1. Introduction The evolutionary p(x)-Laplacian equation,(1)ut=div∇upx-2∇u, x,t∈QT=Ω×0,T,possesses some interesting mechanical properties in the presence of an electromagnetic field [1, 2], the well-posedness of weak solutions to the first initial-boundary value problem of (1) had been researched in [3–7], etc. Here, Ω⊂RN is a bounded domain with the smooth boundary ∂Ω, and p(x)>1 is a C1(Ω¯) function. Sobolev spaces play an important role in the theory of evolutionary p(x)-Laplacian equation. In recent years, the generalized Orlicz-Lebesgue spaces Lp(x) and the corresponding generalized Orlicz-Sobolev spaces W1,p(x) have attracted more and more attention. The spaces Lp(.) are special cases of the generalized Orlicz-Spaces originated by Nakano and developed by Musielak and Orlicz (see [8–10]). We refer to [11–13] for properties of the spaces Lp(.) and Wk,p(.) such as reflexivity, denseness of smooth functions, and Sobolev type embeddings. The study of these spaces has been stimulated by problems of elasticity, fluid dynamics, calculus of variations, and differential equations with p(x)-growth conditions. Roughly speaking, the interest in variable exponent spaces comes not only from their mathematical curiosity but also from their relevance in many applications such as fluid dynamics, elasticity theory, differential equations with nonstandard growth conditions, and image restoration. In addition, the study of the weak solution in other spaces such as Orlicz-Morrey space and B˙∞,∞-1 space is a research problem (see [14–18]).

The so-called anisotropic evolutionary p→(x)-Laplacian equation,(2)ut=∑i=1N∂∂xiuxipix-2uxi,comes closer to the truth than equation (1) [19–21], where p→(x)=(p1(x),p2(x),…,pN(x)). Recently, Zhan et al. [22–24] considered the first initial-boundary value problem to the equation:(3)ut=∑i=1N∂∂xibixuxipix-2uxi,where bi(x)∈C1(Ω¯) (i=1,2,…,N) satisfies(4)bix>0 if x∈Ωand bix=0 if x∈∂Ω.We have shown that this condition may act as the role of the Dirichlet boundary condition(5)ux,t=0, x,t∈∂Ω×0,Tto assure the stability of weak solutions to (3).

In this paper, we will consider the anisotropic parabolic equation(6)vt=∑i=1nbix,tvxipix-2vxixi+fv,x,t, x,t∈QT,with the initial value(7)vx,0=v0x, x∈Ω,and with a partial second boundary value condition(8)∂v∂n=0, x,t∈Γ1×0,T,where bi(x,t)∈C1(QT¯), pi(x)∈C1(Ω¯), pi(x)>1, bi(x,t)≥0, f(v(x,t),x,t)≥0 and at least there is a point x0∈Ω such that f(v(x0,t),x0,t)>0, Ω is a bounded domain with a smooth boundary ∂Ω. We first assume that(9)∂Ω=Γ1∪Γ2,Γ10∩Γ20=∅,where Γ10 and Γ20 are the interior of Γ1 and Γ2, which are relatively open subset of ∂Ω. We mainly assume that(10)bix,t=0, x,t∈Γ2×0,T,denote that p+=maxx∈Ω¯p(x),p-=minx∈Ω¯p(x), for any p(x)∈C1(Ω¯), and let(11)p0=minx∈Ω¯ p1x,p2x,…,pn-1x,pnx,p0=maxx∈Ω¯ p1x,p2x,…,pn-1x,pnx, for any p→(x). As for the anisotropic function spaces and their applications to anisotropic equations, one can refer to [25, 26] and the references therein.

Definition 1. If a function v(x,t) satisfies that(12)v∈L∞QT,∂v∂t∈L2QT,bix,tvxipix∈L20,T;L1Ω,for any function φ∈C1(QT¯), φxi∈L2(0,T;Llocpi(x)(Ω)), φ(x,t)∈Γ2×[0,T]=0,(13)∬QT∂v∂tφ+∑i=1nbix,tvxipix-2vxiφxidxdt=∬QTfv,x,tφdxdt,and, for any ϕ(x)∈C0∞(Ω),(14)limt→0∫Ωux,tϕxdx=∫Ωu0xϕxdx,then we say v(x,t) is a weak solution of equation (6) with the initial value (7) and with the partial second value condition (8).

The main results are the following theorems.

