International Journal of Differential Equations

International Journal of Differential Equations / 2011 / Article

Research Article | Open Access

Volume 2011 |Article ID 941679 | https://doi.org/10.1155/2011/941679

Zhiyong Ma, "Global Existence of the Higher-Dimensional Linear System of Thermoviscoelasticity", International Journal of Differential Equations, vol. 2011, Article ID 941679, 17 pages, 2011. https://doi.org/10.1155/2011/941679

Global Existence of the Higher-Dimensional Linear System of Thermoviscoelasticity

Academic Editor: Stanisław Migórski
Received04 May 2011
Revised28 Jun 2011
Accepted30 Jun 2011
Published08 Sep 2011

Abstract

We obtain a global existence result for the higher-dimensional thermoviscoelastic equations. Using semigroup approach, we will establish the global existence of homogeneous, nonhomogeneous, linear, semilinear, and nonlinear, thermoviscoelastic systems.

1. Introduction

In this paper, we consider global existence of the following thermoviscoelastic model: 𝑢𝑡𝑡𝜇Δ𝑢(𝜆+𝜇)div𝑢+𝜇𝑔Δ𝑢+(𝜆+𝜇)𝑔div𝑢+𝛼𝜃𝑡𝜃=𝑓,(𝑥,𝑡)Ω×(0,),𝑡𝑡Δ𝜃𝑡Δ𝜃+𝛽div𝑢𝑡=,(𝑥,𝑡)Ω×(0,),(1.1) where the sign “” denotes the convolution product in time, which is defined by 𝑔𝑣(𝑡)=𝑡𝑔(𝑡𝑠)𝑣(𝑥,𝑠)𝑑𝑠(1.2) with the initial data 𝑢(𝑥,0)=𝑢0(𝑥),𝑢𝑡(𝑥,0)=𝑢1(𝑥),𝜃(𝑥,0)=𝜃0𝜃(𝑥),𝑥Ω,𝑡(𝑥,0)=𝜃1(𝑥),𝑢(𝑥,0)𝑢(𝑥,𝑠)=𝑤0(𝑥,𝑠),(𝑥,𝑠)Ω×(0,)(1.3) and boundary condition 𝑢=0,𝜃=0,(𝑥,𝑡)Γ×(0,).(1.4) The body Ω is a bounded domain in 𝑅𝑛 with smooth boundary Γ=𝜕Ω (say 𝐶2) and is assumed to be linear, homogeneous, and isotropic. 𝑢(𝑥,𝑡)=(𝑢1(𝑥,𝑡),𝑢2(𝑥,𝑡),,𝑢𝑛(𝑥,𝑡)), and 𝜃(𝑥,𝑡) represent displacement vector and temperature derivations, respectively, from the natural state of the reference configuration at position 𝑥 and time 𝑡. 𝜆,𝜇>0 are Lamé's constants and 𝛼,𝛽>0 the coupling parameters; 𝑔(𝑡) denotes the relaxation function, 𝑤0(𝑥,𝑠) is a specified “history,” and 𝑢0(𝑥),𝑢1(𝑥),𝜃0(𝑥) are initial data. Δ,,div denote the Laplace, gradient, and divergence operators in the space variables, respectively.

We refer to the work by Dafermos [13]. The following basic conditions on the relaxation function 𝑔(𝑡) are (H1)𝑔𝐶1[0,)𝐿1(0,); (H2)𝑔(𝑡)0,𝑔(𝑡)0,𝑡>0; (H3)𝜅=10𝑔(𝑡)𝑑𝑡>0.

In what follows, we denote by the norm of 𝐿2(Ω), and we use the notation 𝑣2=𝑛𝑖=1𝑣𝑖2𝑣,for𝑣=1,𝑣2,,𝑣𝑛.(1.5)

When 𝑓=𝑔==0, system (1.1)–(1.4) is reduced to the thermoelastic system: 𝑢𝑡𝑡𝜇Δ𝑢(𝜆+𝜇)div𝑢+𝛼𝜃𝑡𝜃=0,(𝑥,𝑡)Ω×(0,),𝑡𝑡Δ𝜃𝑡Δ𝜃+𝛽div𝑢𝑡=0,(𝑥,𝑡)Ω×(0,),𝑢=0,𝜃=0,(𝑥,𝑡)Γ×(0,),𝑢(𝑥,0)=𝑢0(𝑥),𝑢𝑡(𝑥,0)=𝑢1(𝑥),𝜃(𝑥,0)=𝜃0(𝑥),𝑥Ω.(1.6) In the one-dimensional space case, there are many works (see e.g., [48]) on the global existence and uniqueness. Liu and Zheng [9] succeeded in deriving in energy decay under the boundary condition (1.4) or𝑢𝑥=0=0,𝜎𝑥=𝑙=0,𝜃𝑥𝑥=0,𝑙=0,(1.7)𝑢𝑥=0=0,𝜎𝑥=𝑙=0,𝜃𝑥=0=0,𝜃𝑥𝑥=𝑙=0,(1.8) or𝑢𝑥=0=0,𝜎𝑥=𝑙=0,𝜃𝑥=0,𝑙=0,(1.9) and Hansen [10] used the method of combining the Fourier series expansion with decoupling technique to solve the exponential stability under the following boundary condition:𝑢𝑥=0=0,𝜎𝑥=𝑙=0,𝜃𝑥𝑥=0=0,𝜃𝑥=𝑙=0,(1.10) where 𝜎=𝑢𝑥𝛾𝜃 is the stress. Zhang and Zuazua [11] studied the decay of energy for the problem of the linear thermoelastic system of type III by using the classical energy method and the spectral method, and they obtained the exponential stability in one space dimension, and in two or three space dimensions for radially symmetric situations while the energy decays polynomially for most domains in two space dimensions.

When 𝛼=𝛽=0, 𝑓==0, system (1.1)-(1.4) is decoupled into the following viscoelastic system: 𝑢𝑡𝑡𝜇Δ𝑢(𝜆+𝜇)div𝑢+𝜇𝑔Δ𝑢+(𝜆+𝜇)𝑔div𝑢=0,(𝑥,𝑡)Ω×(0,),𝑢=0,(𝑥,𝑡)Γ×(0,),𝑢(𝑥,0)=𝑢0(𝑥),𝑢𝑡(𝑥,0)=𝑢1(𝑥),(𝑥,𝑡)Ω,𝑢(𝑥,0)𝑢(𝑥,𝑠)=𝑤0(𝑥),(𝑥,𝑡)Ω×(0,),(1.11) and the wave equation.

