# On Solutions of a Parabolic Equation with Nonstandard Growth Condition

Authors: Huashui Zhan
Journal: Journal of Function Spaces
 (2020)

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## Abstract

A parabolic equation with nonstandard growth condition is considered. A kind of weak solution and a kind of strong solution are introduced, respectively; the existence of solutions is proved by a parabolically regularized method. The stability of weak solutions is based on a natural partial boundary value condition. Two novelty elements of the paper are both the dependence of diffusion coefficientbx,t on the time variable t, and the partial boundary value condition based on a submanifold of ∂Ω×0,T. How to overcome the difficulties arising from the nonstandard growth conditions is another technological novelty of this paper.


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## Body

## 1. Introduction

In this paper, we are concerned with the initial-boundary value problem of a parabolic equation with nonstandard growth condition(1)ut=divbx,turx−1∇u+diva→urx,x,t+fu,x,t,x,t∈QT=Ω×0,Twhere bx,t≥0 is a CQT¯ function, rx>0 is a C1Ω¯ function, a→s,x,t=ais,x,t, ais,x,t∈Cℝ×QT¯, and Ω⊂ℝN is a bounded domain with a smooth boundary ∂Ω. These kinds of equations can be regarded as nonlinear variation of the classical heat equation, the well-known paradigm to explain the diffusion process. In particular, when bx,t≡1, ai=0,rx=r is a constant; equation (1) is with the type
(2)ut=Δur+g∇u,u,x,t,x,t∈QT,which includes the so-called porous medium equation (PME), the fast diffusion equation (FDE) when r>1, and the slow diffusion equation (SDE) when r<1. Correspondingly, if r−=minx∈Ω¯rx>1, then equation (1) is a degenerate parabolic equation and has a slow diffusion characteristic. If r+=maxx∈Ω¯rx<1, then equation (1) is a singular parabolic equation and has a fast diffusion characteristic.In fact, equation (2) itself has a wide number of applications, ranging from plasma physics to filtration in porous media, thin films, Riemannian geometry, relativistic physics, and many others. It has at the same time tested as the testing ground from the development of new methods of analytical investigation, since it offers a variety of surprising phenomena that strongly deviate from the heat equation standard, for example, free boundary, limited regularity, mass loss, and extinction or quenching. There exists an abundant literature on these topics, one can refer to [1–10] and the references therein. In this literature, the nonlinearities are with power types that lead to degenerate or singular parabolicity, and we can gather these collectively under the name “the porous medium equation with standard growth conditions.” Since, in equation (1), the nonlinearity is with variable exponent, so it is called as “with nonstandard growth condition.”From another perspective, sincebx,t≥0, equation (1) is a special case of the usual reaction-diffusion equation
(3)ut=divbu,x,t∇u+diva→u,x,t+fu,x,t,x,t∈QT.Ifbu,x,t is degenerate in the interior of Ω, this equation is a hyperbolic-parabolic mixed type equation theoretically. To ensure the uniqueness of a weak solution to this equation, the entropy condition and the corresponding entropy solution are required, one can refer to references [11–16] for the details. In this paper, we assume that
(4)bx,t>0,x,t∈Ω×0,T.As a consequence, equation (1) is a pure nonlinear parabolic equation and has not the hyperbolic characteristic.The porous medium equation with nonstandard growth conditions was first proposed by Antontsev and Shmarev in [17]. The existence, uniqueness, and localization properties of solutions to equation
(5)ut=divurx∇u+g∇u,u,x,t,had been studied. Another property proved in [17] is the finite speed of propagation, which permits one to consider the free boundary problem. The work [18] by Duque etc. implemented a finite element method with adaptive mesh to approximate the solution of the porous medium equation with nonstandard growth conditions in 2D domains with free boundary. The equidistribution principle deduced by de Boor [19] and the moving mesh for partial differential equations by Huang and Russell [20] were used there. Recently, when bx,t=bx with bx>0,x∈Ω, and
(6)bx=0,x∈∂Ω,the well-posedness of weak solutions to equation (1) has been studied by the author [21]. Roughly speaking, if bx satisfies (6) and
