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: StanisΕ‚aw MigΓ³rski
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: π‘’π‘‘π‘‘βˆ’πœ‡Ξ”π‘’βˆ’(πœ†+πœ‡)βˆ‡div𝑒+πœ‡π‘”βˆ—Ξ”π‘’+(πœ†+πœ‡)π‘”βˆ—βˆ‡div𝑒+π›Όβˆ‡πœƒπ‘‘πœƒ=𝑓,(π‘₯,𝑑)βˆˆΞ©Γ—(0,∞),π‘‘π‘‘βˆ’Ξ”πœƒπ‘‘βˆ’Ξ”πœƒ+𝛽div𝑒𝑑=β„Ž,(π‘₯,𝑑)βˆˆΞ©Γ—(0,∞),(1.1) where the sign β€œβˆ—β€ denotes the convolution product in time, which is defined by ξ€œπ‘”βˆ—π‘£(𝑑)=π‘‘βˆ’βˆžπ‘”(π‘‘βˆ’π‘ )𝑣(π‘₯,𝑠)𝑑𝑠(1.2) with the initial data 𝑒(π‘₯,0)=𝑒0(π‘₯),𝑒𝑑(π‘₯,0)=𝑒1(π‘₯),πœƒ(π‘₯,0)=πœƒ0πœƒ(π‘₯),π‘₯∈Ω,𝑑(π‘₯,0)=πœƒ1(π‘₯),𝑒(π‘₯,0)βˆ’π‘’(π‘₯,βˆ’π‘ )=𝑀0(π‘₯,𝑠),(π‘₯,𝑠)βˆˆΞ©Γ—(0,∞)(1.3) and boundary condition 𝑒=0,πœƒ=0,(π‘₯,𝑑)βˆˆΞ“Γ—(0,∞).(1.4) The body Ξ© is a bounded domain in 𝑅𝑛 with smooth boundary Ξ“=πœ•Ξ© (say 𝐢2) and is assumed to be linear, homogeneous, and isotropic. 𝑒(π‘₯,𝑑)=(𝑒1(π‘₯,𝑑),𝑒2(π‘₯,𝑑),…,𝑒𝑛(π‘₯,𝑑)), and πœƒ(π‘₯,𝑑) represent displacement vector and temperature derivations, respectively, from the natural state of the reference configuration at position π‘₯ and time 𝑑. πœ†,πœ‡>0 are LamΓ©'s constants and 𝛼,𝛽>0 the coupling parameters; 𝑔(𝑑) denotes the relaxation function, 𝑀0(π‘₯,𝑠) is a specified β€œhistory,” and 𝑒0(π‘₯),𝑒1(π‘₯),πœƒ0(π‘₯) are initial data. Ξ”,βˆ‡,div denote the Laplace, gradient, and divergence operators in the space variables, respectively.

We refer to the work by Dafermos [1–3]. The following basic conditions on the relaxation function 𝑔(𝑑) are (H1)π‘”βˆˆπΆ1[0,∞)∩𝐿1(0,∞); (H2)𝑔(𝑑)β‰₯0,π‘”ξ…ž(𝑑)≀0,𝑑>0; (H3)βˆ«πœ…=1βˆ’βˆž0𝑔(𝑑)𝑑𝑑>0.

In what follows, we denote by β€–β‹…β€– the norm of 𝐿2(Ξ©), and we use the notation ‖𝑣‖2=𝑛𝑖=1‖‖𝑣𝑖‖‖2𝑣,for𝑣=1,𝑣2,…,𝑣𝑛.(1.5)

When 𝑓=𝑔=β„Ž=0, system (1.1)–(1.4) is reduced to the thermoelastic system: π‘’π‘‘π‘‘βˆ’πœ‡Ξ”π‘’βˆ’(πœ†+πœ‡)βˆ‡div𝑒+π›Όβˆ‡πœƒπ‘‘πœƒ=0,(π‘₯,𝑑)βˆˆΞ©Γ—(0,∞),π‘‘π‘‘βˆ’Ξ”πœƒπ‘‘βˆ’Ξ”πœƒ+𝛽div𝑒𝑑=0,(π‘₯,𝑑)βˆˆΞ©Γ—(0,∞),𝑒=0,πœƒ=0,(π‘₯,𝑑)βˆˆΞ“Γ—(0,∞),𝑒(π‘₯,0)=𝑒0(π‘₯),𝑒𝑑(π‘₯,0)=𝑒1(π‘₯),πœƒ(π‘₯,0)=πœƒ0(π‘₯),π‘₯∈Ω.(1.6) In the one-dimensional space case, there are many works (see e.g., [4–8]) on the global existence and uniqueness. Liu and Zheng [9] succeeded in deriving in energy decay under the boundary condition (1.4) orπ‘’βˆ£π‘₯=0=0,𝜎∣π‘₯=𝑙=0,πœƒπ‘₯∣π‘₯=0,𝑙=0,(1.7)π‘’βˆ£π‘₯=0=0,𝜎∣π‘₯=𝑙=0,πœƒβˆ£π‘₯=0=0,πœƒπ‘₯∣π‘₯=𝑙=0,(1.8) orπ‘’βˆ£π‘₯=0=0,𝜎∣π‘₯=𝑙=0,πœƒβˆ£π‘₯=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:π‘’βˆ£π‘₯=0=0,𝜎∣π‘₯=𝑙=0,πœƒπ‘₯∣π‘₯=0=0,πœƒβˆ£π‘₯=𝑙=0,(1.10) where 𝜎=𝑒π‘₯βˆ’π›Ύπœƒ 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=β„Ž=0, system (1.1)-(1.4) is decoupled into the following viscoelastic system: π‘’π‘‘π‘‘βˆ’πœ‡Ξ”π‘’βˆ’(πœ†+πœ‡)βˆ‡div𝑒+πœ‡π‘”βˆ—Ξ”π‘’+(πœ†+πœ‡)π‘”βˆ—βˆ‡div𝑒=0,(π‘₯,𝑑)βˆˆΞ©Γ—(0,∞),𝑒=0,(π‘₯,𝑑)βˆˆΞ“Γ—(0,∞),𝑒(π‘₯,0)=𝑒0(π‘₯),𝑒𝑑(π‘₯,0)=𝑒1(π‘₯),(π‘₯,𝑑)∈Ω,𝑒(π‘₯,0)βˆ’π‘’(π‘₯,βˆ’π‘ )=𝑀0(π‘₯),(π‘₯,𝑑)βˆˆΞ©Γ—(0,∞),(1.11) and the wave equation.

