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

1Department of Mathematics, Shanghai University, Shanghai 200444, China

2Department 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:

u_tAt, x, uuft, x, u,∇u, inQ_T,

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 aij_n×nsatisfies regularity conditions onQ_T×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⁻¹j: Γjforj0,1;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∈ Q_T resp., in t, x, y ∈ R ×Γ×Ωfor eachu ∈ Rand continuous inufor a.e.t, x ∈ Q_T 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,J^◦_T:JT\ {0}.α∧β:min{α, β},α∨β:max{α, β}.

iiDD, Y:{φ ∈ C^∞D, φ:D → Y, suppφ ⊂D}forD ⊂ R^l,D1 : {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 Y.

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 ρ

1|ξ|^λ−1

|u−v|, fm, u, ξ−f

m, u, η≤ψ

ρ

1|ξ|^λ−1η^λ−1 ξ−η for|u|, |v| ≤ρ. 2.5

Particularly,fis independent ofξifλ∧λ≤1.

viiW_p^sΩdenotes the Sobolev-Slobodeckii space fors ∈Randp ≥1 with the norm · _Wp^s, especially,W_p⁰Ω LpΩ; and

∂W_p^s:∂W_p^sΓ:W_p^s−1/pΓ0×W_p^s−1−1/pΓ1 p >1

. 2.6

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

W_q,B^s

⎧⎪

⎪⎪

⎪⎪

⎪⎪

⎪⎪

⎪⎨

⎪⎪

⎪⎪

⎪⎪

⎪⎪

⎪⎪

⎩

u∈W_q^s;Bu0

, 1 1

q < s≤2,

u∈W_q^s;γu0 on Γ0

, 1

q < s <11 q,

W_q^s, 0≤s < 1

q,

W_q^−s,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:W¹_pJ,E1, E0:W_p¹J, E0∩LpJ, E1ifE1

→d E0andJ is an interval in R.

xiMRpJT : MRpJT,E1, E₀ denotes all maps B possessing the property of maximal L_pregularity on J_T with respect to E1, E₀, that is, given h ∈ L_pJT, 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

E₁W_p,B^s Ω, E₀W_p,B^s−2Ω, EE_1/p,pW_p,B^s−2/p , 2.9

then,

W¹_pJT,E1, E₀→

⎧⎨

⎩

CJT, E, C

Q_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 inQ_T. 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_∂1

dt ∀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∈C^0,α,1−J_T ×Ω×R,R^n×nwithα1 > s, and there exists aδ0 ∈0,1such that

δ₀|ξ|²≤ξ·at, x, uξ≤ |ξ|²/δ₀, ∀t, x, u∈J_T×Ω×R. 2.13 f0, g₀ ∈ L_pJ, E0 ×∂W_p^sΓ,f₁ ∈ Car_0,λ0,λ1Q_T ×R×Rⁿwith λ₀ ∈ 1, 2, and λ₁<1ps−1/2p1−s.

H3g₁t,·∈C¹⁻W¹_pJT, LrJT, ∂W_p^sΓfor 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→C^s−n/p Ω

, E →C^s−n2/p Ω

. 2.14

The Nemytskii operatora·, uis defined asa·, ut, x:at, x, ut, x. The fact a·, ut, x∈C^{0,α∧s−n2/p}

Q_T

2.15

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∈W¹_pJtwith|u|, |v| ≤ρwe have f₁t, x, u,∇u−f₁t, x, u,∇v

LrJt,E0

≤f₁t, x, u,∇u−f₁t, x, u,∇v

LrJt,LqΩ

≤c

ρ |∇u|^λ1−1|∇u−v|

LrJt,LqΩ

≤c

ρ∇u^λ_L¹⁻¹

^λ1−1^pq/p−qΩ ∇u−v_LpΩ

LrJt.

