# The Well-Posedness of the Solutions Based on theL1 Initial Value Condition

Authors: Huashui Zhan; Zhen Zhou
Journal: Journal of Function Spaces
 (2018)

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## Abstract

The non-Newtonian polytropic filtration equationut=div(a(x)|∇um|p-2∇um) is considered. Only if u0(x)∈L1(Ω), the well-posedness of solutions is studied. If the diffusion coefficient is degenerate on the boundary, then stability of the weak solutions is proved only depending upon the initial value conditions.


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## Body

## 1. Introduction

In this paper, we consider the equation(1)ut=divax∇ump-2∇um,x,t∈QT,with the initial value condition(2)ux,0=u0x,x∈Ω,but without the usual boundary value condition(3)ux,t=0,x,t∈∂Ω×0,T,where Ω⊂RN is a bounded domain with smooth boundary, QT=Ω×(0,T), p>1, m>0, u0∈L1(Ω).This kind of equation is derived from many physical problems [1–5]. In particular, when a(x)=1, m=1, (1) is the evolutionary p-Laplacian equation(4)ut=div∇up-2∇u.While, if a(x)=1, p=2, then (1) is porous media equation(5)ut=△um.Many scholars have made a lot of research on (4) and (5); one can refer to [6–10], etc.Whena(x)=1, (1) becomes usual polytropic infiltration(6)ut=div∇ump-2∇um,x,t∈QT.There are also many papers devoted to this equation. Zhao-Yuan [11] have studied the Cauchy problem of (6) with the initial value u0(x)∈L1(RN); the existence and the uniqueness of the weak solutions were proved and ut∈L1(RN×(τ,T)) was shown for any τ>0. Fan [12] considered the similar problem when u0(x) is just a measure.In [13], Chen-Wnag considered the initial-boundary value problem of the equation(7)ut=divur∇up-2∇u+∇Auwith u0(x)∈Lq(Ω), q≥1. By modifying the usual Morse iteration, imposing some restrictions on the convection function A(s), the local L∞–estimates were made and ut∈L2(RN×(τ,T)) was obtained. In [14], Tsutsumi considered the initial-boundary value problem of the equation(8)uβt=∑i=1N∂∂xi∂u∂xip-2∂u∂xi-λuγ,with the initial value u0β∈L1(Ω). By showing that uβ+1/2t∈L2(RN×(τ,T)), Tsutsumi studied the existence, uniqueness, regularity, and behavior of solutions to (8); we will give more information of [14] at the appendix of this paper. A more general equation was studied by Otto in [15] with the L1 initial value condition. The large time behavior of the solutions of the equations with the type of (1) also has been studied in [16–18], etc. The extinction, positivity, and blow-up of the solutions for a doubly nonlinear degenerate parabolic equation have been studied in [19, 20], etc. Of course, since (1) is one of the most well-known parabolic equations, there are a great deal of papers to study various subjects of this equation; for example, one can refer to [21–27].Ifa(x) is not a constant, the first thing that catches our attention is when a(x)=dist(x,∂Ω)α, m=1, u0(x)∈L∞(Ω), Yin-Wang [28] have made a creative work. They have demonstrated that when α⩾p-1, the uniqueness of the weak solution can be proved even if no boundary value condition is required. However, if α<p-1, based on the initial value (2) and the homogeneous boundary value condition (3), they have proved the uniqueness of the weak solution.In this paper, we assumea(x)∈C1(Ω¯), a(x)>0 in Ω, a(x)=0 on ∂Ω. Different from [28], we only assume that u0(x)∈L1(Ω).Definition 1.
Functionu(x,t) is a weak solution of problem (1)-(2), if (9)u∈C0,T,L1Ω∩L∞Ω×τ,T,a1/p∇um∈LpΩ×τ,T,ut∈L1Ω×τ,T,∬QTutφdxdt+∬QTax∇ump-2∇um·∇φdxdt=0,∀φ∈C0∞QT,limt→0⁡∫Ωux,t-u0xdx=0,where τ∈(0,T).Theorem 2.
Suppose thatm>0, p>1, m(p-1)>1, u0(x)⩾0, u0(x)∈L1(Ω), a(x)|∇u0m|p∈Lp(Ω); then problem (1)-(2) has a solution u in the sense of Definition 1, satisfying(10)u∈CQT∩Lmp-1+p/N-αQT,where α∈(0,m(p-1)).When∫Ω‍[a(x)]-1/p-1dx⩽c, we can define the trace of um (also u) on the boundary. Then the homogeneous boundary value condition (3) can be imposed in the sense of the trace. Nevertheless, the boundary value condition (3) may be redundant.The main aim of this paper is to prove the stability of the weak solutions of (1) without the boundary value condition (3), no matter whether ∫Ω‍a(x)-1/p-1dx⩽c or not.Theorem 3.
Suppose thatm>0, p>1, u(x,t) and v(x,t) are two weak solutions of (1) with different initial value u0(x)∈L1(Ω), and v0(x)∈L1(Ω); if(11)1dp∫Ω∖Ωdax∇apdx<∞,then(12)∫Ωux,t-vx,tdx⩽∫Ωu0x-v0xdx,where d is a small positive constant and Ωd={x∈Ω:a(x)>d}.Theorem 4.
Letm>0, p>1, u and v be two weak solutions of (1) with different initial value u0(x)∈L1(Ω), and v0(x)∈L1(Ω). If(13)∫Ωa-p-1∇apdx<∞,then(14)∫Ωux,t-vx,tdx≤∫Ωu0x-v0xdx,∀t∈0,T.We use some ideas and techniques of [11] to prove Theorem 2. If 1<p<2, m=1, and u0(x)∈L∞(Ω), the stability theorems of the weak solutions independent of the boundary value condition have been proved by Zhan [16, 17].


