# A New Method to Deal with the Stability of the Weak Solutions for a Nonlinear Parabolic Equation

Authors: Huashui Zhan
Journal: Journal of Function Spaces
 (2018)

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## Abstract

Consider the nonlinear parabolic equation∂u/∂t-div(a(x)|∇u|p-2∇u)=f(x,t,u,∇u) with axx∈Ω>0 and a(x)x∈∂Ω=0. Though it is well known that the degeneracy of a(x) may cause the usual Dirichlet boundary value condition to be overdetermined, and only a partial boundary value condition is needed, since the nonlinearity, this partial boundary can not be depicted out by Fichera function as in the linear case. A new method is introduced in the paper; accordingly, the stability of the weak solutions can be proved independent of the boundary value condition.


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## Body

## 1. Introduction and the Main Results

The nonlinear parabolic equation(1)∂u∂t-divax∇up-2∇u=fx,t,u,∇u,x,t∈QT=Ω×0,T,comes from the theory of non-Newtonian fluid and had been studied widely; one can refer to [1–6] and the references therein. Here p>1, Di=∂/∂xi, 0<a(x)∈C1(Ω¯), and Ω⊂RN is a bounded domain with a C2 smooth boundary ∂Ω. If a(x)≥0, (1) with the initial value(2)ux,0=u0x,x∈Ω,and with a partial boundary value condition(3)ux,t=0,x,t∈Σp⊆∂Ω×0,T,has been studied in [7–10]. What catches our attention is since a(x) is degenerate on the boundary, to obtain the regularity of the weak solutions on the boundary becomes difficult, and the trace on the boundary can not be defined in the classical sense. Accordingly, how to construct a suitable function to obtain the stability of the weak solutions becomes formidable. In what follows, we will overcome the difficulty by a new method, and the new method is called the general characteristic function method.For simplicity, we only consider a special case of (1),(4)∂u∂t-divax∇up-2∇u-bixDiu+cx,tu=fx,t,x,t∈QT=Ω×0,T.Certainly, the general characteristic method also can be used to study the stability of weak solutions to (1). We assume that a(x),bi(x), and c(x,t) are C1 functions, and(5)ax>0,x∈Ω,ax=0,x∈∂Ω.Definition 1.
A functionu(x,t) is said to be a weak solution of (4) with the initial value (2), if (6)u∈L∞QT,ut∈L2QT,ax∇up∈L1QT,and for any function g(s)∈C1(R), g(0)=0, φ1∈C01(Ω)¯, φ2∈L∞(0,T;Wloc1,p(Ω)),(7)∬QTutgφ1φ2+ax∇up-2∇u·∇gφ1φ2+ubixixgφ1φ2+bixgxiφ1φ2-cx,tugφ1φ2+fx,tgφ1φ2dxdt=0.The initial value is satisfied in the sense of that (8)limt→0⁡∫Ωux,t-u0xdx=0.The existence of the solution can be proved in a similar way as that of the evolutionaryp–Laplacian equation [4]; we omit the details here. In order to study the stability of the weak solutions, let us introduce a new concept.Definition 2.
A nonnegative continuous functionφ is said to be a general characteristic function of Ω, if and only if that(9)φx>0,x∈Ω,φx=0,x∈∂Ω.
One can see that Definition2 is inspired by the usual characteristic function χ of Ω, which is defined as (10)χx=1,x∈Ω¯,χx=0,x∈RN∖Ω¯.Unlike χ, the general characteristic function φ is not unique. For example, the distance function d(x)=dist(x,∂Ω) and the diffusion function a(x) defined in (5) both are the general characteristic functions. Actually, we only borrow the concept of the characteristic function χ, but no more than that. The main results of the paper are the following theorems.Theorem 3.
Letu and v be two weak solutions of (4) with the initial values u0(x) and v0(x), respectively, p>1 and(11)bix≤ca1/px.If there exists a general characteristic function φ such that(12)∫Ωax∇φφpdx<∞,then (13)∫Ωux,t-vx,tdx≤∫Ωu0x-v0xdx,a.e.t∈0,T.Theorem 4.