2.17

Fromλ1−1q < p−q, we infer that

f₁t, x, u,∇u−f₁t, x, u,∇v

LrJt,E0

≤c

ρ u^λ_W¹⁻¹1

pΩu−v_Wp¹_Ω

LrJt

≤c ρ

u^λ_CI^1−11−θ

t,E · u−v^1−θ_CIt,E·u^λ_E1^1−1θu−v^θ_E1

LrJt

≤c ρ

u^λ_CI^1−11−θ_t,E · u−v^1−θ_CIt,E· u^θλ_L ¹⁻¹

r∗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

LrJt,E0

≤ft, x, u,∇v−ft, x, v,∇v

LrJt,LqΩ

≤c

ρ |∇v|^λ0−1|u−v|

LrJt,LqΩ

≤c ρ

u−v_CJt,E·|∇v|^λ0−1

Lr^Jt,LqΩ.

2.20

Due toq < pandλ0∈1, 2, H ¨older inequality follows that ft, x, u,∇v−ft, x, v,∇v

LrJt,E0≤c ρ

u−v_CJt,E· _t

0

∇v^λ_L^0−1r

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

LrJt,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 u

t, y ϕ

t, x, y dσ

y

−hut, x g0t, x on0, T×Γ, u0, x u0x, onΩ.

3.1

3.1. Local Solvability We assume thatHr

Hr1ϕ∈L_rJT, L_p˙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_pJT^∗for someT^∗>0.

Proof. Note that the embedding2.14holds:

L_p˙Γ→∂W_p^s, L_p˙Γ→W_p^1/p−s Γ

W_p^s−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 matrixa is independent ofusee2, Theorem 1.4. In this paper, we also can get the positivity of the solution and the estimates of the solution inW₂¹Ω 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

ϕ t, x, y

ϕ t, y, x

≥0, ϕt, x :

Γϕ t, x, y

dσ y

<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

∂ν_au₋dσ 1

2

Ω

u²₋

tdx

Ωau∇u−· ∇u−dx

Γ

hut, x−

Γh u

t, y ϕ

t, x, y dσ

y u₋dσ−

Γg0u₋dσ.

3.4

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

Γ×Γh u

t, y ϕ

t, x, y u

t, y

−ut, x dσ

y dσ 1

2

Γ×Γ

hut, x−h u

t, y ϕ

t, y, x

ut, x−u t, y

dσ y

dσ.

3.5

So,

Γ

hut, x−

Γh u

t, y ϕ

t, x, y dσ

y

u₋t, xdσ

Γhut, xu−t, x

1−ϕt, x dσ

Γ×Γh u

t, y ϕ

t, x, y u₋

t, y

−u₋t, x dσ

y dσ

Γhut, xu−t, x

1−ϕt, x dσ 1

2

Γ×Γ

hut, x−h u

t, y

u₋t, x−u₋ t, y

ϕ t, y, x

dσ y

dσ≥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.W₂¹Ω-norm

We denote byJ_maxthe 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_L2Qt ≤C fort≤J_max. 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|²dx

Ωau∇u· ∇u dx−

Γu ∂νau dσ

Ωf0 u dx. 3.10

As similar as the inequality3.6, we have

−

Γu ∂_νau dσ−

Γu

Γh u

t, y ϕ

t, x, y dσ

y

−ht, x g₀

dσ

Γ

1−ϕt, x

hut, xu dσ−

Γg0 u dσ 1

2

Γ×Γ

hut, x−h u

t, y

ut, x−u t, y

ϕ t, x, y

dσ dσ

≥ −

Γg₀ u dσ.

3.11

Hence, 1 2

d dt

Ω|u|²dxδ₀

Ω|∇u|²dx≤

Ωf₀ u dx

Γg₀ u dσ

≤

Ωu²dx

Γu²dσ

1

Ωf₀²dx

Γg₀²dσ

.

3.12

By using the embeddingW_2,B¹ Ω→L₂Γand lettingsmall enough, it is easy to get that

u²_L2Ω∇u²_L2Qt≤C

T;f0, g0, u0

, fort≤Jmax. 3.13

3.4.L_∞Ω-norm

Theorem 3.4. Iff₀∈L_∞Q_Tandg₀∈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 withu^2k−1 k∈Zandk≥2and integrating overΩ, we have

