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: Stanisล‚aw Migรณrski
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: ๐‘ข๐‘ก๐‘กโˆ’๐œ‡ฮ”๐‘ขโˆ’(๐œ†+๐œ‡)โˆ‡div๐‘ข+๐œ‡๐‘”โˆ—ฮ”๐‘ข+(๐œ†+๐œ‡)๐‘”โˆ—โˆ‡div๐‘ข+๐›ผโˆ‡๐œƒ๐‘ก๐œƒ=๐‘“,(๐‘ฅ,๐‘ก)โˆˆฮฉร—(0,โˆž),๐‘ก๐‘กโˆ’ฮ”๐œƒ๐‘กโˆ’ฮ”๐œƒ+๐›ฝdiv๐‘ข๐‘ก=โ„Ž,(๐‘ฅ,๐‘ก)โˆˆฮฉร—(0,โˆž),(1.1) where the sign โ€œโˆ—โ€ denotes the convolution product in time, which is defined by ๎€œ๐‘”โˆ—๐‘ฃ(๐‘ก)=๐‘กโˆ’โˆž๐‘”(๐‘กโˆ’๐‘ )๐‘ฃ(๐‘ฅ,๐‘ )๐‘‘๐‘ (1.2) with the initial data ๐‘ข(๐‘ฅ,0)=๐‘ข0(๐‘ฅ),๐‘ข๐‘ก(๐‘ฅ,0)=๐‘ข1(๐‘ฅ),๐œƒ(๐‘ฅ,0)=๐œƒ0๐œƒ(๐‘ฅ),๐‘ฅโˆˆฮฉ,๐‘ก(๐‘ฅ,0)=๐œƒ1(๐‘ฅ),๐‘ข(๐‘ฅ,0)โˆ’๐‘ข(๐‘ฅ,โˆ’๐‘ )=๐‘ค0(๐‘ฅ,๐‘ ),(๐‘ฅ,๐‘ )โˆˆฮฉร—(0,โˆž)(1.3) and boundary condition ๐‘ข=0,๐œƒ=0,(๐‘ฅ,๐‘ก)โˆˆฮ“ร—(0,โˆž).(1.4) The body ฮฉ is a bounded domain in ๐‘…๐‘› with smooth boundary ฮ“=๐œ•ฮฉ (say ๐ถ2) and is assumed to be linear, homogeneous, and isotropic. ๐‘ข(๐‘ฅ,๐‘ก)=(๐‘ข1(๐‘ฅ,๐‘ก),๐‘ข2(๐‘ฅ,๐‘ก),โ€ฆ,๐‘ข๐‘›(๐‘ฅ,๐‘ก)), and ๐œƒ(๐‘ฅ,๐‘ก) represent displacement vector and temperature derivations, respectively, from the natural state of the reference configuration at position ๐‘ฅ and time ๐‘ก. ๐œ†,๐œ‡>0 are Lamรฉ's constants and ๐›ผ,๐›ฝ>0 the coupling parameters; ๐‘”(๐‘ก) denotes the relaxation function, ๐‘ค0(๐‘ฅ,๐‘ ) is a specified โ€œhistory,โ€ and ๐‘ข0(๐‘ฅ),๐‘ข1(๐‘ฅ),๐œƒ0(๐‘ฅ) are initial data. ฮ”,โˆ‡,div denote the Laplace, gradient, and divergence operators in the space variables, respectively.

We refer to the work by Dafermos [1โ€“3]. The following basic conditions on the relaxation function ๐‘”(๐‘ก) are (H1)๐‘”โˆˆ๐ถ1[0,โˆž)โˆฉ๐ฟ1(0,โˆž); (H2)๐‘”(๐‘ก)โ‰ฅ0,๐‘”๎…ž(๐‘ก)โ‰ค0,๐‘ก>0; (H3)โˆซ๐œ…=1โˆ’โˆž0๐‘”(๐‘ก)๐‘‘๐‘ก>0.

