# Existence of Solutions for the Evolutionp(x)-Laplacian Equation Not in Divergence Form

Authors: Changchun Liu; Junchao Gao; Songzhe Lian
Journal: Journal of Applied Mathematics
 (2012)

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## Abstract

The existence of weak solutions is studied to the initial Dirichlet problem
of the equationut=udiv(|∇u|p(x)−2∇u), with inf p(x)>2. We adopt the method of parabolic regularization. After establishing some necessary uniform estimates on the approximate solutions, we prove the existence of weak solutions.


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## Body

## 1. Introduction

In this paper, we investigate the existence of solutions for thep(x)-Laplacian equation(1.1)∂u∂t=udiv⁡(|∇u|p(x)-2∇u),(x,t)∈QT.
The equation is supplemented the boundary condition(1.2)u(x,t)=0,x∈∂Ω,
and the initial condition(1.3)u(x,0)=u0(x),x∈Ω,
where QT=Ω×(0,T), infp(x)>2, Ω⊂RN is a bounded domain with smooth boundary ∂Ω and 0≤u0(x)∈C(Ω¯)∩W01,p(x)(Ω).In the case whenp is a constant, there have been many results about the existence, localization and extendibility and of weak solutions. We refer the readers to the bibiography given in [1–5] and the references therein.A new interesting kind of fluids of prominent technological interest has recently emerged, the so-called electrorheological fluids. This model includes parabolic equations which are nonlinear with respect to the gradient of the thought solution, and with variable exponents of nonlinearity. The typical case is the so-called evolutionp-Laplace equation with exponent p as a function of the external electromagnetic field (see [6–12] and the references therein). In [6], the authors studied the regularity for the parabolic systems related to a class of non-Newtonian fluids, and the equations involved are nondegenerated.On the other hand, there are also many results to the corresponding ellipticp(x)-Laplace equations [13–15].In the present work, we will study the existence of the solutions to problem (1.1)–(1.3). As we know, when p is a constant, the nondegenerate problems have classical solutions, and hence the weak solutions exist. But in the case of p(x)-Laplace type, there are no results to the corresponding non-degenerate problems. Since (1.1) degenerates whenever u=0 and ∇u=0, we need to regularize the problem in two aspects corresponding to two different degeneracy: the first is the initial and boundary value and the second is the equation. We will first consider the non-degenerate problems. Based on the uniform Schauder estimates and using the method of continuity, we obtain the existence of classical solutions for non-degenerate problems. After establishing some necessary uniform estimates on the approximate solutions, we prove the existence of weak solutions.This paper is arranged as follows. We first state some auxiliary lemmas in Section2, and then we study a general quasilinear equation in Section 3. Subsequently, we discuss the existence of weak solutions in Section 4.


## 2. Preliminaries

Denote that(2.1)p+=esssup⁡Ω¯p(x),p-=essinf⁡Ω¯p(x).Throughout the paper, we assume that(2.2)2<p-≤p(x)≤p+<∞,∀(x,t)∈Ω×[0,T],
where p-,p+ are given constants.To study our problems, we need to introduce some new function spaces. Denote that(2.3)Lp(x)(Ω)={u:uisameasurablereal-valuedfunction,∫Ω|u(x)|p(x)dx<∞},|u|p(x)=inf⁡{λ:λ>0,∫Ω|u(x)λ|p(x)dx≤1},Lp(x)(QT)={u:|u|p(x)∈L1(QT)},Lp(x)(0,T;W01,p(x)(Ω))={u:|u|p(x)∈L1(QT),|∇u|p(x)∈L1(QT),u|∂Ω=0},C2k+α,k+α/2(QT)={u:u∈C2k,k(QT),DtrDxsu∈Cα,α/2,2r+s=2k,0<α≤1},C2k,k(QT)={u:DtrDxsu∈C(QT),0≤2r+s≤2k},W1,p(x)(Ω)={u:u∈Lp(x)(Ω),|∇u|∈Lp(x)(Ω)},|u|1,p(x)=|u|Lp(x)(Ω)+|∇u|Lp(x)(Ω),∀u∈W1,p(x)(Ω).
We use W01,p(x)(Ω) to denote the closure of C0∞(Ω) in W1,p(x).Remark 2.1.
In [16, 17], Zhikov showed
(2.4)W01,p(x)(Ω)≠{v:v∈W1,p(x)(Ω),v|∂Ω=0}.
Hence, the property of the space is different from the case when p is a constant. This will bring us some difficulties in taking the limit of the weak solutions. Luckily, our approximating solutions are in W01,p(x), and hence the limit function is also in W01,p(x) which avoids the above difficulties.We now give the definition of the solutions to our problem.Definition 2.2.
A nonnegative functionu∈L∞(QT), |∇u|∈Lp(x)(QT), and ut∈L2(QT) is said to be a weak solution of (1.1)–(1.3), if for all φ∈C0∞(Q¯T) satisfies the following:
(2.5)∬QT(uφt-u|∇u|p(x)-2∇u∇φ-|∇u|p(x)φ)dxdt=0,lim⁡t→0∫Ω|u(x,t)-u0(x)|dx=0.In the following, we state some of the properties of the function spaces introduced as above.Proposition 2.3 (see [15, 18]).