Letu and v be two nonnegative solutions of (4) with the initial values u0(x) and v0(x), respectively. If p>1,(14)∫Ωa-1/p-1xdx<∞,and there exists a general characteristic function φ such that(15)∫Ω∖Ωλaxφx∇φpdx≤cλp-1,then the stability of the weak solutions is true in the sense of (13). Here, Ωλ={x∈Ω:φ(x)>λ}.A local stability of the weak solutions is given as follows.Theorem 5.
Letu and v be two solutions of (4) with the differential initial values u0(x) and v0(x), respectively. If p>1, and there exists a general characteristic function φ such that (16)a∇φpφ≤c,(17)a-1/p-1xbixp/p-1φ1/p-1x≤c,(18)∫Ωφxixbix2φdx≤c,then (19)∫Ωφxux,t-vx,t2dx≤c∫Ωφxu0x-v0x2dx.From my own perspective, the geometric characteristic of the domainΩ and the degeneracy of the diffusion coefficient a(x) can take place of the usual boundary value condition (20)ux,t=0,x,t∈∂Ω×0,T.The proofs of Theorems 3–5 are based on the general characteristic function, and we call this method as the general characteristic function method.Moreover, if we choose different general characteristic functionsφ, we can obtain different results. For example, if we choose φ=aα(x),α>0, corresponding to Theorems 3–5, we have the following results.Theorem 6.
Letu and v be two weak solutions of (4) with the initial values u0(x) and v0(x), respectively; suppose p>1 and (11) is true, and(21)∫Ωa1-px∇apdx<∞,then the stability (13) is true.Theorem 7.
Letu and v be two nonnegative solutions of (4) with the initial values u0(x) and v0(x), respectively. If p>1, (14) is true, then the stability of the weak solutions is true in the sense of (13).This is due to the fact that only we chooseα≥p-3/p-2, and condition (15) is natural.Theorem 8.
Letu and v be two solutions of (4) with the differential initial values u0(x) and v0(x), respectively. If p>1, then there exists a constant α>0 such that(22)∫Ωaαxux,t-vx,t2dx≤c∫Ωaαxu0x-v0x2dx.This is due to the fact that ifp≥2, condition (16) is natural. When p<2, only we choose 0<α≤2/2-p, and condition (16) is also true. Once more, since φ=aα, condition (17) is natural. At the same time, if p≥2, we can choose α≥2 such that condition (18) is true. When p<2, we can choose 2≤α≤2/2-p such that (18) is true. Then we have the conclusion of Theorem 8.If we chooseφ=dα(x),α>0, corresponding to Theorems 3–5, we have the following corollaries.Corollary 9.
Letu and v be two solutions of (4) with the differential initial values u0(x) and v0(x), respectively. Suppose p>1 and (11) is true, and a(x) satisfies (23)∫Ωaxdpxdx<∞,then the stability (13) is true.
In particular, ifa(x)=d(x), (23) becomes that (24)∫Ωd1-pxdx≤c.Thus, if 1<p<2, (24) is naturally true. Accordingly, only the condition (11) can ensure that the stability (13) is true naturally.Corollary 10.
Letu and v be two solutions of (4) with the differential initial values u0(x) and v0(x), respectively. If p>1, (14) is true, and (25)∫Ω∖Ωλaxdx≤cλp/α,then the stability of the weak solutions is true in the sense of (13). Here Ωλ={x∈Ω:dα(x)>λ}.Corollary 11.
Letu and v be two solutions of (4) with the differential initial values u0(x) and v0(x), respectively. If p>1, and a(x) and bi(x) satisfy(26)a-1/p-1xbixp/p-1dα/p-1x≤c,then(27)∫Ωdαxux,t-vx,t2dx≤c∫Ωdαxu0x-v0x2dx.This is due to the fact that only if we chooseα≥max{1,1/p-1}, conditions (16) and (18) in Theorem 5 are naturally true.As long as you like, you can choose the other general characteristic functions; for example,φ(x)=ea(x)-1 and φ(x)=sina(x)/M, to obtain the corresponding theorems. Here, M=maxx∈Ω¯a(x). By the way, we hope the method introduced in this paper can be beneficial to the well-posedness problems for the evolutionary p(x)–Laplacian equations, overdetermined anisotropic elliptic equations, and the infiltration equations. As we know, there are many papers devoted to these equations; one can refer to [11–23] and the references therein.