There are many works (see, e.g., [9, 12–15]) on exponential stability of energy and asymptotic stability of solution under different assumptions. The notation in this paper will be as follows. 𝐿𝑝,1≀𝑝≀+∞,π‘Šπ‘š,𝑝,π‘šβˆˆπ‘,𝐻1=π‘Š1,2,𝐻10=π‘Š01,2 denote the usual (Sobolev) spaces on Ξ©. In addition, ‖⋅‖𝐡 denotes the norm in the space 𝐡; we also put β€–β‹…β€–=‖⋅‖𝐿2(Ξ©). We denote by πΆπ‘˜(𝐼,𝐡),π‘˜βˆˆβ„•0, the space of k-times continuously differentiable functions from π½βŠ†πΌ into a Banach space 𝐡, and likewise by 𝐿𝑝(𝐼,𝐡),1≀𝑝≀+∞, the corresponding Lebesgue spaces. 𝐢𝛽([0,𝑇],𝐡) denotes the HΓΆlder space of 𝐡-valued continuous functions with exponent π›½βˆˆ(0,1] in variable 𝑑.

2. Main Results

Let the β€œhistory space” 𝐿2(𝑔,(0,∞),(𝐻10(Ξ©))𝑛) consist of ((𝐻10(Ξ©))𝑛)-valued functions 𝑀 on (0,∞) for which ‖𝑀‖2𝐿2𝐻(𝑔,(0,∞),10ξ€Έ(Ξ©)𝑛)=ξ€œβˆž0𝑔(𝑠)‖𝑀(𝑠)β€–2𝐻10ξ€Έ(Ξ©)𝑛𝑑𝑠<∞.(2.1) Put 𝐻ℋ=10ξ€Έ(Ξ©)𝑛×𝐿2ξ€Έ(Ξ©)𝑛×𝐻10(Ξ©)×𝐿2(Ξ©)×𝐿2𝐻𝑔,(0,∞),10ξ€Έ(Ξ©)𝑛(2.2) with the energy norm ‖‖𝑒,𝑣,πœƒ,πœƒπ‘‘ξ€Έβ€–β€–,𝑀ℋ=ξ‚»πœ…β€–π‘’β€–2𝐻10ξ€Έ(Ξ©)𝑛+12‖𝑣‖2+π›Όπ›½β€–β€–πœƒπ‘‘β€–β€–2ξ‚Ά+ξ€œ+β€–βˆ‡πœƒβ€–βˆž0𝑔(𝑠)‖𝑀(𝑠)β€–2𝐻10ξ€Έ(Ξ©)𝑛𝑑𝑠1/2,(2.3) where πœ… denotes the positive constant in (𝐻3), that is, ξ€œπœ…=1βˆ’βˆž0𝑔(𝑑)𝑑𝑑>0.(2.4) Thus we consider the following thermoviscoelastic system:π‘’π‘‘π‘‘βˆ’πœ‡Ξ”π‘’βˆ’(πœ†+πœ‡)βˆ‡div𝑒+πœ‡π‘”βˆ—Ξ”π‘’+(πœ†+πœ‡)π‘”βˆ—βˆ‡div𝑒+π›Όβˆ‡πœƒπ‘‘πœƒ=0,(π‘₯,𝑑)βˆˆΞ©Γ—(0,∞),π‘‘π‘‘βˆ’Ξ”πœƒπ‘‘βˆ’Ξ”πœƒ+𝛽div𝑒𝑑=0,(π‘₯,𝑑)βˆˆΞ©Γ—(0,∞),𝑒=0,πœƒ=0,(π‘₯,𝑑)βˆˆΞ“Γ—(0,∞),𝑒(π‘₯,0)=𝑒0(π‘₯),𝑒𝑑(π‘₯,0)=𝑒1(π‘₯),πœƒ(π‘₯,0)=πœƒ0(π‘₯),πœƒπ‘‘(π‘₯,0)=πœƒ1(π‘₯),π‘₯∈Ω,𝑒(π‘₯,0)βˆ’π‘’(π‘₯,βˆ’π‘ )=𝑀0(π‘₯,𝑠),(π‘₯,𝑑)βˆˆΞ©Γ—(0,∞).(2.5) Let 𝑣(π‘₯,𝑑)=𝑒𝑑(π‘₯,𝑑),𝑀(π‘₯,𝑑,𝑠)=𝑒(π‘₯,𝑑)βˆ’π‘’(π‘₯,π‘‘βˆ’π‘ ).(2.6) Since πœ•ξ€œπœ•πœˆπ‘‘βˆ’βˆžπœ•π‘”(π‘‘βˆ’π‘ )𝑒(𝑠)𝑑𝑠=ξ€œπœ•πœˆβˆž0=ξ€œπ‘”(𝑠)𝑒(π‘‘βˆ’π‘ )π‘‘π‘ βˆž0πœ•π‘”(𝑠)=πœ•πœˆ(𝑒(𝑑)βˆ’π‘€(𝑑,𝑠))𝑑𝑠(1βˆ’πœ…)πœ•π‘’(π‘₯,𝑑)βˆ’ξ€œπœ•πœˆβˆž0𝑔(𝑠)πœ•π‘€(𝑑,𝑠)πœ•πœˆπ‘‘π‘ ,(2.7) System (2.5) can be written as follows: π‘’π‘‘π‘‘βˆ’πœ…πœ‡Ξ”π‘’βˆ’πœ…(πœ†+πœ‡)βˆ‡div𝑒+π›Όβˆ‡πœƒπ‘‘ξ€œβˆ’πœ‡βˆž0ξ€œπ‘”(𝑠)Δ𝑀(𝑑,𝑠)π‘‘π‘ βˆ’(πœ†+πœ‡)∞0πœƒπ‘”(𝑠)βˆ‡div𝑀(𝑑,𝑠)𝑑𝑠=0,(π‘₯,𝑑)βˆˆΞ©Γ—(0,∞),π‘‘π‘‘βˆ’Ξ”πœƒπ‘‘βˆ’Ξ”πœƒ+𝛽div𝑒𝑑𝑒=0,(π‘₯,𝑑)βˆˆΞ©Γ—(0,∞),𝑀(π‘₯,𝑑,𝑠)=𝑒(π‘₯,𝑑)βˆ’π‘’(π‘₯,π‘‘βˆ’π‘ ),(π‘₯,𝑑,𝑠)βˆˆΞ©Γ—(0,∞)Γ—(0,∞),𝑒=0,πœƒ=0,(π‘₯,𝑑)βˆˆΞ“Γ—(0,∞),(π‘₯,0)=𝑒0(π‘₯),𝑒𝑑(π‘₯,0)=𝑒1(π‘₯),πœƒ(π‘₯,0)=πœƒ0(π‘₯),πœƒπ‘‘(π‘₯,0)=πœƒ1(π‘₯),π‘₯∈Ω,𝑀(0,𝑠)=𝑀0(𝑠),(π‘₯,𝑑)βˆˆΞ©Γ—(0,∞).(2.8) We define a linear unbounded operator 𝐴 on β„‹ by 𝐴𝑒,𝑣,πœƒ,πœƒπ‘‘ξ€Έ=ξ€·,𝑀𝑣,𝐡(𝑒,𝑀)βˆ’π›Όβˆ‡πœƒπ‘‘,πœƒπ‘‘,Ξ”πœƒπ‘‘+Ξ”πœƒβˆ’π›½div𝑣,π‘£βˆ’π‘€π‘ ξ€Έ,(2.9) where 𝑀𝑠=πœ•π‘€/πœ•π‘  and ξ€œπ΅(𝑒,𝑀)=πœ…πœ‡Ξ”π‘’+πœ…(πœ†+πœ‡)βˆ‡div𝑒+πœ‡βˆž0ξ€œπ‘”(𝑠)Δ𝑀(𝑠)𝑑𝑠+(πœ†+πœ‡)∞0𝑔(𝑠)βˆ‡div𝑀(𝑠)𝑑𝑠.(2.10) Set 𝑣(π‘₯,𝑑)=𝑒𝑑(π‘₯,𝑑),𝑀(π‘₯,𝑑,𝑠)=𝑒(π‘₯,𝑑)βˆ’π‘’(π‘₯,π‘‘βˆ’π‘ ),Ξ¦=𝑒,𝑣,πœƒ,πœƒπ‘‘ξ€Έ,𝑀,𝐾=(0,𝑓,0,β„Ž,0).(2.11) Then problem (2.8) can be formulated as an abstract Cauchy problem 𝑑Φ𝑑𝑑=𝐴Φ+𝐾,(2.12) on the Hilbert space β„‹ for an initial condition Ξ¦(0)=(𝑒0,𝑒1,πœƒ0,πœƒ1,𝑀0). The domain of A is given by 𝐷(𝐴)=(𝑒,𝑣,πœƒ,𝑀)βˆˆβ„‹βˆΆπœƒβˆˆπ»10(Ξ©),πœƒπ‘‘βˆˆπ»10(Ξ©),πœƒ+πœƒπ‘‘βˆˆπ»2(Ξ©)∩𝐻10𝐻(Ξ©),π‘£βˆˆ10ξ€Έ(Ξ©)π‘›ξ€œ,πœ…π‘’+∞0𝐻𝑔(𝑠)𝑀(𝑠)π‘‘π‘ βˆˆ2(Ξ©)∩𝐻10ξ€Έ(Ξ©)𝑛,𝑀(𝑠)∈𝐻1𝐻𝑔,(0,∞),10(ξ€ΈΞ©)𝑛,,𝑀(0)=0(2.13) where 𝐻1𝐻𝑔,(0,∞),10ξ€Έ(Ξ©)𝑛=ξ€½π‘€βˆΆπ‘€,π‘€π‘ βˆˆπΏ2𝐻𝑔,(0,∞),10ξ€Έ(Ξ©)𝑛.(2.14) It is clear that 𝐷(𝐴) is dense in β„‹.