(i)
The space(Lp(x)(Ω),|·|p(x)) is a separable, uniform convex Banach space, and its conjugate space is Lq(x)(Ω), where 1/p(x)+1/q(x)=1. For any u∈Lp(x)(Ω) and v∈Lq(x)(Ω), we have
(2.6)|∫Ωuvdx|≤(1p-+1q-)|u|p(x)|v|q(x).(ii)
Ifp1,p2∈C+(Ω¯),p1(x)≤p2(x) for any x∈Ω¯, then Lp2(x)↪Lp1(x) and the imbedding continuous.(iii)
There is a constantC>0, such that
(2.7)|u|p(x)≤C|∇u|p(x),∀u∈W01,p(x)(Ω).
This implies that |∇u|p(x) and ∥u∥1,p(x) are equivalent norms of W01,p(x).(iv)
We have∫Ω|u|p(x)dx≥|u|p(x)p--1,for allu∈Lp(x)(Ω).Proposition 2.4 (see [18]).
If we denote(2.8)ρ(u)=∫Ω|u|p(x)dx,∀u∈Lp(x)(Ω),
then(i)
|u|p(x)<1(=1;>1)⇔ρ(u)<1(=1;>1),(ii)
|u|p(x)>1⇒|u|p(x)p-≤ρ(u)≤|u|p(x)p+,|u|p(x)<1⇒|u|p(x)p-≥ρ(u)≥|u|p(x)p+,(iii)
|u|p(x)→0⇔ρ(u)→0;|u|p(x)→∞⇔ρ(u)→∞.Lemma 2.5 (see [4]).
Letθ≥0,A(η)=(A1(η),…,AN(η))=(|η|2+θ)(p-2)/2,η=(η1,…,ηN)∈RN. Then
(2.9)[A(η)-A(η′)]⋅[η-η′]≥C|η-η′|p,∀η,η′∈RN,
where C=C(p) is a positive constant depending on p.


## 3. A General Quasilinear Equation

Here, we will consider the general quasilinear equations(3.1)ut-aij(x,t,u,ux)uxi,xj+a(x,t,u,ux)=0,(3.2)u|ΓT=ψ|ΓT,
where ΓT=∂Ω×(0,T]∪Ω¯×{0}.Proposition 3.1 (see [19, Theorem  2.9 of Chapter I]).
Letu(x,t) be a classical solution of (3.1) in QT. Suppose that the functions aij(x,t,u,p) and a(x,t,u,t) take finite value for any finite u,p, and (x,t)∈Q¯T, and that for (x,t)∈QT and arbitrary u(3.3)aij(x,t,u,0)ξiξj≥0,ua(x,t,u,0)≥-b1u2-b2,
where b1 and b2 are nonnegative constants. Then
(3.4)max⁡QT|u(x,t)|≤M,
where M depends only on b1,b2,T, and maxΓT|u|.We suppose that for(x,t)∈Q¯T,maxQT|u(x,t)|≤M and arbitrary q the functions aij(x,t,u,q),a(x,t,u,q) are continuous in x,t,u,q, continuously differentiable with respect to x,u, and q, and satisfy the inequalities(3.5)ν(1+|q|)m(x,t)-2ξ2≤aij(x,t,u,q)ξiξj≤μ0(1+|q|)m(x,t)-2ξ2,|∂aij∂pk|(1+|q|)3+|a|+|∂a∂qk|(1+|q|)≤μ1(1+|q|)m(x,t),|∂aij∂xk|(1+|q|)2+|∂a∂xk|≤(ɛ+P(|q|))(1+|q|)m(x,t)+1,|∂aij∂u|≤(ɛ+P(|q|))(1+|q|)m(x,t)-2,-∂a∂u≤(ɛ+P(|q|))(1+|q|)m(x,t),
where P(ρ) is a nonnegative continuous function that tends to zero for ρ→∞ and 1<m(x,t)∈C1(Q¯T) is an arbitrary function.Lemma 3.2.
Letu(x,t) be a classical solution of (3.1) in QT. Suppose that the conditions of Proposition 3.1 hold and satisfy (3.5) with a sufficiently small ɛ determined by the numbers M, ν, μ, μ1, and
(3.6)P̂=max⁡ρ≥0P(ρ).
Then
(3.7)max⁡QT|ux(x,t)|≤M1.The proof of Lemma3.2 is quite similar to the Theorem  4.1, chapter VI of [19]; one only has to replace m with m(x,t) and remark that the constants in the proof are depending only on infm(x,t) and supm(x,t); we omit the details.Theorem 3.3.
Suppose that the following conditions are fulfilled.(a)
For(x,t)∈Q¯T and arbitrary u either conditions (3.3) are fulfilled.(b)
For(x,t)∈Q¯T, |u|≤M (where M is taken from estimate (3.4)) and arbitrary p, the functions aij(x,t,u,p) and a(x,t,u,p) are continuous and differentiable with respect to x, u, and p and satisfy inequalities (3.5) with a sufficiently small ε determined by the numbers M, ν, μ, μ1, and
(3.8)P̂=max⁡ρ≥0P(ρ).(c)
For(x,t)∈Q¯T|u|≤M and |p|≤M1 (where M1 is taken from estimate (3.7)), the functions aij(x,t,u,p) and a(x,t,u,p) are continuously differentiable with respect to all of their arguments.(d)
The boundary condition (3.2) is given by a function ψ(x,t) belonging to C2+β,1+β/2(Q¯T) and satisfying on S0={(x,t):x∈∂Ω,t=0} (3.1), that is,
(3.9)ψt-aij(x,0,ψ(x,0),ψx(x,0))ψxixj+a(x,0,ψ(x,0))|x∈∂Ω=0
(in other words, the compatibility conditions of zero and first orders are assumed to be fulfilled).(e)
We have∂Ω∈C2+β.