In what follows, we denote by โ€–โ‹…โ€– the norm of ๐ฟ2(ฮฉ), and we use the notation โ€–๐‘ฃโ€–2=๐‘›๎“๐‘–=1โ€–โ€–๐‘ฃ๐‘–โ€–โ€–2๎€ท๐‘ฃ,for๐‘ฃ=1,๐‘ฃ2,โ€ฆ,๐‘ฃ๐‘›๎€ธ.(1.5)

When ๐‘“=๐‘”=โ„Ž=0, system (1.1)โ€“(1.4) is reduced to the thermoelastic system: ๐‘ข๐‘ก๐‘กโˆ’๐œ‡ฮ”๐‘ขโˆ’(๐œ†+๐œ‡)โˆ‡div๐‘ข+๐›ผโˆ‡๐œƒ๐‘ก๐œƒ=0,(๐‘ฅ,๐‘ก)โˆˆฮฉร—(0,โˆž),๐‘ก๐‘กโˆ’ฮ”๐œƒ๐‘กโˆ’ฮ”๐œƒ+๐›ฝdiv๐‘ข๐‘ก=0,(๐‘ฅ,๐‘ก)โˆˆฮฉร—(0,โˆž),๐‘ข=0,๐œƒ=0,(๐‘ฅ,๐‘ก)โˆˆฮ“ร—(0,โˆž),๐‘ข(๐‘ฅ,0)=๐‘ข0(๐‘ฅ),๐‘ข๐‘ก(๐‘ฅ,0)=๐‘ข1(๐‘ฅ),๐œƒ(๐‘ฅ,0)=๐œƒ0(๐‘ฅ),๐‘ฅโˆˆฮฉ.(1.6) In the one-dimensional space case, there are many works (see e.g., [4โ€“8]) on the global existence and uniqueness. Liu and Zheng [9] succeeded in deriving in energy decay under the boundary condition (1.4) or๐‘ขโˆฃ๐‘ฅ=0=0,๐œŽโˆฃ๐‘ฅ=๐‘™=0,๐œƒ๐‘ฅโˆฃ๐‘ฅ=0,๐‘™=0,(1.7)๐‘ขโˆฃ๐‘ฅ=0=0,๐œŽโˆฃ๐‘ฅ=๐‘™=0,๐œƒโˆฃ๐‘ฅ=0=0,๐œƒ๐‘ฅโˆฃ๐‘ฅ=๐‘™=0,(1.8) or๐‘ขโˆฃ๐‘ฅ=0=0,๐œŽโˆฃ๐‘ฅ=๐‘™=0,๐œƒโˆฃ๐‘ฅ=0,๐‘™=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:๐‘ขโˆฃ๐‘ฅ=0=0,๐œŽโˆฃ๐‘ฅ=๐‘™=0,๐œƒ๐‘ฅโˆฃ๐‘ฅ=0=0,๐œƒโˆฃ๐‘ฅ=๐‘™=0,(1.10) where ๐œŽ=๐‘ข๐‘ฅโˆ’๐›พ๐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 ๐›ผ=๐›ฝ=0, ๐‘“=โ„Ž=0, system (1.1)-(1.4) is decoupled into the following viscoelastic system: ๐‘ข๐‘ก๐‘กโˆ’๐œ‡ฮ”๐‘ขโˆ’(๐œ†+๐œ‡)โˆ‡div๐‘ข+๐œ‡๐‘”โˆ—ฮ”๐‘ข+(๐œ†+๐œ‡)๐‘”โˆ—โˆ‡div๐‘ข=0,(๐‘ฅ,๐‘ก)โˆˆฮฉร—(0,โˆž),๐‘ข=0,(๐‘ฅ,๐‘ก)โˆˆฮ“ร—(0,โˆž),๐‘ข(๐‘ฅ,0)=๐‘ข0(๐‘ฅ),๐‘ข๐‘ก(๐‘ฅ,0)=๐‘ข1(๐‘ฅ),(๐‘ฅ,๐‘ก)โˆˆฮฉ,๐‘ข(๐‘ฅ,0)โˆ’๐‘ข(๐‘ฅ,โˆ’๐‘ )=๐‘ค0(๐‘ฅ),(๐‘ฅ,๐‘ก)โˆˆฮฉร—(0,โˆž),(1.11) and the wave equation.