There are many works (see, e.g., [9, 1215]) on exponential stability of energy and asymptotic stability of solution under different assumptions. The notation in this paper will be as follows. 𝐿𝑝,1𝑝+,𝑊𝑚,𝑝,𝑚𝑁,𝐻1=𝑊1,2,𝐻10=𝑊01,2 denote the usual (Sobolev) spaces on Ω. In addition, 𝐵 denotes the norm in the space 𝐵; we also put =𝐿2(Ω). We denote by 𝐶𝑘(𝐼,𝐵),𝑘0, the space of k-times continuously differentiable functions from 𝐽𝐼 into a Banach space 𝐵, and likewise by 𝐿𝑝(𝐼,𝐵),1𝑝+, the corresponding Lebesgue spaces. 𝐶𝛽([0,𝑇],𝐵) denotes the Hölder space of 𝐵-valued continuous functions with exponent 𝛽(0,1] in variable 𝑡.

2. Main Results

Let the “history space” 𝐿2(𝑔,(0,),(𝐻10(Ω))𝑛) consist of ((𝐻10(Ω))𝑛)-valued functions 𝑤 on (0,) for which 𝑤2𝐿2𝐻(𝑔,(0,),10(Ω)𝑛)=0𝑔(𝑠)𝑤(𝑠)2𝐻10(Ω)𝑛𝑑𝑠<.(2.1) Put 𝐻=10(Ω)𝑛×𝐿2(Ω)𝑛×𝐻10(Ω)×𝐿2(Ω)×𝐿2𝐻𝑔,(0,),10(Ω)𝑛(2.2) with the energy norm 𝑢,𝑣,𝜃,𝜃𝑡,𝑤=𝜅𝑢2𝐻10(Ω)𝑛+12𝑣2+𝛼𝛽𝜃𝑡2++𝜃0𝑔(𝑠)𝑤(𝑠)2𝐻10(Ω)𝑛𝑑𝑠1/2,(2.3) where 𝜅 denotes the positive constant in (𝐻3), that is, 𝜅=10𝑔(𝑡)𝑑𝑡>0.(2.4) Thus we consider the following thermoviscoelastic system:𝑢𝑡𝑡𝜇Δ𝑢(𝜆+𝜇)div𝑢+𝜇𝑔Δ𝑢+(𝜆+𝜇)𝑔div𝑢+𝛼𝜃𝑡𝜃=0,(𝑥,𝑡)Ω×(0,),𝑡𝑡Δ𝜃𝑡Δ𝜃+𝛽div𝑢𝑡=0,(𝑥,𝑡)Ω×(0,),𝑢=0,𝜃=0,(𝑥,𝑡)Γ×(0,),𝑢(𝑥,0)=𝑢0(𝑥),𝑢𝑡(𝑥,0)=𝑢1(𝑥),𝜃(𝑥,0)=𝜃0(𝑥),𝜃𝑡(𝑥,0)=𝜃1(𝑥),𝑥Ω,𝑢(𝑥,0)𝑢(𝑥,𝑠)=𝑤0(𝑥,𝑠),(𝑥,𝑡)Ω×(0,).(2.5) Let 𝑣(𝑥,𝑡)=𝑢𝑡(𝑥,𝑡),𝑤(𝑥,𝑡,𝑠)=𝑢(𝑥,𝑡)𝑢(𝑥,𝑡𝑠).(2.6) Since 𝜕𝜕𝜈𝑡𝜕𝑔(𝑡𝑠)𝑢(𝑠)𝑑𝑠=𝜕𝜈0=𝑔(𝑠)𝑢(𝑡𝑠)𝑑𝑠0𝜕𝑔(𝑠)=𝜕𝜈(𝑢(𝑡)𝑤(𝑡,𝑠))𝑑𝑠(1𝜅)𝜕𝑢(𝑥,𝑡)𝜕𝜈0𝑔(𝑠)𝜕𝑤(𝑡,𝑠)𝜕𝜈𝑑𝑠,(2.7) System (2.5) can be written as follows: 𝑢𝑡𝑡𝜅𝜇Δ𝑢𝜅(𝜆+𝜇)div𝑢+𝛼𝜃𝑡𝜇0𝑔(𝑠)Δ𝑤(𝑡,𝑠)𝑑𝑠(𝜆+𝜇)0𝜃𝑔(𝑠)div𝑤(𝑡,𝑠)𝑑𝑠=0,(𝑥,𝑡)Ω×(0,),𝑡𝑡Δ𝜃𝑡Δ𝜃+𝛽div𝑢𝑡𝑢=0,(𝑥,𝑡)Ω×(0,),𝑤(𝑥,𝑡,𝑠)=𝑢(𝑥,𝑡)𝑢(𝑥,𝑡𝑠),(𝑥,𝑡,𝑠)Ω×(0,)×(0,),𝑢=0,𝜃=0,(𝑥,𝑡)Γ×(0,),(𝑥,0)=𝑢0(𝑥),𝑢𝑡(𝑥,0)=𝑢1(𝑥),𝜃(𝑥,0)=𝜃0(𝑥),𝜃𝑡(𝑥,0)=𝜃1(𝑥),𝑥Ω,𝑤(0,𝑠)=𝑤0(𝑠),(𝑥,𝑡)Ω×(0,).(2.8) We define a linear unbounded operator 𝐴 on by 𝐴𝑢,𝑣,𝜃,𝜃𝑡=,𝑤𝑣,𝐵(𝑢,𝑤)𝛼𝜃𝑡,𝜃𝑡,Δ𝜃𝑡+Δ𝜃𝛽div𝑣,𝑣𝑤𝑠,(2.9) where 𝑤𝑠=𝜕𝑤/𝜕𝑠 and 𝐵(𝑢,𝑤)=𝜅𝜇Δ𝑢+𝜅(𝜆+𝜇)div𝑢+𝜇0𝑔(𝑠)Δ𝑤(𝑠)𝑑𝑠+(𝜆+𝜇)0𝑔(𝑠)div𝑤(𝑠)𝑑𝑠.(2.10) Set 𝑣(𝑥,𝑡)=𝑢𝑡(𝑥,𝑡),𝑤(𝑥,𝑡,𝑠)=𝑢(𝑥,𝑡)𝑢(𝑥,𝑡𝑠),Φ=𝑢,𝑣,𝜃,𝜃𝑡,𝑤,𝐾=(0,𝑓,0,,0).(2.11) Then problem (2.8) can be formulated as an abstract Cauchy problem 𝑑Φ𝑑𝑡=𝐴Φ+𝐾,(2.12) on the Hilbert space for an initial condition Φ(0)=(𝑢0,𝑢1,𝜃0,𝜃1,𝑤0). The domain of A is given by 𝐷(𝐴)=(𝑢,𝑣,𝜃,𝑤)𝜃𝐻10(Ω),𝜃𝑡𝐻10(Ω),𝜃+𝜃𝑡𝐻2(Ω)𝐻10𝐻(Ω),𝑣10(Ω)𝑛,𝜅𝑢+0𝐻𝑔(𝑠)𝑤(𝑠)𝑑𝑠2(Ω)𝐻10(Ω)𝑛,𝑤(𝑠)𝐻1𝐻𝑔,(0,),10(Ω)𝑛,,𝑤(0)=0(2.13) where 𝐻1𝐻𝑔,(0,),10(Ω)𝑛=𝑤𝑤,𝑤𝑠𝐿2𝐻𝑔,(0,),10(Ω)𝑛.(2.14) It is clear that 𝐷(𝐴) is dense in .