Our hypotheses on 𝑓,β„Ž can be stated as follows, which will be used in different theorems: (A1)𝑓=β„Ž=0; (A2)𝑓=𝑓(π‘₯,𝑑)∈𝐢1([0,∞),(𝐿2(Ξ©))𝑛),β„Ž=β„Ž(π‘₯,𝑑)∈𝐢1([0,∞),𝐿2(Ξ©)); (A3)𝑓(π‘₯,𝑑)∈𝐢([0,∞),(𝐻10(Ξ©))𝑛),β„Ž(π‘₯,𝑑)∈𝐢([0,∞),𝐻2(Ξ©)); (A4)𝑓(π‘₯,𝑑)∈𝐢([0,∞),(𝐿2(Ξ©))𝑛),β„Ž(π‘₯,𝑑)∈𝐢([0,∞),𝐿2(Ξ©)), and for any 𝑇>0, π‘“π‘‘βˆˆπΏ1((0,𝑇),(𝐿2(Ξ©))𝑛),β„Žπ‘‘βˆˆπΏ1((0,𝑇),𝐿2(Ξ©)).

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 Ξ¦(0)=(𝑒0,𝑒1,πœƒ0,πœƒ1,𝑀0)∈𝐷(𝐴), there exists a unique global classical solution Ξ¦=(𝑒,𝑣,πœƒ,πœƒπ‘‘,𝑀) to system (2.8) satisfying Ξ¦=(𝑒,𝑣,πœƒ,πœƒπ‘‘,𝑀)∈𝐢1([0,∞),β„‹)∩𝐢([0,∞),𝐷(𝐴)).

Theorem 2.2. Suppose that condition (A2) holds. Relaxation function 𝑔 satisfies (H1)–( H3). Then for any Ξ¦(0)=(𝑒0,𝑒1,πœƒ0,πœƒ1,𝑀0), there exists a unique global classical solution Ξ¦=(𝑒,𝑣,πœƒ,πœƒπ‘‘,𝑀) to system (2.8) satisfying Ξ¦=(𝑒,𝑣,πœƒ,πœƒπ‘‘,𝑀)∈𝐢1([0,∞),β„‹)∩𝐢([0,∞),𝐷(𝐴)), that is, π‘’βˆˆπΆ1ξ€·[𝐻0,∞),10ξ€Έ(Ξ©)𝑛[ξ€·π»βˆ©πΆ0,∞),2(Ξ©)∩𝐻10ξ€Έ(Ξ©)𝑛,π‘£βˆˆπΆ1ξ€·[𝐿0,∞),2(ξ€ΈΞ©)𝑛[ξ€·π»βˆ©πΆ0,∞),10(ξ€ΈΞ©)𝑛,πœƒβˆˆπΆ1ξ€·[0,∞),𝐻10ξ€Έξ€·[(Ξ©)∩𝐢0,∞),𝐻2(Ξ©)∩𝐻10ξ€Έ,πœƒ(Ξ©)π‘‘βˆˆπΆ1ξ€·[0,∞),𝐿2ξ€Έξ€·[(Ξ©)∩𝐢0,∞),𝐻10ξ€Έ,(Ξ©)π‘€βˆˆπΆ1ξ€·[0,∞),𝐿2𝐻𝑔,(0,∞),10ξ€Έ(Ξ©)𝑛[ξ€Έξ€Έβˆ©πΆ0,∞),𝐻1𝐻𝑔,(0,∞),10ξ€Έ(Ξ©)𝑛.ξ€Έξ€Έ(2.15)

Corollary 2.3. Suppose that condition (A3)or (A4) holds. Relaxation function 𝑔 satisfies (H1)–(H3). Then for any Ξ¦(0)=(𝑒0,𝑒1,πœƒ0,πœƒ1,𝑀0)∈D(𝐴), there exists a unique global classical solution Ξ¦=(𝑒,𝑣,πœƒ,πœƒπ‘‘,𝑀)∈𝐢1([0,∞),β„‹)∩𝐢([0,∞),𝐷(𝐴)) to system (2.8).

Corollary 2.4. If 𝑓(π‘₯,𝑑) and β„Ž(π‘₯,𝑑) are Lipschitz continuous functions from [0,𝑇] into (𝐿2(Ξ©))𝑛 and 𝐿2(Ξ©), respectively, then for any Ξ¦=(𝑒,𝑣,πœƒ,πœƒπ‘‘,𝑀)∈𝐷(𝐴), there exists a unique global classical solution Ξ¦=(𝑒,𝑣,πœƒ,πœƒπ‘‘,𝑀)∈𝐢1([0,∞),β„‹)∩𝐢([0,∞),𝐷(𝐴)) to system (2.8).