(7)∫Ωbx−1dx<∞,the uniqueness of weak solution to equation (1) with the initial value
(8)ux,0=u0x,x∈Ω,has been proved in [21]. This result can be comprehended as that, the degeneracy of diffusion coefficient bx on the boundary (6) acts as a role as the usual boundary value condition
(9)ux,t=0,x,t∈∂Ω×0,T.In this paper, different from [21], the diffusion coefficient bx,t is dependent on the time variable t; moreover, we merely assume that bx,t≥0 satisfies (4) and do not restrict the similar condition as (7). As a compensation, a partial boundary value condition
(10)ux,t=0,x,t∈Σ=x,t∈∂Ω×0,T:bx,t>0,is imposed as follows. One can see that, unlike the usual Dirichlet boundary value condition (9), in which ∂Ω×0,T, is a cylinder, Σ⊂∂Ω×0,T appearing in (10) is just a submanifold and generally can not be expressed as a cylinder type as Γ×0,T with Γ⊆∂Ω. The first aim of this paper is to study the stability of weak solutions based on this partial boundary value condition. Another aim of this paper is to study the existence of a strong solution to equation (1). The details are provided below. It is well-known that, for the usual porous medium equation
(11)ut=Δur,x,t∈QT,there holds
(12)∂u∂t∈L2QT,and u is with the Hölder continuity [5]. However, for a porous medium equation with nonstandard growth conditions, weak solutions in [17, 18, 21–30] do not satisfy the regularity as (12). Thus, in this paper, we will try to make up for some gaps in such a field, considering that the weak solutions of equation (1) have the property (12).Definition 1.
Letux,t be a nonnegative function satisfying
(13)u∈L∞QT,ut∈L20,T;W−1,2Ω,bx,turx−1∇u∈L∞0,T;L2Ω.
If for any functionφ∈C01QT, there is
(14)∬QT−∂φ∂tu+bx,turx−1∇u∇φdxdt+∑i=1N∬QTaiurx,x,tφxix,tdxdt=∬QTfu,x,tφdxdt,then ux,t is said to be a weak solution of the initial-boundary value problem of equation (1), provided that initial value (6) is true in the sense
(15)limt→0∫Ωux,t−u0xOxdx=0,Ox∈C0∞Ω,and the partial boundary value condition (10) is true in the sense of the trace.Definition 2.
Letux,t be a nonnegative function satisfying (14) and
(16)u∈L∞QT,urx−1ut∈L2QT,bx,turx−1∇u∈L∞0,T;L2Ω,then ux,t is said to be a strong solution of the initial-boundary value problem of equation (1), provided that initial value (6) is true in the sense of (15), and the partial boundary value condition (10) is true in the sense of the trace.
Sincebx,t satisfies (4), (16) means that utrx and ∇urx exist almost everywhere in QT. This is the reason that we call ux,t is a strong solution of equation (1).
From Definition2, for all φx,t∈L20,T;W01,2Ω, we have
(17)∬QTutφx,tdxdt+∬QTbx,trx∇urx∇φdxdt−∬QTbx,trxurxlogu∇r∇φdxdt+∑i=1N∬QTaiurx,x,tφxidxdt=∬QTfu,x,tφx,tdxdt,which implies that
(18)utL20,T;W−1,2Ω≤c.Thus, ifux,t is a strong solution of equation (1), then it is a weak solution.Theorem 3.
Ifrx is a C1Ω¯ function, rx≥r−>1, ais,x,t∈Cℝ×QT¯, i=1,2,⋯,N, u0x≥0 satisfies
(19)0≤u0∈L∞Ω,bx,0∇u0rx∈L2Ω,
0≤bx,t∈CQT¯ satisfies (4), fs,x,t is a continuous function on ℝ×QT¯ and when s<0(20)fs,x,t>0,x,t∈Ω×0,T,then the initial boundary value problem of equation (1) has a nonnegative weak solution.Theorem 4.
Ifrx is a C1Ω¯ function, rx≥r−>1, ais,x,t∈C1ℝ×QT¯, i=1,2,⋯,N, u0x≥0 satisfies (19), b·,t∈C10,T satisfies (4) and
(21)btx,t≤0,x,t∈Ω×0,T,∂∂saisrx,x,t2bx,t−1≤cM,for any s≤M+1, M=u0xL∞Ω, fs,x,t is a continuous function satisfies (20), then the initial boundary value problem of equation (1) has a nonnegative strong solution.
Throughout this paper, the constantc may be different from one to another, cM represents the constant c depends on M, and cT represents the constant c depends on T.Theorem 5.
Letux,t be a weak solution of equation (1) with the initial value (8) and with the partial homogenous value condition (10),
(22)∫Ωlogu2∇r2dx≤cT.
If 0<r−≤rx∈C1Ω, fs,x,t is a Lipschitz function on ℝ×QT¯, for every given t∈0,T, bx,t is a differential function on the spatial variables xi, i=1,2,⋯,N, ais,x,t∈Cℝ×QT¯ and satisfies
(23)aiurx,x,t−aivrx,x,t≤cMbx,t,(24)∫Ωbx,t−1∇b2dx≤cT,then the solution ux,t is unique.
One can see that there are functions satisfying condition (24). For example, if Γ⊂∂Ω is a relatively open set of ∂Ω, and let dΓx=distx,Γ. Then, for any ft∈C10,T,
(25)bx,t=ftdΓx2is a function satisfying (24).We note that, ifrx≡r is a constant, then condition (22) is naturally true. By the way, just as one reviewer has suggested, one can study Theorem 3 and Theorem 4 when rx≥r−>0 and fs,x,t is just a Carathodory function.