Theorem 2. Suppose that p0≥2, bi(x,t) satisfies (10) and f,g are two C1 functions satisfying(15)fv,s,t≥gv.Let (16)v0x∈L∞Ω,bix,0v0xix∈L20,T;LpixΩ.Then we have a positive constant T1 such that there exists a weak solution of (6) v(x,t)∈L∞(QT1) with the initial condition (7) and with the partial second boundary value condition (8).

We would like to suggest that, since f(v(x,t),x,t)≥0 and at least there is a point x0∈Ω such that f(v(x0,t),x0,t)>0, then the weak solution v(x,t) generally blows up in a finite time [27]. However, the uniqueness of weak solution still may be true.

Theorem 3. If bi(x,t) satisfies (10) and f is a C1 function, and when x is near to Γ2,(17)bix,t≤cdxpi+-1v(x,t) and u(x,t) are two solutions of (6) with the same partial second boundary value condition(18)∂v∂nx,t=∂u∂nx,t=0, x,t∈Γ1×0,T,and with the same initial values(19)v0x=u0x, x∈Ω;then(20)vx,t=ux,t, x,t∈Ω×0,T.

Here T>T1 is the blow-up time of the weak solutions.

At the end of the introduction, we would like to suggest that it is an interesting research problem to study the anisotropic parabolic equation (6) for either (p,q)-Laplacian or p-biharmonic (see [28–30]).

2. The Existence of Solutions Consider the following asymptotic problem:(21)vt=∑i=1nbix,t+εvxi2+εpix-2/2vxixi+fv,x,t, x,t∈QT,with the initial value(22)vx,0=v0εx+ε, x∈Ω,(23)ux,t=ε, x,t∈Γ2×0,T,and a partial second boundary value condition(24)∂v∂n=0, x,t∈Γ1×0,T,where 0<ε<1, v0ε(x)∈C0∞(Ω) such that(25)v0εL∞Ω≤v0L∞Ω,v0εxiLpixΩ≤v0xiLpixΩ,and v0ε(x)+ε→v0(x) in W1,p(x)→(Ω). This is possible; only we assume that p(x)→={pi(x)} and every pi(x) has the logarithmic Hölder continuity [11–13]. Then similar as the usual p-Laplacian equation, one can show that problem (21)-(24) has a classical solution uε by the classical theory for parabolic equations, and vε≥ε>0 provided that f(v(x,t),x,t)≥0 and at least there is a point x0∈Ω such that f(v(x0,t),x0,t)>0 ([31], Theorem 4.1). We first give a lemma in a similar way as Lemma 2.1 in [32].

Lemma 4. If f(v,x,t)≤g(v) and g(v)∈C1(0,∞), then there exists a T1<T such that(26)vεL∞QT1≤cT1,where c=c(T1) represents c dependent on T1.

Proof. Let w(t) be the solution of the ordinary differential equation(27)dwdt=gw,(28)w0=v0x+1L∞Ω.It is well known that there is a local solution w(t), t∈[0,T0]⊂[0,T], where T0=T0(v0x+1L∞(Ω)) ([33], Chapter 5). Let u=vε-w. One has(29)ut-∑i=1nbix,t+εu+wxi2+εpix-2/2u+wxixi=fv,x,t-gw.Since f(u,x,t)≥g(u), one has (30)fvε,x,t-gw≤gvε-gw=vε-w∫01g′θvε+1-θwdθ=cεx,tvε-w.From (29),(31)ut-∑i=1nbix,t+εu+wxi2+εpix-2/2u+wxixi-cεx,tu≤0,with the mixed boundary condition(32)u=vε-w≤ε-u0x+1L∞Ω≤0, x,t∈Γ1×0,T0,(33)∂u∂n=0, x,t∈Γ2×0,T0,and(34)u0x=v0εx+ε-u0x+1L∞Ω≤0, x∈Ω¯.By the Hopf maximum principle, one has (35)ux,t≤0, x,t∈Ω×0,T0.Hence, for any given T1∈(0,T0), one has(36)vεL∞QT1≤cT1.

By multiplying (21) with vε and integrating over Ω, one has(37)bix,t+εvεxiLpixΩ≤cT1.

Lemma 5. If f(v,x,t)≥g(v) and g(v)∈C1(0,∞), then there exists a T1<T such that(38)∂vε∂tL2QT1≤cT1.