There are many works (see, e.g., [9, 12โ€“15]) on exponential stability of energy and asymptotic stability of solution under different assumptions. The notation in this paper will be as follows. ๐ฟ๐‘,1โ‰ค๐‘โ‰ค+โˆž,๐‘Š๐‘š,๐‘,๐‘šโˆˆ๐‘,๐ป1=๐‘Š1,2,๐ป10=๐‘Š01,2 denote the usual (Sobolev) spaces on ฮฉ. In addition, โ€–โ‹…โ€–๐ต denotes the norm in the space ๐ต; we also put โ€–โ‹…โ€–=โ€–โ‹…โ€–๐ฟ2(ฮฉ). We denote by ๐ถ๐‘˜(๐ผ,๐ต),๐‘˜โˆˆโ„•0, the space of k-times continuously differentiable functions from ๐ฝโІ๐ผ into a Banach space ๐ต, and likewise by ๐ฟ๐‘(๐ผ,๐ต),1โ‰ค๐‘โ‰ค+โˆž, the corresponding Lebesgue spaces. ๐ถ๐›ฝ([0,๐‘‡],๐ต) denotes the Hรถlder space of ๐ต-valued continuous functions with exponent ๐›ฝโˆˆ(0,1] in variable ๐‘ก.

2. Main Results

Let the โ€œhistory spaceโ€ ๐ฟ2(๐‘”,(0,โˆž),(๐ป10(ฮฉ))๐‘›) consist of ((๐ป10(ฮฉ))๐‘›)-valued functions ๐‘ค on (0,โˆž) for which โ€–๐‘คโ€–2๐ฟ2๎€ท๐ป(๐‘”,(0,โˆž),10๎€ธ(ฮฉ)๐‘›)=๎€œโˆž0๐‘”(๐‘ )โ€–๐‘ค(๐‘ )โ€–2๎€ท๐ป10๎€ธ(ฮฉ)๐‘›๐‘‘๐‘ <โˆž.(2.1) Put ๎€ท๐ปโ„‹=10๎€ธ(ฮฉ)๐‘›ร—๎€ท๐ฟ2๎€ธ(ฮฉ)๐‘›ร—๐ป10(ฮฉ)ร—๐ฟ2(ฮฉ)ร—๐ฟ2๎€ท๎€ท๐ป๐‘”,(0,โˆž),10๎€ธ(ฮฉ)๐‘›๎€ธ(2.2) with the energy norm โ€–โ€–๎€ท๐‘ข,๐‘ฃ,๐œƒ,๐œƒ๐‘ก๎€ธโ€–โ€–,๐‘คโ„‹=๎‚ป๐œ…โ€–๐‘ขโ€–2๎€ท๐ป10๎€ธ(ฮฉ)๐‘›+12๎‚ตโ€–๐‘ฃโ€–2+๐›ผ๐›ฝโ€–โ€–๐œƒ๐‘กโ€–โ€–2๎‚ถ+๎€œ+โ€–โˆ‡๐œƒโ€–โˆž0๐‘”(๐‘ )โ€–๐‘ค(๐‘ )โ€–2๎€ท๐ป10๎€ธ(ฮฉ)๐‘›๎‚ผ๐‘‘๐‘ 1/2,(2.3) where ๐œ… denotes the positive constant in (๐ป3), that is, ๎€œ๐œ…=1โˆ’โˆž0๐‘”(๐‘ก)๐‘‘๐‘ก>0.(2.4) Thus we consider the following thermoviscoelastic system:๐‘ข๐‘ก๐‘กโˆ’๐œ‡ฮ”๐‘ขโˆ’(๐œ†+๐œ‡)โˆ‡div๐‘ข+๐œ‡๐‘”โˆ—ฮ”๐‘ข+(๐œ†+๐œ‡)๐‘”โˆ—โˆ‡div๐‘ข+๐›ผโˆ‡๐œƒ๐‘ก๐œƒ=0,(๐‘ฅ,๐‘ก)โˆˆฮฉร—(0,โˆž),๐‘ก๐‘กโˆ’ฮ”๐œƒ๐‘กโˆ’ฮ”๐œƒ+๐›ฝdiv๐‘ข๐‘ก=0,(๐‘ฅ,๐‘ก)โˆˆฮฉร—(0,โˆž),๐‘ข=0,๐œƒ=0,(๐‘ฅ,๐‘ก)โˆˆฮ“ร—(0,โˆž),๐‘ข(๐‘ฅ,0)=๐‘ข0(๐‘ฅ),๐‘ข๐‘ก(๐‘ฅ,0)=๐‘ข1(๐‘ฅ),๐œƒ(๐‘ฅ,0)=๐œƒ0(๐‘ฅ),๐œƒ๐‘ก(๐‘ฅ,0)=๐œƒ1(๐‘ฅ),๐‘ฅโˆˆฮฉ,๐‘ข(๐‘ฅ,0)โˆ’๐‘ข(๐‘ฅ,โˆ’๐‘ )=๐‘ค0(๐‘ฅ,๐‘ ),(๐‘ฅ,๐‘ก)โˆˆฮฉร—(0,โˆž).(2.5) Let ๐‘ฃ(๐‘ฅ,๐‘ก)=๐‘ข๐‘ก(๐‘ฅ,๐‘ก),๐‘ค(๐‘ฅ,๐‘ก,๐‘ )=๐‘ข(๐‘ฅ,๐‘ก)โˆ’๐‘ข(๐‘ฅ,๐‘กโˆ’๐‘ ).