Our hypotheses on 𝑓, can be stated as follows, which will be used in different theorems: (A1)𝑓==0; (A2)𝑓=𝑓(𝑥,𝑡)𝐶1([0,),(𝐿2(Ω))𝑛),=(𝑥,𝑡)𝐶1([0,),𝐿2(Ω)); (A3)𝑓(𝑥,𝑡)𝐶([0,),(𝐻10(Ω))𝑛),(𝑥,𝑡)𝐶([0,),𝐻2(Ω)); (A4)𝑓(𝑥,𝑡)𝐶([0,),(𝐿2(Ω))𝑛),(𝑥,𝑡)𝐶([0,),𝐿2(Ω)), and for any 𝑇>0, 𝑓𝑡𝐿1((0,𝑇),(𝐿2(Ω))𝑛),𝑡𝐿1((0,𝑇),𝐿2(Ω)).

We are now in a position to state our main theorems.

Theorem 2.1. Suppose that condition (A1) holds. Relaxation function 𝑔 satisfies (H1)–(H3). Then for any Φ(0)=(𝑢0,𝑢1,𝜃0,𝜃1,𝑤0)𝐷(𝐴), there exists a unique global classical solution Φ=(𝑢,𝑣,𝜃,𝜃𝑡,𝑤) to system (2.8) satisfying Φ=(𝑢,𝑣,𝜃,𝜃𝑡,𝑤)𝐶1([0,),)𝐶([0,),𝐷(𝐴)).

Theorem 2.2. Suppose that condition (A2) holds. Relaxation function 𝑔 satisfies (H1)–( H3). Then for any Φ(0)=(𝑢0,𝑢1,𝜃0,𝜃1,𝑤0), there exists a unique global classical solution Φ=(𝑢,𝑣,𝜃,𝜃𝑡,𝑤) to system (2.8) satisfying Φ=(𝑢,𝑣,𝜃,𝜃𝑡,𝑤)𝐶1([0,),)𝐶([0,),𝐷(𝐴)), that is, 𝑢𝐶1[𝐻0,),10(Ω)𝑛[𝐻𝐶0,),2(Ω)𝐻10(Ω)𝑛,𝑣𝐶1[𝐿0,),2(Ω)𝑛[𝐻𝐶0,),10(Ω)𝑛,𝜃𝐶1[0,),𝐻10[(Ω)𝐶0,),𝐻2(Ω)𝐻10,𝜃(Ω)𝑡𝐶1[0,),𝐿2[(Ω)𝐶0,),𝐻10,(Ω)𝑤𝐶1[0,),𝐿2𝐻𝑔,(0,),10(Ω)𝑛[𝐶0,),𝐻1𝐻𝑔,(0,),10(Ω)𝑛.(2.15)

Corollary 2.3. Suppose that condition (A3)or (A4) holds. Relaxation function 𝑔 satisfies (H1)–(H3). Then for any Φ(0)=(𝑢0,𝑢1,𝜃0,𝜃1,𝑤0)D(𝐴), there exists a unique global classical solution Φ=(𝑢,𝑣,𝜃,𝜃𝑡,𝑤)𝐶1([0,),)𝐶([0,),𝐷(𝐴)) to system (2.8).

Corollary 2.4. If 𝑓(𝑥,𝑡) and (𝑥,𝑡) are Lipschitz continuous functions from [0,𝑇] into (𝐿2(Ω))𝑛 and 𝐿2(Ω), respectively, then for any Φ=(𝑢,𝑣,𝜃,𝜃𝑡,𝑤)𝐷(𝐴), there exists a unique global classical solution Φ=(𝑢,𝑣,𝜃,𝜃𝑡,𝑤)𝐶1([0,),)𝐶([0,),𝐷(𝐴)) to system (2.8).

Theorem 2.5. Suppose relaxation function 𝑔 satisfies (H1)–(H3), 𝑓=𝑓(Φ), and =(Φ),Φ=(𝑢,𝑣,𝜃,𝜃𝑡,𝑤), and 𝐾=(0,𝑓,0,,0) satisfies the global Lipschitz condition on ; that is, there is a positive constant 𝐿 such that for all Φ1,Φ2, 𝐾Φ1Φ𝐾2Φ𝐿1Φ2.(2.16) Then for any Φ(0)=(𝑢0,𝑢1,𝜃0,𝜃1,𝑤0), there exists a global mild solution Φ to system (2.8) such that Φ𝐶([0,),), that is, [𝐻𝑢𝐶0,),1Γ1(Ω)𝑛[,𝜃𝐶0,),𝐻10(Ω),𝜃𝑡[𝐶0,),𝐿2,[𝐿(Ω)𝑣𝐶0,),2(Ω)𝑛[,𝑤𝐶(0,),𝐿2𝐻𝑔,(0,),10(Ω)𝑛.(2.17)

Theorem 2.6. Suppose 𝑓=𝑓(Φ) and =(Φ),Φ=(𝑢,𝑣,𝜃,𝜃𝑡,𝑤), and 𝐾=(0,𝑓,0,,0) is a nonlinear operator from 𝐷(𝐴) into 𝐷(𝐴) and satisfies the global Lipschitz condition on 𝐷(𝐴); that is, there is a positive constant 𝐿 such that for all Φ1,Φ2𝐷(𝐴), 𝐾Φ1Φ𝐾2𝐷(𝐴)Φ𝐿1Φ2𝐷(𝐴).(2.18) Then for any Φ(0)=(𝑢0,𝑢1,𝜃0,𝜃1,𝑤0)𝐷(𝐴), there exists a unique global classical solution Φ=(𝑢,𝑣,𝜃,𝜃𝑡,𝑤)𝐶1([0,),)𝐶([0,),𝐷(𝐴)) to system (2.8).

