Boundary Value Problems
Volume 2009 (2009), Article ID 484879, 17 pages
doi:10.1155/2009/484879
Research Article

The Existence and Behavior of Solutions for Nonlocal Boundary Problems

Yuandi Wang1 and Shengzhou Zheng2

1Department of Mathematics, Shanghai University, Shanghai 200444, China
2Department of Mathematics, Beijing Jiaotong University, Beijing 100044, China

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

Academic Editor: Pavel Drabek

Copyright © 2009 Yuandi Wang and Shengzhou 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.

Abstract

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.

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 Sections 3 and 4 to 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 problem (IBVP for short):

where Ω is a bounded strictly Lipschitz domain with its boundary Γ = 𝜕 Ω = Γ 0 ∪ Γ 1 and Γ 0 ∩ Γ 1 = ∅ , 𝑄 𝑇 = [ 0 , 𝑇 ) × Ω ,

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

The coefficient matrix 𝐚 = ( 𝑎 𝑖 𝑗 ) 𝑛 × 𝑛 satisfies regularity conditions on 𝑄 𝑇 × ℝ , respectively. The directional derivative 𝜕 𝜈 𝐚 𝑢 ∶ = 𝛾 𝐚 ∇ 𝑢 ⋅ 𝜈 , 𝜈 is the outer unit-normal vector on Γ ; the function 𝛿 ∶ Γ → { 0 , 1 } is defined as 𝛿 − 1 ( 𝑗 ) ∶ = Γ 𝑗 for 𝑗 = 0 , 1 ; 𝛾 denotes the trace operator.

We introduce precise assumptions:

where ( ℝ + ∶ = [ 0 , + ∞ ) ) are Carathéodory functions; that is, 𝑓 (resp., 𝑔 ) is measurable in ( 𝑡 , 𝑥 ) ∈ 𝑄 𝑇 (resp., in ( 𝑡 , 𝑥 , 𝑦 ) ∈ ℝ + × Γ × Ω ) for each 𝑢 ∈ ℝ and continuous in 𝑢 for a.e. ( 𝑡 , 𝑥 ) ∈ 𝑄 𝑇 (resp., ( 𝑡 , 𝑥 ) ∈ ℝ + × Γ ) . More general, the function 𝑔 can be a nonlocal function, for example, ∫ 𝑔 ( 𝑡 , 𝑥 , 𝑢 ) = Ω 0 𝑥 0 2 0 0 𝑑 𝑘 ( 𝑡 , 𝑥 , 𝑦 , 𝑢 ) 𝑑 𝑦 or ∫ 𝑔 ( 𝑡 , 𝑥 , 𝑢 ) = Γ 0 𝑥 0 2 0 0 𝑑 𝑘 ( 𝑡 , 𝑥 , 𝑢 ) 𝑑 𝜎 .

Let 𝑋 and 𝑌 be Banach spaces, we introduce some notations as follows:

(i) 𝐽 𝑇 ∶ = [ 0 , 𝑇 ) , ∘ 𝐽 𝑇 ∶ = 𝐽 𝑇 ⧵ { 0 } . 𝛼 ⋀ 𝛽 ∶ = m i n { 𝛼 , 𝛽 } , 𝛼 ∨ 𝛽 ∶ = m a x { 𝛼 , 𝛽 } .(ii) 𝒟 ( 𝐷 , 𝑌 ) ∶ = { 𝜙 ∈ 𝐶 ∞ ( 𝐷 ) , 𝜙 ∶ 𝐷 ↦ 𝑌 , s u p p 𝜙 ⊂ 𝐷 } for 𝐷 ⊂ ℝ 𝑙 , 𝒟 1 ∶ = { 𝑣 ∈ 𝒟 ( 𝐽 𝑇 × Ω ) } ∩ { 𝑣 | Γ 0 = 0 } .(iii) ℒ ( 𝑋 , 𝑌 ) ∶ = { all continuous linear operators from 𝑋 into 𝑌 } , and ℒ ( 𝑋 ) ∶ = ℒ ( 𝑋 , 𝑋 ) .(iv) 𝑓 ( 𝑡 , 𝑢 ) denotes the Nemytskii operator induced by 𝑓 ( 𝑡 , 𝑥 , 𝑢 ( 𝑡 , 𝑥 ) , ∇ 𝑢 ( 𝑡 , 𝑥 ) ) .(v) 𝐶 1 − ( 𝑋 , 𝑌 ) denotes the set of all locally Lipschitz-continuous functions from 𝑋 into 𝑌 .(vi) C a r 0 , 𝜆 , 𝜆 ( 𝑀 × ℝ × ℝ 𝑛 , ℝ ) , 𝜆 , and 𝜆 ≥ 1 , denotes the set of all Carathéodory functions 𝑓 on 𝑚 ∈ 𝑀 such that 𝑓 ( 𝑚 , 0 ) = 0 , and there exists a nondecreasing function 𝜓 ≥ 0 with