## 2. The Proof of Theorems 3-4

In this section, we give the proof of the existence theorems.Proof of Theorem 3.
By the monotone convergent method [5], since fs,x,t>0 when s<0, there is a nonnegative weak solution unx,t of the following initial-boundary value problem
(26)unt=divbx,t+1nunrx−1∇u+divaunrx,x,t+fun,x,t,x,t∈QT,unx,t=0,x,t∈∂Ω×0,T,unx,0=u0x,x∈Ω,which satisfies
(27)un∈L∞QT,0≤unx,t≤u0xL∞Ω+1,(28)bx,t+1n12unrx−12∇un∈L2QT.
Multiplying (26) by ϕ=unrx, then we obtain
(29)∬QTuntunrxdxdt=∬QTdivbx,t+1nunrx−1∇ununrxdxdt+∑i=1N∬QT∂aiunrx,x,t∂xiunrxdxdt+∬QTfun,x,tunrxdxdt.
In the first place, by a similar calculation as that in [21], we can extrapolate that
(30)∬QTbx,t+1nunrx−1∇un2dxdt≤c,(31)∫Ωbx,t+1n∇unrx2dx≤cT,and show that
(32)bx,t+1nunrx−112∇un⇀bx,t12urx−1∇u,weakly in L2QT.
In the second place, for allφx,t∈L20,T;W01,2Ω, we have
(33)∬QTuntφx,tdxdt+∬QTbx,t+1/nrx∇unrx∇φdxdt−∬QTbx,t+1/nrxunrxlogun∇r∇φdxdt+∑i=1N∬QTaiunrx,x,tφxidxdt=∬QTfun,x,tφx,tdxdt,from this equality, by (27) and (31), it is easy to get
(34)untL20,T;W−1,2Ω≤c.
Moreover, from (27), there is a nonnegative function, un⇀u weakly star in L∞QT. For any ϕ∈C01Ω,0≤ϕ≤1, by (34), we can show that
(35)ϕunrxtL20,T;W−1,2Ω≤c.
SinceH0sΩ↪W1,2Ω when s>N/2+1, then W−1,2Ω↪H−sΩ. Accordingly,
(36)ϕunrxtL20,T;H−sΩ≤c.
In addition, we have(37)∬QT∇ϕunrx2dxdt≤cϕ1+∫0T∫Ωϕ∇unrx2dxdt≤cϕ,where Ωϕ=suppϕ. Thus,
(38)ϕunrxL20,T;W01,2Ω≤c.
SinceW01,2Ω↪L2Ω↪Ω, Aubin’s compactness theorem in [31] yields that there is a function v such that ϕunrx⟶ϕv strongly in L2QT. Then, ϕunrx⟶ϕv almost everywhere in QT, by that un⇀u weakly star in L∞QT, we know that v=urx. The arbitraries of ϕ yields un⟶u almost everywhere in QT.
Thus, sinceais,x,t and fu,s,t are continuous functions on ℝ×QT¯), we have
(39)aiun,x,t⟶aiu,x,t,a.e.inQT,i=1,2,⋯,N,(40)fun,x,t⟶fu,x,t,a.e.inQT.
Letn⟶∞ in (29). By (32), (39), and (40), we know ux,t satisfies (13) and (14).
The last but not the least, by a process of limit, in a similar way as that in [17], we can choose the test function φx,t=χt1,t2ϕx in which ϕx∈C0∞Ω and χt1,t2 is the characteristic function of t1,t2⊂0,T. Then
(41)∫t1t2∫Ωbx,turx−1∇u∇ϕ+∑i=1Naiurx,x,tϕxi−fu,x,tϕxdxdt=∫Ωux,t2−ux,t1ϕxdx,
Lett=t2 and t1⟶0. Then we have (15).
Finally, sincebx,t>0 when x∈Σ in (10), by (32), we know that the trace of ux,t on Σ is well defined, the details are given in Section 4 as follows. Till now, we have shown that u is a weak solution of equation (1) with the initial value (6) and the partial boundary value condition (10).Proof of Theorem 4.