(2.6) Since ๐œ•๎€œ๐œ•๐œˆ๐‘กโˆ’โˆž๐œ•๐‘”(๐‘กโˆ’๐‘ )๐‘ข(๐‘ )๐‘‘๐‘ =๎€œ๐œ•๐œˆโˆž0=๎€œ๐‘”(๐‘ )๐‘ข(๐‘กโˆ’๐‘ )๐‘‘๐‘ โˆž0๐œ•๐‘”(๐‘ )=๐œ•๐œˆ(๐‘ข(๐‘ก)โˆ’๐‘ค(๐‘ก,๐‘ ))๐‘‘๐‘ (1โˆ’๐œ…)๐œ•๐‘ข(๐‘ฅ,๐‘ก)โˆ’๎€œ๐œ•๐œˆโˆž0๐‘”(๐‘ )๐œ•๐‘ค(๐‘ก,๐‘ )๐œ•๐œˆ๐‘‘๐‘ ,(2.7) System (2.5) can be written as follows: ๐‘ข๐‘ก๐‘กโˆ’๐œ…๐œ‡ฮ”๐‘ขโˆ’๐œ…(๐œ†+๐œ‡)โˆ‡div๐‘ข+๐›ผโˆ‡๐œƒ๐‘ก๎€œโˆ’๐œ‡โˆž0๎€œ๐‘”(๐‘ )ฮ”๐‘ค(๐‘ก,๐‘ )๐‘‘๐‘ โˆ’(๐œ†+๐œ‡)โˆž0๐œƒ๐‘”(๐‘ )โˆ‡div๐‘ค(๐‘ก,๐‘ )๐‘‘๐‘ =0,(๐‘ฅ,๐‘ก)โˆˆฮฉร—(0,โˆž),๐‘ก๐‘กโˆ’ฮ”๐œƒ๐‘กโˆ’ฮ”๐œƒ+๐›ฝdiv๐‘ข๐‘ก๐‘ข=0,(๐‘ฅ,๐‘ก)โˆˆฮฉร—(0,โˆž),๐‘ค(๐‘ฅ,๐‘ก,๐‘ )=๐‘ข(๐‘ฅ,๐‘ก)โˆ’๐‘ข(๐‘ฅ,๐‘กโˆ’๐‘ ),(๐‘ฅ,๐‘ก,๐‘ )โˆˆฮฉร—(0,โˆž)ร—(0,โˆž),๐‘ข=0,๐œƒ=0,(๐‘ฅ,๐‘ก)โˆˆฮ“ร—(0,โˆž),(๐‘ฅ,0)=๐‘ข0(๐‘ฅ),๐‘ข๐‘ก(๐‘ฅ,0)=๐‘ข1(๐‘ฅ),๐œƒ(๐‘ฅ,0)=๐œƒ0(๐‘ฅ),๐œƒ๐‘ก(๐‘ฅ,0)=๐œƒ1(๐‘ฅ),๐‘ฅโˆˆฮฉ,๐‘ค(0,๐‘ )=๐‘ค0(๐‘ ),(๐‘ฅ,๐‘ก)โˆˆฮฉร—(0,โˆž).(2.8) We define a linear unbounded operator ๐ด on โ„‹ by ๐ด๎€ท๐‘ข,๐‘ฃ,๐œƒ,๐œƒ๐‘ก๎€ธ=๎€ท,๐‘ค๐‘ฃ,๐ต(๐‘ข,๐‘ค)โˆ’๐›ผโˆ‡๐œƒ๐‘ก,๐œƒ๐‘ก,ฮ”๐œƒ๐‘ก+ฮ”๐œƒโˆ’๐›ฝdiv๐‘ฃ,๐‘ฃโˆ’๐‘ค๐‘ ๎€ธ,(2.9) where ๐‘ค๐‘ =๐œ•๐‘ค/๐œ•๐‘  and ๎€œ๐ต(๐‘ข,๐‘ค)=๐œ…๐œ‡ฮ”๐‘ข+๐œ…(๐œ†+๐œ‡)โˆ‡div๐‘ข+๐œ‡โˆž0๎€œ๐‘”(๐‘ )ฮ”๐‘ค(๐‘ )๐‘‘๐‘ +(๐œ†+๐œ‡)โˆž0๐‘”(๐‘ )โˆ‡div๐‘ค(๐‘ )๐‘‘๐‘ .(2.10) Set ๐‘ฃ(๐‘ฅ,๐‘ก)=๐‘ข๐‘ก๎€ท(๐‘ฅ,๐‘ก),๐‘ค(๐‘ฅ,๐‘ก,๐‘ )=๐‘ข(๐‘ฅ,๐‘ก)โˆ’๐‘ข(๐‘ฅ,๐‘กโˆ’๐‘ ),ฮฆ=๐‘ข,๐‘ฃ,๐œƒ,๐œƒ๐‘ก๎€ธ,๐‘ค,๐พ=(0,๐‘“,0,โ„Ž,0).(2.11) Then problem (2.8) can be formulated as an abstract Cauchy problem ๐‘‘ฮฆ๐‘‘๐‘ก=๐ดฮฆ+๐พ,(2.12) on the Hilbert space โ„‹ for an initial condition ฮฆ(0)=(๐‘ข0,๐‘ข1,๐œƒ0,๐œƒ1,๐‘ค0). The domain of A is given by ๎€ฝ๐ท(๐ด)=(๐‘ข,๐‘ฃ,๐œƒ,๐‘ค)โˆˆโ„‹โˆถ๐œƒโˆˆ๐ป10(ฮฉ),๐œƒ๐‘กโˆˆ๐ป10(ฮฉ),๐œƒ+๐œƒ๐‘กโˆˆ๐ป2(ฮฉ)โˆฉ๐ป10๎€ท๐ป(ฮฉ),๐‘ฃโˆˆ10๎€ธ(ฮฉ)๐‘›๎€œ,๐œ…๐‘ข+โˆž0๎€ท๐ป๐‘”(๐‘ )๐‘ค(๐‘ )๐‘‘๐‘ โˆˆ2(ฮฉ)โˆฉ๐ป10๎€ธ(ฮฉ)๐‘›,๐‘ค(๐‘ )โˆˆ๐ป1๎€ท๎€ท๐ป๐‘”,(0,โˆž),10(๎€ธฮฉ)๐‘›๎€ธ๎€พ,,๐‘ค(0)=0(2.13) where ๐ป1๎€ท๎€ท๐ป๐‘”,(0,โˆž),10๎€ธ(ฮฉ)๐‘›๎€ธ=๎€ฝ๐‘คโˆถ๐‘ค,๐‘ค๐‘ โˆˆ๐ฟ2๎€ท๎€ท๐ป๐‘”,(0,โˆž),10๎€ธ(ฮฉ)๐‘›๎€ธ๎€พ.(2.14) It is clear that ๐ท(๐ด) is dense in โ„‹.