Theorem 2.5. Suppose relaxation function 𝑔 satisfies (H1)–(H3), 𝑓=𝑓(Ξ¦), and β„Ž=β„Ž(Ξ¦),Ξ¦=(𝑒,𝑣,πœƒ,πœƒπ‘‘,𝑀), and 𝐾=(0,𝑓,0,β„Ž,0) satisfies the global Lipschitz condition on β„‹; that is, there is a positive constant 𝐿 such that for all Ξ¦1,Ξ¦2βˆˆβ„‹, ‖‖𝐾Φ1ξ€Έξ€·Ξ¦βˆ’πΎ2‖‖ℋ‖‖Φ≀𝐿1βˆ’Ξ¦2β€–β€–β„‹.(2.16) Then for any Ξ¦(0)=(𝑒0,𝑒1,πœƒ0,πœƒ1,𝑀0)βˆˆβ„‹, there exists a global mild solution Ξ¦ to system (2.8) such that Φ∈𝐢([0,∞),β„‹), that is, ξ‚€[ξ‚€π»π‘’βˆˆπΆ0,∞),1Ξ“1(Ξ©)𝑛[,πœƒβˆˆπΆ0,∞),𝐻10ξ€Έ(Ξ©),πœƒπ‘‘ξ€·[∈𝐢0,∞),𝐿2ξ€Έ,ξ€·[𝐿(Ξ©)π‘£βˆˆπΆ0,∞),2ξ€Έ(Ξ©)𝑛[,π‘€βˆˆπΆ(0,∞),𝐿2𝐻𝑔,(0,∞),10ξ€Έ(Ξ©)𝑛.(2.17)

Theorem 2.6. Suppose 𝑓=𝑓(Ξ¦) and β„Ž=β„Ž(Ξ¦),Ξ¦=(𝑒,𝑣,πœƒ,πœƒπ‘‘,𝑀), and 𝐾=(0,𝑓,0,β„Ž,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 Ξ¦1,Ξ¦2∈𝐷(𝐴), ‖‖𝐾Φ1ξ€Έξ€·Ξ¦βˆ’πΎ2‖‖𝐷(𝐴)‖‖Φ≀𝐿1βˆ’Ξ¦2‖‖𝐷(𝐴).(2.18) Then for any Ξ¦(0)=(𝑒0,𝑒1,πœƒ0,πœƒ1,𝑀0)∈𝐷(𝐴), there exists a unique global classical solution Ξ¦=(𝑒,𝑣,πœƒ,πœƒπ‘‘,𝑀)∈𝐢1([0,∞),β„‹)∩𝐢([0,∞),𝐷(𝐴)) to system (2.8).

3. Some Lemmas

In this section in order to complete proofs of Theorems 2.1–2.6, we need first Lemmas 3.1–3.5. For the abstract initial value problem, 𝑑𝑒𝑑𝑑+𝐡𝑒=𝐾,𝑒(0)=𝑒0,(3.1) where 𝐡 is a maximal accretive operator defined in a dense subset 𝐷(𝐡) of a Banach space 𝐻. We have the following.

Lemma 3.1. Let 𝐡 be a linear operator defined in a Hilbert space 𝐻,𝐡∢𝐷(𝐡)βŠ‚π»β†¦π». Then the necessary and sufficient conditions for 𝐡 being maximal accretive are (i)Re(𝐡π‘₯,π‘₯)β‰₯0,forallπ‘₯∈𝐷(𝐡), (ii)𝑅(𝐼+𝐡)=𝐻.

Proof. We first prove the necessity. 𝐡 is an accretive operator, so we have (π‘₯,π‘₯)=β€–π‘₯β€–2≀‖π‘₯+πœ†π΅π‘₯β€–2=(π‘₯,π‘₯)+2πœ†Re(𝐡π‘₯,π‘₯)+πœ†2‖𝐡π‘₯β€–2.(3.2) Thus, for all πœ†>0, πœ†Re(𝐡π‘₯,π‘₯)β‰₯βˆ’2‖𝐡π‘₯β€–2.(3.3) Letting πœ†β†’0, we get (i). Furthermore, (ii) immediately follows from the fact that 𝐡 is m-accretive.
We now prove the sufficiency. It follows from (i) that for all πœ†>0, β€–π‘₯βˆ’π‘¦β€–2≀Re(π‘₯βˆ’π‘¦,π‘₯βˆ’π‘¦+πœ†π΅(π‘₯βˆ’π‘¦))≀‖π‘₯βˆ’π‘¦β€–β€–π‘₯βˆ’π‘¦+πœ†(𝐡π‘₯βˆ’π΅π‘¦)β€–.(3.4) Now it remains to prove that 𝐡 is densely defined. We use a contradiction argument. Suppose that it is not true. Then there is a nontrivial element π‘₯0 belonging to orthogonal supplement of 𝐷(𝐡) such that for all π‘₯∈𝐷(𝐡), ξ€·π‘₯,π‘₯0ξ€Έ=0.(3.5) It follows from (ii) that there is π‘₯βˆ—βˆˆπ·(𝐡) such that π‘₯βˆ—+𝐡π‘₯βˆ—=π‘₯0.(3.6) Taking the inner product of (3.5) with π‘₯βˆ—, we deduce that ξ€·π‘₯βˆ—+𝐡π‘₯βˆ—,π‘₯βˆ—ξ€Έ=0.(3.7) Taking the real part of (3.7), we deduce that π‘₯βˆ—=0, and by (3.6), π‘₯0=0, which is a contradiction. Thus the proof is complete.

Lemma 3.2. Suppose that 𝐡 is π‘š-accretive in a Banach space 𝐻, and 𝑒0∈𝐷(𝐡). Then problem (3.1) has a unique classical solution 𝑒 such that π‘’βˆˆπΆ1([[0,∞),𝐻)∩𝐢(0,∞),𝐷(𝐡)).(3.8)

Lemma 3.3. Suppose that 𝐾=𝐾(𝑑), and 𝐾(𝑑)∈𝐢1([0,∞),𝐻),𝑒0∈𝐷(𝐡).(3.9) Then problem (3.1) admits a unique global classical solution 𝑒 such that π‘’βˆˆπΆ1([[0,∞),𝐻)∩𝐢(0,∞),𝐷(𝐡))(3.10) which can be expressed as 𝑒(𝑑)=𝑆(𝑑)𝑒0+ξ€œπ‘‘0𝑆(π‘‘βˆ’πœ)𝐾(𝜏)π‘‘πœ.(3.11)

