The Existence and Behavior of Solutions for Nonlocal Boundary Problems

Volume 2009, Article ID 484879,17pages doi:10.1155/2009/484879

Research Article

The Existence and Behavior of Solutions for

Nonlocal Boundary Problems

Yuandi Wang

_{and Shengzhou Zheng}

1_{Department of Mathematics, Shanghai University, Shanghai 200444, China}

2_{Department of Mathematics, Beijing Jiaotong University, Beijing 100044, China}

Correspondence should be addressed to Yuandi Wang,yuandi [email protected]

Received 16 October 2008; Revised 19 March 2009; Accepted 23 March 2009

Recommended by Pavel Drabek

The purpose of this work is to investigate the uniqueness and existence of local solutions for the boundary value problem of a quasilinear parabolic equation. The result is obtained via the abstract theory of maximal regularity. Applications are given to some model problems in nonstationary

radiative heat transfer and reaction-diffusion equation with nonlocal boundary flux conditions.

Copyrightq2009 Y. Wang and S. Zheng. 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

The existence of solutions for quasilinear parabolic equation with boundary conditions and initial conditions can be discussed by maximal regularity, and more and more works on this field show that the maximal regularity method is efficient. Here we will use some of recently results developed by H. Amann to investigate a specific boundary value problems and then apply the existence theorem to two nonlocal problems.

This paper consists of three parts. In the next section we present and prove the existence and unique theorem of an abstract boundary problem. Then we give some applications of the results in Sections3and4to two reaction-diffusion model problems that arise from nonstationary radiative heat transfer in a system of moving absolutely black bodies and a reaction-diffusion equation with nonlocal boundary flux conditions.

2. Notations and Abstract Result

We consider the following quasilinear parabolic initial boundary value problemIBVP for short:

Bt, x, uuδgt, x, u,∇u, on∂Ω,

ux,0 u0x, onΩ,

2.1

whereΩ is a bounded strictly Lipschitz domain with its boundary Γ ∂Ω Γ0∪Γ1 and

Γ0∩Γ1∅,QT 0, T×Ω,

At, x, uu−∇at, x, u∇u, 2.2

and−Ais a second-order strongly elliptic differential operator with the boundary operator given by

Bt, x, uu:δ∂νau 1−δγu. 2.3

The coefficient matrixa aijn×nsatisfies regularity conditions onQT×R, respectively. The directional derivative∂νau : γa∇u·ν,ν is the outer unit-normal vector onΓ; the function δ: Γ → {0, 1}is defined asδ−1_{j: Γ}

jforj0,1;γdenotes the trace operator. We introduce precise assumptions:

f:Q_T×Rn1−→R,

g:R×Γ×Rn1−→R,

2.4

whereR: 0,∞are Carath´eodory functions; that is,fresp.,gis measurable int, x∈ QT resp., in t, x, y ∈ R ×Γ×Ωfor eachu ∈ Rand continuous inufor a.e.t, x ∈ QT

resp.,t, x∈R×Γ. More general, the functiongcan be a nonlocal function, for example,

gt, x, u _Ωkt, x, y, udyorgt, x, u _Γkt, x, udσ.

LetXandYbe Banach spaces, we introduce some notations as follows:

iJT : 0, T,

◦

JT:JT\ {0}.α∧β:min{α, β},α∨β:max{α, β}.

iiDD, Y:{φ ∈ C∞D, φ:D → Y, suppφ ⊂D}forD ⊂ Rl_,D

1 : {v∈ DJT ×

Ω} ∩ {v|Γ00}.

iiiLX, Y:{all continuous linear operators fromXintoY}, andLX:LX, X.

ivft, udenotes the Nemytskii operator induced byft, x, ut, x,∇ut, x.

vC1−_{X, Ydenotes the set of all locally Lipschitz-continuous functions fromX into}

viCar_0,λ,λM×R×Rn_{,R,λ, andλ≥1, denotes the set of all Carath´eodory functions}

fonm∈Msuch thatfm,0 0, and there exists a nondecreasing functionψ ≥0 with

_{fm, u, ξ−fm, v, ξ≤ψ}

ρ1|ξ|λ−1|u−v|,

fm, u, ξ−fm, u, η≤ψρ1|ξ|λ−1ηλ−1 ξ−η

for|u|, |v| ≤ρ. 2.5

Particularly,fis independent ofξifλ∧λ≤1.