We consider the regularized problem (26) and obtain (27)–(32) as in the proof Theorem 3.
Let us multiplyuntrx on both sides of equation (26), integrate it over Qt=Ω×0,t. Then
(42)∬Qtrxurx−1unt2dxdt=∬Qtdivbx,t+1nunrx−1∇ununtrxdxdt+∬Qtfun,x,tuntrxdxdt+∑i=1N∬Qt∂aiunrx,x,t∂xiuntrxdxdt.
Sinceun≥0,
(43)unrx−1∇un=1rx∇unrx−∇rrxunrxlnun,∬Qtdivbx,t+1nunrx−1∇ununtrxdxdt=∬Qtdivbx,t+1n1rx∇unrx−∇rrxunrxlnununtrxdxdt=−∬Qtbx,t+1n12rx∂∂t∇unrx2dxdt+∬Qtbx,t+1n∇rrxunrxlnun∂∂t∇unrxdxdt,
we have(44)∬Qtrxunrx−1unt2dxdt+∬Qtbx,t+1n12rx∂∂t∇unrx2dxdt=∬Qtbx,t+1n∇rrxunrxlnun∂∂t∇unrxdxdt+∬Qtfun,x,tuntrxdxdt.
Sincebtx,t≤0,
(45)−∬Qtbtx,t∇rrxunrxlnun∇unrxdxdt≤12∬Qt−btx,t2rx∇unrx2+4∇runrxlnun2dxdt≤12∬Qt−btx,t2rx∇unrx2dxdt+c,by the assumption rx≥r−>1,
(46)−∬Qtbx,t+1n∇rrxrxunrx−1lnun+unrx−1untdxdt≤14∬Qtrxunrx−1unt2dxdt+4∬Qtunrx−1rxbx,t+1n∇rrxrxlnun+12dxdt≤14∬Qtrxunrx−1unt2dxdt+c,we have
(47)∬Qtbx,t+1n∇rrxunrxlnun∂∂t∇unrxdxdt=∫Ωbx,t+1n∇rrxunrxlnun∇unrxdx0t−∬Qtbtx,t∇rrxunrxlnun∇unrxdxdt−∬Qtbx,t+1n∇rrxrxunrx−1lnun+unrx−1untdxdt≤∫Ωbx,t+1n∇rrxunrx+1lnun∇unrxdx0t+12∬Qt−btx,trx∇unrx2dxdt+14∬Qtrxunrx−1unt2dxdt+c.
By the assumption∂/∂unaiunrx,x,t2bx,t−1≤c and
(48)∂∂xiaiunrx,x,t≤cMwhen un≤M+1, we have
(49)∬QT∂aiunrx,x,t∂xiuntrxdxdt=∬QTrxunrx−1∂∂unaiunrx,x,tunxirx+∂∂xiaiunrx,x,tuntdxdt≤4∬QT∂∂unaiunrx,x,t2bx,t−1rxunrx−1bx,tunxirx2dxdt+14∬QTrxunrx−1unt2dxdt+4∬QT∂∂xiaiunrx,x,t2rxunrx−1dxdt+14∬QTrxunrx−1unt2dxdt≤c+c∬QTbx,t∇unrx2dxdt+14∬QTrxunrx−1unt2dxdt.
Moreover, we have(50)∬QTfun,x,tuntdxdt≤c+14∬QTrxunrx−1unt2dxdt,(51)∬Qtbx,t+1n12rx∂∂t∇unrx2dxdt=12∫Ωbx,t+1n1rx∇unrx2dx0t−12∬Qtbtx,trx∇unrx2dxdt.
Now, combining with (44)–(51), we have
(52)∬Qtrxunrx−1unt2dxdt≤c.
Then(53)∬Qtuntrx2dxdt≤c.
Sincebx,t>0 when x∈Ω, by Sobolev embedding theorem, (32) and (53) implies that unrx⟶u1 a.e. in QT. Since un⇀u weakly star in L∞QT, by the uniqueness property of the weak convergence, we know that u1=urx.
Accordingly,ais,x,t∈C1ℝ×QT¯, we have
(54)limn→∞aiunrx,x,t=aiurx,x,t,i=1,2,⋯,N,as well as
(55)limn→∞funrx,x,t=furx,x,t,by that fu,x,t is a contunuous function.
Moreover, for anyφx,t∈C01QT,
(56)limn→0∬QTunrx−1unt−urx−1utφx,tdxdt=limn→0∬QT2rx+1untrx−12+1−utrx−12+1φx,tdxdt=−limn→0∬QT2rx+1unrx−12+1−urx−12+1φtx,tdxdt=0.
Thus,(57)unrx−1unt⇀urx−1ut,inL2QT.
Letn⟶∞ in (29). By (32), (35), (39), (56), and (57), we know ux,t satisfies (14) and (16). The initial value (6) and the partial boundary value condition have the same meaning as those in Theorem 3.