Proof. Since 𝑆(𝑑)𝑒0 satisfies the homogeneous equation and nonhomogeneous initial condition, it suffices to verify that 𝑀(𝑑) given by ξ€œπ‘€(𝑑)=𝑑0𝑆(π‘‘βˆ’πœ)𝐾(𝜏)π‘‘πœ(3.12) belongs to 𝐢1([0,∞),𝐻)∩𝐢([0,∞),𝐷(𝐡)) and satisfies the nonhomogeneous equation. Consider the following quotient of difference 𝑀(𝑑+β„Ž)βˆ’π‘€(𝑑)β„Ž=1β„Žξ‚΅ξ€œ0𝑑+β„Žπ‘†ξ€œ(𝑑+β„Žβˆ’πœ)𝐾(𝜏)π‘‘πœβˆ’π‘‘0𝑆=1(π‘‘βˆ’πœ)𝐾(𝜏)π‘‘πœβ„Žξ€œπ‘‘π‘‘+β„Ž1𝑆(𝑑+β„Žβˆ’πœ)𝐾(𝜏)π‘‘πœ+β„Žξ€œπ‘‘0=1(𝑆(𝑑+β„Žβˆ’πœ)βˆ’π‘†(π‘‘βˆ’πœ))𝐾(𝜏)π‘‘πœβ„Žξ€œπ‘‘π‘‘+β„Ž1𝑆(𝑧)𝐾(𝑑+β„Žβˆ’π‘§)𝑑𝑧+β„Žξ€œπ‘‘0𝑆(𝑧)(𝐾(𝑑+β„Žβˆ’π‘§)βˆ’πΎ(π‘‘βˆ’π‘§))𝑑𝑧.(3.13) When β„Žβ†’0, the terms in the last line of (3.13) have limits: ξ€œπ‘†(𝑑)𝐾(0)+𝑑0[𝑆(𝑧)𝐾′(π‘‘βˆ’π‘§)π‘‘π‘§βˆˆπΆ(0,∞),𝐻).(3.14) It turns out that π‘€βˆˆπΆ1([0,∞),𝐻) and the terms in the third line of (3.13) have limits too, which should be 𝑆(0)𝐾(𝑑)βˆ’π΅π‘€(𝑑)=𝐾(𝑑)βˆ’π΅π‘€(𝑑).(3.15) Thus the proof is complete.

Lemma 3.4. Suppose that 𝐾=𝐾(𝑑), and [𝐾(𝑑)∈𝐢(0,∞),𝐷(𝐡)),u0∈𝐷(𝐡).(3.16) Then problem (3.1) admits a unique global classical solution.

Proof. From the proof of Lemma 3.2, we can obtain 𝑀(𝑑+β„Ž)βˆ’π‘€(𝑑)β„Ž=1β„Žξ€œπ‘‘π‘‘+β„Žπ‘†1(𝑑+β„Žβˆ’πœ)𝐾(𝜏)π‘‘πœ+β„Žξ€œπ‘‘0=1(𝑆(𝑑+β„Žβˆ’πœ)βˆ’π‘†(π‘‘βˆ’πœ))𝐾(𝜏)π‘‘πœβ„Žξ€œπ‘‘π‘‘+β„Ž1𝑆(𝑑+β„Žβˆ’πœ)𝐾(𝜏)π‘‘πœ+β„Žξ€œπ‘‘0𝑆(π‘‘βˆ’πœ)𝑆(β„Ž)βˆ’πΌβ„Žξ‚ΆπΎ(𝜏)π‘‘πœ.(3.17) When β„Žβ†’0, the last terms in the line of (3.17) have limits: ξ€œπ‘†(0)𝐾(𝑑)βˆ’π‘‘0ξ€œπ‘†(π‘‘βˆ’πœ)𝐡𝐾(𝜏)π‘‘πœ=𝑆(0)𝐾(𝑑)βˆ’π΅π‘‘0𝑆(π‘‘βˆ’πœ)𝐾(𝜏)π‘‘πœ=𝐾(𝑑)βˆ’π΅π‘€(𝑑).(3.18) Combining the results of Lemma 3.3 proves the lemma.

Lemma 3.5. Suppose that 𝐾=𝐾(𝑑), and [𝐾(𝑑)∈𝐢(0,∞),𝐻),𝑒0∈𝐷(𝐡),(3.19) and for any 𝑇>0, πΎπ‘‘βˆˆπΏ1([]0,𝑇,𝐻).(3.20) Then problem (3.1) admits a unique global classical solution.

Proof. We first prove that for any 𝐾1∈𝐿1([0,𝑇],𝐻), the function 𝑀 given by the following integral: ξ€œπ‘€(𝑑)=𝑑0𝑆(π‘‘βˆ’πœ)𝐾1π‘‘πœ(3.21) belongs to 𝐢([0,𝑇],𝐻). Indeed, we infer from the difference π‘€ξ€œ(𝑑+β„Ž)βˆ’π‘€(𝑑)=0𝑑+β„Žπ‘†(𝑑+β„Žβˆ’πœ)𝐾1ξ€œ(𝜏)π‘‘πœβˆ’π‘‘0𝑆(π‘‘βˆ’πœ)𝐾1ξ€œ(𝜏)π‘‘πœ=(𝑆(β„Ž)βˆ’πΌ)𝑀(𝑑)+𝑑𝑑+β„Žπ‘†(𝑑+β„Žβˆ’πœ)𝐾1(𝜏)π‘‘πœ(3.22) that as β„Žβ†’0, ξ€œβ€–π‘€(𝑑+β„Ž)βˆ’π‘€(𝑑)‖≀‖(𝑆(β„Ž)βˆ’πΌ)𝑀(𝑑)β€–+𝑑𝑑+β„Žβ€–πΎ1(𝜏)β€–π‘‘πœβŸΆ0,(3.23) where we have used the strong continuity of 𝑆(𝑑) and the absolute continuity of integral for ‖𝐾1β€–βˆˆπΏ1[0,𝑑].
Now it can be seen from the last line of (3.13) that for almost every π‘‘βˆˆ[0,𝑇],𝑑𝑀/𝑑𝑑 exists, and it equals ξ€œπ‘†(𝑑)𝐾(0)+𝑑0ξ€œπ‘†(𝑧)𝐾′(π‘‘βˆ’π‘§)𝑑𝑧=𝑆(𝑑)𝐾(0)+𝑑0[]𝑆(π‘‘βˆ’πœ)𝐾′(𝜏)π‘‘πœβˆˆπΆ(0,𝑇,𝐻).(3.24) Thus, for almost every 𝑑, 𝑑𝑀𝑑𝑑=βˆ’π΅π‘€+𝐾.(3.25) Since 𝑀 and 𝐾 both belong to 𝐢([0,𝑇],𝐻), it follows from (3.25) that for almost every 𝑑, 𝐡𝑀 equals a function belonging to 𝐢([0,𝑇],𝐻). Since 𝐡 is a closed operator, we conclude that []π‘€βˆˆπΆ(0,𝑇,𝐷(𝐡))∩𝐢1([]0,𝑇,𝐻)(3.26) and (3.25) holds for every 𝑑. 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 π‘“βˆΆ[0,∞)→𝑅 is uniformly continuous and is in 𝐿1(0,∞), then limπ‘‘β†’βˆžπ‘“(𝑑)=0.(3.27)

Lemma 3.7. Suppose that the relaxation function 𝑔 satisfies (𝐻1) and (𝐻2). If π‘€βˆˆπ»1(𝑔,(0,∞),(𝐻10(Ξ©))𝑛) and 𝑀(0)=0, then π‘”ξ…žβ€–(𝑠)‖𝑀(𝑠)2(𝐻10(Ξ©))π‘›βˆˆπΏ1(0,∞),limπ‘ β†’βˆžπ‘”(𝑠)‖𝑀(𝑠)β€–2(𝐻10(Ξ©))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, 0∈𝜌(𝐴), where 𝜌(𝐴) is the resolvent of the operator 𝐴.