## 2. Prior Estimate

Consider the regularized problem(15)ut=div⁡ax+δ∇um2+ηp-2/2∇um,x,t∈QT,(16)ux,0=u0ε+ε,x∈Ω,(17)ux,t=ε,x∈∂Ω×0,T,where 0<δ,ε,η<1, u0ε∈C0∞(Ω), u0ε⩾0, and(18)limε→0⁡∫Ωu0εx-u0xdx=0.It is well known that problem (15)-(17) has a nonnegative classical solution uδεη, and(19)ε⩽uδεη⩽Mε.Constant C in this paper is independent of δ, ε, η. Subscripts of uδεη are omitted in the following proofs.Lemma 5.
The solution of problem (15)-(17) satisfies that(20)∫Ωuδεηx,tdx⩽C1+u0L1Ω,t∈0,T.Proof.
Noting that∂um/∂ν⩽0 on ∂Ω, multiplying (15) by u/u+r and integrating it by parts over Ω×(0,t), we have(21)∫Ωu2x,tux,t+rdx⩽∫Ωu2x,0ux,0+rdx+∫0t∫Ωruutu+r2dxdτ-∫0t∫Ωax+δ∇um2+ηp-2/2∇u2rmum-1u+r2dxdτ.Let r→0, we have (22)∫Ωux,tdx⩽∫Ωux,0dx+C.Lemma 6.
u δ ε η satisfies that(23)∫0T∫Ωax+δuδεηα-m1+uδεηα2∇uδεηm2+ηp-2/2∇um2dxdt⩽Cα,u0L1Ω,(24)∫0T∫Ωuδεηmp-1+p/N-αdxdt⩽Cα,u0L1Ω,where α∈(0,m(p-1)).Proof.
Multiplying (15) by uα/1+uα and integrating it over Ω×(0,t), we have(25)∫Ω∫0ux,tsαsα+1dsdx+αm∫0t∫Ωax+δ∇um2+ηp-2/2∇um2uα-m1+uα2dxdτ⩽∫Ω∫0u0sαsα+1dsdx,that is, (23).
Letξ∈C01(Ω); 0⩽ξ⩽1; ξ=1 in Ω′, Ω′={x∈Ω∣dist(x,∂Ω)>d}; denote u1=max⁡{u,1}.
By Sobolev inequalities, whenp⩽N,(26)∬QTξpu1mp-1+p/N-αdxdt⩽C∬QT∇ξu1mp-1-α/ppdxdt·supt∈0,T⁡∫Ωu1dxp/N.∬QT∇ξu1mp-1-α/ppdxdt⩽C∬QT∇ξpu1mp-1-α+ξp∇u1mp-1-α/ppdxdt⩽C∬QT∇ξpump-1-αdxdt+C∬QTξpax+δ∇um2+ηp-2/2∇um2uα-m1+uα2dxdt+C.By (23) and Lemma 5, we have(27)∬QTξpump-1+p/N-αdxdt⩽Cα,u0L1Ω1+∬QT∇ξpump-1-αdxdt.Let ξ=ψb, ψ∈C0∞(Ω), 0⩽ψ⩽1, b=N/p(m(p-1)+p/N-α). Applying also the Hölder inequality, we have(28)∬QTξpump-1+p/N-αdxdt⩽Cα,u0L1Ω1+∬QTξpump-1+p/N-αdxdtmp-1-α/mp-1+p/N-α.Letting d→0 yields (24).
Whenp>N, by (23) and Sobolev inequalities, (24) still holds.Lemma 7.
u δ ε η satisfies that(29)supQρ⁡uδεη⩽Cρ,γ∬Q2ρuδεη1mp-1-1+γdxdt1/γ,where (30)Qρ=Ω×T-ρp,T,T>2ρp,mp-1>1,γ∈12,1+pN.Proof.