## 3. The Stability Based on the Partial Boundary Value Condition

In this section, we will prove Theorem5.Lemma 6.
Letut∈Lp′0,T;W−1,p′Ω for any p>1, p′=p/p−1. For any continuous function hs, Hs=∫0shsds, a.e.t1,t2∈0,T,
(58)∫t1t2∫Ωhuutdxdt=∫ΩHux,t2−Hux,t1dx.This Lemma can be seen as a generalization of Lemma 2.2 in [32]. We postpone the reader to it for details.Theorem 7.
Let0<r−≤rx∈C1Ω¯, fs,x,t be a Lipschitz function on ℝ×QT¯, bx,t∈CQT with that bx,t>0 if x∈Ω and for every given t∈0,T, bx,t be a differential function on the spatial variables xi,i=1,2,⋯,N, ais,x,t∈Cℝ×QT¯ and satisfy
(59)∫Ωbx,t−1∇b2dx≤c,(60)aiurx,x,t−aivrx,x,t≤cgix,t,∫Ωbxix,tgix,tbx,tdx<∞.
Ifux,t,vx,t are two nonnegative weak solutions of equation (1) with the initial value u0x,v0x, respectively, with the same homogeneous partial boundary value condition
(61)ux,t=vx,t=0,x,t∈Σ=x,t∈∂Ω×0,T:bx,t>0,and satisfying
(62)∫Ωlogu2∇r2dx≤c,∫Ωlogv2∇r2dx≤c,then
(63)∫Ωux,t−vx,tdx≤c∫Ωu0x−v0xdx.Proof.
Letgns=∫0shnτdτ, hns=2n1−ns+. Here and the after, n is a positive integer.
(64)limn→∞gns=sgns,limn→∞shns=0.
Supposed thatχτ,st is the characteristic function of τ,s⊂0,T, since ux,t and vx,t are with the same homogeneous partial boundary value condition (61), we then choose
(65)χτ,stgnbx,turx−vrxas the test function. Thus
(66)∫τs∫Ωgnbx,turx−vrx∂u−v∂tdxdt+∫τs∫Ωbx,trx∇urx−∇vrx∇urx−vrxbx,t·hnbx,turx−vrxdxdt+∫τs∫Ωbx,trx∇urx−∇vrx·∇burx−vrx·hnbx,turx−vrxdxdt−∫τs∫Ωbx,trxurxlogu−vrxlogv∇r·∇urx−vrxbx,t·hnbx,turx−vrxdxdt−∫τs∫Ωbx,trxurxlogu−vrxlogv∇r·∇burx−vrx·hnbx,turx−vrxdxdt+∑i=1N∫τs∫Ωaiurx,x,t−bivrx,x,tbxix,turx−vrx·hnbx,turx−vrx+bx,turx−vrxxihnbx,turx−vrxdxdt=∫τs∫Ωfu,x,t−fv,x,tgnbx,turx−vrxdxdt.
In the first place(67)∫Ωbx,t2rx∇urx−∇vrx·∇urx−vrxhnbx,turx−vrxdx≥0.
In the second place, by the Hölder inequality, using (59) and (64), we obtain