Proof. By multiplying (21) with ∂uε/∂t and integrating over QT1, one has(39)vεt=-∑i=1n∬QT1bix,t+εvxi2+εpix-2/2vxixi∂vεxi∂tdxdt+∬QT1fvε,x,t∂vε∂tdxdt.Since(40)bix,t+εvxi2+εpix-2/2vxixi∂vεxi∂t=12bix,t+ε∂∂t∫0vεxi2s+εpix-2/2ds,accordingly(41)∬QT1bix,t+εvxi2+εpix-2/2vxixi∂vεxi∂tdxdt=12∬QT1bix,t+ε∂∂t∫0vεxi2s+εpix-2/2dsdxdt=12∬QT1∂∂t∫0vεxi2bix,t+εs+εpix-2/2dsdxdt-12∬QT1bitx,t∫0vεxi2s+εpix-2/2dsdxdt=12∫Ω∫0vεxix,T12bix,T1+εs+εpix-2/2dsdx-12∫Ω∫0v0εxi2bix,0+εs+εpix-2/2dsdx-12∬QT1bitx,t∫0vεxi2s+εpix-2/2dsdxdt.Thus by (37), using Young’s inequality, one has(42)∬QT1∂vε∂t2dxdt=∑i=1N∫Ω1pixbix,T1+εvxix,T12+εpix/2dx-∑i=1N∫Ω1pixbix,0+εv0εxi2+εpix/2dx-∑i=1N∫Ω1pixbitx,tvεxi2+εpix/2dx+∬QT1fvε,x,t∂vε∂tdxdt≤c.

Proof of Theorem <xref ref-type="statement" rid="thm1.1">2</xref>. By multiplying (21) with vε and integrating over QT1, one has(43)∬QT1εpix-2/2bix,t+εvxi2dxdt≤cT1,(44)∬QT1bix,t+εvxi2+εpix-2/2dxdt≤cT1.By (37), (39), (43), and (44), there is a function v that satisfies(45)vε→v, a.e. QT1,fvε,x,t→fv,x,t, a.e. QT1,bix,t1/pixvεxi⇀bix,t1/pixvxi, in L10,T;LpixΩ∂vε∂t⇀∂v∂t, in L2QT1.In addition, if one notices that (46)bix,t+εpix-1/pixvεxi2+εpix/pix-1≤bix,t+εvεxi2+εpixpix-2/2pix-1vεxipix/pix-1≤cbix,t+εvεxipix+εpixpix-2/2pix-1bix,t+εvεxipix/pix-1;by (43) and (44) and by that pi(x)/pi(x)-1≤2, one has(47)∬QT1bix,t+εpix-1/pixvεxi2+εpix/pix-1dxdt≤c∬QT1bix,t+εvεxipixdxdt+∬QT1 + εpixpix-2/2pix-1bix,t+εvεxipix/pix-1dxdt≤c∬QT1bix,t+εvεxipixdxdt+c∬QT1 + εpixpix-2/2pix-1bix,t+εvεxi2+1dxdt≤c∬QT1bix,t+εvεxipixdxdt+c≤cT1.Then, there exists wi∈L1(0,T1;Lpi(x)/pi(x)-1(Ω)), i=1,2,…,n such that(48)bix,t+εpix-1/pixvεxi2+εpix-2/2⇀wi, in L10,T1;Lpix/pix-1Ω.Using the similar method as that of the evolutionary p-Laplacian equation [34], we can deduce that(49)limε→0∬QT∑i=1Nvεxipix-2vεxiξxdxdt=∬QT∑i=1Nwiξxidxdt=∬QT∑i=1Nvxipix-2vxiξxidxdt,for any ξ(x,t)∈C01(QT).

At last, the initial value in the sense of (14) can be found in [3]. Consequently, vε is the solution of (6).

3. The Stability One can refer to [11–13] for the definitions of the exponent variable spaces, Lq(x)(Ω),·Lq(x)(Ω), W1,q(x)(Ω),·W1,q(x)(Ω), and W01,q(x)(Ω). Also, one can find other details and recent applications to partial differential equations in [35–39].

Lemma 6 (see [<xref ref-type="bibr" rid="B11">11</xref>–<xref ref-type="bibr" rid="B13">13</xref>]). If p(x) and q(x) are real functions with 1/p(x)+1/q(x)=1 and q(x)>1, then, for any v∈Lp(x)(Ω) and u∈Lq(x)(Ω), we have(50)∫Ωvudx≤2vLpxΩuLqxΩ.Moreover, (51)if vLqxΩ=1, then ∫Ωvqxdx=1,if vLqxΩ>1, then vLqxq-≤∫Ωvqxdx≤vLqxq+,if vLqxΩ<1, then vLqxq+≤∫Ωvqxdx≤vLqxq-.