Letξ be the cut-off function of Qρ, 0⩽ξ⩽1. Let φ=ξpu2γ-1. Multiplying (15) by φ and integrating it over Ω×(0,t), we have(31)12γ∫Ωξpu2γx,Tdx+2γ-1m∬Q2ρξpax+δ∇um2+ηp-2/2∇um2u2γ-1-mdxdt⩽p∬Q2ρξp-1∇ξax+δ∇um+ηp-2/2∇umu2γ-1dxdt+p2γ∬Q2ρξp-1ξtu2γdxdt.When 1<p<2,(32)∇um2+ηp-2/2∇um2⩾∇um2+ηp/2-ηp/2,∇um2+ηp-2/2∇um⩽∇um2+ηp-2/2.When 2⩽p,(33)∇um2+ηp-2/2∇um2⩾∇ump,∇um2+ηp-2/2∇um⩽C∇ump-1+1.By Young’s inequality, when 1<p<2,(34)supT-2ρp<t<T⁡∫Ωξpu2γdx+∬Q2ρax+δ∇ξump-1+2γ-1/ppdxdt⩽C∬Q2ρ∇ξpax+δu2γ-1+mp-1dxdt+C∬Q2ρξtu2γdxdt+Cηp/2∬Q2ρξpu2γ-1-mdxdt,when η small enough, since ε⩽u,(35)ηp/2∬Q2ρξpu2γ-1-mdxdt⩽∬Q2ρξpu2γ-1dxdt;when 2⩽p,(36)supT-2ρp<t<T∫Ωξpu2γdx+∬Q2ρax+δ∇ξump-1+2γ-1/ppdxdt⩽C∬Q2ρ∇ξpax+δu2γ-1+mp-1dxdt+C∬Q2ρξtu2γdxdt+C∬Q2ρξp-1∇ξu2γ-1ax+δdxdt.Let β=max⁡1,2γ-1+m(p-1)/2γ, h=p1+2γp/N(2γ-1+m(p-1)). By the embedding inequality,(37)∬Q2ρξphump-1+2γp/N+2γ-1dxdt⩽C∬Q2ρ∇ξβump-1+2γ-1/ppdxdt·supT-2ρp<t<T⁡∫Ωξβ2γp/2γ-1+mp-1u2γdxp/N⩽CsupT-2ρp<t<T∫Ωξβ2γp/2γ-1+mp-1u2γdx+∬Q2ρ∇ξβump-1+2γ-1/ppdxdt1+p/N.Let τ∈[1/2,1), ρι=2ρτ+1-τ/2ι, ι=0,1,2,3,…, and ξι(x,t) be truncation function of Qρι, ξι=1 on Qρι+1. K=1+p/N, Kι=2γ. We have(38)∬Qρι+1ump-1-1+Kι+1dxdt⩽∬Qριξιphump-1-1+Kι+1dxdt⩽c1-τρp∬Qριump-1-1+KιdxdtK.By Morse iteration,(39)supQ2ρτ⁡u⩽c1-τρp+N∬Q2ρump-1-1+Kdxdt1/K,that is (40)supQ2ρτ⁡u⩽supQ2ρ⁡uK-γ/kc1-τρp+N∬Q2ρump-1-1+γdxdt1/K.
By the Young inequality,(41)supQ2ρτ⁡u⩽12supQ2ρ⁡u+c1-τρp+N∬Q2ρump-1-1+γdxdt1/γ,from [29], we have(42)supQ2ρτ⁡u⩽Cρ,γ∬Q2ρump-1-1+γdxdt1/γ.Lemma 8.
u δ ε η satisfies that(43)∫t1t2∫Ωax+δ∇uδεηm2+ηp-2/2∇uδεηm2dxdt⩽Cτ,(44)∫t1t2∫Ωuδεηm-1uδεηt2dxdt⩽Cτ,(45)∫0t2∫Ωuδεηt2dxdt⩽Cε,for any t1,t2∈[τ,T).Proof.
Multiplying (15) by um and integrating it over Ω×(t1,t2),(46)∫t1t2∫Ωax+δ∇um2+ηp-2/2∇um2dxdt⩽1m+1∫Ωum+1x,t2dx+1m+1∫Ωum+1x,t1dx.By Lemmas 5–7, uLq(q⩾1,t⩾τ) is uniformly bounded, so (43) holds.
Multiplying (15) by utm and integrating it,(47)∫t1t2∫Ωmum-1ut2dxdt+12∫Ωax+δ∫0∇umx,t2+ηsp-2/2dsdxt=t1t=t2=0.Thus, (44) holds.
When integral domain of the above equation isΩ×(0,t2), since ε⩽u, (45) holds.