viiWs

pΩdenotes the Sobolev-Slobodeckii space fors ∈Randp ≥1 with the norm

· Ws

p, especially,W

0

pΩ LpΩ; and

∂W_ps:∂W_psΓ:Wps−1/pΓ0×Wps−1−1/pΓ1

p >1. 2.6

viiiWs

p,BΩ,s∈−2, 2\ {Z1/p}Zis the set of integral numbers, is defined as

Ws q,B

⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩

u∈Ws q;Bu0

, 1 1

q < s≤2,

u∈Ws

q;γu0 on Γ0

, 1

q < s <1

1

q, Ws

q, 0≤s < 1

q,

W−s q,B

, −2≤s <0, s /∈Z1 q,

2.7

whereqq/q−1,Xis the dual space ofX, andB is the formally adjoint operator.

xW1

pJ:W1pJ,E1, E0:Wp1J, E0∩LpJ, E1ifE1

d

→E0andJ is an interval in

R.

xiMRpJT : MRpJT,E1, E0 denotes all maps B possessing the property of

maximal Lpregularity on JT with respect to E1, E0, that is, given h ∈ LpJT, E0,

the initial problem

˙

vBvh, inJ◦_T, v0 0 2.8

has a unique solutionv∈W1

pJT,E1, E0.

Now we turn to discuss the local existence result. We write

then,

W1

pJT,E1, E0→

⎧ ⎨ ⎩

CJT, E,

CQ_T, ifp > n2. 2.10

Exactly,W1

pJT,E1, E0→BUCQTasp > n2, where BUCQTdenotes the Banach space of all functions being bounded and uniformly continuous inQT. So, we will not emphasize it in the following.

Aweaksolutionuof IBVP2.1is defined as aW1

pJT,E1, E0functionu,s∈1,1

1/p, satisfying

_T

0

−v, u˙ ∇v,at, u∇udt− v0, u0

T

0

v, ft, uγv, gt, u_∂1dt ∀v∈ D1,

2.11

where·,·and·,·_∂j denote the obvious duality pairings onΩandΓj, respectively. Set

ft, u:f0t f1t, u, gt, u:g0t g1t, u with f1t,0 g1t,0 0. 2.12

After these preparations we introduce the following hypotheses:

H1p > n2 ands∈1,11/p.

H2at, x, u∈C0,α,1−_J

T ×Ω×R,Rn×nwithα1 > s, and there exists aδ0 ∈0,1such

that

δ0|ξ|2≤ξ·at, x, uξ≤ |ξ|2/δ0, ∀t, x, u∈JT×Ω×R. 2.13

f0, g0 ∈ LpJ, E0 ×∂WpsΓ,f1 ∈ Car0,λ0,λ1QT ×R×R

n_{with λ}

0 ∈ 1, 2, and

λ1<1ps−1/2p1−s.

H3g1t,·∈C1−Wp1JT, LrJT, ∂WpsΓfor somer > p.

Theorem 2.1. Let assumptions (H1)–(H3) be satisfied. Then for eachu0∈Ethe quasilinear problem

2.1possesses a unique weak solutionut, x∈W1

pJT∗,E₁, E₀for someT∗>0.

Proof. Recall that

E1→Cs−n/p

Ω, E →Cs−n2/pΩ. 2.14

The Nemytskii operatora·, uis defined asa·, ut, x:at, x, ut, x. The fact

a·, ut, x∈C0,α∧s−n2/p_Q T

shows the maximal regularity of the operatorA. By 1, Theorem 2.1, if, for t ≤ T,f1 ∈

C1−_W1

pJt, LrJt, E0 for some r > p, then the existence and the uniqueness of a local

solution will be proved.

The remain work is to check the Lipschitz-continuity. Set

q: np

n 2−sp,

θ:1p 21−s.

2.16

ThenLqΩ→E0. So, foru, v∈W1pJtwith|u|, |v| ≤ρwe have

f1t, x, u,∇u−f1t, x, u,∇v_LrJt,E0

≤f1t, x, u,∇u−f1t, x, u,∇v_LrJt,LqΩ

≤cρ |∇u|λ1−1|∇u−v|

LrJt,LqΩ

≤cρ∇uλ_L1−1

λ1−1pq/p−qΩ ∇u−vLpΩ

LrJt .