Then there exists a unique solution of problem (3.1) and (3.2) in the space C2+β,1+β/2(Q¯T). This solution has derivatives utxi from L2(QT).Proof.
We consider problem (3.1) and (3.2) along with a one-parameter family of problems of the same type
(3.10)Lτu=ut-[τaij(x,t,u,ux)+(1-τ)(1+ux2)m(x)/2-1δij]uxixj+τa(x,t,u,ux)-(1-τ)[ψt-(1+ψx2)m(x)/2-1Δψ]=0,u|ΓT=ψ|ΓT,0≤τ≤1,assumingψ∈C2+β,1+β/2(Q¯T).
Define the Banach space
(3.11)X={w∈C1+α,(1+α)/2(QT¯)|w|ΓT=0}.
For any w∈X, let v=w+ψ. Using Schauder theory, the linear problem
(3.12)vt-[τaij(x,t,w,wx)+(1-τ)(1+wx2)m(x)/2-1δij]vxixj+τa(x,t,w,wx)-(1-τ)[ψt-(1+ψx2)m(x)/2-1Δψ]=0,u|ΓT=ψ|ΓT
admits a unique solution u∈C2+α,1+α/2(Q¯T). Let z=u-ψ, clearly z∈X, and define the map G:X→X such that z=G(w). By [19], we know that G is continuous and compact. By Proposition 3.1, Lemma 3.2, and the Leray-Schauder fixed point principle, the operator G has a fixed point u.


## 4. Existence

In this section, we are going to prove the existence of solutions of the problem (1.1)–(1.3).Theorem 4.1.
Assume thatp(x)∈C1(Ω¯),infp(x)>2, and 0≤u0(x)∈C(Ω¯)∩W01,p(x)(Ω). Then the problem (1.1)–(1.3) admits a weak solution u.Consider the following problem:(4.1)∂uɛ,η∂t=uɛ,ηdiv⁡((|∇uɛ,η|2+η)(p(x)-2)/2∇uɛ,η),(4.2)uɛ,η|ST=ɛ,uɛ,η|t=0=u0+ε,
where ST=∂Ω×(0,T], ε∈(0,1), and η∈(0,ɛ). Roughly speaking, here we use to regularize the initial-boundary value and use η to regularize the equation. Thus, we have to carry out two limit processes, that is, first let η→0 (along a certain subsequence) and then let ɛ→0.We first change (4.1) into the form(4.3)ut-aij(x,t,u,ux)uxi,xj+a(x,t,u,ux)=0,
where(4.4)aij(x,t,u,ux)=uδij(|∇u|2+η)(p(x)-2)/2+u(p(x)-2)(|∇u|2+η)(p(x)-4)/2∂iu∂ju,a(x,t,u,ux)=u12(|∇u|2+η)(p(x)-2)/2∂ip(x)∂iuln⁡(|∇u|2+η).
It is easily seen that (4.4) satisfies (3.3) and (3.5), where p(x) instead of m(x,t). By Theorem 3.3, we know that (4.1)-(4.2) has a classical solution uɛ,η.Proposition 4.2.
We have(4.5)ɛ≤uɛ,η≤|u0|∞+ε,uε1,η≤uε2,η,forɛ1≤ɛ2.Proof.
By the maximum principle, we know thatɛ≤uɛ,η≤|u0|∞+ε.
A simple calculation shows that(4.6)∂uɛ1,η∂t-aij(x,t,uε1,η,∇uε1,η)∂2uε1,η∂xi∂xj+a(x,t,uε1,η,∇uε1,η)=0,∂uε2,η∂t-aij(x,t,uε2,η,∇uε2,η)∂2uε2,η∂xi∂xj+a(x,t,uε2,η,∇uε2,η)=0,
where aij(x,t,u,ux) and a(x,t,u,ux) are defined as (4.4).
It is easy to prove that(4.7)ξiaij(x,t,uεi,η,∇uεi,η)ξj≥εiη(p(x)-2)/2|ξ|2,∀ξ∈RN.
Hence, we have
(4.8)∂uε1,η∂t-∂uε2,η∂t=aij(x,t,uε1,η,∇uε1,η)∂2(uε1,η-uε2,η)∂xi∂xj+[aij(x,t,uε1,η,∇uε1,η)-aij(x,t,uε2,η,∇uε2,η)]∂2uε2,η∂xi∂xj+[a(x,t,uε1,η,∇uε1,η)-a(x,t,uε2,η,∇uε2,η)].