Our hypotheses on ๐‘“,โ„Ž can be stated as follows, which will be used in different theorems: (A1)๐‘“=โ„Ž=0; (A2)๐‘“=๐‘“(๐‘ฅ,๐‘ก)โˆˆ๐ถ1([0,โˆž),(๐ฟ2(ฮฉ))๐‘›),โ„Ž=โ„Ž(๐‘ฅ,๐‘ก)โˆˆ๐ถ1([0,โˆž),๐ฟ2(ฮฉ)); (A3)๐‘“(๐‘ฅ,๐‘ก)โˆˆ๐ถ([0,โˆž),(๐ป10(ฮฉ))๐‘›),โ„Ž(๐‘ฅ,๐‘ก)โˆˆ๐ถ([0,โˆž),๐ป2(ฮฉ)); (A4)๐‘“(๐‘ฅ,๐‘ก)โˆˆ๐ถ([0,โˆž),(๐ฟ2(ฮฉ))๐‘›),โ„Ž(๐‘ฅ,๐‘ก)โˆˆ๐ถ([0,โˆž),๐ฟ2(ฮฉ)), and for any ๐‘‡>0, ๐‘“๐‘กโˆˆ๐ฟ1((0,๐‘‡),(๐ฟ2(ฮฉ))๐‘›),โ„Ž๐‘กโˆˆ๐ฟ1((0,๐‘‡),๐ฟ2(ฮฉ)).