2.17

Fromλ1−1q < p−q, we infer that

f1t, x, u,∇u−f1t, x, u,∇v_LrJt,E0

≤cρ uλ_W1−11

pΩu−vWp1Ω

LrJt

≤cρ u_Cλ1−_I11−θ

t,E · u−v

1−θ CIt,E·

u_Eλ1−1θ

1 u−v

θ E1

LrJt

≤cρ u_Cλ1−_I11−θ

t,E · u−v

1−θ CIt,E· u

θλ1−1

Lr∗Jt,E1· u−v

θ LpJt,E1,

2.18

wherer∗: λ1−1prθ/p−θr. Note thatλ1θ <1, we can chooser > psuch that

_{f1t, x, u,∇u−f1t, x, u,∇v}

LrJt,E0≤c

u_W1

pJt

· u−v_W1

pJt. 2.19

On the other hand, the hypotheses guarantee that

ft, x, u,∇v−ft, x, v,∇v_L rJt,E0

≤ft, x, u,∇v−ft, x, v,∇v_L

rJt,LqΩ

≤cρ |∇v|λ0−1|u−v|

LrJt,LqΩ

≤cρu−v_CJt,E·|∇v|λ0−1

LrJt,LqΩ .

Due toq < pandλ0∈1, 2, H ¨older inequality follows that

ft, x, u,∇v−ft, x, v,∇v_L

rJt,E0≤c

ρu−v_CJt,E· t

0

∇v_Lλ0−1r pΩ dτ

1/r

. 2.21

The hypothesis ofλ0means that one can find anr > psuch that

_{ft, x, u,∇v−ft, x, v,∇v}

LrJt,E0 ≤c

ρ,v_W1

pJt

· u−v_W1

pJt. 2.22

Obviously, if λ0 1, the above inequality is followed from 2.20 immediately. Hence it

follows from2.19and2.22that

ft, x, u,∇u−ft, x, v,∇v_L

rJt,E0≤c

u, v_W1

pJt

· u−v_W1

pJt. 2.23

This ends the proof.

We apply the above theorem to the following two examples in next sections. For this, in the remainder we suppose that hypothesesH1-H2hold and that

Γ Γ1Γ0∅, p˙:

n−1p

n 1−sp. 2.24

3. A Radiative Heat Transfer Problem

We see a nonlinear initial-boundary value problem, which, in particular, describes a nonstationary radiative heat transfer in a system of absolutely black bodiese.g., refer to

2. A problem is

ut− ∇at, x, u∇u f0t, x inQT,

∂νauu

Γh

ut, yϕt, x, ydσy−hut, x g0t, x on0, T×Γ,

u0, x u0x, onΩ.

3.1

3.1. Local Solvability

We assume thatHr

Hr1ϕ∈LrJT, Lp˙Γ×Γ;

Hr2his locally Lipschitz continuous andh0 0.

Theorem 3.1. Let assumptions (H1)-(H2) and (Hr) be satisfied. Then problem3.1, for allu0 ∈E,

has a uniqueu∈W1

Proof. Note that the embedding2.14holds:

Lp˙Γ→∂Wps, Lp˙Γ→W_p1/p−sΓ

Wps−1/pΓ

. 3.2

HenceTheorem 2.1implies the result immediately.

In fact, Amosov proved in 2005 the uniqueness of the solution for a simple case, that is, problem in which the matrixais independent ofusee2, Theorem 1.4. In this paper, we also can get the positivity of the solution and the estimates of the solution inW1

2Ω

andL∞Ωin this part. We have tried to achieve the global existence, but it is still an open problem.

In the rest of this section, we always assume thatH1-H2andHrhold.

3.2. Positivity

Assume that

Hhuis nondecreasing with h0 0, and

ϕt, x, yϕt, y, x≥0,

ϕt, x:

Γϕ

t, x, ydσy<1.

3.3

Theorem 3.2. Let assumption (H) be satisfied. Iff0, g0, u0is nonnegative, then the solutionuof

problem3.1is also nonnegative.