Let w=uε1,η-uε2,η,aij(x,t)=aij(x,t,uε1,η,∇uε1,η). Using the mean value theorem, we have
(4.9)[aij(x,t,uε1,η,∇uε1,η)-aij(x,t,uε2,η,∇uε2,η)]∂2uε2,η∂xi∂xj=[aij(x,t,uε1,η,∇uε1,η)-aij(x,t,uε1,η,∇uε2,η)]∂2uε2,η∂xi∂xj+[aij(x,t,uε1,η,∇uε2,η)-aij(x,t,uε2,η,∇uε2,η)]∂2uε2,η∂xi∂xj=dk(x,t)∂w∂xk+e(x,t)w.
Similarly, we get
(4.10)a(x,t,uε1,η,∇uε1,η)-a(x,t,uε2,η,∇uε2,η)=fk(x,t)∂w∂xk+g(x,t)w,
where dk(x,t),e(x,t),fk(x,t), and g(x,t) are bounded functions. Hence, we see that
(4.11)∂w∂t=aij(x,t)∂2w∂xi∂xj+bk(x,t)∂w∂xk+c(x,t)w.
Since w≤0 on ΓT, by comparison principle of linear parabolic equation, we have w≤0.Lemma 4.3.
For allα∈[0,1) and η∈(0,ε), there hold
(4.12)(1)∬QT(|∇uɛ,η|2+η)p(x)/2uε,ηαdxdt≤C,(2)∬QT(∂uɛ,η∂t)2dxdt≤C.Proof.
Multiplying (4.1) by uɛ,η-α, integrating both sides of the equality over QT and integrating by parts, we derive
(4.13)∬QT∂uε,η∂tuɛ,η-αdxdt=11-α∫Ω(uε,η1-α(x,T)-uɛ,η1-α(x,0))dx=∬QTuɛ,η1-αdiv⁡((|∇uɛ,η|2+η)(p(x)-2)/2∇uɛ,η)dxdt=∫0T∫∂Ω[uɛ,η1-α(|∇uɛ,η|2+η)(p(x)-2)/2∂uɛ,η∂ν]dσdt-(1-α)∬QT(|∇uɛ,η|2+η)(p(x)-2)/2|∇uɛ,η|2uɛ,η-αdxdt,
where ν denotes the outward normal to ∂Ω×(0,T). Since from (4.5), uɛ,η≥ε, we have ∂uɛ,η/∂ν≤0 on ∂Ω×(0,T). Hence
(4.14)∬QT(|∇uɛ,η|2+η)(p(x)-2)/2|∇uɛ,η|2uɛ,η-αdxdt≤1(1-α)2∫Ωuɛ,η1-α(x,0)dx≤C,
where C=C(α,Ω,|u0|∞).
Usinguɛ,η≥ε>η and Young’s inequality, we have
(4.15)η∬QT(|∇uε,η|2+η)(p(x)-2)/2uε,η-αdxdt≤∬QT(|∇uε,η|2+η)(p(x)-2)/2uε,η(p(x)-2)α/p(x)η2α/p(x)uε,η2α/p(x)dxdt≤12∬QT(|∇uε,η|2+η)p(x)/2uε,η-αdxdt+C(p-,p+,T,Ω).
Combining (4.14) with (4.15) yields
(4.16)∬QT(|∇uε,η|2+η)p(x)/2uε,η-αdxdt=∬QT(|∇uε,η|2+η)(p(x)-2)/2uε,η-α(|∇uε,η|2+η)dxdt≤η∬QT(|∇uε,η|2+η)(p(x)-2)/2uε,η-αdxdt+C≤12∬QT(|∇uε,η|2+η)p(x)/2uε,η-αdxdt+C(p,T,Ω).
Hence,
(4.17)∬QT(|∇uε,η|2+η)p(x)/2uε,η-αdxdt≤C.
Multiplying (4.1) by (∂uɛ,η/∂t)uε,η-1, integrating both sides of the equality over QT and integrating by parts and noticing that (uε,η)t=0 on ∂Ω×(0,T), we derive
(4.18)∬QT(∂uε,η∂t)2uɛ,η-1dxdt=∬QTdiv⁡((|∇uɛ,η|2+η)(p(x)-2)/2∇uɛ,η)∂uɛ,η∂tdxdt=∬QTdiv⁡((|∇uɛ,η|2+η)(p(x)-2)/2∇uɛ,η∂uɛ,η∂t)dxdt-∬QT((|∇uɛ,η|2+η)(p(x)-2)/2∇uɛ,η)∇(∂uɛ,η∂t)dxdt=-∬QT((|∇uɛ,η|2+η)(p(x)-2)/2∇uɛ,η)∇(∂uɛ,η∂t)dxdt=-∫Ω1p(x)(|∇uɛ,η(x,T)|2+η)p(x)/2dx+∫Ω1p(x)(|∇uɛ,η(x,0)|2+η)p(x)/2dx≤∫Ω1p(x)(|∇u0|2+η)p(x)/2dx.Equation (4.5), Lemma 4.3, and Proposition 2.3 imply that, for any ɛ∈(0,1), there exists a subsequence of uɛ,η, denoted by uɛ,ηk, and a function uɛ∈L∞(QT)  ∇u∈Lp(x)(QT), such that, as η=ηk→0,(4.19)uɛ,η⟶uɛ,a.e.inQT,(4.20)∇uɛ,η⟶∇uɛ,weakly inLp(x)(QT),(4.21)∂uɛ,η∂t⟶∂uɛ∂t,weakly inL2(QT).Lemma 4.4.