Ωut− ∇au∇uu^2k−1dx

Ωf0 u^2k−1dx. 3.14 That is,

2^−k d dt

Ω|u|^2kdx

Ωau∇u· ∇u^2k−1dx−

Γu^2k−1 ∂νau dσ

Ωf0 u^2k−1dx. 3.15

But,

Ωau∇u· ∇u^2k−1dx

2^k−1

Ωu^2k−2 au∇u· ∇u dx

2^k−1 2^2−2k

Ωau∇u^2k−1· ∇u^2k−1dx,

−

Γu^2k−1∂_νau dσ−

Γu^2k−1

Γh u

t, y ϕ

t, x, y dσ

y

−ht, x g₀

dσ

Γ

1−ϕt, x

u^2k−1 hut, xu dσ−

Γg₀u^2k−1dσ 1

2

Γ×Γ

hut, x−h u

t, y

u^2k−1t, x−u^2k−1 t, y

ϕ t, x, y

dσ²,

≥ −

Γg0u^2k−1dσ.

3.16

Therefore,

1 2^k

d dt

Ω|u|^2kdxδ0

2^k−1

Ω|u|^2k−2|∇u|²dx 1

2^k d dt

Ω|u|^2kdxδ₀2^k−1 4^k−1

Ω

∇u^2k−12dx

≤

Ωf u^2k−1dx

Γg0 u^2k−1dσ

≤

1−2^−k

Ωu^2kdx

Γu^2kdσ

^1−2k2^−k

Ωf₀^2kdx

Γg₀^2kdσ

,

3.17

where Young’s inequality,αβ≤/rα^r−r/r/rβ^rα, β≥0,0< ε <1, has been used at the last inequality. We apply the embeddingW_2,B¹ Ω→L₂Γagain withvu^2k−1and choose small enough, then we attain the following inequality:

d dt

Ωu^2kdx−

C2^k−

Ωu^2kdx≤^1−2k

Ωf₀^2kdx

Γg₀^2kdσ

. 3.18

By Gronwall’s inequality, the inequality3.18becomes

Ωu^2kdx≤e^C2k−εt·

Ωu²₀^kdx^1−2k _t

0

e^C2k−t−τ

Ωf₀^2kdx

Γg₀^2kdσ

dτ. 3.19

SetB_∞:1u0_L∞Ωf0_L∞Q

Tg0_L∞0,T×Γ, then we deduce that u_L

2kΩ≤ B_∞e^εCt ε

e^−t

Ω

εu0^2k B_∞^2k dx

_t

0 Ω

f₀^2k B²_∞^kdx

Γ

g₀^2k B_∞^2kdσ

! dτ

"1/2^k

≤ε⁻¹B_∞e^εCt|Ω|t|Ω||Γ|^1/2k.

3.20

Letk → ∞,the inequality3.20implies ut,·_L∞Ω≤e^Ctu0_L∞ΩC

T;f0

L∞QT,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_pJT^∗for someT^∗>0.

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

u_tAt, x, uuft, x, u,∇u, inQ_T, Bt, x, uuκut, x g1t, x, u g0, on∂Ω,

ux,0 u₀x, onΩ,

4.2

where

κut, x:

Ωk

t, x, y, u t, y

,∇u t, y

dy, 4.3

and

HkThe function k satisfies the Carath´eodory condition on t, x, y ∈ QT : 0, T×Γ× Ω,k|_u00 and fk∈Car_0,λ

2,λ2QT×R×Rⁿwith 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≤∇u^λ_L²⁻¹

λ2−1pΩ· ∇u−v_LpΩ

LrJt

≤c u^λ_W²⁻¹1

p · u−v_Wp¹ 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^λ_CJ^2−11−θ_t,E u−v^1−θ_CJt,E·u^λ_E1^{2−1pθ/p−θr}

LrJt

u−v^θ_E1

LrJt

≤c

u_Wp¹_Jt

· u−v_W¹_pJt.

4.6

Similarly, fromλ2≤p1 we have

Ω|∇u|^λ2−1|u−v|dy LrJt

≤c u−v_CJt,E· _t

0

u^r_W1 pdτ

_1/r

≤c

u_Wp¹_Jt

· u−v_Wp¹_Jt.

4.7

Combining two inequalities4.6and4.7, we obtain that κu−κv_LrJt,∂Wp^s

≤

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

≤c

u_Wp¹_Jt

· u−v_Wp¹_Jt.

4.8

The claim follows immediately fromTheorem 4.1.