Proof. Putu₋:0∧u. Multiplying the equation withu₋and integrating overΩ, we have

Ωf0u−dx

1 2

Ω

u2₋

tdx−

Ω∇au∇uu−dx

1

2

Ω

u2₋

tdx

Ωau∇u· ∇u−dx−

Γ

∂u ∂νau−dσ

1

2

Ω

u2₋

tdx

Ωau∇u−· ∇u−dx

Γ

hut, x−

Γh

ut, yϕt, x, ydσyu−dσ−

Γg0u−dσ.

3.4

By using the assumption ofH, we can get following equality:

Γ×Γh

ut, yϕt, x, yut, y−ut, xdσydσ

1

2

Γ×Γ

hut, x−hut, yϕt, y, xut, x−ut, ydσydσ.

So,

Γ

hut, x−

Γh

ut, yϕt, x, ydσyu₋t, xdσ

Γhut, xu−t, x

1−ϕt, xdσ

Γ×Γh

ut, yϕt, x, yu−t, y−u−t, xdσydσ

Γhut, xu−t, x

1−ϕt, xdσ

1

2

Γ×Γ

hut, x−hut, yu₋t, x−u₋t, yϕt, y, xdσydσ≥0.

3.6

At the last inequality, the monotonity ofhonuand the restrictionϕ < 1 are used. Therefore,

d dtu−

2

L2Ω≤

Ωf0u−dx

Γg0u−dσ≤0. 3.7

Ifu0≥0, thenu−t, x≡0. The assertion follows.

3.3.

W

2

Ω-norm

We denote byJmaxthe maximal interval of the solution of problem3.1.

Lemma 3.3. There exists a constantC CT;f0, g0, u0such that the solutionuof problem3.1

satisfies

ut,·_L2Ω∇u2_L

2Qt ≤C fort≤Jmax. 3.8

Proof. Multiplying byuand integrating overΩ, we have

Ωut− ∇au∇uudx

Ωf0 u dx. 3.9

That is,

1 2

d dt

Ω|u| 2_dx

Ωau∇u· ∇u dx−

Γu ∂νau dσ

As similar as the inequality3.6, we have

−

Γu ∂νau dσ−

Γu

Γh

ut, yϕt, x, ydσy−ht, xg0

dσ

Γ

1−ϕt, xhut, xu dσ−

Γg0 u dσ

1

2

Γ×Γ

hut, x−hut, yut, x−ut, yϕt, x, ydσ dσ

≥ −

Γg0 u dσ.

3.11

Hence,

1 2

d dt

Ω|u| 2_dxδ

0

Ω|∇u| 2_dx≤

Ωf0 u dx

Γg0 u dσ

≤

Ωu 2_dx

Γu 2_dσ

1

Ωf 2 0dx

Γg 2 0dσ

.

3.12

By using the embeddingW1

2,BΩ→L2Γand lettingsmall enough, it is easy to get that

u2_L2Ω∇u2_L2Qt≤CT;f0, g0, u0

, fort≤Jmax. 3.13

3.4.

L

Ω-norm

Theorem 3.4. Iff0∈L∞QTandg0∈L∞0, T×Γ, then the solutionut, xof problem3.1is

bounded with itsL∞-norm for allt∈Jmax.

Proof. From the hypothesisH1and embedding 2.10, one has thatu ∈ CQ_Tandun _∈

W₂1Ω, n1,2, . . .. By multiplying withu2k−1 k∈Zandk≥2and integrating overΩ, we have