We let gm(s) be an odd function, and(52)gms=1,s>1m,m2s2e1-m2s2,0≤s⩽1m.Then,(53)limm→∞ gms=sgns, s∈-∞,+∞.

Let φ(x) be a C1(Ω¯) function satisfying(54)φxx∈Γ2=0,φxx∈Ω¯∖Γ2>0,and(55)Ωm=x∈Ω:φx>1m.Define(56)φmx=1,if x∈Ωm,mφx,if x∈Ω∖Ωm.Then φm(x)|x∈Γ2=0 and(57)φmxix=0,if x∈Ωm,mφxix,if x∈Ω∖Ωm.

Theorem 7. Let v(x,t)∈L∞(QT) and u(x,t)∈L∞(QT) be two solutions of (6) with the same partial second boundary value condition (8) and with the initial values v0(x) and u0(x). If ∂Ω=Γ1∪Γ2 satisfies (9), bi satisfies (10),(58)fv,x,t-fu,x,t≤cv-u,and, for large enough m,(59)messsupt∈0,T∫Ω∖Ωmbix,tφxxipixdx1/pi+≤c;then(60)∫Ωvx,t-ux,tdx≤∫Ωvx,0-ux,0dx, a.e. t∈0,T.

Proof. Let χ[τ,s] be the characteristic function of [τ,s)⊆[0,T). By a process of limit, we can choose the test function as χ[τ,s]φmgm(v-u). Then(61)∫τs∫Ωφmgmv-u∂v-u∂tdxdt+∑i=1n∫τs∫Ωbixvxipix-2vxi-uxipix-2uxivxi-uxigm′v-uφmxdxdt+∑i=1n∫τs∫Ωbixvxipix-2vxi-uxipix-2uxivxi-uxigmv-uφmxidxdt=∫τs∫Ωfv,x,t-fu,x,tφmgmv-udxdt.

First of all,(62)∫τs∫Ωbixvxipix-2vxi-uxipix-2uxivxi-uxigm′v-uφmxdxdt≥0.

Secondly, since vt∈L2(QT), ut∈L2(QT), by the Lebesgue dominated convergence theorem, we have(63)limm→∞∫τs∫Ωφmxgmv-u∂v-u∂tdxdt=∫Ωv-ux,sdx-∫Ωv-ux,τdx.

By (59), we have (64)∫τs∫Ωbix,tvxipix-2vxi-uxipix-2uxiφmxigmv-udxdt=∫τs∫Ω∖Ωmbix,tvxipix-2vxi-uxipix-2uxiφmxigmv-udxdt≤m∫τs∫Ω∖Ωmbix,tvxipix-1+uxipix-1φxigmv-udxdt≤cm∫τs∫Ω∖Ωmbix,tvxipix+uxipixdx1/qi+∫Ω∖Ωmbix,tφxipixdx1/pi+dt≤c∫τs∫Ω∖Ωmbix,tvxipixdx1/qi++∫Ω∖Ωmbix,tuxipixdx1/qi+·m∫Ω∖Ωmbix,tφxipixdx1/pi+dt≤c∫τs∫Ω∖Ωmbix,tvxipixdx1/qi+dt+c∫τs∫Ω∖Ωmbix,tuxipixdx1/qi+dt,where qi(x)=pi(x)/pi(x)-1, qi+=maxx∈Ω¯qi(x).

Then we have(65)limm→∞∫τs∫Ωbix,tvxipix-2vxi-uxipix-2uxiφmxigmv-udxdt≤climm→∞∫Ω∖Ωmbix,tvxipixdx1/qi++∫Ω∖Ωmbix,tuxipixdx1/qi+=0.

In addition, by (58)(66)limm→∞∫τs∫Ωfv,x,t-fu,x,tφmgmv-udxdt≤∫τs∫Ωvx,t-ux,tdxdt.

At last, let m→∞ in (61). Then(67)∫Ωvx,s-ux,sdx⩽∫Ωvx,τ-ux,τdx.By the arbitrary of τ, we have the conclusion.

Proof of Theorem <xref ref-type="statement" rid="thm1.2">3</xref>. We only need to choose(68)φx=dx;in Theorem 7, the conclusion is clear.