(68)limn→0∫τs∫Ωbx,trx∇urx−∇vrx∇burx−vrxhnbx,turx−vrxdxdt≤limn→0∫τs∫Ωbx,trx∇urx+∇vrx∇burx−vrx·hnbx,turx−vrxdxdt≤climn→0∫τs∫Ωbx,t∇urx2+∇vrx2dx12·∫Ωbx,t∇bbx,tbx,turx−vrxhnbx,turx−vrx2dx12dt=0.
In the third place, for simplism, let us denote that(69)Dn=Ω:urx−vrx<1n,D0=x∈Ω:u−v=0as a consequence
(70)limn→∞Dn=D0.
Since condition (62) implies
(71)limn→0∫τs∫Dnbx,turxlogurx−vrxlogvrx2hnbx,turx−vrx2∇r2dxdt≤limn→0∫τs∫Dnurxlogurx−vrxlogvrx2urx−vrx2∇r2dxdt≤∫τs∫D01+rxlogv2∇r2dxdt<∞,using the Lebesgue dominated convergence theorem, we have
(72)limn→∞∫τs∫Dnbx,turxlogurx−vrxlogvrx2hnbx,turx−vrx2∇r2dxdt=0,which yields
(73)limn→∞∫τs∫Ωbx,trxurx+1logu−vrxlogv·∇rbx,t∇urx−vrxhnbx,turx−vrxdxdt≤limn→∞∫τs∫Dnbx,t∇urx−∇vrx2dx12·c∫Dnbx,turxlogurx−vrxlogvrx2hnbx,turx−vrx2∇r2dx12dt=0.
In the fourth place, since∣∇b∣≤c, using (64) and the Lebesgue dominated convergence theorem,
(74)limn→∞∫τs∫Ωbx,trxurxlogu−vrxlogv∇r·∇burx−vrx·hnbx,turx−vrxdxdt=0,is obviously.
In the fifth place, since(75)aiurx,x,t−aivrx,x,t≤cgix,t,i=1,2,⋯,N,satisfies (60), using the Lebesgue dominated convergence theorem, we can extrapolate
(76)limn→∞∫τs∫Ωaiurx,x,t−aivrx,x,tbxix,turx−vrx·hnbx,turx−vrxdxdt=limn→∞∫τs∫Ωaiurx,x,t−aivrx,x,tbxix,tbx,tbx,turx−vrx·hnbx,turx−vrxdxdt=0.
Also by the Hölder inequality, we are easily to obtain(77)limn→∞∫τs∫Ωaiurx,x,t−aivrx,x,t·bx,turx−vrxxihnbx,turx−vrxdxdt=0.
In the sixth place,(78)limn→∞∫τs∫Ωfu,x,t−fv,x,tgnbx,turx−vrxdxdt≤c∫τs∫Ωux,t−vx,tdxdt.
At last, by Lemma6, choosing p=p′=2, by (64), we have
(79)limn→∞∫τs∫Ωgnburx−vrx∂u−v∂tdx=∫τs∫Ωux,s−vx,sdx−∫Ωux,τ−vx,τdxdt.
Then, lettingn→∞ in (66), we have
(80)∫Ωux,s−vx,sdx≤c∫Ωux,τ−vx,τdx+∫τs∫Ωux,s−vx,sdx
By the Gronwall inequality, we have(81)∫Ωux,s−vx,sdx≤c∫Ωux,τ−vx,τdx.
By the arbitraries ofτ, we have
(82)∫Ωux,s−vx,sdx⩽∫Ωu0x−v0xdx.
At last, since conditions (22), (23), and (24) imply all the assumptions in Theorem 7 are true, we have Theorem 4.