Particularly, 𝑓 is independent of 𝜉 if 𝜆 ⋀ 𝜆 ≤ 1 .

(vii) 𝑊 𝑠 𝑝 ( Ω ) denotes the Sobolev-Slobodeckii space for 𝑠 ∈ ℝ and 𝑝 ≥ 1 with the norm ‖ ⋅ ‖ 𝑊 𝑠 𝑝 , especially, 𝑊 0 𝑝 ( Ω ) = 𝐿 𝑝 ( Ω ) ; and (viii) 𝑊 𝑠 𝑝 , ℬ ( Ω ) , 𝑠 ∈ [ − 2 , 2 ] ⧵ { ℤ + 1 / 𝑝 } ( ℤ is the set of integral numbers), is defined as

where 𝑞 = 𝑞 / ( 𝑞 − 1 ) , 𝑋 is the dual space of 𝑋 , and ℬ 0 𝑥 0 2 2 𝑑 5 is the formally adjoint operator.

(x) 𝕎 1 𝑝 ( 𝐽 ) ∶ = 𝕎 1 𝑝 ( 𝐽 , ( 𝐸 1 , 𝐸 0 ) ) ∶ = 𝑊 1 𝑝 ( 𝐽 , 𝐸 0 ) ∩ 𝐿 𝑝 ( 𝐽 , 𝐸 1 ) if 𝐸 1 𝑑 ↪ 𝐸 0 and 𝐽 is an interval in ℝ . (xi) ℳ 𝑅 𝑝 ( 𝐽 𝑇 ) ∶ = ℳ 𝑅 𝑝 ( 𝐽 𝑇 , ( 𝐸 1 , 𝐸 0 ) ) denotes all maps 𝐵 possessing the property of maximal   𝐿 𝑝   regularity on 𝐽 𝑇 with respect to ( 𝐸 1 , 𝐸 0 ) , that is, given ℎ ∈ 𝐿 𝑝 ( 𝐽 𝑇 , 𝐸 0 ) , the initial problem

has a unique solution 𝑣 ∈ 𝕎 1 𝑝 ( 𝐽 𝑇 , ( 𝐸 1 , 𝐸 0 ) ) .

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

then, Exactly, 𝕎 1 𝑝 ( 𝐽 𝑇 , ( 𝐸 1 , 𝐸 0 ) ) ↪ B U C ( 𝑄 𝑇 ) as 𝑝 > 𝑛 + 2 , where B U C ( 𝑄 𝑇 ) denotes the Banach space of all functions being bounded and uniformly continuous in 𝑄 𝑇 . So, we will not emphasize it in the following.

A (weak )  solution    𝑢 of IBVP (2.1) is defined as a 𝕎 1 𝑝 ( 𝐽 𝑇 , ( 𝐸 1 , 𝐸 0 ) ) function 𝑢 , 𝑠 ∈ [ 1 , 1 + 1 / 𝑝 ) , satisfying where ⟨ ⋅ , ⋅ ⟩ and ⟨ ⋅ , ⋅ ⟩ 𝜕 𝑗 denote the obvious duality pairings on Ω and Γ 𝑗 , respectively.