3. Some Lemmas

In this section in order to complete proofs of Theorems 2.12.6, we need first Lemmas 3.13.5. For the abstract initial value problem, 𝑑𝑢𝑑𝑡+𝐵𝑢=𝐾,𝑢(0)=𝑢0,(3.1) where 𝐵 is a maximal accretive operator defined in a dense subset 𝐷(𝐵) of a Banach space 𝐻. We have the following.

Lemma 3.1. Let 𝐵 be a linear operator defined in a Hilbert space 𝐻,𝐵𝐷(𝐵)𝐻𝐻. Then the necessary and sufficient conditions for 𝐵 being maximal accretive are (i)Re(𝐵𝑥,𝑥)0,forall𝑥𝐷(𝐵), (ii)𝑅(𝐼+𝐵)=𝐻.

Proof. We first prove the necessity. 𝐵 is an accretive operator, so we have (𝑥,𝑥)=𝑥2𝑥+𝜆𝐵𝑥2=(𝑥,𝑥)+2𝜆Re(𝐵𝑥,𝑥)+𝜆2𝐵𝑥2.(3.2) Thus, for all 𝜆>0, 𝜆Re(𝐵𝑥,𝑥)2𝐵𝑥2.(3.3) Letting 𝜆0, we get (i). Furthermore, (ii) immediately follows from the fact that 𝐵 is m-accretive.
We now prove the sufficiency. It follows from (i) that for all 𝜆>0, 𝑥𝑦2Re(𝑥𝑦,𝑥𝑦+𝜆𝐵(𝑥𝑦))𝑥𝑦𝑥𝑦+𝜆(𝐵𝑥𝐵𝑦).(3.4) Now it remains to prove that 𝐵 is densely defined. We use a contradiction argument. Suppose that it is not true. Then there is a nontrivial element 𝑥0 belonging to orthogonal supplement of 𝐷(𝐵) such that for all 𝑥𝐷(𝐵), 𝑥,𝑥0=0.(3.5) It follows from (ii) that there is 𝑥𝐷(𝐵) such that 𝑥+𝐵𝑥=𝑥0.(3.6) Taking the inner product of (3.5) with 𝑥, we deduce that 𝑥+𝐵𝑥,𝑥=0.(3.7) Taking the real part of (3.7), we deduce that 𝑥=0, and by (3.6), 𝑥0=0, which is a contradiction. Thus the proof is complete.

Lemma 3.2. Suppose that 𝐵 is 𝑚-accretive in a Banach space 𝐻, and 𝑢0𝐷(𝐵). Then problem (3.1) has a unique classical solution 𝑢 such that 𝑢𝐶1([[0,),𝐻)𝐶(0,),𝐷(𝐵)).(3.8)

Lemma 3.3. Suppose that 𝐾=𝐾(𝑡), and 𝐾(𝑡)𝐶1([0,),𝐻),𝑢0𝐷(𝐵).(3.9) Then problem (3.1) admits a unique global classical solution 𝑢 such that 𝑢𝐶1([[0,),𝐻)𝐶(0,),𝐷(𝐵))(3.10) which can be expressed as 𝑢(𝑡)=𝑆(𝑡)𝑢0+𝑡0𝑆(𝑡𝜏)𝐾(𝜏)𝑑𝜏.(3.11)

Proof. Since 𝑆(𝑡)𝑢0 satisfies the homogeneous equation and nonhomogeneous initial condition, it suffices to verify that 𝑤(𝑡) given by 𝑤(𝑡)=𝑡0𝑆(𝑡𝜏)𝐾(𝜏)𝑑𝜏(3.12) belongs to 𝐶1([0,),𝐻)𝐶([0,),𝐷(𝐵)) and satisfies the nonhomogeneous equation. Consider the following quotient of difference 𝑤(𝑡+)𝑤(𝑡)=10𝑡+𝑆(𝑡+𝜏)𝐾(𝜏)𝑑𝜏𝑡0𝑆=1(𝑡𝜏)𝐾(𝜏)𝑑𝜏𝑡𝑡+1𝑆(𝑡+𝜏)𝐾(𝜏)𝑑𝜏+𝑡0=1(𝑆(𝑡+𝜏)𝑆(𝑡𝜏))𝐾(𝜏)𝑑𝜏𝑡𝑡+1𝑆(𝑧)𝐾(𝑡+𝑧)𝑑𝑧+𝑡0𝑆(𝑧)(𝐾(𝑡+𝑧)𝐾(𝑡𝑧))𝑑𝑧.(3.13) When 0, the terms in the last line of (3.13) have limits: 𝑆(𝑡)𝐾(0)+𝑡0[𝑆(𝑧)𝐾(𝑡𝑧)𝑑𝑧𝐶(0,),𝐻).(3.14) It turns out that 𝑤𝐶1([0,),𝐻) and the terms in the third line of (3.13) have limits too, which should be 𝑆(0)𝐾(𝑡)𝐵𝑤(𝑡)=𝐾(𝑡)𝐵𝑤(𝑡).(3.15) Thus the proof is complete.

Lemma 3.4. Suppose that 𝐾=𝐾(𝑡), and [𝐾(𝑡)𝐶(0,),𝐷(𝐵)),u0𝐷(𝐵).(3.16) Then problem (3.1) admits a unique global classical solution.

Proof. From the proof of Lemma 3.2, we can obtain 𝑤(𝑡+)𝑤(𝑡)=1𝑡𝑡+𝑆1(𝑡+𝜏)𝐾(𝜏)𝑑𝜏+𝑡0=1(𝑆(𝑡+𝜏)𝑆(𝑡𝜏))𝐾(𝜏)𝑑𝜏𝑡𝑡+1𝑆(𝑡+𝜏)𝐾(𝜏)𝑑𝜏+𝑡0𝑆(𝑡𝜏)𝑆()𝐼𝐾(𝜏)𝑑𝜏.(3.17) When 0, the last terms in the line of (3.17) have limits: 𝑆(0)𝐾(𝑡)𝑡0𝑆(𝑡𝜏)𝐵𝐾(𝜏)𝑑𝜏=𝑆(0)𝐾(𝑡)𝐵𝑡0𝑆(𝑡𝜏)𝐾(𝜏)𝑑𝜏=𝐾(𝑡)𝐵𝑤(𝑡).(3.18) Combining the results of Lemma 3.3 proves the lemma.

