On Initial Boundary Value Problems with Equivalued Surface for Nonlinear Parabolic Equations

Volume 2009, Article ID 739097,23pages doi:10.1155/2009/739097

Research Article

On Initial Boundary Value Problems

with Equivalued Surface for Nonlinear

Parabolic Equations

Fengquan Li

Department of Applied Mathematics, Dalian University of Technology, Dalian 116024, China

Correspondence should be addressed to Fengquan Li,[email protected]

Received 6 January 2009; Revised 12 March 2009; Accepted 22 May 2009

Recommended by Sandro Salsa

We will use the concept of renormalized solution to initial boundary value problems with equivalued surface for nonlinear parabolic equations, discuss the existence and uniqueness of renormalized solution, and give the relation between renormalized solutions and weak solutions.

Copyrightq2009 Fengquan Li. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

1. Introduction

LetΩ ⊂ RN_{N ≥ 2be a bounded domain with Lipschitz boundary ∂Ω Γ. T is a fixed}

positive constant,Q Ω×0, T. We consider the following nonlinear parabolic boundary value problems with equivalued surface:

∂u ∂t −

N

i,j1

∂ ∂xi

aijx, u∂u ∂xj

fx, t inQ,

uCta function oftto be determined onΓ×0, T,

Γ

∂u

∂nLdsAt ∀

a.e. t∈0, T,

ux,0 0 in Ω,

P

wheref ∈L2_QandA∈L2_{0, T,n n}

1, . . . , nNdenotes the unit outward normal vector

onΓand

∂u ∂nL

N

i,j1

aijx, u∂u

∂xjni.

There are many concrete physical sources for problem P, for example, in the petroleum exploitation, udenotes the oil pressure, andAt is the rate of total oil flux per unit length of the well at the timet; in the combustion theory,udenotes the temperature, for any fixed timet, the temperature distribution on the boundary is a constant to be determined, while, the total heatAtthrough the boundary is givencf.1–7. For linear equations, the existence, uniqueness of solution to the corresponding problem are well understoodcf.1–

3, for the purpose, the Galerkin method was used. For semilinear equations, the existence of global smooth solution was obtained in7in which a comparison principle was established. Ifaijx, uis locally Lipschitz continuous with respect to the second variable, the existence

and uniqueness of bounded weak solution to problemPhave been discussed in8under the hypotheses off ∈ LqQandA ∈ Lr0, Twith q > N/2 1,r > N 2. However, if

f ∈L2_{QandA ∈L}2_{0, T, we cannot get a bounded weak solution. In order to deal with}

this situation, we will introduce the concept of renormalized solution to problem Pand discuss the existence and uniqueness of renormalized solution.

The paper is organized as follows. In Section 2, we introduce the concept of renormalized solution and prove the existence of renormalized solution to problemP. In

Section 3, uniqueness and a comparison principle of renormalized solution to problemPare established. InSection 4, we discuss the relation between renormalized solutions and weak solutions for problemP.

2. Existence of Renormalized Solution to Problem

P

In order to prove the existence of renormalized solution to problem P, we make the following assumptions.

Letaij : Ω×R → Rbe Carath´eodory functions with 1 ≤ i, j ≤ N. We assume that aij·,0∈L∞Ωand for any givenM >0 there existdM∈L∞Ωand a positive constantλ0

such that for everys,s1,s2∈R,ξ ξ1, . . . , ξN∈RN, and a.e.x∈Ω,

_{aijx, s1−aijx, s2≤dMx|s1−s2|, |sk| ≤M, k}1,2, 2.1

N

i,j1

aijx, sξiξj≥λ0|ξ|2. 2.2

Set

V v∈H1 _Ω|v|

Γconstant

. 2.3

Under hypotheses2.1-2.2andf ∈ L2_{Q,A ∈ L}2_{0, T, we cannot obtain an L}∞

estimate on the determined functionCt; thus, we cannot prove the existence of bounded weak solutions to problem P, hence aij·, uDju may not belong to L2Q. In order to

As usual, fork >0,Tkdenotes the truncation function defined by

Tkv

⎧ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎩

k, ifv > k,

v, if|v| ≤k,

−k, ifv <−k.