Ωut− ∇au∇uu 2k₋₁

dx

Ωf0 u 2k₋₁

dx. 3.14

That is,

2−k d

dt

Ω|u| 2k

dx

Ωau∇u· ∇u 2k₋₁

dx−

Γu 2k₋₁

∂νau dσ

Ωf0 u 2k₋₁

But,

Ωau∇u· ∇u 2k₋₁

dx2k−1

Ωu 2k₋₂

au∇u· ∇u dx

2k−122−2k

Ωau∇u 2k−1

· ∇u2k−1dx,

−

Γu 2k₋₁

∂νau dσ−

Γu 2k₋₁

Γh

ut, yϕt, x, ydσy−ht, x g0

dσ Γ

1−ϕt, xu2k₋₁

hut, xu dσ−

Γg0u 2k₋₁

dσ

1

2

Γ×Γ

hut, x−hut, yu2k−1t, x−u2k−1t, yϕt, x, ydσ2,

≥ −

Γg0u 2k₋₁

dσ. 3.16 Therefore, 1 2k d dt

Ω|u| 2k

dxδ0

2k−1

Ω|u| 2k₋₂

|∇u|2dx

1

2k

d dt

Ω|u| 2k

dxδ0

2k₋₁ 4k−1

Ω

∇u2k−1

2dx

≤

Ωf u 2k₋₁

dx

Γg0 u 2k₋₁

dσ

≤1−2−k

Ωu 2k dx Γu 2k dσ

1−2k2−k

Ωf 2k 0 dx Γg 2k 0 dσ , 3.17

where Young’s inequality,αβ≤/rαr−r/r_/rβr_{α, β≥0,0< ε <1, has been used at the} last inequality. We apply the embeddingW1

2,BΩ→L2Γagain withvu2 k−1

and choose

small enough, then we attain the following inequality:

d dt

Ωu 2k

dx−C2k−

Ωu 2k

dx≤1−2k Ωf 2k 0 dx Γg 2k 0 dσ

. 3.18

By Gronwall’s inequality, the inequality3.18becomes

Ωu 2k

dx≤eC2k_−εt

·

Ωu 2k

0 dx1−2

k _t

0

eC2k_−t−τ

Ωf 2k 0 dx Γg 2k 0 dσ

SetB∞:1u0L∞Ωf0L∞QTg0L∞0,T×Γ, then we deduce that

u_L

2kΩ≤

B∞eεCt

ε

e−t

Ω

εu02

k

B2k

∞

dx _t

0 Ω

f₀2k B2k

∞

dx

Γ

g₀2k B2k

∞

dσ !

dτ "1/2k

≤ε−1_B

∞eεCt|Ω|t|Ω||Γ|1/2

k .

3.20

Letk → ∞,the inequality3.20implies

ut,·_L∞Ω≤eCtu0L∞ΩC

T;f0_L∞Q

T, _g0

L∞0,T×Γ

fort∈Jmax. 3.21

The claim follows.

One immediate consequence of the above theorem is.

Corollary 3.5. The L_∞-norm of the solution u, that is, ut,·_L∞Ω, of problem 3.1 is nonincreasing iff0g00.

4. A Nonlocal Boundary Value Problem

We now consider the problem2.1with the following boundary value condition:

Bt, x, u gt, x, u,∇u Φut, x g1t, x, u g0. 4.1

The functionΦin4.1can be in nonlocal form.

IBVP2.1with a nonlocal term stands, for example, for a model problem arising from quasistatic thermoelasticity. Results on linear problems can be found in3–5. As far as we know, this kind of nonlocal boundary condition appeared first in 1952 in a paper6by W. Feller who discussed the existence of semigroups. There are other problems leading to this boundary condition, for example, control theory see7–12 etc.. Some other fields such as environmental science 13 and chemical diffusion14 also give rise to such kinds of problems. We do not give further comments here.

Carl and Heikkil¨a 15 proved the existence of local solutions of the semilinear problem by using upper and lower solutions and pseudomonotone operators. But their results based on the monotonicity hypotheses off,g, andΦwith respect tou.

In this section, we assume thatH1andH2always hold and assume that

Hn1 Φ0 0 andΦ∈C1−_W1

pJT, LrJT, Lp˙Γfor somer > p;

Hn2g1t, x,0 0,g1 satisfies the Carath´eodory condition on t, x and g1t, x,· ∈

C1−_R.

By the embedding theorem andTheorem 2.1, we get immediately.