## 3. Proof of Theorem2

### 3.1.η→0

From Lemmas5–8, for δ>0, ε>0, there exist a subsequence of {uδεη}, still denoted by {uδεη}, and a function uδε; when η→0, we have(48)uδεη→uδεinCΩ′×τ,T,(49)∇uδεηm⇀∇uδεminLpΩ×τ,T,(50)uδεηt⇀uδεtinL2QT,where Ω′⊂⊂Ω is an open subset such that Ω′¯⊂Ω.Letφ∈C01(QT). A weak solution of problem (15)-(17), uδεη, satisfies(51)∬QTuδεηtφdxdt+∬QTax+δ∇uδεηm2+ηp-2/2∇uδεηm∇φdxdt=0,where(52)∬QTax+δ∇uδεηm2+ηp-2/2∇uδεηm∇φdxdt=∬QTax+δ∇uδεηmp-2∇uδεηm∇φdxdt+p-22η∬QTax+δ∫01∇uδεηm2+sηp-4/2ds∇uδεηm∇φdxdt.limη→0⁡p-22η∬QTax+δ∫01∇uδεηm2+sηp-4/2ds∇uδεηm∇φdxdt=0.Let φ=ψ(uδεηm-uδεm), ψ∈C01(QT).(53)∬QTax+δ∇uδεηmp-2∇uδεηm-∇uδεmp-2∇uδεm·∇φdxdt=∬QTax+δ∇uδεηmp-2∇uδεηm-∇uδεmp-2∇uδεm·∇ψuδεηm-uδεmdxdt+∬QTax+δ∇uδεηmp-2∇uδεηm-∇uδεmp-2∇uδεm·∇uδεηm-uδεmψdxdt.By Hölder’s inequality and (43)-(48), we have(54)limη→0⁡∬QTax+δ∇uδεηmp-2∇uδεηm-∇uδεmp-2∇uδεm·∇ψuδεηm-uδεmdxdt=0.By (49)(55)limη→0∬QTax+δ∇uδεmp-2∇uδεm∇uδεηm-uδεmψdxdt=0.Meanwhile,(56)∬QTax+δ∇uδεηmp-2∇uδεηm∇uδεηm-uδεmψdxdt⩽∬QTax+δ∇uδεηm2+ηp-2/2∇uδεηm·∇uδεηm-uδεmψdxdt⩽-∬QTφdiv⁡ax+δ∇uδεηm2+ηp-2/2∇uδεηmuδεηm-uδεmdxdt-∬QT∇φax+δ∇uδεηm2+ηp-2/2·∇uδεηmuδεηm-uδεmdxdt⩽∬QTφuδεηtuδεηm-uδεmdxdt+C∬QT∇φax+δ∇uδεηmp-2·∇uδεηmuδεηm-uδεmdxdt+Cηp-2/2∬QT∇φax+δ·∇uδεηmuδεηm-uδεmdxdt.By Hölder’s inequality and (48)-(50), we have(57)limη→0⁡∬QTφuδεηtuδεηm-uδεmdxdt+C∬QT∇φax+δ∇uδεηmp-2·∇uδεηmuδεηm-uδεmdxdt+Cηp-2/2∬QT∇φax+δ·∇uδεηmuδεηm-uδεmdxdt=0,and notice that(58)0⩽∬QTax+δ∇uδεηmp-2∇uδεηm-∇uδεmp-2∇uδεm·∇uδεηm-uδεmψdxdt,so(59)limη→0⁡∬QTax+δ∇uδεηmp-2∇uδεηm∇φdxdt=∬QTax+δ∇uδεmp-2∇uδεm∇φdxdt.By (49), uδε=ε, a.e. on ∂Ω×(0,T).Sinceuδεt∈L2(QT), for any 0<t1<t2<T, using the Hölder inequality, we have(60)∫Ωuδεx,t2-uδεx,t1dxdt≤t2-t1∫01∫Ω∂∂tuδεx,st2+1-st1dxds≤t2-t1∫0T∫Ω∂∂tuδεx,tdxdt≤ct2-t1uδεtL2QT,so uδε∈C(0,T;L1(Ω)), and(61)limt→0⁡∫Ωuδεx,t-u0ε+εdx=0.By the above discussion, we can deduce that uδε is a weak solution of problem(62)ut=divax+δ∇ump-2∇um,x,t∈QT,ux,t=ε,x,t∈∂Ω×0,T,ux,0=u0ε+ε,x∈Ω.