2.4

Set

W ξ∈C∞Q|ξT 0, ξt|_ΓCtan arbitrary function oft. 2.5

Definition 2.1. A renormalized solution to problemPis a measurable functionu:Q → R, satisfyingu∈L2_{0, T;V∩L}∞_{0, T;L}2_{Ωand for allh∈C}1

cR,ξ∈W,

−

Q ξt

_u

0

hrdrdxdt

Q N

i,j1

aijx, uDjuDihuξdxdt

Q

fhuξdxdt

_T

0

Athut|_Γξt|_Γdt,

2.6

lim

m→ ∞

{x,t∈Q:m≤|ux,t|≤m 1}

N

i,j1

aijx, uDjuDiudxdt0. 2.7

Remark 2.2. Each term in2.6and2.7is well defined. Indeed, the first term on the left side of2.6is welldefined as|u₀hrdr| ≤ hL∞|u|andu ∈L2Q. The second term on the left

side of2.6should be understood as

{x,t∈Q:|u|<k}

N

i,j1

aijx, TkuDjTkuDihTkuξdxdt, 2.8

for k > 0 such that supph ⊂ −k, k. Since u ∈ L2_{0, T;V, it is the same for huξ and}

hut|_Γξt|_Γ. The integral in2.7should be understood as

{x,t∈Q:m≤|ux,t|≤m 1}

N

i,j1

aijx, Tm 1uDjTm 1uDiTm 1udxdt. 2.9

Remark 2.3. Note that if u is a renormalized solution of problem P, we get Bhu

_u

0hrdr ∈ L20, T;V,Bhut ∈ L20, T;V L1Q; thus,Bhu ∈ C0, T;L1Ω, hence Bhu·,0 0 makes sense.

Remark 2.4. By approximation,2.6holds for anyh∈W1,∞_{Rwith compact support and all} ξ∈ {ξ∈L2_{0, T;V|ξ}

Now we can state the existence result for prolemPas follows.

Theorem 2.5. Under hypotheses2.1-2.2andf ∈ L2_{Q,A ∈ L}2_{0, T, problemPadmits a} renormalized solutionu∈L2_{0, T;V∩L}∞_{0, T;L}2_{Ωin the sense ofDefinition 2.1.}

In order to proveTheorem 2.5, we will consider the following problem:

∂un ∂t −

N

i,j1

∂ ∂xi

a_ijnx, un∂un ∂xj

f inQ,

un|_Γ×0,TCnt a function oftto be determined onΓ×0, T,

Γ

∂un

∂nLdsAt ∀

a.e. t∈0, T,

unx,0 0 inΩ,

Pn

wherea_ijnx, u aijx, Tnu,i, j1,2, . . . , N.

Then problemPnadmits a unique weak solutionun ∈L20, T;V∩C0, T;L2Ω

such thatun∈L20, T;Vand satisfies

unt, v

V_,V

Ω

N

i,j1

a_ijnx, unDjunDivdx

Ωfx, tvxdx Atv|Γ, a.e. t∈0, T, ∀v∈V,

2.10

unx,0 0 a.e. x∈Ω. 2.11

In fact, here we can prove the existence of weak solution for problem Pn via Galerkin method. Let us consider the operator

B:L2Ω−→V,

F−→v,

2.12

wherevis the weak solution of the following problem:

−Δv vF inΩ,

vCa constant to be determined onΓ,

Γ

∂v ∂nLds

0.

E

L2_{Ω. By Riesz-Schauder’s theory, there is a completed orthogonal eigenvalues sequence}

{wk_{}of the operatorB. Here we may take the special orthogonal system{w}k_}.

DefineA:V → V,

Aw, vV_,V

Ω

N

i,j1

an_ijx, wDjwDivdx,

Ft, v_V,V

Ωfx, tvxdx Atv|Γ, a.e. t∈0, T, ∀w, v∈V.

2.13

Letum nx, t

m

k1Φkmn wk,then Galerkin equations can be written as

um

nt, wk

Aumnt, wk

Ft, wk_{, a.e. t∈0, T,}

umnx,0 0 a.e. x∈Ω.

2.14

By using the same arguments as13, Lemma 30.4, we get a solutionum

n ∈L20, T;Vto the

above Galerkin equations such thatum_n ∈ L20, T;V. Moreover, we can easily prove the following estimates:

umnL∞_0,T;L2_Ω≤C0,

umnL2_0,T;V≤C0,

AumnL2_0,T;V≤C0,

_um

nL2_0,T;V≤C0,

2.15

whereC0is a positive constant independent ofm.