Theorem 4.1. Suppose hypotheses of (Hn) satisfy. Then problem2.1, for allu0 ∈E, withgdefined

in4.1has a uniqueu∈W1

For the simplicity in expression, we turn to consider a problem with nonlocal boundary value

utAt, x, uuft, x, u,∇u, inQT,

Bt, x, uuκut, x g1t, x, u g0, on∂Ω,

ux,0 u0x, onΩ,

4.2

where

κut, x:

Ωk

t, x, y, ut, y,∇ut, ydy, 4.3

and

HkThe function k satisfies the Carath´eodory condition on t, x, y ∈ QT : 0, T×Γ×

Ω,k|_u00 andfk∈Car_0,λ

2,λ2QT×R×R

n_with

λ2<1

ps−1

2p1−s, λ2< p1. 4.4

Theorem 4.2. Let assumption (Hk) be satisfied. Then Problem4.2, for anyu0 ∈ E, has a unique

solutionu∈W1

pJT∗for someT∗>0. Proof. First, we see that

Ω|∇u|

λ2−1_|∇u−v|dy

LrJt

≤∇uλ_L2−1

λ₂−1pΩ· ∇u−vLpΩ

LrJt

≤c uλ_W2−11

p · u−vW

1

p

LrJt .

4.5

Chooseθ ∈ 0, 1such thatλ1θ < 1, thenλ2−1θ/1−θ < 1. Consequently, there exists

r > psuch that

Ω|∇u|

λ2−1_|∇u−v|dy

LrJt

≤cu_Cλ2−_J11−θ t,E u−v

1−θ CJt,E·

u_Eλ2−1pθ/p−θr

1

LrJt

u−vθ_E1

LrJt

≤cu_W1

pJt

· u−v_W1

pJt.

Similarly, fromλ2≤p1 we have

Ω|∇u|

λ2−1_|u−v|dy

LrJt

≤c u−v_CJt,E· t

0

ur_W1

pdτ 1/r

≤cu_W1

pJt

· u−v_W1

pJt.

4.7

Combining two inequalities4.6and4.7, we obtain that

κu−κv_LrJt,∂Ws p

≤

Ωkt, x, y, u,∇u−kt, x, y, v,∇vdy

LrJt,Lp˙Γ

≤c

Ωψρ

1|∇u|λ2−1|∇v|λ2−1| ∇u−v|1|∇v|λ2−1|u−v|dy

LrJt

≤cu_W1

pJt

· u−v_W1

pJt.

4.8

The claim follows immediately fromTheorem 4.1.

A special case of problem4.2is

ut− ∇au∇u ft, x, u, inQT,

∂νauu

Ωk

t, x, y, udygt, x, u,

u0, x u0x, onΩ.

on0, T×Γ, 4.9

That is,fandkin4.9are independent of gradient∇u.

4.1.

W

2

Ω-norm

In order to discuss the global existence of solution, in the rest of this section we assume the following.

HklSuppose there exists a continuous functionφ:R → R such that

f1t, x, u, k

Lemma 4.3. There exists a constant C CT;f0, g0, u0 such that the solution of problem4.9

satisfies

u2_L2Ω∇u2_L2Qt≤C, fort≤Jmax. 4.11

Proof. We multiply the first equation in4.9withuand then integrate overΩ, and we find that

1 2

d dtu

2

L2Ωδ0∇u

2

L2Ω

≤

Ωft, x, uu dx

Γu

Ωk

t, x, y, udygt, x, u

dσx

≤φtu2_L

2ΩuL1Γ

u_L1Ωu_L1Γ ε1

u2_L

2Ωu

2

L2Γ

1

ε1

f02_L2Ωg02_L2Γ

.

4.12

SinceWθ

2Ω→L2Γforθ∈1/2,1, by interpolation inequality and Young’s inequality we

have that

u_L2Γ≤Cu_Wθ

2Ω

≤Cuθ_W1

2Ωu

1−θ L2Ω

≤ε2uW1

2ΩCε2uL2Ω.

4.13

Apply Young’s inequality again and then chooseεj small enoughj 1,2; it is not difficult to get

1 2

d dtu

2

L2Ω−Cεφtu

2

L2Ω

δ0−ε

φt 1∇u2_L2Ω

≤ 1 ε1

f02_L2Ωg02_L2Γ

,

4.14

whereδ0 −εφt 1 > 0 fort ∈ 0, T. Therefore, by multiplying withe−2Cε

t

0φτdτ and

integrating over0, t, the inequality4.14follows the claim.

4.2.