Set

After these preparations we introduce the following hypotheses:

(H1) 𝑝 > 𝑛 + 2 and 𝑠 ∈ [ 1 , 1 + 1 / 𝑝 ) .(H2) 𝐚 ( 𝑡 , 𝑥 , 𝑢 ) ∈ 𝐶 0 , 𝛼 , 1 − ( 𝐽 𝑇 × Ω × ℝ , ℝ 𝑛 × 𝑛 )   with   𝛼 + 1 > 𝑠 , and there exists a   𝛿 0 ∈ ( 0 , 1 )   such that ( 𝑓 0 , 𝑔 0 ) ∈ 𝐿 𝑝 ( 𝐽 , 𝐸 0 × 𝜕 𝑊 𝑠 𝑝 ( Γ ) ) , 𝑓 1 ∈ C a r 0 , 𝜆 0 , 𝜆 1 ( 𝑄 𝑇 × ℝ × ℝ 𝑛 ) with 𝜆 0 ∈ [ 1 , 2 ) , and 𝜆 1 < 1 + 𝑝 ( 𝑠 − 1 ) / ( 2 + 𝑝 ( 1 − 𝑠 ) ) .(H3) 𝑔 1 ( 𝑡 , ⋅ ) ∈ 𝐶 1 − ( 𝕎 1 𝑝 ( 𝐽 𝑇 ) , 𝐿 𝑟 ( 𝐽 𝑇 , 𝜕 𝑊 𝑠 𝑝 ( Γ ) ) ) for some 𝑟 > 𝑝 .

Theorem 2.1. Let assumptions (H1)-(H3) be satisfied. Then for each 𝑢 0 ∈ 𝐸 the quasilinear problem (2.1) possesses a unique weak solution 𝑢 ( 𝑡 , 𝑥 ) ∈ 𝕎 1 𝑝 ( 𝐽 𝑇 ∗ , ( 𝐸 1 , 𝐸 0 ) ) for some 𝑇 ∗ > 0 .

Proof. Recall that The Nemytskii operator 𝐚 ( ⋅ , 𝑢 ) is defined as 𝐚 ( ⋅ , 𝑢 ) ( 𝑡 , 𝑥 ) ∶ = 𝐚 ( 𝑡 , 𝑥 , 𝑢 ( 𝑡 , 𝑥 ) ) . The fact shows the maximal regularity of the operator 𝒜 . By [1, Theorem  2.1], if, for 𝑡 ≤ 𝑇 , 𝑓 1 ∈ 𝐶 1 − ( 𝕎 1 𝑝 ( 𝐽 𝑡 ) , 𝐿 𝑟 ( 𝐽 𝑡 , 𝐸 0 ) ) 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
Then 𝐿 𝑞 ( Ω ) ↪ 𝐸 0 . So, for 𝑢 , 𝑣 ∈ 𝕎 1 𝑝 ( 𝐽 𝑡 ) with | 𝑢 | , | 𝑣 | ≤ 𝜌 we have From ( 𝜆 1 − 1 ) 𝑞 < 𝑝 − 𝑞 , we infer that where 𝑟 ∗ ∶ = ( 𝜆 1 − 1 ) 𝑝 𝑟 𝜃 / ( 𝑝 − 𝜃 𝑟 ) . Note that 𝜆 1 𝜃 < 1 , we can choose 𝑟 > 𝑝 such that
On the other hand, the hypotheses guarantee that
Due to 𝑞 < 𝑝 and 𝜆 0 ∈ [ 1 , 2 ) , Hölder inequality follows that The hypothesis of 𝜆 0 means that one can find an 𝑟 > 𝑝 such that Obviously, if 𝜆 0 = 1 , the above inequality is followed from (2.20) immediately. Hence it follows from (2.19) and (2.22) that 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 hypotheses (H1)-(H2) hold and that

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 bodies (e.g., refer to [2]). A problem is

3.1. Local Solvability

We assume that (Hr)

(Hr1) 𝜑 ∈ 𝐿 𝑟 ( 𝐽 𝑇 , 𝐿 ̇ 𝑝 ( Γ × Γ ) ) ;(Hr2) ℎ   is locally Lipschitz continuous and   ℎ ( 0 ) = 0 .