### 3.2.ε→0

By Lemmas5–8, {uδε} is uniformly bounded and uniformly continuous on any compact subset of QT, so it has a subsequence still denoted by {uδε}, when ε→0,(63)uδε→uδinCΩ′×τ,T,∇uδεm⇀∇uδminLpΩ×τ,T,uδεt⇀uδtinL1Ω×τ,T.From [29] and similar to the previous proof, uδ is a weak solution of problem(64)ut=div⁡ax+δ∇ump-2∇um,x,t∈QT,ux,t=0,x,t∈∂Ω×0,T,ux,0=u0x,x∈Ω.


### 3.3.δ→0

According to Lemmas5–8, {uδ} has a convergent subsequence still denoted by {uδ},(65)uδ→uinCΩ′×τ,T,ax∇uδmp-2∇uδm⇀ζ→inLp/p-1Ω×τ,T,uδt⇀utinL1Ω×τ,T.By the Hölder inequality and (43),(66)∫t1t2∫Ωδ∇uδmp-2∇uδm·∇φdxdt≤δ1/p∫t1t2∫Ωδ∇uδmpdxdtp-1/p∬suppφ∇φpdxdt1/p≤cδ,that is, (67)limδ→0⁡∫t1t2∫Ωδ∇uδmp-2∇uδm·∇φdxdt=0.Similar to the previous proof, we have(68)∬QTax∇ump-2∇um·∇φdxdt=∬QTζ→·∇φdxdt.At last, the initial value condition (2) satisfied in the sense(69)limt→0⁡ux,t-u0xdx=0,can be proved in a similar way to that for the evolutionary p–Laplacian equation [29]; we omit the details here. Thus u is a weak solution of problem (1)-(2).Lemma 9.
If∫Ω‍a(x)-1/p-1dx⩽c, then ∫Ω‍|∇uδm|dx⩽c.Proof.
(70) ∫ Ω ∇ u δ m d x = ∫ x ∈ Ω : a 1 / p - 1 ∇ u δ m ⩽ 1 ∇ u δ m d x + ∫ x ∈ Ω : a 1 / p - 1 ∇ u δ m > 1 ∇ u δ m d x ⩽ ∫ Ω a x - 1 / p - 1 d x + ∫ Ω a x ∇ u δ m p d x ⩽ C ,where C is a positive constant independent of δ.By this Lemma, we can define the trace ofum (also u) on the boundary. Thus u(x,t) is a weak solution of the initial-boundary value problem of (1).