Proof. By a straightforward calculation, it follows from Lemma 3.7 that 𝐴𝑒,𝑣,πœƒ,πœƒπ‘‘ξ€Έ,ξ€·,𝑀𝑒,𝑣,πœƒ,πœƒπ‘‘,𝑀ℋ=πœ…(𝑣,𝑒)(𝐻10(Ξ©))𝑛+12𝐡(𝑒,𝑀)βˆ’π›Όβˆ‡πœƒπ‘‘ξ€Έ+𝛼,𝑣2π›½βˆ‡πœƒπ‘‘ξ€Έ+𝛼,βˆ‡πœƒξ€·2π›½Ξ”πœƒπ‘‘+Ξ”πœƒβˆ’div𝑣,πœƒπ‘‘ξ€Έ+ξ€·π‘£βˆ’π‘€π‘ ξ€Έ,𝑀𝐿2(𝑔,(0,∞),(𝐻10(Ξ©))𝑛)𝛼=βˆ’β€–β€–2π›½βˆ‡πœƒπ‘‘β€–β€–2+ξ€œβˆž0π‘”ξ…žβ€–(𝑠)𝑀(𝑠)β€–2(𝐻10(Ξ©))𝑛𝑑𝑠≀0.(3.29) Thus, 𝐴 is dissipative.
To prove that 0∈𝜌(𝐴), for any 𝐺=(𝑔1,𝑔2,𝑔3,𝑔4,𝑔5)βˆˆβ„‹, consider 𝐴Φ=𝐺,(3.30) that is, 𝑣=𝑔1𝐻,in10ξ€Έ(Ξ©)𝑛,(3.31)𝐡(𝑒,𝑀)βˆ’π›Όβˆ‡πœƒπ‘‘=𝑔2𝐿,in2(ξ€ΈΞ©)π‘›πœƒ,(3.32)𝑑=𝑔3,in𝐿2(Ξ©),(3.33)Ξ”πœƒπ‘‘+Ξ”πœƒβˆ’π›½div𝑣=𝑔4,in𝐿2(Ξ©),(3.34)π‘£βˆ’π‘€π‘ =𝑔5,in𝐿2𝐻𝑔,(0,∞),10ξ€Έ(Ξ©)𝑛.(3.35) Inserting 𝑣=𝑔1 and πœƒπ‘‘=𝑔3 obtained from (3.31), (3.33) into (3.34), we obtain Ξ”πœƒ=𝑔4+𝛽div𝑔1βˆ’Ξ”π‘”3∈𝐿2(Ξ©).(3.36) By the standard theory for the linear elliptic equations, we have a unique πœƒβˆˆπ»2(Ξ©)∩𝐻10(Ξ©) satisfying (3.36).
We plug 𝑣=𝑔1 obtained from (3.31) into (3.35) to get 𝑀𝑠=𝑔1βˆ’π‘”5∈𝐿2𝐻𝑔,(0,∞),1Ξ“1(Ξ©)𝑛.(3.37) Applying the standard theory for the linear elliptic equations again, we have a unique π‘€βˆˆπ»1(𝑔,(0,∞),(𝐻10(Ξ©))𝑛) satisfying (3.37). Then plugging πœƒ and 𝑀 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 π‘’βˆˆπ·(𝐴) for (3.32) and such that βˆ«πœ…π‘’+∞0𝑔(𝑠)𝑀(𝑠)π‘‘π‘ βˆˆ(𝐻2(Ξ©)∩𝐻10(Ξ©))𝑛. Thus the unique solvability of (3.30) follows. It is clear from the regularity theory for the linear elliptic equations that ‖Φ‖ℋ≀𝐾‖𝐺‖ℋ with 𝐾 being a positive constant independent of Ξ¦. 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 (𝑒𝑛,𝑣𝑛,πœƒπ‘›,πœƒπ‘›π‘‘,𝑀𝑛)∈𝐷(𝐴) be such that 𝑒𝑛,𝑣𝑛,πœƒπ‘›,πœƒπ‘›π‘‘,π‘€π‘›ξ€ΈβŸΆξ€·π‘’,𝑣,πœƒ,πœƒπ‘‘ξ€Έπ΄ξ€·π‘’,𝑀inβ„‹,𝑛,𝑣𝑛,πœƒπ‘›,πœƒπ‘‘π‘›,π‘€π‘›ξ€ΈβŸΆ(π‘Ž,𝑏,𝑐,𝑑,𝑒)inβ„‹.(3.38) Then we have π‘’π‘›ξ€·π»βŸΆπ‘’in10ξ€Έ(Ξ©)𝑛,𝑣(3.39)π‘›ξ€·πΏβŸΆπ‘£in2ξ€Έ(Ξ©)𝑛,πœƒ(3.40)π‘›βŸΆπœƒin𝐻10πœƒ(Ξ©),(3.41)π‘›π‘‘βŸΆπœƒπ‘‘in𝐿2𝑀(Ξ©),(3.42)π‘›βŸΆπ‘€in𝐿2𝐻𝑔,(0,∞),10ξ€Έ(Ξ©)𝑛𝑣,(3.43)π‘›ξ€·π»βŸΆπ‘Žin10ξ€Έ(Ξ©)𝑛𝐡𝑒,(3.44)𝑛,π‘€π‘›ξ€Έβˆ’π›Όβˆ‡πœƒπ‘›π‘‘ξ€·πΏβŸΆπ‘in2ξ€Έ(Ξ©)π‘›πœƒ,(3.45)π‘›π‘‘βŸΆπ‘in𝐻10(Ξ©),(3.46)Ξ”πœƒπ‘›π‘‘+Ξ”πœƒπ‘›βˆ’π›½divπ‘£π‘›βŸΆπ‘‘in𝐿2𝑣(Ξ©),(3.47)π‘›βˆ’π‘€π‘›π‘ βŸΆπ‘’in𝐿2𝐻𝑔,(0,∞),10ξ€Έ(Ξ©)𝑛.(3.48) By (3.40) and (3.44), we deduce π‘£π‘›ξ€·π»βŸΆπ‘£in10ξ€Έ(Ξ©)𝑛,𝐻(3.49)𝑣=π‘Žβˆˆ10ξ€Έ(Ξ©)𝑛.(3.50) By (3.42) and (3.46), we deduce πœƒπ‘›π‘‘βŸΆπœƒπ‘‘in𝐻10πœƒ(Ξ©),(3.51)𝑑=π‘βˆˆπ»10(Ξ©).(3.52) By (3.47) and (3.49), we deduce Ξ”πœƒπ‘›π‘‘+Ξ”πœƒπ‘›βŸΆπ‘‘+𝛽div𝑣in𝐿2(Ξ©),(3.53) and consequently, it follows from (3.41), that πœƒπ‘›π‘‘+πœƒπ‘›βŸΆπœƒπ‘‘+πœƒin𝐻2(Ξ©)∩𝐻10(Ξ©),(3.54) since Ξ” is an isomorphism from 𝐻2(Ξ©)∩𝐻10(Ξ©) onto 𝐿2(Ξ©). It therefore follows from (3.47) and (3.54) that 𝑑=Ξ”πœƒπ‘›π‘‘+Ξ”πœƒπ‘›βˆ’π›½div𝑣,πœƒπ‘‘+πœƒβˆˆπ»2(Ξ©)∩𝐻10(Ξ©).(3.55) By (3.43), (3.48), and (3.49), we deduce π‘€π‘›βŸΆπ‘€in𝐻1𝐻𝑔,(0,∞),1Ξ“1(Ξ©)𝑛,(3.56)𝑒=π‘£βˆ’π‘€π‘ ,π‘€βˆˆπ»1𝐻𝑔,(0,∞),10ξ€Έ(Ξ©)𝑛,𝑀(0)=0.(3.57) In addition, it follows from (3.39), (3.43), (3.51) that 𝐡𝑒𝑛,π‘€π‘›ξ€Έβˆ’π›Όβˆ‡πœƒπ‘›π‘‘βŸΆπ΅(𝑒,𝑀)βˆ’π›Όβˆ‡πœƒπ‘‘(3.58) in the distribution. It therefore follows from (3.45) and (3.58) that 𝑏=𝐡(𝑒,𝑀)βˆ’π›Όβˆ‡πœƒπ‘‘ξ€·πΏ,𝐡(𝑒,𝑀)∈2ξ€Έ(Ξ©)𝑛,(3.59) and consequently, ξ€œπœ…π‘’+∞0𝐻𝑔(𝑠)𝑀(𝑠)π‘‘π‘ βˆˆ2(Ξ©)∩𝐻10ξ€Έ(Ξ©)𝑛,(3.60) since πœ‡Ξ”+(πœ†+πœ‡)βˆ‡div is an isomorphism from 𝐻2(Ξ©)∩𝐻10(Ξ©) onto 𝐿2(Ξ©). Thus, by (3.50), (3.52), (3.55), (3.57), (3.59), (3.60), we deduce 𝐴𝑒,𝑣,πœƒ,πœƒπ‘‘ξ€Έ=ξ€·,𝑀(π‘Ž,𝑏,𝑐,𝑑,𝑒),𝑒,𝑣,πœƒ,πœƒπ‘‘ξ€Έ,π‘€βˆˆπ·(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 0∈𝜌(𝐴), the resolvent set of 𝐴, then 𝐴 is the infinitesimal generator of a 𝐢0-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 𝐢0-semigroup of contractions on 𝐻 if and only if 𝐴 is dissipative and 𝑅(πΌβˆ’π΄)=𝐻.