Asη=ηk→0, we have
(4.22)(1)∬QT|∇uɛ,η-∇uɛ|p(x)dxdt⟶0,(2)∬QT||∇uɛ,η|p(x)-|∇uɛ|p(x)|dxdt⟶0,(3)∬QT|(|∇uɛ,η|2+η)(p(x)-2)/2|∇uɛ,η|2-|∇uɛ|p(x)|dxdt⟶0,(4)∬QT|uɛ,η(|∇uɛ,η|2+η)(p(x)-2)/2∇uɛ,η-uɛ|∇uɛ|p(x)-2∇uɛ|dxdt⟶0.Proof.
Observe that(uɛ,η-uɛ)/uɛ,η∈Lp(x)(0,T;W01,p(x)(Ω)). Multiplying (4.1) by (uɛ,η-uɛ)/uɛ,η, integrating both sides of the equality over QT and integrating by parts, we derive
(4.23)∬QT(∂uɛ,η∂tuɛ,η-uɛuɛ,η+(|∇uɛ,η|2+η)(p(x)-2)/2∇uɛ,η∇(uɛ,η-uɛ))dxdt=0.
By Hölder inequality and Lemma 4.3, we obtain
(4.24)∬QT∂uɛ,η∂tuɛ,η-uɛuɛ,η⟶0,(η⟶0).
Hence,
(4.25)∬QT(|∇uɛ,η|2+η)(p(x)-2)/2∇uɛ,η∇(uɛ,η-uɛ)dxdt⟶0.
We divide the integral in (4.25) in the following way:
(4.26)∬QT(|∇uɛ,η|2+η)(p(x)-2)/2∇uɛ,η∇(uɛ,η-uɛ)dxdt=∬QT[(|∇uɛ,η|2+η)(p(x)-2)/2∇uɛ,η-(|∇uɛ|2+η)(p(x)-2)/2∇uɛ]⋅∇(uɛ,η-uɛ)dxdt+∬QT[(|∇uɛ|2+η)(p(x)-2)/2-|∇uɛ|(p(x)-2)]∇uɛ∇uɛ,ηdxdt+∬QT[|∇uɛ|(p(x)-2)-(|∇uɛ|2+η)(p(x)-2)/2]|∇uɛ|2dxdt+∬QT|∇uɛ|(p(x)-2)∇uɛ∇(uɛ,η-uɛ)dxdt=I1+I2+I3+I4.
From (4.20), we see that
(4.27)I4⟶0,(η⟶0).
Using Lemma 4.3, we have
(4.28)|I3|≤∬QT[(∇uɛ|2+η)p(x)/2-|∇uɛ|p(x)]dxdt≤η2∬QTp(x)(∇uɛ|2+η)(p(x)-2)/2dxdt⟶0,(η⟶0).
Now we estimate I2. If p(x)∈(2,3], then (p(x)-1)/2∈(0,1]. Using |ar-br|≤|a-b|r (r∈[0,1],a,b≥0) gives
(4.29)|I2|≤∬QT|(|∇uɛ|2+η)(p(x)-1)/2-|∇uɛ|p(x)-1||∇uɛ,η|dxdt≤η(p--1)/2∬QT|∇uɛ,η|dxdt⟶0,(η⟶0).
If p(x)>3, we obtain
(4.30)|I2|≤η2∬QT(p(x)-1)(|∇uɛ|2+η)(p-3)/2|∇uɛ,η|dxdt⟶0,(η⟶0).
By (4.25), (4.26), and I2,I3,I4→0, we obtain
(4.31)I1=∬QT[(|∇uɛ,η|2+η)(p(x)-2)/2∇uɛ,η-(|∇uɛ|2+η)(p(x)-2)/2∇uɛ]⋅∇(uɛ,η-uɛ)dxdt⟶0,(η⟶0).
Again by Lemma 2.5, we get
(4.32)I1=∬QT[(|∇uɛ,η|2+η)(p(x)-2)/2∇uɛ,η-(|∇uɛ|2+η)(p(x)-2)/2∇uɛ]⋅∇(uɛ,η-uɛ)dxdt≥C∬QT|∇uɛ,η-∇uɛ|p(x)dxdt.
Letting η→0, we obtain (1). Again noticing that
(4.33)∬QT||∇uɛ,η|p(x)-|∇uɛ|p(x)|dxdt≤∬QTp(x)(|∇uɛ,η|+|∇uɛ|)p(x)-1||∇uɛ,η|-|∇uɛ||dxdt≤∬QTp(x)(|∇uɛ,η|+|∇uɛ|)p-1|∇uɛ,η-∇uɛ|dxdt≤Cp+||∇uɛ,η|+|∇uɛ||p(x)|∇uɛ,η-∇uɛ|p(x)≤C|∇uɛ,η-∇uɛ|p(x),
by Proposition 2.4, we see that (2) holds. To prove (3), we have
(4.34)∬QT|(|∇uɛ,η|2+η)(p(x)-2)/2|∇uɛ,η|2-|∇uɛ|p(x)|dxdt≤∬QT|(|∇uɛ,η|2+η)(p(x)-2)/2-|∇uɛ,η|p(x)-2||∇uɛ,η|2dxdt+∬QT||∇uɛ,η|p(x)-|∇uɛ|p(x)|dxdt≤∬QT|(|∇uɛ,η|2+η)p(x)/2-|∇uɛ,η|p(x)|dxdt+∬QT||∇uɛ,η|p(x)-|∇uɛ|p(x)|dxdt≤η2∬QTp(x)(|∇uɛ,η|2+η)(p(x)-2)/2dxdt+∬QT||∇uɛ,η|p(x)-|∇uɛ|p(x)|dxdt.