## 3.1.η→0

From Lemmas5–8, for δ>0, ε>0, there exist a subsequence of {uδεη}, still denoted by {uδεη}, and a function uδε; when η→0, we have(48)uδεη→uδεinCΩ′×τ,T,(49)∇uδεηm⇀∇uδεminLpΩ×τ,T,(50)uδεηt⇀uδεtinL2QT,where Ω′⊂⊂Ω is an open subset such that Ω′¯⊂Ω.Letφ∈C01(QT). A weak solution of problem (15)-(17), uδεη, satisfies(51)∬QTuδεηtφdxdt+∬QTax+δ∇uδεηm2+ηp-2/2∇uδεηm∇φdxdt=0,where(52)∬QTax+δ∇uδεηm2+ηp-2/2∇uδεηm∇φdxdt=∬QTax+δ∇uδεηmp-2∇uδεηm∇φdxdt+p-22η∬QTax+δ∫01∇uδεηm2+sηp-4/2ds∇uδεηm∇φdxdt.limη→0⁡p-22η∬QTax+δ∫01∇uδεηm2+sηp-4/2ds∇uδεηm∇φdxdt=0.Let φ=ψ(uδεηm-uδεm), ψ∈C01(QT).(53)∬QTax+δ∇uδεηmp-2∇uδεηm-∇uδεmp-2∇uδεm·∇φdxdt=∬QTax+δ∇uδεηmp-2∇uδεηm-∇uδεmp-2∇uδεm·∇ψuδεηm-uδεmdxdt+∬QTax+δ∇uδεηmp-2∇uδεηm-∇uδεmp-2∇uδεm·∇uδεηm-uδεmψdxdt.By Hölder’s inequality and (43)-(48), we have(54)limη→0⁡∬QTax+δ∇uδεηmp-2∇uδεηm-∇uδεmp-2∇uδεm·∇ψuδεηm-uδεmdxdt=0.By (49)(55)limη→0∬QTax+δ∇uδεmp-2∇uδεm∇uδεηm-uδεmψdxdt=0.Meanwhile,(56)∬QTax+δ∇uδεηmp-2∇uδεηm∇uδεηm-uδεmψdxdt⩽∬QTax+δ∇uδεηm2+ηp-2/2∇uδεηm·∇uδεηm-uδεmψdxdt⩽-∬QTφdiv⁡ax+δ∇uδεηm2+ηp-2/2∇uδεηmuδεηm-uδεmdxdt-∬QT∇φax+δ∇uδεηm2+ηp-2/2·∇uδεηmuδεηm-uδεmdxdt⩽∬QTφuδεηtuδεηm-uδεmdxdt+C∬QT∇φax+δ∇uδεηmp-2·∇uδεηmuδεηm-uδεmdxdt+Cηp-2/2∬QT∇φax+δ·∇uδεηmuδεηm-uδεmdxdt.By Hölder’s inequality and (48)-(50), we have(57)limη→0⁡∬QTφuδεηtuδεηm-uδεmdxdt+C∬QT∇φax+δ∇uδεηmp-2·∇uδεηmuδεηm-uδεmdxdt+Cηp-2/2∬QT∇φax+δ·∇uδεηmuδεηm-uδεmdxdt=0,and notice that(58)0⩽∬QTax+δ∇uδεηmp-2∇uδεηm-∇uδεmp-2∇uδεm·∇uδεηm-uδεmψdxdt,so(59)limη→0⁡∬QTax+δ∇uδεηmp-2∇uδεηm∇φdxdt=∬QTax+δ∇uδεmp-2∇uδεm∇φdxdt.By (49), uδε=ε, a.e. on ∂Ω×(0,T).Sinceuδεt∈L2(QT), for any 0<t1<t2<T, using the Hölder inequality, we have(60)∫Ωuδεx,t2-uδεx,t1dxdt≤t2-t1∫01∫Ω∂∂tuδεx,st2+1-st1dxds≤t2-t1∫0T∫Ω∂∂tuδεx,tdxdt≤ct2-t1uδεtL2QT,so uδε∈C(0,T;L1(Ω)), and(61)limt→0⁡∫Ωuδεx,t-u0ε+εdx=0.By the above discussion, we can deduce that uδε is a weak solution of problem(62)ut=divax+δ∇ump-2∇um,x,t∈QT,ux,t=ε,x,t∈∂Ω×0,T,ux,0=u0ε+ε,x∈Ω.