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

4. Proofs of Theorems 2.1–2.5

Proof of Theorem 2.1. By (2.2), it is clear that β„‹ is a Hilbert space. By Lemmas 3.8–3.10, we can deduce that the operator 𝐴 is the infinitesimal generator of a 𝐢0-semigroup of contractions on Hilbert space β„‹. 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 𝐢0-semigroup of contractions on Hilbert space β„‹. Applying the result and Lemma 3.11, we can conclude that 𝑅(πΌβˆ’π΄)=𝐻. If we choose operator 𝐡=βˆ’π΄, we can obtain 𝐷(𝐡)=𝐷(𝐴) and 𝐷(𝐡) is dense in β„‹. Noting that by (𝐴2), we know that 𝐾=(0,𝑓,0,β„Ž,0)∈𝐢1([0,∞),β„‹); therefore, applying Lemma 3.1, we can conclude the operator 𝐡 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,β„Ž,0)∈𝐢([0,∞),𝐷(𝐴)) or 𝐾∈𝐢([0,∞),β„‹), and for any 𝑇>0,πΎπ‘‘βˆˆπΏ1((0,𝑇),β„‹). Noting that 𝐡=βˆ’π΄ 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(π‘₯,𝑑)=(0,𝑓,0,β„Ž,0) are Lipschitz continuous functions from [0,𝑇] into β„‹. Moreover, by (2.2), it is clear that β„‹ is a reflexive Banach space. Therefore, πΎπ‘‘βˆˆπΏ1([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 𝐡=βˆ’π΄ is the maximal accretive operator of a 𝐢0 semigroup 𝑆(𝑑). On the other hand, 𝐾=(0,𝑓,0,β„Ž,0) satisfies the global Lipschitz condition on β„‹. 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 πœ™(Ξ¦)=𝑆(𝑑)Ξ¦0+ξ€œπ‘‘0ξ‚»[𝑆(π‘‘βˆ’πœ)𝐾(Ξ¦(𝜏))π‘‘πœ,(4.1)Ξ©=Φ∈𝐢(0,+∞),𝐻)∣sup𝑑β‰₯0ξ€·β€–Ξ¦(𝑑)β€–π‘’βˆ’π‘˜π‘‘ξ€Έξ‚Ό<∞,(4.2) where π‘˜ is a positive constant such that π‘˜>𝐿. In Ξ©, we introduce the following norm: β€–Ξ¦β€–Ξ©=sup𝑑β‰₯0ξ€·β€–Ξ¦(𝑑)β€–π‘’βˆ’π‘˜π‘‘ξ€Έ.(4.3) Clearly, Ξ© is a Banach space. We now show that the nonlinear operator πœ™ defined by (4.1) maps Ξ© into itself, and the mapping is a contraction. Indeed, for Φ∈Ω, we have β€–πœ™(Ξ¦)‖≀‖𝑆(𝑑)Ξ¦0ξ€œβ€–+𝑑0‖𝑆(π‘‘βˆ’πœ)‖‖𝐾(Ξ¦)β€–π‘‘πœβ‰€β€–Ξ¦0ξ€œβ€–+𝑑0‖𝐾(Ξ¦)β€–π‘‘πœβ‰€β€–Ξ¦0ξ€œβ€–+𝑑0(𝐿‖Φ(𝜏)β€–+‖𝐾(0)β€–)π‘‘πœβ‰€β€–Ξ¦0β€–+𝐢0𝑑+𝐿sup𝑑β‰₯0β€–Ξ¦(𝑑)β€–π‘’βˆ’π‘˜π‘‘ξ€œπ‘‘0π‘’π‘˜πœπ‘‘πœβ‰€β€–Ξ¦0β€–+𝐢0𝐿𝑑+π‘˜π‘’π‘˜π‘‘β€–Ξ¦β€–Ξ©,(4.4) where 𝐢0=‖𝐾(0)β€–. Thus, β€–β€–πœ™(Ξ¦)Ω≀sup𝑑β‰₯0ξ€Ίξ€·β€–Ξ¦0β€–+𝐢0π‘‘ξ€Έπ‘’βˆ’π‘˜π‘‘ξ€»+πΏπ‘˜β€–Ξ¦β€–Ξ©<∞.(4.5) that is, πœ™(Ξ¦)∈Ω.
For Ξ¦1,Ξ¦2∈Ω, we have β€–β€–πœ™ξ€·Ξ¦1ξ€Έξ€·Ξ¦βˆ’πœ™2ξ€Έβ€–β€–Ξ©=sup𝑑β‰₯0π‘’βˆ’π‘˜π‘‘β€–β€–β€–ξ€œπ‘‘0𝐾Φ𝑆(π‘‘βˆ’πœ)1(ξ€Έξ€·Ξ¦πœ)βˆ’πΎ2(β€–β€–β€–πœ)ξ€Έξ€Έπ‘‘πœβ‰€sup𝑑β‰₯0π‘’βˆ’π‘˜π‘‘πΏξ€œπ‘‘0β€–Ξ¦1βˆ’Ξ¦2β€–π‘‘πœβ‰€sup𝑑β‰₯0ξ‚€π‘’βˆ’π‘˜π‘‘β‹…πΏπ‘˜β‹…ξ€·π‘’π‘˜π‘‘ξ€Έξ‚β€–β€–Ξ¦βˆ’11βˆ’Ξ¦2β€–β€–Ξ©β‰€πΏπ‘˜β€–Ξ¦1βˆ’Ξ¦2β€–Ξ©.(4.6) Therefore, by the contraction mapping theorem, the problem has a unique solution in Ξ©. To show that the uniqueness also holds in 𝐢([0,∞),𝐻), let Ξ¦1,Ξ¦2∈𝐢([0,∞),𝐻) be two solutions of the problem and let Ξ¦=Ξ¦1βˆ’Ξ¦2. Then ξ€œΞ¦(𝑑)=𝑑0𝐾Φ𝑆(π‘‘βˆ’πœ)1ξ€Έξ€·Ξ¦βˆ’πΎ2ξ€œξ€Έξ€Έπ‘‘πœ,β€–Ξ¦(𝑑)‖≀𝐿𝑑0β€–Ξ¦(𝜏)β€–π‘‘πœ.(4.7) By the Gronwall inequality, we immediately conclude that Ξ¦(𝑑)=0; that is, the uniqueness in 𝐢([0,∞),𝐻) follows. Thus the proof is complete.