The above estimates imply that there exists a subsequence of{um

n}still be denoted by

{um

n} such that

umn un weak∗ in L∞

0, T;L2Ω,

umn un weakly in L20, T;V,

um

n un weakly in L2

0, T;V,

Aumn Aun weakly in L2

0, T;V.

2.16

To deal with the time derivative of truncation function, we introduce a time regularization of a functionu∈L2_{0, T;V. Let}

uνx, t

t

−∞νux, se

νs−t_{ds, ux, s ux, sχ}

0,Ts, 2.17

whereχ0,T denotes the characteristic function of a set0, Tand ν > 0. This convolution

function has been first used in14 see also10, and it enjoys the following properties:uν

belongs toC0, T;V,uνx,0 0,anduvconverges strongly touinL20, T;Vasνtends

to the infinity. Moreover, we have

uνtνu−uν, 2.18

and finally ifu∈L∞Q, thenuν∈L∞Qand

uνL∞_Q≤ u_L∞_Q, ∀ν >0. 2.19

Takingvuntin2.10, then integrating over0, τwithτ∈0, T, we have

_τ

0

Ω

d

dt|unx, t|

2_dxdt

_τ

0

Ω

N

i,j1

a_ijnx, unDjunDiundxdt

τ

0

Ωfundxdt

τ

0

Atunt|Γdt.

2.20

By 2.2, trace theorem, H ¨older’s inequality, Young’s inequality and Gronwall’s inequality, we get

unL∞0,T;L2_Ω≤C1, 2.21

unL2_0,T;V≤C1, 2.22

whereC1is a positive constant depending only onfL2_Q,A_L2_0,T,λ₀, but independent of nandun.

By2.21and2.22, there is a subsequence of{un}still denoted by{un}such that

un u weak∗ in L∞

0, T;L2Ω,

un u weakly in L20, T;V

un|_Γ u|_Γ weakly in L20, T.

2.23

Using the same method as10, we can obtain

un−→u a.e.inQ

Thus for any givenk >0,

Tkun Tku weakly inL20, T;V, strongly inL2Q,a.e. in Q. 2.25

By15, Lemma 2 and Lemma 3, we have

un−→u strongly in LqQ, ∀1≤q <2

4

N, 2.26

un−→u strongly inLrΓ×0, T, ∀2≤r <2

2

N. 2.27

impling that

un|Γ−→u|Γ, a.e.in 0, T. 2.28

For any givenk >0, it follows from2.27-2.28and Vitali’s theorem that

Tkun|Γ−→Tku|Γstrongly inL20, T. 2.29

Set

ηνu Tkuν. 2.30

Similar to 10, this function has the following properties:ηνut νTku − ηνu, ηνu0 0,|ηνu| ≤k,

ηνu−→Tku strongly inL20, T;V,asν tends to the infinity. 2.31

For any fixedhandkwithh > k >0, let

wnT2k

un−Thun Tkun−ηνu

. 2.32

Then we have the following lemma.

Lemma 2.6. Under the previous assumptions, we have

T

0

<unt, wn> dt≥ωn, ν, h, 2.33

wherelimh→ ∞limν→ ∞limn→ ∞ωn, ν, h 0.

Lemma 2.7. Under the previous assumptions, for any givenk >0, we have

Tkun−→Tku strongly inL20, T;V. 2.34

Proof. Takingvwntin2.10, then integrating over0, T, byLemma 2.6, we have

Q N

i,j1

a_ijnx, unDjunDiwndxdt≤

Q

fwndxdt

T

0

Atwnt|Γdt ωn, ν, h. 2.35

Now note thatDwn0 if|un|> h 4k; then if we setMh 4k, splitting the integral

on the left side of2.35on the sets{x, t∈Q:|unx, t|> k}and{x, t∈Q:|unx, t| ≤k},

∀n > M, we get

Q N

i,j1

a_ijnx, unDjunDiwndxdt

Q N

i,j1

aijx, TMunDjTMunDiwndxdt

≥

Q N

i,j1

aijx, TkunDjTkunDi

Tkun−ηνu

dxdt

−

{|un|>k} N

i,j1

_{aijx, TMunDjTMunDiηνudxdt.}

2.36

While,

{|un|>k} N

i,j1

_{aijx, TMunDjTMunDiηνudxdt}

≤

{|un|>k} N

i,j1

_{aijx, TMunDjTMun|DiTku|dxdt}

Q N

i,j1

_{aijx, TMunDjTMunDiηνu−DiTkudxdt.}

2.37

For any fixedh >0,2.1and2.22imply thataijx, TMunDjTMunis bounded inL2Q

with respect ton, while|DiTku|χ{|un|>k} strongly converges to zero inL2Q. Moreover it

follows from2.31that

{|un|>k} N

i,j1

where limν→ ∞limn→ ∞ωn, ν 0. Equations2.38,2.36, and2.35imply that

Q N

i,j1

aijx, TkunDjTkunDi

Tkun−ηνu

dxdt

≤

Q

fwndxdt

_T

0

Atwnt|Γdt ωn, ν ωn, ν, h.