L

Ω-norm

Lemma 4.4. Let assumptions ofLemma 4.3be satisfied. Iff0, g0∈L∞QT×0, T×Γ, then the solutionuof problem4.9satisfies

u_L∞Ω≤CT;f0, g0, u0

Proof. We multiply the first equation in4.9withu2k₋₁

and integrate overΩ, then we reach that

Jt: 1 2k

d dtu

2k L₂kΩ

δ0

2k₋₁ 4k−1

∇u2k−12

L2Ω

≤

Ωft, x, uu 2k₋₁

dx

Γu 2k₋₁

Ωk

t, x, y, udygt, x, u

dσx

≤φtu2_Lk

2kΩu

2k₋₁ L₂kΓ

u_L1Ω|Γ|2−k u_L

2kΓ

Ωf0u 2k₋₁

dx

Γg0u 2k₋₁

dσ.

4.16

As the same as the inequality4.13, we have

u2_Lk

2kΩ≤ε1

∇u2k−12

L2Ω

Cε1u2

k L₂kΩ,

u2_Lk−1

2kΩ· uL1Ω≤

ε1∇u2

k−1

2

L2Ω

Cε1u2

k L₂kΩ

1−2−k

· u_L1Ω

≤ε1−2−k

1 ∇u2

k−1

21−2−k

L2Ω ·

u_L1Ω

Cε1u2

k L₂kΩ.

4.17

Hence,

Jt≤φt1Cε1 Cε2

u2_Lk

2kΩ

ε1ε2∇u2

k−1

2

L2Ω

ε1₁−2−k|Γ|2−k·∇u2k−121−2− k L2Ω Cε22−k

f02

k

L₂kΩg0

2k L₂kΓ

.

4.18

We might as well assume thatu_L2Ω>0, so,

∇u2k−1−2

1−k

L2Ω

41−k2−k

Ωu 2k₋₂

|∇u|2dx −2−k

−→ u−_L1_∞Ω ask−→∞. 4.19

The boundedness of solutionu_L2Ω≤Cfort≤Jmaxis used in above deduction.

Letεjj 1,2small enough, then we have

2−kd

dtu

2k

L2kΩ−Cεφtu

2k

L2kΩ≤Cε2

−k_f

02

k

L₂kΩg0

2k L₂kΓ

Multiplying with 2k_·e−Cε2kt

0φτdτ, then integrating over0, t, we obtain that

u_L

2kΩe

−Cεt0φτdτ

2k

≤ u02

k

L₂kΩCε t

0

e−Cε2ks0φτdτf₀2

k

L₂kΩg0

2k L₂kΓ

ds.

4.21

By a similar limitation process as in3.21, we get

u_L∞Ω≤CT;f0, g0, u0

fort≤Jmax. 4.22

This closes the end of proof.

4.3. Decay Behavior

In order to investigate the decay behavior of solution for problem4.9, we assume that

Hkdthere are two continuous functionϕt>0 andt≥0 (t≥0) such that

f1t, x, uu≤ −ϕtu2,

g1t, x, uu≤0,

_k

t, x, y, u≤t|u|

4.23

for allt, x, y∈R×Ω×Γ.

Theorem 4.5. Let the assumption (Hkd) be satisfied and, ube the solution of problem 4.9with

f0, g0 0. ThenuL2Ωdecay to zero ast → ∞for some small functionst.

Proof. We useuto multiply the first equation in the system4.9and then integrate overΩ. Thus, we get that

1 2

d dtu

2

L2Ωδ0∇u

2

L2Ω

≤

Ωft, x, uu dx

Γu

Ωk

t, x, y, udygt, x, u

dσ

≤ −ϕtu2_L

2ΩtuL1ΓuLΩ

≤

Ct #

|Γ||Ω| −ϕt

· u2_L2ΩCt #

|Γ||Ω|∇u2_W1 2Ω.

4.24

In the above process the inequality4.13is used. If we choosetas

t≤ 1

C$|Γ||Ω|·min

δ0, min

t∈0, Tϕt

then

u_L2Ω≤ u0L2Ω·e −t

0ϕτdτ, t∈0, T. 4.26

This ends the proof.

Moreover, one can verify thatu_LpΩalso decay to zeroast → ∞ifp≥2.

Acknowledgments

The first author wishes to thank Professor Herbert Amann for many useful discussions concerning the problem of this paper. The author also want to thank the referees’ suggestions. This work is supported partly by the National NSF of China Grant nos. 10572080 and 10671118and by Shanghai Leading Academic Discipline Projectno. J50101.

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