Theorem 3.1. Let assumptions (H1)-(H2) and (Hr) be satisfied. Then problem (3.1), for all 𝑢 0 ∈ 𝐸 , has a unique 𝑢 ∈ 𝕎 1 𝑝 ( 𝐽 𝑇 ∗ ) for some 𝑇 ∗ > 0 .

Proof. Note that the embedding (2.14) holds: Hence Theorem 2.1 implies the result immediately.

In fact, Amosov proved in 2005 the uniqueness of the solution for a simple case, that is, problem in which the matrix 𝐚 is independent of 𝑢 (see [2, Theorem  1.4]). In this paper, we also can get the positivity of the solution and the estimates of the solution in 𝑊 1 2 ( Ω ) and 𝐿 ∞ ( Ω ) in 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 that (H1)-(H2) and (Hr) hold.

3.2. Positivity

Assume that

(H+) ℎ ( 𝑢 )   is nondecreasing with   ℎ ( 0 ) = 0 , and

Theorem 3.2. Let assumption ( 𝐻 + ) be satisfied. If ( 𝑓 0 , 𝑔 0 , 𝑢 0 ) is nonnegative, then the solution 𝑢 of problem (3.1) is also nonnegative.

Proof. Put 𝑢 − ⋀ 𝑢 ∶ = 0 . Multiplying the equation with 𝑢 − and integrating over Ω , we have By using the assumption of ( 𝐻 + ), we can get following equality: So, At the last inequality, the monotonity of ℎ on 𝑢 and the restriction 𝜑 < 1 are used. Therefore, If 𝑢 0 ≥ 0 , then 𝑢 − ( 𝑡 , 𝑥 ) ≡ 0 . The assertion follows.

3.3. 𝑊 1 2 ( Ω ) -norm

We denote by 𝐽 m a x the maximal interval of the solution of problem (3.1).

Lemma 3.3. There exists a constant 𝐶 = 𝐶 ( 𝑇 ; 𝑓 0 , 𝑔 0 , 𝑢 0 ) such that the solution 𝑢 of problem (3.1) satisfies

Proof. Multiplying by 𝑢 and integrating over Ω , we have That is, As similar as the inequality (3.6), we have Hence, By using the embedding 𝑊 1 2 , ℬ ( Ω ) ↪ 𝐿 2 ( Γ ) and letting 𝜖 small enough, it is easy to get that

3.4. 𝐿 ∞ ( Ω ) -norm

Theorem 3.4. If 𝑓 0 ∈ 𝐿 ∞ ( 𝑄 𝑇 ) and 𝑔 0 ∈ 𝐿 ∞ ( [ 0 , 𝑇 ] × Γ ) , then the solution 𝑢 ( 𝑡 , 𝑥 ) of problem (3.1) is bounded with its 𝐿 ∞ -norm for all 𝑡 ∈ 𝐽 m a x .

Proof. From the hypothesis (H1) and embedding (2.10), one has that 𝑢 ∈ 𝐶 ( 𝑄 𝑇 ) and 𝑢 𝑛 ∈ 𝑊 1 2 ( Ω ) , 𝑛 = 1 , 2 , … . By multiplying with 𝑢 2 𝑘 − 1 ( 𝑘 ∈ ℤ + and 𝑘 ≥ 2 ) and integrating over Ω , we have That is, But, Therefore, where Young's inequality, 𝛼 𝛽 ≤ ( 𝜖 / 𝑟 ) 𝛼 𝑟 + 𝜖 − 𝑟 ′ / 𝑟 / 𝑟 𝛽 𝑟 ′ ( 𝛼 , 𝛽 ≥ 0 , 0 < 𝜀 < 1 ) , has been used at the last inequality. We apply the embedding 𝑊 1 2 , ℬ ( Ω ) ↪ 𝐿 2 ( Γ ) again with 𝑣 = 𝑢 2 𝑘 − 1 and choose 𝜖 small enough, then we attain the following inequality: By Gronwall's inequality, the inequality (3.18) becomes Set 𝐵 ∞ ∶ = 1 + ‖ 𝑢 0 ‖ 𝐿 ∞ ( Ω ) + ‖ 𝑓 0 ‖ 𝐿 ∞ ( 𝑄 𝑇 ) + ‖ 𝑔 0 ‖ 𝐿 ∞ ( [ 0 , 𝑇 ] × Γ ) , then we deduce that Let 𝑘 → + ∞ , the inequality (3.20) implies The claim follows.