2.39

By2.25,2.31, and2.39, we get

Q N

i,j1

aijx, TkunDjTkunDiTkun−Tkudxdt

≤

Q

fwndxdt

_T

0

Atwnt|Γdt ωn, ν ωn, ν, h.

2.40

By2.24-2.25and the Lebesgue dominated convergence theorem, we have

Q

fwndxdt

Q fT2k

u−Thu Tku−ηνu

dxdt ωn, 2.41

where limn→ ∞ωn 0.2.31and2.41imply that

Q

fwndxdt

Q

fT2ku−Thudxdt ωn, ν; 2.42

thus, we get

Q

fwndxdtωn, ν, h. 2.43

Similarly to the proof of2.43, we also have

_T

0

Atwnt|Γdtωn, ν, h. 2.44

Therefore we get

Q N

i,j1

aijx, Tkun

DjTkun−DjTku

DiTkun−Tkudxdt

≤ωn, ν, h−

Q N

i,j1

aijx, TkunDjTkuDiTkun−Tkudxdt.

Letn, ν,then andhtend to the infinity, respectively, we get

lim

n→ ∞

Q N

i,j1

aijx, TkunDjTkun−TkuDiTkun−Tkudxdt0. 2.46

Using2.2,2.25, and2.46, we obtain2.34.

Proof ofTheorem 2.5. For any givenξ ∈ W,h ∈C1

cR, suppose that supph⊂ −k, k, taking vhuntξtin2.10and integrating over0, T, we have

T

0

unt, hunξdt

Q N

i,j1

a_ijnx, unDjunDihunξdxdt

T

0

Athunt|_Γξt|_Γdt

Q

fhunξdxdt.

2.47

By12, Lemma 1.4, we have

_T

0

unt, hunξdt−

Q ξt

_un

0

hrdrdxdt. 2.48

However

−

Q ξt

_un

0

hrdrdxdt−→ −

Q ξt

_u

0

hrdrdxdt. 2.49

In fact, Noting2.26we get

−

Q ξt

_un

0

hrdrdxdt

Q ξt

_u

0

hrdrdxdt

Q ξt

_un

u

hrdrdxdt

≤ ξtL2_Qh_L∞_Run−uL2_Q−→0, asn−→ ∞.

Asn > k, we have

Q N

i,j1

a_ijnx, unDjunDihunξdxdt

Q N

i,j1

aijx, TkunDjTkunDiTkunhunξdxdt

Q N

i,j1

aijx, TkunDjTkunhunDiξdxdt.

2.51

Equations2.472.25and2.34imply that

Q N

i,j1

aijx, TkunDjTkunDiTkunhunξdxdt

−→

Q N

i,j1

aijx, TkuDjTkuDiTkuhuξdxdt,

Q N

i,j1

aijx, TkunDjTkunDiξhundxdt

−→

Q N

i,j1

aijx, TkuDjTkuDiξhudxdt.

2.52

Thus we get

Q N

i,j1

a_ijnx, unDjunDihunξdxdt

−→

Q N

i,j1

aijx, TkuDjTkuDihuξdxdt

Q N

i,j1

aijx, uDjuDihuξdxdt.

2.53

It follows from2.28that

_T

0

Athunt|Γξt|Γdt−→

_T

0

Equation2.24yields

Q

fhunξdxdt−→

Q

fhuξdxdt. 2.55

Takingn → ∞in2.47, by2.48,2.49, and2.53–2.55, we obtain

−

Q ξt

_u

0

hrdrdxdt

Q N

i,j1

aijx, uDjuDihuξdxdt

_T

0

Athut|_Γξt|_Γdt

Q

fhuξdxdt.