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

Theorem 2.1. Suppose that condition (A1) holds. Relaxation function ๐‘” satisfies (H1)โ€“(H3). Then for any ฮฆ(0)=(๐‘ข0,๐‘ข1,๐œƒ0,๐œƒ1,๐‘ค0)โˆˆ๐ท(๐ด), there exists a unique global classical solution ฮฆ=(๐‘ข,๐‘ฃ,๐œƒ,๐œƒ๐‘ก,๐‘ค) to system (2.8) satisfying ฮฆ=(๐‘ข,๐‘ฃ,๐œƒ,๐œƒ๐‘ก,๐‘ค)โˆˆ๐ถ1([0,โˆž),โ„‹)โˆฉ๐ถ([0,โˆž),๐ท(๐ด)).

Theorem 2.2. Suppose that condition (A2) holds. Relaxation function ๐‘” satisfies (H1)โ€“( H3). Then for any ฮฆ(0)=(๐‘ข0,๐‘ข1,๐œƒ0,๐œƒ1,๐‘ค0), there exists a unique global classical solution ฮฆ=(๐‘ข,๐‘ฃ,๐œƒ,๐œƒ๐‘ก,๐‘ค) to system (2.8) satisfying ฮฆ=(๐‘ข,๐‘ฃ,๐œƒ,๐œƒ๐‘ก,๐‘ค)โˆˆ๐ถ1([0,โˆž),โ„‹)โˆฉ๐ถ([0,โˆž),๐ท(๐ด)), that is, ๐‘ขโˆˆ๐ถ1๎€ท[๎€ท๐ป0,โˆž),10๎€ธ(ฮฉ)๐‘›๎€ธ๎€ท[๎€ท๐ปโˆฉ๐ถ0,โˆž),2(ฮฉ)โˆฉ๐ป10๎€ธ(ฮฉ)๐‘›๎€ธ,๐‘ฃโˆˆ๐ถ1๎€ท[๎€ท๐ฟ0,โˆž),2(๎€ธฮฉ)๐‘›๎€ธ๎€ท[๎€ท๐ปโˆฉ๐ถ0,โˆž),10(๎€ธฮฉ)๐‘›๎€ธ,๐œƒโˆˆ๐ถ1๎€ท[0,โˆž),๐ป10๎€ธ๎€ท[(ฮฉ)โˆฉ๐ถ0,โˆž),๐ป2(ฮฉ)โˆฉ๐ป10๎€ธ,๐œƒ(ฮฉ)๐‘กโˆˆ๐ถ1๎€ท[0,โˆž),๐ฟ2๎€ธ๎€ท[(ฮฉ)โˆฉ๐ถ0,โˆž),๐ป10๎€ธ,(ฮฉ)๐‘คโˆˆ๐ถ1๎€ท[0,โˆž),๐ฟ2๎€ท๎€ท๐ป๐‘”,(0,โˆž),10๎€ธ(ฮฉ)๐‘›๎€ท[๎€ธ๎€ธโˆฉ๐ถ0,โˆž),๐ป1๎€ท๎€ท๐ป๐‘”,(0,โˆž),10๎€ธ(ฮฉ)๐‘›.๎€ธ๎€ธ(2.15)

Corollary 2.3. Suppose that condition (A3)or (A4) holds. Relaxation function ๐‘” satisfies (H1)โ€“(H3). Then for any ฮฆ(0)=(๐‘ข0,๐‘ข1,๐œƒ0,๐œƒ1,๐‘ค0)โˆˆD(๐ด), there exists a unique global classical solution ฮฆ=(๐‘ข,๐‘ฃ,๐œƒ,๐œƒ๐‘ก,๐‘ค)โˆˆ๐ถ1([0,โˆž),โ„‹)โˆฉ๐ถ([0,โˆž),๐ท(๐ด)) to system (2.8).