## 2. The Proof of Theorem3

For smallη>0, let(28)Sηs=∫0shητdτ,hηs=2η1-sη+.Obviouslyhη(s)∈C(R), and (29)hηs≥0,shηs≤1,Sηs≤1;limη→0⁡Sηs=sgn⁡s,limη→0⁡sSη′s=0.Proof of Theorem3.
Letu and v be two solutions of (4) with the different initial values u0(x) and v0(x), respectively. We choose Sη(φ(u-v)) as the test function in Definition 1. Let φ be a general characteristic function. Then(30)∫ΩSηφu-v∂u-v∂tdx+∫Ωaxφx∇up-2∇u-∇vp-2∇v·∇u-vSη′φu-vdx+∫Ωax∇up-2∇u-∇vp-2∇v·∇φu-vSη′φu-vdx+∫ΩDibixu-vSηφu-v+φbixu-v·u-vxiSη′φu-vdx+∫Ωbixu-v·φxiu-vSη′φu-vdx+∫Ωcx,tu-vSηφu-vdxdt=0.
Let us analyze every term in (30).(31)limη→0⁡∫ΩSηφu-v∂u-v∂tdx=ddt∫Ωux,t-vx,tdx,(32)∫Ωaxφx∇up-2∇u-∇vp-2∇v·∇u-vSη′φu-vdx≥0,(33)∫Ωau-vSη′φu-v∇up-2∇u-∇vp-2∇v·∇φdx=∫Ω:φu-v<ηa-p-1/pau-vSη′φu-vap-1/p∇up-2∇u-∇vp-2∇v·∇φdx≤∫Ω:φu-v<ηa1/pu-vSη′φu-v·∇φpdx1/p·∫Ω:φu-v<ηax∇up+∇vpdxp-1/p.If {x∈Ω:u-v=0} has 0 measure, since(34)∫Ωax∇φφpdx<∞,consequently (35)∫Ω:φu-v<ηa1/p∇φφφu-vSη′φu-vpdx≤c,(36)limη→0⁡∫Ω:φu-v<ηax∇up+∇vpdxp-1/p=∫Ω:u-v=0ax∇up+∇vpdxp-1/p=0.If {x∈Ω:u-v=0} has a positive measure, then by the Lebesgue dominated convergence theorem(37)limη→0⁡∫Ω:φu-v<ηa1/p∇φφφu-vSη′φu-vpdx1/p=∫Ω:u-v=0ax∇φφplimη→0⁡φu-vSη′u-vφpdx1/p=0.By (29) and condition (12), using the Lebesgue dominated convergence theorem, in both cases, we have(38)limη→0⁡∫Ωaxφu-vSη′φu-v∇up-2∇u-∇vp-2∇v∇φdx=0.
At the same time, by (11), |bi(x)|≤ca1/p(x),(39)∫Ωbixu-v·φxiu-vSη′φu-vdx≤∫Ωbixu-vSη′φu-vφu-vφxiφdx≤c∫ΩSη′φu-vφu-vbixφxiφdx≤c∫ΩSη′φu-vφu-vp/p-1dxp-1/p·∫Ωax∇φφpdx1/p.Then (40)limη→0⁡∫Ωbixu-vφxiu-vSη′φu-vdx=0.
Moreover, bybix≤ca1/p(x),(41)∫Ωbixφxu-vSη′φu-vu-vxidx≤c∫Ωax∇up+∇vpdx1/p∫Ωφxu-vSη′φu-vp/p-1p-1/p.Therefore, we have(42)limη→0⁡∫Ωbixu-vSη′aβu-vu-vxiaβxdx=0.(43)limη→0⁡∫ΩDibixu-vSηaβu-v+∫Ωcx,tu-vSηaβu-vdx≤c∫Ωux,t-vx,tdx.
Now, letη→0 in (30). Then(44)ddt∫Ωux,t-vx,tdx⩽∫Ωux,t-vx,tdx.By the Gronwall inequality, we have (45)∫Ωux,t-vx,tdx⩽cT∫Ωu0-v0dx,∀t∈0,T.Theorem 3 is proved.