2.56

For any givenm >0, takingv T1unt−Tmuntin2.10, then integrating over0, T,

we get

T

0

<unt, T1un−Tmun> dt

Q N

i,j1

a_ijnx, unDjunDiT1un−Tmundxdt

T

0

AtT1unt|_Γ−Tmunt|_Γdt

Q

fT1un−Tmundxdt.

2.57

Setting S1s

s

0T1t−Tmtdt, then 0 ≤ S1s ≤ |s|sign0|s| −m, for alls ∈ R, where

sign₀s 1 ifs >0, sign₀s 0 ifs≤0. Thus we get

{m≤|un|≤m 1} N

i,j1

aijx, unDjunDiundxdt≤

{t∈0,T:|unt|Γ|≥m}

|At|dt

{|un|≥m}

_fdxdt.

2.58

Letn, mtend to the infinity in2.58, respectively, then one can deduce thatusatisfies2.7. Thusuis a renormalized solution to problemPin the sense ofDefinition 2.1. This finishes the proof ofTheorem 2.5.

Remark 2.8. Using the same approach as before, we can deal with the nonzero initial value

u0/0. In fact, we only replaceηνu Tkuνbyηνu Tkuν e−νtTku0in2.30.

3. Uniqueness of Renormalized Solution to Problem

P

Only simply modifying12, Lemma 3.1, we can obtain the following result.

Lemma 3.1. Letube a renormalized solution to problemPfor the dataf, A. Then

Q

ξtsign₀u

u

0

hrdrdxdt

T

0

sign₀ut|_ΓAtξt|_Γhut|_Γdt

Q

sign₀ufhuξdxdt

≥

Q

sign₀u

N

i,j1

aijx, uDjuDihuξdxdt,

3.1

Q

ξtsign₀−u

u

0

hrdrdxdt

T

0

sign₀−ut|_ΓAtξt|_Γhut|_Γdt

Q

sign₀−ufhuξdxdt

≤

Q

sign₀−u

N

i,j1

aijx, uDjuDihuξdxdt,

3.2

for anyh∈C1

cR, h≥0, ξ∈W, ξ≥0.

Let sign denote the multivalued function defined by sign r 0 if r < 0, and sign 0⊂0,1, sign r 1 ifr >0.

Lemma 3.2. Fori1,2, letfi∈L2Q,Ai∈L20, T,uibe a renormalized solution to problemP

for the datafi, Ai. Then there existK1∈sign u1−u2andK2 ∈sign u1|Γ−u2|Γsuch that

−

Q

ξtsign0u1−u2

_u1

u2

hrdrdxdt

Q

sign₀u1−u2

N

i,j1

hu1aijx, u1Dju1−hu2aijx, u2Dju2

Diξdxdt

Q N

i,j1

sign₀u1−u2

hu1aijx, u1Dju1Diu1−hu2aijx, u2Dju2Diu2

ξdxdt

≤

Q

K1

hu1f1−hu2f2

ξdxdt

T

0

K2hu1t|_ΓA1t−hu2t|_ΓA2tξ|_Γdt,

∀h∈C1cR, h≥0, ∀ξ∈W, ξ≥0.

3.3

Proof. Letξ∈W,ξ≥0,ρlbe a sequence of mollifiers in R with suppρl⊂−2/l,0andρl≥0.

Define

Note that forlsufficiently large,

x, s−→ξlx, t, s∈W, ∀t∈0, T, x, t−→ξlx, t, s∈W, ∀s∈0, T.

3.5

Leth ∈ C1

cR,h ≥ 0,Hε ∈ W1,∞Rbe defined byHεr Hr/ε, whereH ∈ W1,∞R, Hr 0 forr ≤ 0,Hr rfor 0< r <1 andHr 1 ifr ≥1. Asu1,u2 are renormalized

solutions , according to2.6, for a.e.t∈0, T, we have

Q ξls

u1

0

hrHεr−u2t, xdrdxds

Q

f1hu1Hεu1s, x−u2t, xξldxds

_T

0

A1shu1s|ΓHεu1s|Γ−u2t|Γξls|Γds

Q N

i,j1

aijx, u1Dju1Dihu1Hεu1s, x−u2t, xξldxds

3.6

and for a.e.s∈0, T, we have

Q ξlt

u2

0

hrHεu1s, x−rdrdxdt

Q

f2hu2Hεu1s, x−u2t, xξldxdt

_T

0

A2thu2t|ΓHεu1s|Γ−u2t|Γξlt|Γdt

Q N

i,j1

aijx, u2Dju2Dihu2Hεu1s, x−u2t, xξldxdt.