Using Lemma 4.3 and (2), we see that (3) holds.
Finally, we prove (4). We have(4.35)∬QT|uɛ,η(|∇uɛ,η|2+η)(p(x)-2)/2∇uɛ,η-uɛ|∇uɛ|p(x)-2∇uɛ|dxdt≤∬QT|uɛ,η-uɛ|(|∇uɛ,η|2+η)(p(x)-2)/2|∇uɛ,η|dxdt+∬QTuɛ(|∇uɛ,η|2+η)(p(x)-2)/2|∇uɛ,η-∇uɛ|dxdt+∬QTuɛ|(|∇uɛ,η|2+η)(p(x)-2)/2-|∇uɛ|p(x)-2||∇uɛ|dxdt=Ia+Ib+Ic.
Equation (4.19) implies that
(4.36)Ia⟶0,(η⟶0).
To estimate Ib, notice that
(4.37)Ib≤C||∇uɛ,η|2+η|(p(x)-2)p(x)/2(p(x)-1)|∇uɛ,η-∇uɛ|p(x)⟶0,(η⟶0).
As p(x)∈(2,3], we have (p(x)-2)/2∈(0,1],
(4.38)Ic≤C∬QT||∇uɛ,η|2-|∇uɛ|2+η|(p(x)-2)/2|∇uɛ|dxdt≤C∬QT2(p(x)-2)/2(||∇uɛ,η|2-|∇uɛ|2|(p(x)-2)/2+η|(p-2)/2)|∇uɛ|dxdt≤C∬QT2(p(x)-2)/2|∇uɛ,η-∇uɛ|(p(x)-2)/2(|∇uɛ,η|+|∇uɛ|)p(x)/2dxdt+C∬QT(2η)(p(x)-2)/2|∇uɛ|dxdt.
By Hölder inequality, we have
(4.39)Ic⟶0,(η⟶0).
If p(x)>3, we have
(4.40)Ic≤C∬QT(p(x)-2)((|∇uɛ,η|2+η)(p-3)/2+|∇uɛ|p(x)-3)⋅|(|∇uɛ,η|2+η)1/2-|∇uɛ|||∇uɛ|dxdt≤C∬QT(p(x)-2)((|∇uɛ,η|2+η)(p-3)/2+|∇uɛ|p(x)-3)⋅||∇uɛ,η|2+η-|∇uɛ|2|1/2|∇uɛ|dxdt≤C21/2∬QT(p(x)-2)((|∇uɛ,η|2+η)(p-3)/2+|∇uɛ|p(x)-3)⋅(|∇uɛ,η|+|∇uɛ|)3/2|∇(uɛ,η-uɛ)|dxdt+C(2η)1/2∬QT(p(x)-2)((|∇uɛ|2+η)(p(x)-3)/2+|∇uɛ|p(x)-3)|∇uɛ|dxdt.
Hence,
(4.41)Ic⟶0,(η⟶0).
Thus (4) is proved, and the proof of Lemma 4.4 is complete.Proposition 4.5.
We obtain thatuɛ is a weak solution of the problem
(4.42)∂u∂t=udiv⁡(|∇u|p(x)-2∇u),u|ST=ɛ,u|t=0=u0+ɛ,
then
(4.43)ɛ≤uɛ≤|u0|∞+ɛ,uɛ1≤uɛ2,(ɛ1≤ɛ2),a.e.inQT,(4.44)∬QT|∇uɛ|pdxdt≤C,∬QT(∂uɛ∂t)2dxdt≤C,
where C is independent of ɛ.Proof.
Obviously, for allɛ∈(0,1), uɛ-ɛ∈Lp(x)(0,T;W1,p(x)(Ω)). By Proposition 4.2 and (4.19)–(4.21), we know that (4.43) holds. (4.44) follows from (4.5), (4.19)–(4.21), and Lemma 4.3. To prove that u satisfies the integral equality in Definition 2.2, we multiply (4.1) by φ∈C0∞(QT), integrate both sides of the equality on QT, and integrate by parts to derive
(4.45)∬QT[-uɛ,ηφt+uɛ,η(|∇uɛ,η|2+η)(p(x)-2)/2∇uɛ,η∇φ+(|∇uɛ,η|2+η)(p(x)-2)/2|∇uɛ,η|2φ]dxdt=0.
Letting η=ηk→0 to pass to limit and using (4.19) and Lemma 4.4 show that
(4.46)∬QT[-uɛφt+uɛ|∇uɛ|(p(x)-2)∇uɛ∇φ+|∇uɛ|p(x)φ]dxdt=0.