## 3.2.ε→0

By Lemmas5–8, {uδε} is uniformly bounded and uniformly continuous on any compact subset of QT, so it has a subsequence still denoted by {uδε}, when ε→0,(63)uδε→uδinCΩ′×τ,T,∇uδεm⇀∇uδminLpΩ×τ,T,uδεt⇀uδtinL1Ω×τ,T.From [29] and similar to the previous proof, uδ is a weak solution of problem(64)ut=div⁡ax+δ∇ump-2∇um,x,t∈QT,ux,t=0,x,t∈∂Ω×0,T,ux,0=u0x,x∈Ω.


## 3.3.δ→0

According to Lemmas5–8, {uδ} has a convergent subsequence still denoted by {uδ},(65)uδ→uinCΩ′×τ,T,ax∇uδmp-2∇uδm⇀ζ→inLp/p-1Ω×τ,T,uδt⇀utinL1Ω×τ,T.By the Hölder inequality and (43),(66)∫t1t2∫Ωδ∇uδmp-2∇uδm·∇φdxdt≤δ1/p∫t1t2∫Ωδ∇uδmpdxdtp-1/p∬suppφ∇φpdxdt1/p≤cδ,that is, (67)limδ→0⁡∫t1t2∫Ωδ∇uδmp-2∇uδm·∇φdxdt=0.Similar to the previous proof, we have(68)∬QTax∇ump-2∇um·∇φdxdt=∬QTζ→·∇φdxdt.At last, the initial value condition (2) satisfied in the sense(69)limt→0⁡ux,t-u0xdx=0,can be proved in a similar way to that for the evolutionary p–Laplacian equation [29]; we omit the details here. Thus u is a weak solution of problem (1)-(2).Lemma 9.
If∫Ω‍a(x)-1/p-1dx⩽c, then ∫Ω‍|∇uδm|dx⩽c.Proof.
(70) ∫ Ω ∇ u δ m d x = ∫ x ∈ Ω : a 1 / p - 1 ∇ u δ m ⩽ 1 ∇ u δ m d x + ∫ x ∈ Ω : a 1 / p - 1 ∇ u δ m > 1 ∇ u δ m d x ⩽ ∫ Ω a x - 1 / p - 1 d x + ∫ Ω a x ∇ u δ m p d x ⩽ C ,where C is a positive constant independent of δ.By this Lemma, we can define the trace ofum (also u) on the boundary. Thus u(x,t) is a weak solution of the initial-boundary value problem of (1).