3.7

Integrating the above two equalities int, respectively, s over0, Tand taking their difference, we get

_T

0

Q

ξls

_u1

0

hrHεr−u2t, xdr−ξlt

_u2

0

hrHεu1s, x−rdr

dxdsdt

T

0

Q

f1hu1Hεu1s, x−u2t, x−f2hu2Hεu1s, x−u2t, x

ξldxdsdt

_T

0

_T

0

A1shu1s|ΓHεu1s|Γ−u2t|Γξls|Γ

_T 0 Q N

i,j1

aijx, u1Dju1hu1DiHεu1s, x−u2t, xξl

−aijx, u2Dju2hu2DiHεu1s, x−u2t, xξl

dxdsdt T 0 Q N

i,j1

hu1aijx, u1Dju1Diu1Hεu1s, x−u2t, x

−hu2aijx, u2Dju2Diu2Hεu1s, x−u2t, x

ξldxdsdt.

3.8

Denote the three integrals on the left-hand side byI1,I2,I3, the two integrals on the

right-hand side byI4,I5.

It is easy to prove that

lim

l→ ∞εlim→0I5

Q N

i,j1

sign₀u1−u2ξ

hu1aijx, u1Dju1Diu1−hu2aijx, u2Dju2Diu2

dxdt.

3.9

Similarly to the estimates forI2in12,c.f. page 102, we can obtain

lim

l→ ∞εlim→0I2≤

Q

K1

f1hu1−f2hu2

ξdxdt,

lim

l→ ∞εlim→0I3≤

_T

0

ξt|_ΓK2hu1t|ΓA1t−hu2t|ΓA2tdt,

3.10

where K1 χ{u1>u2} χ{u1u2}sign0f1 − f2, K2 χ{t∈0,T:u1t|Γ>u2t|Γ} sign0A1 −

A2χ{t∈0,T:u1t|Γu2t|Γ}.

As forI1, recall that suppρl⊂−2/l,0, hence

lim

ε→0I1

_T

0

Q

sign₀u1x, s−u2x, t

ξls

_u1x,s

u2x,t

hrdr ξlt

_u1x,s

u2x,t hrdr

dxdsdt

Q

ξlx, t,0sign0−u2x, t

₀

u2x,t

hrdrdxdt

Q

ξlx,0, ssign0u1x, s

_u1x,s

0 hrdrdxds _T 0 Q

sign₀u1x, s−u2x, tξtx, tρlt−s

_u1x,s

u2x,thrdrdsdt

Q

ξx,0ρl−ssign0u1x, s

_u1x,s

0

hrdrdxdsI11 I12.

We have

lim

l→ ∞I11

Q

sign₀u1−u2ξt

_u1

u2

hrdrdxdt. 3.12

Consider the function

φlx, s

_T

s

ρl−rdrξx,0

_2/l

inf{s,2/l}ρl−rdrξx,0. 3.13

Note thatξ ∈ W, thus forl sufficiently large,φl ∈ W. Applying3.1with u u1,ξ φl, ff1,At A1s, andts, we have

I12 −

Q

φl

ssign0u1

_u1

0

hrdrdxds

≤

T

0

sign₀u1s|_ΓA1sφls|_Γhu1s|_Γds

Q

sign₀u1φlfhu1dxds−

Q

sign₀u1

N

i,j1

aijx, u1Dju1Di

φlhu1

dxds.

3.14

It is easy to prove that the integrals on the right-hand side of3.14converge to 0 asl → ∞. Thus we get

lim

l→ ∞I12 ≤0. 3.15

It remains to considerI4. We have

I4

T

0

Q N

i,j1

hu1aijx, u1Dju1Hεu1−u2t,·

−hu2aijx, u2Dju2Hεu1s,·−u2

Diξldxdsdt

1

ε

0,T×Q∩{0<|u1x,s−u2x,t|<ε}

N

i,j1

hu1aijx, u1Dju1Diu1−u2t,·

−hu2aijx, u2Dju2Diu1s,·−u2

ξldxdsdt

I41 I42.

It is easy to see that

lim

l→ ∞εlim→0I41

Q

sign₀u1−u2

N

i,j1

hu1aijx, u1Dju1−hu2aijx, u2Dju2

Diξdxdt.