## 3. The Proof of Theorem4

Proof of Theorem4.
By Definition1, for any φ1∈C01(Ω)¯, φ2∈L∞(0,T;Wloc1,p(Ω)), g(s)=s, we have(46)∬QT∂u-v∂tφ1φ2+ax∇up-2∇u-∇vp-2∇v·∇φ1φ2+u-vbixixφ1φ2+bixφ1φ2xi-cx,tu-vφ1φ2dxdt=0.Let φ be a general characteristic function, Ωλ={x∈Ω:φ(x)>λ} as before, and let (47)ϕλx=1,ifx∈Ωλ,1λφx,ifx∈Ω∖Ωλ.
Now, we chooseφ1=ϕλ(x),φ2=χ[τ,s]Sη(u-v) in (46); accordingly,(48)∫τs∫ΩϕλxSηu-v∂u-v∂tdxdt+∫τs∫Ωϕλxax∇up-2∇u-∇vp-2∇v·∇u-vSη′u-vdxdt+∫τs∫Ωax∇up-2∇u-∇vp-2∇v·∇ϕλxSηu-vdxdt+∫τs∫Ωu-vbixixϕλxSηu-v+bixϕλxSηu-vxidxdt-∫τs∫Ωcx,tu-vϕλxSηu-vdxdt=0.Clearly,(49)∫Ωϕλxax∇up-2∇u-∇vp-2∇v·∇u-vSη′u-vdx≥0.(50)∫Ωax∇up-2∇u-∇vp-2∇v·∇ϕλxSηu-vdx≤∫Ω∖Ωλax∇up-2∇u-∇vp-2∇v·∇ϕλxSηu-vdx≤∫Ω∖Ωλax∇up-2∇u-∇vp-2∇v∇ϕλxdx≤cλ∫Ω∖Ωλax∇up-1∇adx+∫Ω∖Ωλax∇vp-1∇adx.Since (15), (51)∫Ω∖Ωλaxφx∇φpdx≤cλp-1,then (52)cλ∫Ω∖Ωλφxaxφx∇φpdx1/p≤cλλ∫Ω∖Ωλaxφx∇φpdx1/p≤c.By (50), (52), using the Hölder inequality, we have(53)
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.Thus(54)limλ→0⁡limη→0⁡∫Ωax∇up-2∇u-∇vp-2∇v·∇ϕλxSηu-vdx=0.Moreover, by ∫Ω‍a-1/p-1(x)dx<c, using (29) and Lebesgue dominated convergence theorem, (55)limη→0⁡∫Ωϕλbixu-vSη′u-vu-vxidx=limη→0⁡∫Ωϕλa1/pxa1/pxbixu-vSη′u-vu-vxidx≤limη→0⁡∫Ωbiϕλa1/pSη′u-vu-vp/p-1dxp-1/p∫Ωax∇up+∇vpdx1/p=0.The last equality of (55) is based on the following analysis:(56)∫Ωbiϕλa1/pSη′u-vu-vp/p-1dx=∫Ωλbia1/pSη′u-vu-vp/p-1dx+1λ∫Ω∖Ωλbiφa1/pSη′u-vu-vp/p-1dxwhile, since a(x)≥c(λ), x∈Ωλ, using Lebesgue dominated convergence theorem, we have (57)limη→0∫Ωλbia1/pSη′u-vu-vp/p-1dx=0,and by Sη′(u-v)(u-v)≤c, ∫Ω‍a-1/p-1(x)dx<∞,(58)limη→0⁡1λ∫Ω∖Ωλbiφa1/pSη′u-vu-vp/p-1dx=0.Moreover,(59)limλ→0⁡limη→0⁡∫Ωϕλxibixu-vSηu-vdx≤limλ→0⁡cλ∫Ω∖Ωλ∇φdx≤limλ→0⁡cλ∫Ω∖Ωλax∇φpdx1/p∫Ω∖Ωλa-1/p-1xdxp-1/p=0,by (52) and ∫Ω‍a-1/p-1(x)dx<c, while (60)limη→0⁡∫Ωu-vbixixϕλxSηu-vdx≤∫Ωu-vdxis obviously true.
By (55), (59), and (60),(61)limη→0⁡∫Ωu-vbixixϕλxSηu-v+bixϕλxSηu-vxidx≤∫Ωu-vdx.
At last,(62)limλ→0⁡limη→0⁡∫τs∫ΩϕλxSηu-v∂u-v∂tdxdt=∫τsddt∫Ωux,t-vx,tdxdt.
Now, after lettingη→0, let λ→0 in (47). Then, using (49), (54), (55), (59), (60), (61), and (62), and by the Gronwall inequality, we have(63)∫Ωux,t-vx,tdx⩽∫Ωu0-v0dx.