## 4. Proof of Theorem3

Letφ∈C01(QT); we have(71)∬QTu-vtφdxdt+∬QTax∇ump-2∇um-∇vmp-2∇vm·∇φdxdt=0.Let(72)gns=1s>1n,ns0⩽s⩽1n,0s<0,ξdx=1ax>d,axdax⩽d.DenoteΩd={x∈Ω:a(x)>d}, χj(t)∈C01(0,T), 0⩽χj⩽1, χj→χ(j→∞); χ is characteristic function of (t1,t2).Letφ=ξdχjgn(um-vm) in (71); since u,v∈C(0,T;L1(Ω), a(x)|∇um|∈Lp(Ω×(τ,T)), one has(73)∬QTξdχjgnum-vm∂∂tu-vdxdt+∬QTax∇ump-2∇um-∇vmp-2∇vm∇ξdχjgnum-vmdxdt+∬QTax∇ump-2∇um-∇vmp-2∇vmξdχjgn′∇um-vmdxdt=0.(74)∬QTax∇ump-2∇um-∇vmp-2∇vmξdχjgn′·∇um-vmdxdt⩾0.(75)∬QTax∇ump-2∇um-∇vmp-2∇vm·∇ξdχjgnum-vmdxdt⩽∬QTax∇ump-1+∇vmp-1∇ξdχjgndxdt⩽c∫0T∫Ω∖Ωdax∇ump+ax∇vmp+ax∇ξdpdxdt.Since (76)0⩽limd→0∫Ω∖Ωdax∇ξdpdx⩽limd→0⁡cdp∫Ω∖Ωdax∇apdx=0,one has(77)limd→0⁡∬QTax∇ump-2∇um-∇vmp-2∇vm·∇ξdχjgnum-vmdxdt=0.Letd→0, n→∞ in (73); we have(78)∬QTχjtsgn+u-v∂∂tu-vdxdt⩽0,that is,(79)∬QTχj′tu-v+dxdt⩾0,so(80)∬QTχjt∂∂tu-v+dxdt⩽0.Let j→∞; we have(81)∫t1t2∫Ω∂∂tu-v+dxdt⩽0,so(82)∫Ωux,t2-vx,t2+dx⩽∫Ωux,t1-vx,t1+dx.Since u and v are symmetrical,(83)∫Ωvx,t2-ux,t2+dx⩽∫Ωvx,t1-ux,t1+dx.Let t1→0; we have(84)∫Ωux,t-vx,tdx⩽∫Ωu0x-v0xdx.


## 5. Proof of Theorem4

Proof.
For a small positive constantd>0, define(85)ξdx=1,ifx∈Ωd,axd,ifx∈Ω∖Ωd,as before, where (86)Ωd=x∈Ω:ax>d.Then (87)∇ξd=1d∇a,x∈Ω∖Ωd.
Ifu and v are two weak solutions of (1) with the different initial values u0(x) and v0(x), respectively, we choose gn(ξd(um-vm)) as the test function. One has(88)∫Ωgnξdum-vm∂u-v∂tdx+∫Ωax∇ump-2∇um-∇vmp-2∇vm·ξd∇um-vmhnξdum-vmdx+∫Ωax∇ump-2∇um-∇vmp-2∇vm·∇ξdum-vmhnξdum-vmdx=0.For the terms on the left hand side of (88),(89)limn→∞⁡limd→0⁡∫Ωgnξdum-vm∂u-v∂tdx=limn→∞⁡∫Ωgnum-vm∂u-v∂tdx=∫Ωsignum-vm∂u-v∂tdx=∫Ωsignu-v∂u-v∂tdx=ddt∫Ωu-vdx,(90)∫Ωax∇ump-2∇um-∇vmp-2∇vm·ξd∇um-vmgn′ξdum-vmdx≥0.By the fact that(91)um-vmgn′ξdum-vm=ξdum-vmhnξdum-vm1ξd≤cϕ,(92)∇ξdξd=∇aa,x∈Ω∖Ωd.Using the Young inequality, by (91) (92), one has(93)∫Ωax∇ump-2∇um-∇vmp-2∇vm·∇ξdum-vmhnξdum-vmdx≤∫Ω∖Ωdax∇ump-1+∇vmp-1∇ξdξdξdum-vmhnξdum-vmdx≤c∫Ω∖Ωd∇aaxax∇ump-1+∇vmp-1ξdum-vmhnξdum-vmdx≤c∫Ω∖Ωda1-p-1/px∇aaxap-1/px∇ump-1+∇vmp-1dx≤c∫Ω∖Ωdax∇ump+∇vmp+a-p-1x∇apdx,which goes to 0 as d→0, since it is assumed that (94)∫Ωa-p-1x∇apdx<∞.
Now, after lettingd→0, let n→∞ in (88). Then, by (89), (90), and (93), one has (95)ddt∫Ωu-vdx⩽0,one has(96)∫Ωux,t-vx,tdx⩽∫Ωu0x-v0xdx,∀t∈0,T,and thus Theorem 4 is proved.



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*Source: 6525637-2018-09-16.xml*