Applying Lemma 4.3, we derive
(4.47)∫Ω|uɛ,η-u0(x)-ɛ|dx≤Ct1/2,
where C is independent of ɛ and η. Hence,
(4.48)∫Ω|uɛ-u0(x)-ɛ|dx⟶0,(t⟶0).From (4.43), we see that u is bounded and increasing in ɛ, which implies the existence of a function u, such that, as ɛ→0,(4.49)uɛ⟶u,a.e.inQT,(4.50)∇uɛ⟶∇u,weakly inLp(x)(QT),(4.51)∂uɛ∂t⟶∂u∂t,weakly inL2(QT).Lemma 4.6.
Asɛ→0, we have
(4.52)(1)∬QT|∇uɛ2-∇u2|p(x)dxdt⟶0,(2)∬Qcɛ|∇uɛ-∇u|p(x)dxdt⟶0,Qcɛ={(x,t)∈QT,uɛ≥c,c>0},(3)∬Qc|∇uɛ-∇u|p(x)dxdt⟶0,Qc={(x,t)∈QT,u≥c,c>0}.Proof.
We may takeφ=uɛp(x)-2(uɛ2-ɛ2-u2), in the integral equality satisfied by u. Then it is easy to see that
(4.53)∬QT∂uɛ∂tuɛp(x)-2(uɛ2-ɛ2-u2)dxdt=-∬QTuɛ|∇uɛ|p(x)-2∇uɛ∇[uɛp(x)-2(uɛ2-ɛ2-u2)]dxdt-∬QT|∇uɛ|p(x)uɛp(x)-2(uɛ2-ɛ2-u2)dxdt=-∬QT(p(x)-1)|∇uɛ|p(x)uɛp(x)-2(uɛ2-u2)dxdt+ɛ2∬QT(p(x)-1)|∇uɛ|p(x)uɛp(x)-2dxdt-∬QT|∇uɛ|p(x)-2∇uɛ∇(uɛ2-u2)uɛp(x)-1dxdt-∬QT|∇uɛ|p(x)-2∇uɛuɛp(x)-1∇p(x)ln⁡uɛ(uɛ2-ɛ2-u2)dxdt.
Hence, by (4.43), we have
(4.54)∬QT∂uɛ∂tuɛp(x)-2(uɛ2-ɛ2-u2)dxdt≤ɛ2∬QT(p(x)-1)|∇uɛ|p(x)uɛp(x)-2dxdt-∬QT|∇uɛ|p(x)-2∇uɛ∇(uɛ2-u2)uɛp(x)-1dxdt-∬QT|∇uɛ|p(x)-2∇uɛuɛp(x)-1∇p(x)ln⁡uɛ(uɛ2-ɛ2-u2)dxdt.
Notice that
(4.55)∬QT|∇uɛ|p(x)-2∇uɛ∇(uɛ2-u2)uɛp(x)-1dxdt=∬QT(12)p(x)-1|∇uɛ2|p-2∇uɛ2∇(uɛ2-u2)dxdt.
So
(4.56)∬QT∂uɛ∂tuɛp(x)-2(uɛ2-ɛ2-u2)dxdt+∬QT(12)p(x)-1|∇uɛ2|p(x)-2∇uɛ2∇(uɛ2-u2)dxdt≤ɛ2∬QT(p(x)-1)|∇uɛ|p(x)uɛp(x)-2dxdt-∬QT|∇uɛ|p(x)-2∇uɛuɛp(x)-1∇p(x)ln⁡uɛ(uɛ2-ɛ2-u2)dxdt.
Hence,
(4.57)∬QT∂uɛ∂tuɛp(x)-2(uɛ2-ɛ2-u2)dxdt+∬QT(12)p(x)-1[|∇uɛ2|p(x)-2∇uɛ2-|∇u2|p(x)-2∇u2]∇(uɛ2-u2)dxdt≤-∬QT(12)p(x)-1|∇u2|p(x)-2∇u2∇(uɛ2-u2)dxdt+ɛ2∬QT(p(x)-1)|∇uɛ|p(x)uɛp(x)-2dxdt-∬QT|∇uɛ|p(x)-2∇uɛuɛp(x)-1∇p(x)ln⁡uɛ(uɛ2-ɛ2-u2)dxdt≤-∬QT(12)p(x)-1|∇u2|p(x)-2∇u2∇(uɛ2-u2)dxdt+ɛ2∬QT(p(x)-1)|∇uɛ|p(x)uɛp(x)-2dxdt+C∬QT|∇uɛ|p(x)-1uɛp(x)(uɛ2-ɛ2-u2)dxdt.
By (4.43), (4.44), and (4.49), we see that as ɛ→0,
(4.58)∇uɛ2⟶∇u2,weakly inLp(x)(QT),∬QT∂uɛ∂tuɛp(x)-2(uɛ2-ɛ2-u2)dxdt⟶0,∬QT∂uɛ∂t|∇uɛ|p(x)-1uɛp(x)(uɛ2-ɛ2-u2)dxdt⟶0,(ɛ⟶0).
Therefore,
(4.59)lim⁡ɛ→0¯∬QT[|∇uɛ2|p(x)-2∇uɛ2-|∇u2|p(x)-2∇u2]∇(uɛ2-u2)dxdt≤0.
By Lemma 2.5,
(4.60)∬QT[|∇uɛ2|p(x)-2∇uɛ2-|∇u2|p(x)-2∇u2]∇(uɛ2-u2)dxdt≥∬QT|∇uɛ2-∇u2|p(x)dxdt.