Lemma 3.5. Suppose that 𝐾=𝐾(𝑡), and [𝐾(𝑡)𝐶(0,),𝐻),𝑢0𝐷(𝐵),(3.19) and for any 𝑇>0, 𝐾𝑡𝐿1([]0,𝑇,𝐻).(3.20) Then problem (3.1) admits a unique global classical solution.

Proof. We first prove that for any 𝐾1𝐿1([0,𝑇],𝐻), the function 𝑤 given by the following integral: 𝑤(𝑡)=𝑡0𝑆(𝑡𝜏)𝐾1𝑑𝜏(3.21) belongs to 𝐶([0,𝑇],𝐻). Indeed, we infer from the difference 𝑤(𝑡+)𝑤(𝑡)=0𝑡+𝑆(𝑡+𝜏)𝐾1(𝜏)𝑑𝜏𝑡0𝑆(𝑡𝜏)𝐾1(𝜏)𝑑𝜏=(𝑆()𝐼)𝑤(𝑡)+𝑡𝑡+𝑆(𝑡+𝜏)𝐾1(𝜏)𝑑𝜏(3.22) that as 0, 𝑤(𝑡+)𝑤(𝑡)(𝑆()𝐼)𝑤(𝑡)+𝑡𝑡+𝐾1(𝜏)𝑑𝜏0,(3.23) where we have used the strong continuity of 𝑆(𝑡) and the absolute continuity of integral for 𝐾1𝐿1[0,𝑡].
Now it can be seen from the last line of (3.13) that for almost every 𝑡[0,𝑇],𝑑𝑤/𝑑𝑡 exists, and it equals 𝑆(𝑡)𝐾(0)+𝑡0𝑆(𝑧)𝐾(𝑡𝑧)𝑑𝑧=𝑆(𝑡)𝐾(0)+𝑡0[]𝑆(𝑡𝜏)𝐾(𝜏)𝑑𝜏𝐶(0,𝑇,𝐻).(3.24) Thus, for almost every 𝑡, 𝑑𝑤𝑑𝑡=𝐵𝑤+𝐾.(3.25) Since 𝑤 and 𝐾 both belong to 𝐶([0,𝑇],𝐻), it follows from (3.25) that for almost every 𝑡, 𝐵𝑤 equals a function belonging to 𝐶([0,𝑇],𝐻). Since 𝐵 is a closed operator, we conclude that []𝑤𝐶(0,𝑇,𝐷(𝐵))𝐶1([]0,𝑇,𝐻)(3.26) and (3.25) holds for every 𝑡. Thus the proof is complete.

To prove that the operator 𝐴 defined by (2.14) is dissipative, we need the following lemma.

Lemma 3.6. If the function 𝑓[0,)𝑅 is uniformly continuous and is in 𝐿1(0,), then lim𝑡𝑓(𝑡)=0.(3.27)

Lemma 3.7. Suppose that the relaxation function 𝑔 satisfies (𝐻1) and (𝐻2). If 𝑤𝐻1(𝑔,(0,),(𝐻10(Ω))𝑛) and 𝑤(0)=0, then 𝑔(𝑠)𝑤(𝑠)2(𝐻10(Ω))𝑛𝐿1(0,),lim𝑠𝑔(𝑠)𝑤(𝑠)2(𝐻10(Ω))𝑛=0.(3.28)

Proof. See, for example, the work by Liu in [16].

Lemma 3.8. Suppose relaxation function 𝑔 satisfies (H1)–(H3). The operator 𝐴 defined by (2.13) is dissipative; furthermore, 0𝜌(𝐴), where 𝜌(𝐴) is the resolvent of the operator 𝐴.

Proof. By a straightforward calculation, it follows from Lemma 3.7 that 𝐴𝑢,𝑣,𝜃,𝜃𝑡,,𝑤𝑢,𝑣,𝜃,𝜃𝑡,𝑤=𝜅(𝑣,𝑢)(𝐻10(Ω))𝑛+12𝐵(𝑢,𝑤)𝛼𝜃𝑡+𝛼,𝑣2𝛽𝜃𝑡+𝛼,𝜃2𝛽Δ𝜃𝑡+Δ𝜃div𝑣,𝜃𝑡+𝑣𝑤𝑠,𝑤𝐿2(𝑔,(0,),(𝐻10(Ω))𝑛)𝛼=2𝛽𝜃𝑡2+0𝑔(𝑠)𝑤(𝑠)2(𝐻10(Ω))𝑛𝑑𝑠0.(3.29) Thus, 𝐴 is dissipative.
To prove that 0𝜌(𝐴), for any 𝐺=(𝑔1,𝑔2,𝑔3,𝑔4,𝑔5), consider 𝐴Φ=𝐺,(3.30) that is, 𝑣=𝑔1𝐻,in10(Ω)𝑛,(3.31)𝐵(𝑢,𝑤)𝛼𝜃𝑡=𝑔2𝐿,in2(Ω)𝑛𝜃,(3.32)𝑡=𝑔3,in𝐿2(Ω),(3.33)Δ𝜃𝑡+Δ𝜃𝛽div𝑣=𝑔4,in𝐿2(Ω),(3.34)𝑣𝑤𝑠=𝑔5,in𝐿2𝐻𝑔,(0,),10(Ω)𝑛.(3.35) Inserting 𝑣=𝑔1 and 𝜃𝑡=𝑔3 obtained from (3.31), (3.33) into (3.34), we obtain Δ𝜃=𝑔4+𝛽div𝑔1Δ𝑔3𝐿2(Ω).(3.36) By the standard theory for the linear elliptic equations, we have a unique 𝜃𝐻2(Ω)𝐻10(Ω) satisfying (3.36).
We plug 𝑣=𝑔1 obtained from (3.31) into (3.35) to get 𝑤𝑠=𝑔1𝑔5𝐿2𝐻𝑔,(0,),1Γ1(Ω)𝑛.(3.37) Applying the standard theory for the linear elliptic equations again, we have a unique 𝑤𝐻1(𝑔,(0,),(𝐻10(Ω))𝑛) satisfying (3.37). Then plugging 𝜃 and 𝑤 just obtained from solving (3.36), (3.37), respectively, into (3.32) and applying the standard theory for the linear elliptic equations again yield the unique solvability of 𝑢𝐷(𝐴) for (3.32) and such that 𝜅𝑢+0𝑔(𝑠)𝑤(𝑠)𝑑𝑠(𝐻2(Ω)𝐻10(Ω))𝑛. Thus the unique solvability of (3.30) follows. It is clear from the regularity theory for the linear elliptic equations that Φ𝐾𝐺 with 𝐾 being a positive constant independent of Φ. Thus the proof is completed.