Proof of Theorem 2.6. Since 𝐡 is the maximal accretive operator, 𝐾=(0,𝑓,0,β„Ž,0) satisfies the global Lipschitz condition on 𝐷(𝐴). Let 𝐴1=𝐷(𝐡),𝐡1=𝐡2ξ€·π΅βˆΆπ·1𝐡=𝐷2ξ€ΈβŸΌπ΄1.(4.8) Then 𝐴1 is a Banach space, and 𝐡1=𝐡2 is a densely defined operator from 𝐷(𝐡2) into 𝐴1. In what follows we prove that 𝐡1 is π‘š-accretive in 𝐴1=𝐷(B).
Indeed, for any π‘₯,π‘¦βˆˆπ·(𝐡2), since 𝐡 is accretive in 𝐻, we have β€–β€–π‘₯βˆ’π‘¦+πœ†(𝐡π‘₯βˆ’π΅π‘¦)𝐷(𝐡)=ξ‚€β€–β€–π‘₯βˆ’π‘¦+πœ†(𝐡π‘₯βˆ’π΅π‘¦)2+‖‖𝐡𝐡π‘₯βˆ’π΅π‘¦+πœ†2π‘₯βˆ’π΅2𝑦‖‖21/2β‰₯ξ€·β€–π‘₯βˆ’π‘¦β€–2+‖𝐡π‘₯βˆ’π΅π‘¦β€–2ξ€Έ1/2=β€–π‘₯βˆ’π‘¦β€–π·(𝐡).(4.9) that is, 𝐡1 is accretive in 𝐴1. Furthermore, since 𝐡 is π‘š-accretive in 𝐻, for any π‘¦βˆˆπ», there is a unique π‘₯∈𝐷(𝐡) such that π‘₯+𝐡π‘₯=𝑦.(4.10) Now for any π‘¦βˆˆπ΄1=𝐷(𝐡), (4.10) admits a unique solution π‘₯∈𝐷(𝐡). It turns out that 𝐡π‘₯=π‘¦βˆ’π‘₯∈𝐷(𝐡).(4.11) Thus π‘₯∈𝐷(𝐡2); that is, 𝐡1 is m-accretive in 𝐴1. Let 𝑆1(𝑑) be the semigroup generated by 𝐡1. If Ξ¦0∈𝐷(𝐡2)=𝐷(𝐡1), then Ξ¦(𝑑)=𝑆1(𝑑)Ξ¦0ξ€·[ξ€·π΅βˆˆπΆ0,+∞),𝐷2ξ€Έξ€Έβˆ©πΆ1([0,+∞),𝐷(𝐡))(4.12) is unique classical solution of the problem. On the other hand, Ξ¦(𝑑)=𝑆1(𝑑)Ξ¦0 is also a classical solution in [𝐢(0,+∞),𝐷(𝐡))∩𝐢1([0,+∞),𝐻).(4.13) This implies that 𝑆1(𝑑) is a restriction of 𝑆(𝑑) on 𝐴1. By virtue of the proof of Theorem 2.5, there exists a unique mild solution Φ∈𝐢([0,+∞),𝐴1). Since 𝑆1(𝑑) is a restriction of 𝑆(𝑑) on 𝐷(𝐡), and moreover, we infer from 𝐾(Ξ¦) being an operator from 𝐷(𝐡) to 𝐷(𝐡) and Lemma 3.4 that Ξ¦ 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, β€œOn 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, β€œAsymptotic 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, β€œAn 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, β€œGlobal 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, β€œEnergy decay rates in linear thermoelasticity,” Funkcialaj Ekvacioj, vol. 35, no. 1, pp. 19–30, 1992. View at: Google Scholar | Zentralblatt MATH
  6. Y. Shibata, β€œNeumann 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, β€œGlobal 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, β€œGlobal 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, β€œOn 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, β€œExponential 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, β€œDecay 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, β€œExponential 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, β€œSemigroup 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, β€œAn 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, β€œOn the type of C0-semigroup associated with the abstract linear viscoelastic system,” Zeitschrift fΓΌr Angewandte Mathematik und Physik, vol. 47, no. 1, pp. 1–15, 1996. View at: Publisher Site | Google Scholar
  16. W. Liu, β€œThe exponential stabilization of the higher-dimensional linear system of thermoviscoelasticity,” Journal de MathΓ©matiques Pures et AppliquΓ©es, 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 Β© 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.


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