Hence,
(4.61)lim⁡ɛ→0¯∬QT|∇uɛ2-∇u2|p(x)dxdt≤0,
that is,
(4.62)∬QT|∇uɛ2-∇u2|p(x)dxdt⟶0,(ɛ⟶0).
Applying (a+b)p≤2p(ap+bp), (a,b>0), and 2uɛ∇(uɛ-u)=∇(uɛ2-u2)-2(uɛ-u)∇u, we have
(4.63)∬QTuɛp(x)|∇uɛ-∇u|p(x)dxdt≤∬QT|∇uɛ2-∇u2|pdxdt+∬QT2p(x)|∇uɛ-∇u|p(x)|∇u|p(x)dxdt.
Equation (4.50) and (1) imply that the right side tends to zero as ɛ→0. Since uɛ≥c in Qcɛ, (2) is proved. (3) is an immediate consequence of (2).Lemma 4.7.
For anyα∈[0,1), we have
(4.64)∬QT|∇uɛ|p(x)uɛ-αdxdt≤C,
where C is independent of ɛ.Proof.
From Lemmas4.3 and 4.4, it is easily seen that
(4.65)∬QT|∇uɛ,η|p(x)uɛ,η-αdxdt≤C,(4.66)∬QT||∇uɛ,η|p(x)-|∇uɛ|p(x)|dxdt⟶0,(η=ηk⟶0).
Using uɛ,η,uɛ≥ɛ, (4.66), and Proposition 4.5, we have
(4.67)∬QT||∇uɛ,η|p(x)uɛ,η-α-|∇uɛ|p(x)uɛ-α|dxdt≤∬QT||∇uɛ,η|p(x)-|∇uɛ|p(x)|uɛ,η-αdxdt+∬QT|∇uɛ|p(x)|uɛ,η-α-uɛ-α|uɛ,η-αuɛ-αdxdt≤1ɛα∬QT||∇uɛ,η|p(x)-|∇uɛ|p(x)|dxdt+1ɛ2α∬QT|∇uɛ|p(x)|uɛ,η-α-uɛ-α|dxdt⟶0.
The proof of Lemma 4.7 is completed by combining (4.67) with (4.65).Lemma 4.8.
Asɛ→0, we have(4.68)(1)∬QT||∇uɛ|p(x)-|∇u|p(x)|dxdt⟶0,(2)∬QT|uɛ|∇uɛ|p(x)-2∇uɛ-u|∇u|p(x)-2∇u|dxdt⟶0.Proof.
Letχρ and χɛρ be the characteristic functions of {(x,t)∈QT;u(x,t)<ρ} and {(x,t)∈QT;uɛ(x,t)<ρ}, respectively. Then
(4.69)∬QT||∇uɛ|p(x)-|∇u|p(x)|dxdt≤∬QT||∇uɛ|p(x)χɛρ-|∇u|p(x)χρ|dxdt+∬QT||∇uɛ|p(x)(1-χɛρ)-|∇u|p(x)(1-χρ)|dxdt≤∬QT|∇uɛ|p(x)χɛρdxdt+∬QT|∇u|p(x)χρdxdt+∬QT|∇u|p(x)(χɛρ-χρ)dxdt+∬QT||∇uɛ|p(x)-|∇u|p(x)|(1-χɛρ)dxdt=I1+I2+I3+I4.
Taking α=1/2 in Lemma 4.7, we obtain that I1≤Cρ1/2. Since I2→0 (ρ→0), for any δ>0, we can choose ρ>0 such that I1+I2<δ/3. For fixed ρ>0, χɛρ→χρ (ɛ→0) a.e. in QT, so there exists ɛ1>0 such that I3<δ/3 as ɛ<ɛ1. By Lemma 4.6, I4→0 (ɛ→0), so there exists ɛ2∈(0,ɛ1) such that I4<δ/3 as ɛ∈(0,ɛ2). Summing up, we have
(4.70)∬QT||∇uɛ|p(x)-|∇u|p(x)|dxdt<δ,∀ɛ<ɛ2,
thus (1) holds. To prove (2), observe that
(4.71)∬QT|uɛ|∇uɛ|p(x)-2∇uɛ-u|∇u|p(x)-2∇u|dxdt≤12∬QT|∇uɛ|p(x)-2|∇uɛ2-∇u2|dxdt+12∬QT||∇uɛ|p(x)-2-|∇u|p(x)-2||∇u2|dxdt=12(Ia+Ib).
Using Höder’s inequality and Lemma 4.7, we obtain that
(4.72)Ia≤|∇uɛ|p(x)(p(x)-2)/(p(x)-1)|∇uɛ2-∇u2|p(x)⟶0.
By means of the inequality |ar-br|≤|a-b|r(r∈[0,1],a,b≥0), Höder’s inequality and (1), we have
(4.73)Ib=∬QT|(|∇uɛ|p(x))(p(x)-2)/p(x)-(|∇u|p(x))(p(x)-2)/p(x)||∇u2|dxdt≤C∬QT||∇uɛ|p(x)-|∇u|p(x)|(p(x)-2)/p(x)|∇u2|dxdt≤C||∇uɛ|p(x)-|∇u|p(x)||∇u2|p(x)/2⟶0.
Thus the proof of Theorem 4.1 is complete.



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*Source: 835495-2012-02-16.xml*