Lemma 3.9. The operator 𝐴 defined by (2.13) is closed.

Proof. To prove that 𝐴 is closed, let (𝑢𝑛,𝑣𝑛,𝜃𝑛,𝜃𝑛𝑡,𝑤𝑛)𝐷(𝐴) be such that 𝑢𝑛,𝑣𝑛,𝜃𝑛,𝜃𝑛𝑡,𝑤𝑛𝑢,𝑣,𝜃,𝜃𝑡𝐴𝑢,𝑤in,𝑛,𝑣𝑛,𝜃𝑛,𝜃𝑡𝑛,𝑤𝑛(𝑎,𝑏,𝑐,𝑑,𝑒)in.(3.38) Then we have 𝑢𝑛𝐻𝑢in10(Ω)𝑛,𝑣(3.39)𝑛𝐿𝑣in2(Ω)𝑛,𝜃(3.40)𝑛𝜃in𝐻10𝜃(Ω),(3.41)𝑛𝑡𝜃𝑡in𝐿2𝑤(Ω),(3.42)𝑛𝑤in𝐿2𝐻𝑔,(0,),10(Ω)𝑛𝑣,(3.43)𝑛𝐻𝑎in10(Ω)𝑛𝐵𝑢,(3.44)𝑛,𝑤𝑛𝛼𝜃𝑛𝑡𝐿𝑏in2(Ω)𝑛𝜃,(3.45)𝑛𝑡𝑐in𝐻10(Ω),(3.46)Δ𝜃𝑛𝑡+Δ𝜃𝑛𝛽div𝑣𝑛𝑑in𝐿2𝑣(Ω),(3.47)𝑛𝑤𝑛𝑠𝑒in𝐿2𝐻𝑔,(0,),10(Ω)𝑛.(3.48) By (3.40) and (3.44), we deduce 𝑣𝑛𝐻𝑣in10(Ω)𝑛,𝐻(3.49)𝑣=𝑎10(Ω)𝑛.(3.50) By (3.42) and (3.46), we deduce 𝜃𝑛𝑡𝜃𝑡in𝐻10𝜃(Ω),(3.51)𝑡=𝑐𝐻10(Ω).(3.52) By (3.47) and (3.49), we deduce Δ𝜃𝑛𝑡+Δ𝜃𝑛𝑑+𝛽div𝑣in𝐿2(Ω),(3.53) and consequently, it follows from (3.41), that 𝜃𝑛𝑡+𝜃𝑛𝜃𝑡+𝜃in𝐻2(Ω)𝐻10(Ω),(3.54) since Δ is an isomorphism from 𝐻2(Ω)𝐻10(Ω) onto 𝐿2(Ω). It therefore follows from (3.47) and (3.54) that 𝑑=Δ𝜃𝑛𝑡+Δ𝜃𝑛𝛽div𝑣,𝜃𝑡+𝜃𝐻2(Ω)𝐻10(Ω).(3.55) By (3.43), (3.48), and (3.49), we deduce 𝑤𝑛𝑤in𝐻1𝐻𝑔,(0,),1Γ1(Ω)𝑛,(3.56)𝑒=𝑣𝑤𝑠,𝑤𝐻1𝐻𝑔,(0,),10(Ω)𝑛,𝑤(0)=0.(3.57) In addition, it follows from (3.39), (3.43), (3.51) that 𝐵𝑢𝑛,𝑤𝑛𝛼𝜃𝑛𝑡𝐵(𝑢,𝑤)𝛼𝜃𝑡(3.58) in the distribution. It therefore follows from (3.45) and (3.58) that 𝑏=𝐵(𝑢,𝑤)𝛼𝜃𝑡𝐿,𝐵(𝑢,𝑤)2(Ω)𝑛,(3.59) and consequently, 𝜅𝑢+0𝐻𝑔(𝑠)𝑤(𝑠)𝑑𝑠2(Ω)𝐻10(Ω)𝑛,(3.60) since 𝜇Δ+(𝜆+𝜇)div is an isomorphism from 𝐻2(Ω)𝐻10(Ω) onto 𝐿2(Ω). Thus, by (3.50), (3.52), (3.55), (3.57), (3.59), (3.60), we deduce 𝐴𝑢,𝑣,𝜃,𝜃𝑡=,𝑤(𝑎,𝑏,𝑐,𝑑,𝑒),𝑢,𝑣,𝜃,𝜃𝑡,𝑤𝐷(𝐴).(3.61) Hence, 𝐴 is closed.

Lemma 3.10. Let 𝐴 be a linear operator with dense domain 𝐷(𝐴) in a Hilbert space 𝐻. If 𝐴 is dissipative and 0𝜌(𝐴), the resolvent set of 𝐴, then 𝐴 is the infinitesimal generator of a 𝐶0-semigroup of contractions on 𝐻.

Proof. See, for example, the work by Liu and Zheng in [17] and by Pazy in [18].

Lemma 3.11. Let 𝐴 be a densely defined linear operator on a Hilbert space 𝐻. Then 𝐴 generates a 𝐶0-semigroup of contractions on 𝐻 if and only if 𝐴 is dissipative and 𝑅(𝐼𝐴)=𝐻.

Proof. See, for example, the work by Zheng in [19].

4. Proofs of Theorems 2.12.5

Proof of Theorem 2.1. By (2.2), it is clear that is a Hilbert space. By Lemmas 3.83.10, we can deduce that the operator 𝐴 is the infinitesimal generator of a 𝐶0-semigroup of contractions on Hilbert space . Applying the result and Lemma 3.2, we can obtain our result.