## 4. The Proof of Theorem5

Proof of Theorem5.
Letu and v be two solutions of (4) with the initial values u0(x) and v0(x), respectively. From the definition of the weak solution, if g(s)=s, for any φ1∈C01(Ω)¯, φ2∈L∞(0,T;Wloc1,p(Ω)),(64)∬QTφ1φ2∂u-v∂tdxdt=-∬QTax∇up-2∇u-∇vp-2∇v∇φ1φ2dxdt-∬QTu-vbixiφ1φ2+bixφ1φ2xidxdt+∬QTu-vcx,tφ1φ2dxdt.In particular, we choose (65)φ1=φ,φ2=χτ,su-vwhere χ[τ,s] is the characteristic function on [τ,s] and φ is a general characteristic function. Denoting Qτs=Ω×[τ,s], then(66)∬Qτsax∇up-2∇u-∇vp-2∇v·∇u-vφdxdt=∬Qτsaφ∇up-2∇u-∇vp-2∇v·∇u-vdxdt+∬Qτsax∇up-2∇u-∇vp-2∇vu-v∇φdxdt.Clearly, (67)∬Qτsaφ∇up-2∇u-∇vp-2∇v∇u-vdxdt≥0.In the first place, for the second term on the right-hand side of (66), we have (68)∬Qτsu-vax∇up-2∇u-∇vp-2∇v∇φdxdt≤∬Qτsu-vax∇up-1+∇vp-1∇φdxdt≤c∫τs∫Ωax∇up+∇vpdxdtp-1/p·∫τs∫Ωax∇φpu-vpdxdt1/p≤c∫τs∫Ωa∇φpu-vpdxdt1/p.Now, since a|∇φ|p/φ≤c, we have (69)∬Qτsu-va∇up-2∇u-∇vp-2∇v∇φdxdt≤c∫τs∫Ωφu-vpdxdt1/p.If p≥2,(70)∫τs∫Ωφu-vpdxdt1/p≤c∫τs∫Ωφu-v2dxdt1/p.If 1<p<2, by Hölder inequality (71)∫τs∫Ωφu-vpdxdt1/p≤c∫τs∫Ωφu-v2dxdt1/2.In the second place, by (17)-(18)(72)a-1/p-1xbixp/p-1φ1/p-1x≤c,∫Ωφxibi2φdx≤c;since u,v∈L∞(QT), we easily see that(73)∬Qτsbixu-vu-vφxidxdt=∬Qτsbixu-v2φxidxdt+∬Qsbixu-vu-vxiφdxdt≤c∫τs∫Ωλu-vφxibixdx+∫τs∫Ωa-1/pp′φbiu-vp′dxdt1/p′∫τs∫Ωax∇up+∇vpdxdt1/p≤c∫τs∫Ωφxibiφ2dxdt1/2∫τs∫Ωφu-v2dxdt1/2+c∫τs∫Ωa-1/p-1φ1/p-1bip/p-1dxdt1/p′∫τs∫Ωax∇up+∇vpdxdt1/p≤c∫τs∫Ωφu-v2dxdt1/2+c∫τs∫Ωφu-vp′dxdt1/p′.In the third place, we have (74)∬Qτsbixixu-v2+cx,tu-v2φdxdt≤c∫τs∫Ωφu-v2dxdt1/2,(75)∬Qτsu-vφ∂u-v∂tdxdt=∫Ωφux,s-vx,s2dx-∫Ωφux,τ-vx,τ2dx.
At last, by (66), (67), (68), (69), (70), (71), (73), (74), and (75), we let λ→0 in (64). Then (76)∫Ωφux,s-vx,s2dx-∫Ωφux,τ-vx,τ2dx≤c∫0s∫Ωφux,t-vx,t2dxdtq,where q<1. By (76), we easily show that(77)∫Ωφux,s-vx,s2dx≤∫Ωφux,τ-vx,τ2dx.Thus, by the arbitrary of τ, we have(78)∫Ωφux,s-vx,s2dx≤∫Ωφu0-v02dx.The proof is complete.



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