3.17

Since

I42

1

ε

0,T×Q∩{0<|u1x,s−u2x,t|<ε}ξl

N

i,j1

hu1x, s−hu2x, t

×aijx, u1Dju1Diu1−u2t,·dxdsdt

1

ε

0,T×Q∩{0<|u1x,s−u2x,t|<ε}ξl

N

i,j1

hu2x, t

×aijx, u1Dju1−aijx, u2Dju2

Diu1−u2dxdsdt

I₄₂1 I₄₂2,

3.18

letk >0 such that supph⊂−k, k, forε <1,we have

I₄₂1≤

0,T×Q∩{0<|u1x,s−u2x,t|<ε}

N

i,j1

|ξl|h|dk 1x|k 1 aijx,0

×2|DTk 1u1|2 |DTk 1u2|2

dxdsdt.

3.19

Noting that the right side integral of3.19belongs toL1_{0, T×Q, we get}

lim

ε→0I

1

42 0. 3.20

It follows from2.1and2.2that

I₄₂2 1 ε

0,T×Q∩{0<|u1x,s−u2x,t|<ε} ξl

×N

i,j1

hu2x, t

aijx, u1Dju1−aijx, u1Dju2

Diu1−u2dxdsdt

1

ε

×N

i,j1

hu2x, t

aijx, u1−aijx, u2

Dju2Diu1−u2dxdsdt

≥ −1

ε

0,T×Q∩{0<|u1x,s−u2x,t|<ε}ξl

×N

i,j1

hu2|dk 1x|

|DTk 1u1|2 2|DTk 1u2|2

|u1−u2|dxdsdt.

3.21

Using the same approach as in3.20, we get

lim

ε→0

1

ε

0,T×Q∩{0<|u1x,s−u2x,t|<ε} ξl

×N

i,j1

hu2|dk 1x|

|DTk 1u1|2 2|DTk 1u2|2

|u1−u2|dxdsdt0.

3.22

By3.20–3.22and3.18we have

lim

ε→0I42≥0. 3.23

Equations 3.8–3.12, 3.15–3.18, 3.20, and 3.23 imply that 3.3 holds. Thus

Lemma 3.2is proved.

Remark 3.3. In fact, by the density result,3.3 is satisfied by any given ξ ∈ W1 {ξ ∈

W1,∞Q|ξT 0, ξt|_ΓCt an arbitary function of t}andξ≥0.

Now we state the uniqueness and comparison principle of renormalized solution to problemPas follows.

Theorem 3.4. Under hypotheses2.1and2.2, fori1,2, letfi ∈L2Q,Ai ∈L20, T,uibe a

renormalized solution to problemPfor the datafi, Ai. Then there existK1∈sign u1−u2and

K2∈sign u1|_Γ−u2|_Γsuch that for a.e.0< τ < T,

Ωu1τ−u2τ dx≤

τ

0

ΩK1

f1−f2

dxdt

τ

0

K2A1−A2dt. 3.24

In particular, for any givenA∈L2_{0, Tandf∈L}2_{Q, the renormalized solutionuto}

Proof. For any givenτ∈0, Tand any givenε >0 sufficiently small, letαεsbe defined by

αεt

⎧ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎩

1, if 0≤t≤τ−ε, τ−t

ε , ifτ−ε < t < τ,

0, ifT ≥t≥τ.

3.25

Definingξx, t αεt,∀x, t∈Q, it is easy to see thatξ∈W1andξ≥0.

Takingξαεin3.3, we get

−

Q

αεtsign0u1−u2

u1

u2

hrdrdxdt

Q N

i,j1

sign₀u1−u2αε

hu1aijx, u1Dju1Diu1−hu2aijx, u2Dju2Diu2

dxdt

≤

Q

K1

hu1f1−hu2f2

αεtdxdt

_T

0

K2hu1t|ΓA1t−hu2t|ΓA2tαεtdt.

3.26

Defininghmr infm 1− |r| ,1and replacinghwithhm in3.26, then lettingm →

∞, we obtain

−

Q

αεtu1−u2 dxdt≤

Q

αεtK1

f1−f2

dxdt

_T

0

K2A1t−A2tαεtdt. 3.27

In fact, by the Lebesgue dominated convergence theorem, we have

−

Q

αεtsign0u1−u2

_u1

u2

hmrdrdxdt−→ −

Q

αεtu1−u2 dxdt,

Q

K1

hmu1f1−hmu2f2

αεtdxdt−→

Q

αεtK1

f1−f2

dxdt,

_T

0

K2hmu1t|ΓA1t−hmu2t|ΓA2tαεtdt,−→

_T

0

As for the second term in3.26, we have

Q N

i,j1

sign₀u1−u2αε

hmu1aijx, u1Dju1Diu1−hmu2aijx, u2Dju2Diu2

dxdt

Q N

i,j1

sign₀u1−u2αεhmu1aijx, u1Dju1Diu1dxdt

−

Q N

i,j1

sign₀u1−u2αεhmu2aijx, u2Dju2Diu2dxdtJ1 J2.