One immediate consequence of the above theorem is.

Corollary 3.5. The 𝐿 ∞ -norm of the solution 𝑢 , that is, ‖ 𝑢 ( 𝑡 , ⋅ ) ‖ 𝐿 ∞ ( Ω ) , of problem (3.1) is nonincreasing if 𝑓 0 = 𝑔 0 = 0 .

4. A Nonlocal Boundary Value Problem

We now consider the problem (2.1) with the following boundary value condition:

The function Φ in (4.1) can be in nonlocal form.

IBVP (2.1) with a nonlocal term stands, for example, for a model problem arising from quasistatic thermoelasticity. Results on linear problems can be found in [3–5]. As far as we know, this kind of nonlocal boundary condition appeared first in 1952 in a paper [6] by W. Feller who discussed the existence of semigroups. There are other problems leading to this boundary condition, for example, control theory (see [7–12] etc.). Some other fields such as environmental science [13] and chemical diffusion [14] also give rise to such kinds of problems. We do not give further comments here.

Carl and Heikkilä [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 of 𝑓 , 𝑔 , and Φ with respect to 𝑢 .

In this section, we assume that (H1) and (H2) always hold and assume that

(Hn1) Φ ( 0 ) = 0 and Φ ∈ 𝐶 1 − ( 𝕎 1 𝑝 ( 𝐽 𝑇 ) , 𝐿 𝑟 ( 𝐽 𝑇 , 𝐿 ̇ 𝑝 ( Γ ) ) ) for some 𝑟 > 𝑝 ;(Hn2) 𝑔 1 ( 𝑡 , 𝑥 , 0 ) = 0 ,    𝑔 1 satisfies the Carathéodory condition on ( 𝑡 , 𝑥 ) and 𝑔 1 ( 𝑡 , 𝑥 , ⋅ ) ∈ 𝐶 1 − ( ℝ ) .

By the embedding theorem and Theorem 2.1, we get immediately.

Theorem 4.1. Suppose hypotheses of (Hn) satisfy. Then problem (2.1), for all 𝑢 0 ∈ 𝐸 , with 𝑔 defined in (4.1) has a unique 𝑢 ∈ 𝕎 1 𝑝 ( 𝐽 𝑇 ∗ ) for some 𝑇 ∗ > 0 .

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

where and

(Hk) The function    𝑘    satisfies the Carathéodory condition on    ( 𝑡 , 𝑥 , 𝑦 ) ∈ 𝒬 𝑇 ∶ = [ 0 , 𝑇 ] × Γ × Ω , 𝑘 | 𝑢 = 0 = 0    and f 𝑘 ∈ C a r 0 , 𝜆 2 , ̃ 𝜆 2 ( 𝒬 𝑇 × ℝ × ℝ 𝑛 ) with

Theorem 4.2. Let assumption (Hk) be satisfied. Then Problem (4.2), for any 𝑢 0 ∈ 𝐸 , has a unique solution 𝑢 ∈ 𝕎 1 𝑝 ( 𝐽 𝑇 ∗ ) for some 𝑇 ∗ > 0 .

Proof. First, we see that Choose 𝜃 ∈ ( 0 , 1 ) such that 𝜆 1 𝜃 < 1 , then ( 𝜆 2 − 1 ) 𝜃 / ( 1 − 𝜃 ) < 1 . Consequently, there exists 𝑟 > 𝑝 such that Similarly, from ̃ 𝜆 2 ≤ 𝑝 + 1 we have Combining two inequalities (4.6) and (4.7), we obtain that
The claim follows immediately from Theorem 4.1.

