upxdx<∞,∫Ωax∇vpxdx<∞.Here, we had used the fact (48)1ηa1/px∇aLpxΩ∖Ωη≤1η∫Ω∖Ωηax∇apxdx1/p+≤η1/p+-1∫Ω∖Ωη∇apxdx1/p+≤c,by (6).
For any givent∈0,T, let Dη={x∈Ω:u-v<η}. Then (49)∫Ωbiu-biv·u-vxiSη′u-vaηxdx=∫x∈Ω:u-v<ηbiu-biv·u-vxiSη′u-vaηxdx≤a1/pxbiu-bivSη′u-vaηxLqxDηa1/pxu-vxiLpxDη,which goes to zero when η→0 by the assumption that (50)∫Ωa-1/px-1xdx<∞.At the same time, (51)limη→0⁡∫Ωbiu-biv·Sηu-vaηxixdx≤climη→0⁡1η∫Ω∖Ωηbiu-bivSηu-vaxixdx≤climη→0⁡1η∫Ω∖Ωηu-vSηu-vaxixdx≤climη→0⁡1η∫Ω∖Ωηu-vdx≤csupx,t∈Σp×0,T⁡u-v.
Now, letη→0 in (41). Then (52)ddtu-v1≤csupx,t∈Σp×0,T⁡u-v.It implies that (53)∫Ωux,t-vx,tdx⩽∫Ωu0-v0dx+csupx∈Σp⁡u-v,∀t∈0,T.Proof of Theorem2.
From Definition7, if α=0, one can see that condition (7) is naturally true. In other words, condition (7) is not necessary.Proof of Theorem3.
Letu, v be two solutions of (1) with the initial values u0(x), v0(x), respectively. For a small positive constant, β≥2, we may choose χ[τ,s](u-v)aβ as a test function. Then(54)∬Qτsu-vξλ∂u-v∂tdxdt=-∬Qτsaxuαx∇upx-2∇u-∇vpx-2∇v∇u-vaβdxdt-∬Qτsaxuαx-vαx∇vpx-2∇v∇u-vaβ-∬Qτsbiu-bivu-vaβxidxdt.
We have(55)∬Qτsa1+βxuαx∇upx-2∇u-∇vpx-2∇v∇u-vdxdt≥0,∬Qτsaxuαx∇upx-2∇u-∇vpx-2∇vu-v∇aβdxdt≤∬Qτsu-vaxuαx∇upx-1+∇vpx-1∇aβdxdt≤c∫τs∫Ωax∇upx+∇vpxdxdt1/q1·∫τs∫Ωa1+pxβ-1u-vpxdxdt1/p1≤c∫τs∫Ωa1+pxβ-1u-vpxdxdt1/p1.Here, we have used the fact that ∇a≤c. Now, since β≥2, by Ho¨lder inequality, we have (56)∬Qτsu-vaxuαx∇upx-2∇u-∇vpx-2∇v∇aβdxdt≤c∫τs∫Ωaβu-vpxdxdt1/p1.Let Ω1={x∈Ω:p(x)≥2}, Ω2={x∈Ω:1<p(x)<2}; then (57)∫τs∫Ω1aβu-vpxdxdt1/p1≤c∫τs∫Ω1aβu-v2dxdt1/p1.By Ho¨lder inequality(58)∫τs∫Ω2aβu-vpxdxdt1/p1≤c∫τs∫Ωaβu-v2dxdt1/p11/p12,where p12=2/px+ or p12=2/px-. By (54)–(58), we have (59)∬Qτsu-vaxuαx∇upx-2∇u-∇vpx-2∇v∇aβdxdt≤c∫τs∫Ωaβu-v2dxdt1/l,where l>1.
At the same time, we have(60)∬Qτsaxuαx-vαx∇vpx-2∇v∇u-vaβdxdt=∬Qτsaxuαx-vαx∇vpx-2∇v∇u-vaβdxdt+∬Qτsax,tuαx-vαx∇vpx-2∇v∇au-vβaβ-1dxdt.
By condition (11), a(x)∇up(x)≤c, a(x)∇vp(x)≤c, using Young inequality; by (60), it is easy to show that (61)∬Qτsaxuαx-vαx∇vp-2∇v∇u-vaβdxdt≤c∫τs∫Ωaβu-v2dxdt1/k,where k>1.
At last(62)∬Qτsbiu-bivu-vaβxidxdt=∬Qτsbiu-bivu-vaxiβdxdt+∬Qsbiu-bivu-vxiaβdxdt,and since bi is a Lipschitz function, u,v∈L∞(QT), we have (63)∬Qτsbiu-bivu-vaxiβdxdt=β∫τs∫Ωbiu-bivu-vaβ-1axidx≤c∫τs∫Ωu-v2aβ-1dxdt≤c∫τs∫Ωaβ-1u-vdxdt=c∫τs∫Ωaβ/2u-vaβ/2-1dxdt≤c∫τs∫Ωaβu-v2dx1/2∫τs∫Ωaβ-2dx1/2≤c∫τs∫Ωaβu-v2dx1/2,only if β≥2. (64)∬Qτsbiu-bivu-vxiaβdxdt≤∫τsaβ-α/pxbiu-bivLqxΩa1/pxu-vxiLpxΩdt≤∫τs∫Ωaβ-1/pxqxbiu-bivqxdx1/q1∫Ωax∇upx+∇vpxdx1/p1dt≤∫τs∫Ωaβ-1/pxqxbiu-bivqxdx1/q1dt≤c∫τs∫Ωaβ-1/pxqxbiu-bivqxdxdt1/q1≤c∫τs∫Ωaβ-1/pxqxu-vqx-1u-vdxdt1/q1≤c∫τs∫Ωaku-vdxdt1/q1≤c∫τs∫Ωaβu-v2dxdt1/2q1.Here (65)k=β-1px+1pxpx-1-1=βpxpx,and clearly 2k≥β.
Since(66)∬Qτsu-vaβ∂u-v∂tdxdt=∫Ωaβux,s-vx,s2dx-∫Ωaβux,τ-vx,τ2dx,then, by (54)–(66), we have (67)∫Ωaβux,s-vx,s2dx-∫Ωaβux,τ-vx,τ2dx≤c∫0s∫Ωaβux,t-vx,t2dxdtq,where q<1. By (67), it is easy to obtain the local stability (12), and we omit the details here.


## 4. Conclusions

The equation considered in the paper comes from electrorheological fluids, which may be double degenerate or singular. Moreover, the diffusion coefficient is degenerate on the boundary; then the solutions generally lack the regularity to define the trace on the boundary. The facts make it difficult to obtain the stability of the weak solutions. By introducing a new kind of the weak solution, the paper successfully overcomes the difficulty. Moreover, importantly, the main result (Theorem1) shows that the electrorheological fluid theory must be complicated compared to the non-Newtonian fluid theory.



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*Source: 2397474-2017-06-01.xml*