Proof of Theorem 2.2. we have known that the operator 𝐴 is the infinitesimal generator of a 𝐶0-semigroup of contractions on Hilbert space . Applying the result and Lemma 3.11, we can conclude that 𝑅(𝐼𝐴)=𝐻. If we choose operator 𝐵=𝐴, we can obtain 𝐷(𝐵)=𝐷(𝐴) and 𝐷(𝐵) is dense in . Noting that by (𝐴2), we know that 𝐾=(0,𝑓,0,,0)𝐶1([0,),); therefore, applying Lemma 3.1, we can conclude the operator 𝐵 is the maximal accretive operator. Then we can complete the proof of Theorem 2.2 in term of Lemma 3.3.

Proof of Corollary 2.3. By (𝐴3) or (𝐴4), we derive that 𝐾=(0,𝑓,0,,0)𝐶([0,),𝐷(𝐴)) or 𝐾𝐶([0,),), and for any 𝑇>0,𝐾𝑡𝐿1((0,𝑇),). Noting that 𝐵=𝐴 is the maximal accretive operator, we use Lemmas 3.4 and 3.5 to prove the corollary.

Proof of Corollary 2.4. We know that 𝐾(𝑥,𝑡)=(0,𝑓,0,,0) are Lipschitz continuous functions from [0,𝑇] into . Moreover, by (2.2), it is clear that is a reflexive Banach space. Therefore, 𝐾𝑡𝐿1([0,𝑇],𝐻). Hence applying Lemma 3.5, we may complete the proof of the corollary.

Proof of Theorem 2.5. By virtue of the proof of Theorem 2.2, we know that 𝐵=𝐴 is the maximal accretive operator of a 𝐶0 semigroup 𝑆(𝑡). On the other hand, 𝐾=(0,𝑓,0,,0) satisfies the global Lipschitz condition on . Therefore, we use the contraction mapping theorem to prove the present theorem. Two key steps for using the contraction mapping theorem are to figure out a closed set of the considered Banach space and an auxiliary problem so that the nonlinear operator defined by the auxiliary problem maps from this closed set into itself and turns out to be a contraction. In the following we proceed along this line.
Let 𝜙(Φ)=𝑆(𝑡)Φ0+𝑡0[𝑆(𝑡𝜏)𝐾(Φ(𝜏))𝑑𝜏,(4.1)Ω=Φ𝐶(0,+),𝐻)sup𝑡0Φ(𝑡)𝑒𝑘𝑡<,(4.2) where 𝑘 is a positive constant such that 𝑘>𝐿. In Ω, we introduce the following norm: ΦΩ=sup𝑡0Φ(𝑡)𝑒𝑘𝑡.(4.3) Clearly, Ω is a Banach space. We now show that the nonlinear operator 𝜙 defined by (4.1) maps Ω into itself, and the mapping is a contraction. Indeed, for ΦΩ, we have 𝜙(Φ)𝑆(𝑡)Φ0+𝑡0𝑆(𝑡𝜏)𝐾(Φ)𝑑𝜏Φ0+𝑡0𝐾(Φ)𝑑𝜏Φ0+𝑡0(𝐿Φ(𝜏)+𝐾(0))𝑑𝜏Φ0+𝐶0𝑡+𝐿sup𝑡0Φ(𝑡)𝑒𝑘𝑡𝑡0𝑒𝑘𝜏𝑑𝜏Φ0+𝐶0𝐿𝑡+𝑘𝑒𝑘𝑡ΦΩ,(4.4) where 𝐶0=𝐾(0). Thus, 𝜙(Φ)Ωsup𝑡0Φ0+𝐶0𝑡𝑒𝑘𝑡+𝐿𝑘ΦΩ<.(4.5) that is, 𝜙(Φ)Ω.
For Φ1,Φ2Ω, we have 𝜙Φ1Φ𝜙2Ω=sup𝑡0𝑒𝑘𝑡𝑡0𝐾Φ𝑆(𝑡𝜏)1(Φ𝜏)𝐾2(𝜏)𝑑𝜏sup𝑡0𝑒𝑘𝑡𝐿𝑡0Φ1Φ2𝑑𝜏sup𝑡0𝑒𝑘𝑡𝐿𝑘𝑒𝑘𝑡Φ11Φ2Ω𝐿𝑘Φ1Φ2Ω.(4.6) Therefore, by the contraction mapping theorem, the problem has a unique solution in Ω. To show that the uniqueness also holds in 𝐶([0,),𝐻), let Φ1,Φ2𝐶([0,),𝐻) be two solutions of the problem and let Φ=Φ1Φ2. Then Φ(𝑡)=𝑡0𝐾Φ𝑆(𝑡𝜏)1Φ𝐾2𝑑𝜏,Φ(𝑡)𝐿𝑡0Φ(𝜏)𝑑𝜏.(4.7) By the Gronwall inequality, we immediately conclude that Φ(𝑡)=0; that is, the uniqueness in 𝐶([0,),𝐻) follows. Thus the proof is complete.

Proof of Theorem 2.6. Since 𝐵 is the maximal accretive operator, 𝐾=(0,𝑓,0,,0) satisfies the global Lipschitz condition on 𝐷(𝐴). Let 𝐴1=𝐷(𝐵),𝐵1=𝐵2𝐵𝐷1𝐵=𝐷2𝐴1.(4.8) Then 𝐴1 is a Banach space, and 𝐵1=𝐵2 is a densely defined operator from 𝐷(𝐵2) into 𝐴1. In what follows we prove that 𝐵1 is 𝑚-accretive in 𝐴1=𝐷(B).
Indeed, for any 𝑥,𝑦𝐷(𝐵2), since 𝐵 is accretive in 𝐻, we have 𝑥𝑦+𝜆(𝐵𝑥𝐵𝑦)𝐷(𝐵)=𝑥𝑦+𝜆(𝐵𝑥𝐵𝑦)2+𝐵𝐵𝑥𝐵𝑦+𝜆2𝑥𝐵2𝑦21/2𝑥𝑦2+𝐵𝑥𝐵𝑦21/2=𝑥𝑦𝐷(𝐵).(4.9) that is, 𝐵1 is accretive in 𝐴1. Furthermore, since 𝐵 is 𝑚-accretive in 𝐻, for any 𝑦𝐻, there is a unique 𝑥𝐷(𝐵) such that 𝑥+𝐵𝑥=𝑦.(4.10) Now for any 𝑦𝐴1=𝐷(𝐵), (4.10) admits a unique solution 𝑥𝐷(𝐵). It turns out that 𝐵𝑥=𝑦𝑥𝐷(𝐵).(4.11) Thus 𝑥𝐷(𝐵2); that is, 𝐵1 is m-accretive in 𝐴1. Let 𝑆1(𝑡) be the semigroup generated by 𝐵1. If Φ0𝐷(𝐵2)=𝐷(𝐵1), then Φ(𝑡)=𝑆1(𝑡)Φ0[𝐵𝐶0,+),𝐷2𝐶1([0,+),𝐷(𝐵))(4.12) is unique classical solution of the problem. On the other hand, Φ(𝑡)=𝑆1(𝑡)Φ0 is also a classical solution in [𝐶(0,+),𝐷(𝐵))𝐶1([0,+),𝐻).(4.13) This implies that 𝑆1(𝑡) is a restriction of 𝑆(𝑡) on 𝐴1. By virtue of the proof of Theorem 2.5, there exists a unique mild solution Φ𝐶([0,+),𝐴1). Since 𝑆1(𝑡) is a restriction of 𝑆(𝑡) on 𝐷(𝐵), and moreover, we infer from 𝐾(Φ) being an operator from 𝐷(𝐵) to 𝐷(𝐵) and Lemma 3.4 that Φ is a classical solution to the problem. Thus the proof is complete.