A special case of problem4.2is

u_t− ∇au∇u ft, x, u, inQ_T,

∂ν_auu

Ωk

t, x, y, u

dygt, x, u, u0, x u0x, onΩ.

on0, T×Γ, 4.9

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

4.1.W₂¹Ω-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

t, x, y, u , g₁ t, x, u≤φt|u|. 4.10

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≤J_max. 4.11

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

1 2

d

dtu²_L2Ωδ0∇u²_L2Ω

≤

Ωft, x, uu dx

Γu

Ωk

t, x, y, u

dygt, x, u

dσx

≤φt

u²_L2Ωu_L1Γ

u_L1Ωu_L1Γ

ε₁

u²_L2Ωu²_L2Γ

1 ε1

f₀²

L2Ωg₀²

L2Γ

.

4.12

SinceW₂^θΩ→L₂Γforθ∈1/2,1, by interpolation inequality and Young’s inequality we have that

u_L2Γ≤Cu_W^θ

2Ω

≤Cu^θ_W1

2Ωu^1−θ_L2Ω

≤ε₂u_W¹

2ΩCε2u_L2Ω.

4.13

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

1 2

d

dtu²_L2Ω−Cεφtu²_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^{−2Cεt0φτdτ} 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_∞Q_T×0, T×Γ, then the solutionuof problem4.9satisfies

u_L∞Ω≤C

T;f₀, g₀, u₀

, fort≤J_max. 4.15

Proof. We multiply the first equation in4.9withu^2k−1and integrate overΩ, then we reach that

Jt: 1 2^k

d dtu²_L^k

2kΩ δ₀ 2^k−1 4^k−1

∇u^2k−12

L2Ω

≤

Ωft, x, uu^2k−1dx

Γu^2k−1

Ωk

t, x, y, u

dygt, x, u

dσx

≤φt u²_L^k

2kΩu²_L^k−1

2kΓ

u_L1Ω|Γ|^2−k u_L

2kΓ

Ωf₀u^2k−1dx

Γg₀u^2k−1dσ.

4.16

As the same as the inequality4.13, we have u²_L^k

2kΩ≤ε1∇u^2k−12

L2ΩCε1u²_L^k

2kΩ, u²_L^k−1

2kΩ· u_L1Ω≤

ε1∇u^2k−12

L2ΩCε1u²_L^k

2kΩ 1−2^−k

· u_L1Ω

≤ε¹⁻²₁ ^−k∇u^{2k−121−2−k}

L2Ω · u_L1Ω Cε1u²_L^k

2kΩ.

4.17

Hence,

Jt≤

φt1Cε1 Cε2 u²_L^k

2kΩ

ε₁ε₂∇u^2k−12

L2Ωε¹⁻²₁ ^−k|Γ|^2−k·∇u^{2k−121−2−k}

L2Ω

Cε22^−kf0^2k

L₂kΩg0^2k

L₂kΓ

.

4.18

We might as well assume thatu_L2Ω>0, so, ∇u^{2k−1−21−k}

L2Ω4^1−k2−k

Ωu^2k−2|∇u|²dx ₋₂^−k

−→ u⁻¹_L∞Ω ask−→∞. 4.19

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

Letε_jj 1,2small enough, then we have 2^−kd

dtu²_L^k

2kΩ−Cεφtu²_L^k

2kΩ≤Cε2^−kf₀^2k

L₂kΩg₀^2k

L₂kΓ

. 4.20

Multiplying with 2^k·e^{−Cε2kt0φτdτ}, then integrating over0, t, we obtain that u_L

2kΩe^{−Cεt0φτdτ}2^k

≤ u0²_L^k

2kΩCε

_t

0

e^{−Cε2ks0φτdτ}f0^2k

L₂kΩg0^2k

L₂kΓ

ds.

4.21

By a similar limitation process as in3.21, we get u_L∞Ω≤C

T;f₀, g₀, u₀

fort≤J_max. 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. Thenu_L2Ωdecay 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

dtu²_L2Ωδ₀∇u²_L2Ω

≤

Ωft, x, uu dx

Γu

Ωk

t, x, y, u

dygt, x, u

dσ

≤ −ϕtu²_L2Ωtu_L1Γu_LΩ

≤

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

, t∈0, T, 4.25

then

u_L2Ω≤ u0_L2Ω·e^−t0ϕτdτ , 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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