Corollary 2.4. If ๐‘“(๐‘ฅ,๐‘ก) and โ„Ž(๐‘ฅ,๐‘ก) are Lipschitz continuous functions from [0,๐‘‡] into (๐ฟ2(ฮฉ))๐‘› and ๐ฟ2(ฮฉ), respectively, then for any ฮฆ=(๐‘ข,๐‘ฃ,๐œƒ,๐œƒ๐‘ก,๐‘ค)โˆˆ๐ท(๐ด), there exists a unique global classical solution ฮฆ=(๐‘ข,๐‘ฃ,๐œƒ,๐œƒ๐‘ก,๐‘ค)โˆˆ๐ถ1([0,โˆž),โ„‹)โˆฉ๐ถ([0,โˆž),๐ท(๐ด)) to system (2.8).

Theorem 2.5. Suppose relaxation function ๐‘” satisfies (H1)โ€“(H3), ๐‘“=๐‘“(ฮฆ), and โ„Ž=โ„Ž(ฮฆ),ฮฆ=(๐‘ข,๐‘ฃ,๐œƒ,๐œƒ๐‘ก,๐‘ค), and ๐พ=(0,๐‘“,0,โ„Ž,0) satisfies the global Lipschitz condition on โ„‹; that is, there is a positive constant ๐ฟ such that for all ฮฆ1,ฮฆ2โˆˆโ„‹, โ€–โ€–๐พ๎€ทฮฆ1๎€ธ๎€ทฮฆโˆ’๐พ2๎€ธโ€–โ€–โ„‹โ€–โ€–ฮฆโ‰ค๐ฟ1โˆ’ฮฆ2โ€–โ€–โ„‹.(2.16) Then for any ฮฆ(0)=(๐‘ข0,๐‘ข1,๐œƒ0,๐œƒ1,๐‘ค0)โˆˆโ„‹, there exists a global mild solution ฮฆ to system (2.8) such that ฮฆโˆˆ๐ถ([0,โˆž),โ„‹), that is, ๎‚€[๎‚€๐ป๐‘ขโˆˆ๐ถ0,โˆž),1ฮ“1๎‚(ฮฉ)๐‘›๎‚๎€ท[,๐œƒโˆˆ๐ถ0,โˆž),๐ป10๎€ธ(ฮฉ),๐œƒ๐‘ก๎€ท[โˆˆ๐ถ0,โˆž),๐ฟ2๎€ธ,๎€ท[๎€ท๐ฟ(ฮฉ)๐‘ฃโˆˆ๐ถ0,โˆž),2๎€ธ(ฮฉ)๐‘›๎€ธ[,๐‘คโˆˆ๐ถ(0,โˆž),๐ฟ2๎€ท๎€ท๐ป๐‘”,(0,โˆž),10๎€ธ(ฮฉ)๐‘›๎€ธ.(2.17)

Theorem 2.6. Suppose ๐‘“=๐‘“(ฮฆ) and โ„Ž=โ„Ž(ฮฆ),ฮฆ=(๐‘ข,๐‘ฃ,๐œƒ,๐œƒ๐‘ก,๐‘ค), and ๐พ=(0,๐‘“,0,โ„Ž,0) is a nonlinear operator from ๐ท(๐ด) into ๐ท(๐ด) and satisfies the global Lipschitz condition on ๐ท(๐ด); that is, there is a positive constant ๐ฟ such that for all ฮฆ1,ฮฆ2โˆˆ๐ท(๐ด), โ€–โ€–๐พ๎€ทฮฆ1๎€ธ๎€ทฮฆโˆ’๐พ2๎€ธโ€–โ€–๐ท(๐ด)โ€–โ€–ฮฆโ‰ค๐ฟ1โˆ’ฮฆ2โ€–โ€–๐ท(๐ด).(2.18) Then for any ฮฆ(0)=(๐‘ข0,๐‘ข1,๐œƒ0,๐œƒ1,๐‘ค0)โˆˆ๐ท(๐ด), there exists a unique global classical solution ฮฆ=(๐‘ข,๐‘ฃ,๐œƒ,๐œƒ๐‘ก,๐‘ค)โˆˆ๐ถ1([0,โˆž),โ„‹)โˆฉ๐ถ([0,โˆž),๐ท(๐ด)) to system (2.8).