Acknowledgment

This work was supported in part by Foundation of Shanghai Second Polytechnic University of China (no. A20XQD210006).

References

  1. C. M. Dafermos, “On the existence and the asymptotic stability of solutions to the equations of linear thermoelasticity,” Archive for Rational Mechanics and Analysis, vol. 29, pp. 241–271, 1968. View at: Publisher Site | Google Scholar | Zentralblatt MATH
  2. C. M. Dafermos, “Asymptotic stability in viscoelasticity,” Archive for Rational Mechanics and Analysis, vol. 37, pp. 297–308, 1970. View at: Google Scholar | Zentralblatt MATH
  3. C. M. Dafermos, “An abstract Volterra equation with applications to linear viscoelasticity,” Journal of Differential Equations, vol. 7, pp. 554–569, 1970. View at: Publisher Site | Google Scholar | Zentralblatt MATH
  4. R. Racke, Y. Shibata, and S. M. Zheng, “Global solvability and exponential stability in one-dimensional nonlinear thermoelasticity,” Quarterly of Applied Mathematics, vol. 51, no. 4, pp. 751–763, 1993. View at: Google Scholar | Zentralblatt MATH
  5. J. E. M. Rivera, “Energy decay rates in linear thermoelasticity,” Funkcialaj Ekvacioj, vol. 35, no. 1, pp. 19–30, 1992. View at: Google Scholar | Zentralblatt MATH
  6. Y. Shibata, “Neumann problem for one-dimensional nonlinear thermoelasticity,” in Partial Differential Equations, vol. 2, pp. 457–480, Polish Academy of Sciences, Warsaw, Poland, 1992. View at: Google Scholar
  7. M. Slemrod, “Global existence, uniqueness, and asymptotic stability of classical smooth solutions in one-dimensional nonlinear thermoelasticity,” Archive for Rational Mechanics and Analysis, vol. 76, no. 2, pp. 97–133, 1981. View at: Publisher Site | Google Scholar | Zentralblatt MATH
  8. S. Zheng and W. Shen, “Global solutions to the Cauchy problem of quasilinear hyperbolicparabolic coupled systems,” Scientia Sinica, vol. 4, pp. 357–372, 1987. View at: Google Scholar
  9. Z. Liu and S. Zheng, “On the exponential stability of linear viscoelasticity and thermoviscoelasticity,” Quarterly of Applied Mathematics, vol. 54, no. 1, pp. 21–31, 1996. View at: Google Scholar
  10. S. W. Hansen, “Exponential energy decay in a linear thermoelastic rod,” Journal of Mathematical Analysis and Applications, vol. 167, no. 2, pp. 429–442, 1992. View at: Publisher Site | Google Scholar | Zentralblatt MATH
  11. X. Zhang and E. Zuazua, “Decay of solutions of the system of thermoelasticity of type III,” Communications in Contemporary Mathematics, vol. 5, no. 1, pp. 25–83, 2003. View at: Publisher Site | Google Scholar | Zentralblatt MATH
  12. W. Desch and R. K. Miller, “Exponential stabilization of Volterra integral equations with singular kernels,” Journal of Integral Equations and Applications, vol. 1, no. 3, pp. 397–433, 1988. View at: Publisher Site | Google Scholar | Zentralblatt MATH
  13. R. H. Fabiano and K. Ito, “Semigroup theory and numerical approximation for equations in linear viscoelasticity,” SIAM Journal on Mathematical Analysis, vol. 21, no. 2, pp. 374–393, 1990. View at: Publisher Site | Google Scholar | Zentralblatt MATH
  14. R. H. Fabiano and K. Ito, “An approximation framework for equations in linear viscoelasticity with strongly singular kernels,” Quarterly of Applied Mathematics, vol. 52, no. 1, pp. 65–81, 1994. View at: Google Scholar | Zentralblatt MATH
  15. K. Liu and Z. Liu, “On the type of C0-semigroup associated with the abstract linear viscoelastic system,” Zeitschrift für Angewandte Mathematik und Physik, vol. 47, no. 1, pp. 1–15, 1996. View at: Publisher Site | Google Scholar
  16. W. Liu, “The exponential stabilization of the higher-dimensional linear system of thermoviscoelasticity,” Journal de Mathématiques Pures et Appliquées, vol. 77, no. 4, pp. 355–386, 1998. View at: Publisher Site | Google Scholar | Zentralblatt MATH
  17. Z. Liu and S. Zheng, Semigroups Associated with Dissipative Systems, vol. 398 of Research Notes in Mathematics, Chapman & Hall/CRC, Boca Raton, Fla, 1999.
  18. A. Pazy, Semigroups of Linear Operators and Applications to Partial Differential Equations, vol. 44 of Applied Mathematical Sciences, Springer, New York, NY, USA, 1983.
  19. S. Zheng, Nonlinear Evolution Equations, vol. 133 of Monographs and Surveys in Pure and Applied Mathematics, CRC Press, Boca Raton, Fla, USA, 2004. View at: Publisher Site

Copyright © 2011 Zhiyong Ma. 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.

Related articles

No related content is available yet for this article.
 PDF Download Citation Citation
 Download other formatsMore
 Order printed copiesOrder
Views667
Downloads896
Citations

Related articles

No related content is available yet for this article.

Article of the Year Award: Outstanding research contributions of 2021, as selected by our Chief Editors. Read the winning articles.