## 4. The Explanation of the Partial Boundary Value Condition

In [21], condition (7) guarantees
(83)∫Ω∇urxdx<∞,then we can define the trace ofurx on ∂Ω, and the homogeneous boundary value condition (9) is valid. But in this paper, we do not assume
(84)∫Ωbx,t−1dx≤cT,which is similar as condition (7), how to guarantee the partial boundary value condition (10) is true in the sense of the trace? Actually, by the character of ux,t defined in Definition 1(85)bx,turx−1∇u∈L∞0,T;L2Ω,it is not difficult to show that
(86)bx,t∇urx2∈L1QT.Now, for a given small constantδ and for any t∈0,T, if we denote
(87)Ω1δλt=x∈Ω:bx,t>δ⋂x∈Ω:dx<λwhere dx=distx,∂Ω is the distance function from the boundary ∂Ω, then
(88)limλ→0Ω1δλt=Σ1δt=x∈∂Ω:bx,t>δ.If we denote that(89)Σ0=x,t∈∂Ω×0,T:bx,t=0,Σ=x,t∈∂Ω×0,T:bx,t>0,∑0⋂Σ=∂Σ,then
(90)⋃δ>0Σ1δt×0,T=Σ⋃∂Σ.Sincebx,t satisfies (4) and (86), we have
(91)∫Ω1δt∇urxdx≤cδ∫Ω1δtbx,t∇vrxdx<∞.Then, we can define the trace ofurx on Σ1δt. By the arbitrariness of δ, we can define the trace of urx on Σ1δt. In other words, no matter whether condition (84) is true or not, we can always define the trace of urx (also u) on Σ1δt due to that bx,t>0 on Σ, and so the partial homogenous value condition (10) is always valid. Certainly, we should point out that, on the remained part of the boundary
(92)Σ1=∂Ω×0,T\Σ=x,t∈∂Ω×0,T:bx,t=0whether the trace of urx can be defined or not is unknown, unless condition (84) is supplied.


## 5. Conclusion

Since the beginning of this century, the evolutionarypx-Laplacian equations ut=divbu,x,t∇upx−2∇u+g∇u,u,x,t have been studied by many mathematicians in [32–37]. So far, however, there has been little discussion about the porous medium equation with nonstandard growth conditions. Therefore, this study makes a major contribution to research in the related fields by imposing a reasonable partial boundary value condition matching up with the equation. The unusual is that this partial boundary value condition is based on a submanifold Σ⊂∂Ω×0,T. In addition, since the equation is with nonstandard growth conditions, how to deal with the nonlinearity becomes difficult, especially when we try to prove that urx−1ut∈L2QT.



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*Source: 9397620-2020-06-02.xml*