A special case of problem (4.2) is

That is, 𝑓 and 𝑘 in (4.9) are independent of gradient ∇ 𝑢 .

4.1. 𝑊 1 2 ( Ω ) -norm

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

(Hkl)Suppose there exists a continuous function   𝜙 ∶ ℝ + → ℝ +   such that

Lemma 4.3. There exists a constant 𝐶 = 𝐶 ( 𝑇 ; 𝑓 0 , 𝑔 0 , 𝑢 0 ) such that the solution of problem (4.9) satisfies

Proof. We multiply the first equation in (4.9) with 𝑢 and then integrate over Ω , and we find that Since 𝑊 𝜃 2 ( Ω ) ↪ 𝐿 2 ( Γ ) for 𝜃 ∈ ( 1 / 2 , 1 ) , by interpolation inequality and Young's inequality we have that Apply Young's inequality again and then choose 𝜀 𝑗 small enough ( 𝑗 = 1 , 2 ); it is not difficult to get where 𝛿 0 − 𝜀 ( 𝜙 ( 𝑡 ) + 1 ) > 0 for 𝑡 ∈ [ 0 , 𝑇 ] . Therefore, by multiplying with 𝑒 ∫ − 2 𝐶 ( 𝜀 ) 𝑡 0 0 𝑥 0 2 0 0 𝑑 𝜙 ( 𝜏 ) 𝑑 𝜏 and integrating over [ 0 , 𝑡 ] , the inequality (4.14) follows the claim.

4.2. 𝐿 ∞ ( Ω ) -norm

Lemma 4.4. Let assumptions of Lemma 4.3 be satisfied. If ( 𝑓 0 , 𝑔 0 ) ∈ 𝐿 ∞ ( 𝑄 𝑇 × ( [ 0 , 𝑇 ] × Γ ) ) , then the solution 𝑢 of problem (4.9) satisfies

Proof. We multiply the first equation in (4.9) with 𝑢 2 𝑘 − 1 and integrate over Ω , then we reach that As the same as the inequality (4.13), we have Hence, We might as well assume that ‖ 𝑢 ‖ 𝐿 2 ( Ω ) > 0 , so, The boundedness of solution ‖ 𝑢 ‖ 𝐿 2 ( Ω ) ≤ 𝐶 for 𝑡 ≤ 𝐽 m a x is used in above deduction.
Let 𝜀 𝑗 ( 𝑗 = 1 , 2 ) small enough, then we have
Multiplying with 2 𝑘 ⋅ 𝑒 − 𝐶 ( 𝜀 ) 2 𝑘 ∫ 𝑡 0 0 𝑥 0 2 0 0 𝑑 𝜙 ( 𝜏 ) 𝑑 𝜏 , then integrating over [ 0 , 𝑡 ] , we obtain that By a similar limitation process as in (3.21), we get This closes the end of proof.

4.3. Decay Behavior

In order to investigate the decay behavior of solution for problem (4.9), we assume that

(Hkd) there are two continuous function 𝜑 ( 𝑡 ) > 0 and 𝜖 ( 𝑡 ) ≥ 0    ( 𝑡 ≥ 0 ) such that

for all ( 𝑡 , 𝑥 , 𝑦 ) ∈ ℝ + × Ω × Γ .

Theorem 4.5. Let the assumption (Hkd) be satisfied and, 𝑢 be the solution of problem (4.9) with ( 𝑓 0 , 𝑔 0 ) = 0 . Then ‖ 𝑢 ‖ 𝐿 2 ( Ω ) decay to zero as 𝑡 → ∞ for some small functions 𝜖 ( 𝑡 ) .

Proof. We use 𝑢 to multiply the first equation in the system (4.9) and then integrate over Ω . Thus, we get that In the above process the inequality (4.13) is used. If we choose 𝜖 ( 𝑡 ) as then This ends the proof.
Moreover, one can verify that ‖ 𝑢 ‖ 𝐿 𝑝 ( Ω ) also decay to zero (as 𝑡 → ∞ ) if 𝑝 ≥ 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 10671118) and by Shanghai Leading Academic Discipline Project (no. J50101).

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