3.29

Moreover,

|Jk| ≤

{m≤|uk|≤m 1} N

i,j1

aijx, ukDjukDiukdxdt, k1,2. 3.30

Asu1,u2are renormalized solutions, noting2.7, we prove

lim

m→ ∞J1mlim→ ∞J20. 3.31

By3.29and3.31, the second term in3.26tends to zero.

Lettingεtend to zero in3.27,3.24follows from3.25and3.27.

4. The Relation between Weak Solutions and

Renormalized Solutions for Problem

P

In this section, we will see that the concept of renormalized solution is an extension of the concept of weak solution. The main result in this section is the following theorem.

Theorem 4.1. iAssume thatu∈L20, T;V∩L∞0, T;L2Ωandaij·, uDju∈L2Q,i, j

1,2, . . . N. Thenuis a weak solution to problemPif and only ifuis a renormalized solution to problemP.

iiIfu∈L20, T;V∩L∞Q, thenuis a weak solution to problemPif and only ifuis a renormalized solution to problemP.

Proof. iIfuis a weak solution to problemP, we have

_T

0

ut, vV,Vdt

Q N

i,j1

aijx, uDjuDivdxdt

Q

fvdxdt

_T

0

Atvt|_Γdt, ∀v∈L2_{0, T;V,}

ux,0 0. 4.2

Noting thatu ∈ L2_{0, T;V,a}

ijx, uDju ∈ L2Q, i, j 1,2. . . N, we haveu ∈ L20, T;V,

henceu∈C0, T;L2_{Ω, for any givenh∈C}1

cRandξ∈W. Takingvξhuin4.1, we

have

_T

0

ut, ξhudt−

Q N

i,j1

aijx, uDjuDiξhudxdt

Q

fξhudxdt

_T

0

Athut|_Γξt|Γdt.

4.3

By12, Lemma 1.4, we have

_T

0

ut, ξhudt−

Q ξt

_u

0

hrdrdxdt. 4.4

Equations4.3and4.4imply that

−

Q ξt

_u

0

hrdr

Q N

i,j1

aijx, uDjuDiξhudxdt

Q

fξhudxdt

_T

0

Athut|_Γξt|_Γdt. ∀ξ∈W, h∈Cc1R.

4.5

For any givenm, letSmr

_r

0Tm 1s−Tmsds, it is easy to see that 0≤Smr≤ |r|. Taking

vTm 1u−Tmuin4.1, we get

ΩSmux, Tdx

Q N

i,j1

aijx, uDjuDiTm 1u−Tmudxdt

Q

fTm 1u−Tmudxdt

T

0

AtTm 1ut|Γ−Tmut|Γdt.

4.6

By Lebesgue’s dominated convergence theorem, we obtain the first term on the left side, and all terms on the right side of4.6converge to zero asm → ∞. Then it follows from4.6

that

lim

m→ ∞

{m≤|u|≤m 1}

N

i,j1

aijx, uDjuDiudxdt

lim

m→ ∞

Q N

i,j1

aijx, uDjuDiTm 1u−Tmudxdt0.

Conversely, assume that u is a renormalized solution. Applying2.6 with hu

Hn 1− |u|,where H ∈ C∞R,H ≥ 0,H 0 on−∞,0,and H 1 on1, ∞, as

n → ∞, we get

−

Q

ξtudxdt

Q N

i,j1

aijx, uDjuDiξdxdt

Q fξdxdt

T

0

Atξt|_Γdt. 4.8

Henceuis a weak solution to problemP.

iiDue tou∈L∞Q, assumption3.1and the definition of a renormalized solution to problemP,iiis an immediate consequence ofi.

Remark 4.2. Theorems2.5and3.4improve those results of1,3,8.

Acknowledgments

This work is supported by NCET of Chinano: 060275and Foundation of Maths X of DLUT.

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