3. Some Lemmas

In this section in order to complete proofs of Theorems 2.1โ€“2.6, we need first Lemmas 3.1โ€“3.5. For the abstract initial value problem, ๐‘‘๐‘ข๐‘‘๐‘ก+๐ต๐‘ข=๐พ,๐‘ข(0)=๐‘ข0,(3.1) where ๐ต is a maximal accretive operator defined in a dense subset ๐ท(๐ต) of a Banach space ๐ป. We have the following.

Lemma 3.1. Let ๐ต be a linear operator defined in a Hilbert space ๐ป,๐ตโˆถ๐ท(๐ต)โŠ‚๐ปโ†ฆ๐ป. Then the necessary and sufficient conditions for ๐ต being maximal accretive are (i)Re(๐ต๐‘ฅ,๐‘ฅ)โ‰ฅ0,forall๐‘ฅโˆˆ๐ท(๐ต), (ii)๐‘…(๐ผ+๐ต)=๐ป.

Proof. We first prove the necessity. ๐ต is an accretive operator, so we have (๐‘ฅ,๐‘ฅ)=โ€–๐‘ฅโ€–2โ‰คโ€–๐‘ฅ+๐œ†๐ต๐‘ฅโ€–2=(๐‘ฅ,๐‘ฅ)+2๐œ†Re(๐ต๐‘ฅ,๐‘ฅ)+๐œ†2โ€–๐ต๐‘ฅโ€–2.(3.2) Thus, for all ๐œ†>0, ๐œ†Re(๐ต๐‘ฅ,๐‘ฅ)โ‰ฅโˆ’2โ€–๐ต๐‘ฅโ€–2.(3.3) Letting ๐œ†โ†’0, we get (i). Furthermore, (ii) immediately follows from the fact that ๐ต is m-accretive.
We now prove the sufficiency. It follows from (i) that for all ๐œ†>0, โ€–๐‘ฅโˆ’๐‘ฆโ€–2โ‰คRe(๐‘ฅโˆ’๐‘ฆ,๐‘ฅโˆ’๐‘ฆ+๐œ†๐ต(๐‘ฅโˆ’๐‘ฆ))โ‰คโ€–๐‘ฅโˆ’๐‘ฆโ€–โ€–๐‘ฅโˆ’๐‘ฆ+๐œ†(๐ต๐‘ฅโˆ’๐ต๐‘ฆ)โ€–.(3.4) Now it remains to prove that ๐ต is densely defined. We use a contradiction argument. Suppose that it is not true. Then there is a nontrivial element ๐‘ฅ0 belonging to orthogonal supplement of ๐ท(๐ต) such that for all ๐‘ฅโˆˆ๐ท(๐ต), ๎€ท๐‘ฅ,๐‘ฅ0๎€ธ=0.(3.5) It follows from (ii) that there is ๐‘ฅโˆ—โˆˆ๐ท(๐ต) such that ๐‘ฅโˆ—+๐ต๐‘ฅโˆ—=๐‘ฅ0.(3.6) Taking the inner product of (3.5) with ๐‘ฅโˆ—, we deduce that ๎€ท๐‘ฅโˆ—+๐ต๐‘ฅโˆ—,๐‘ฅโˆ—๎€ธ=0.(3.7) Taking the real part of (3.7), we deduce that ๐‘ฅโˆ—=0, and by (3.6), ๐‘ฅ0=0, which is a contradiction. Thus the proof is complete.

Lemma 3.2. Suppose that ๐ต is ๐‘š-accretive in a Banach space ๐ป, and ๐‘ข0โˆˆ๐ท(๐ต). Then problem (3.1) has a unique classical solution ๐‘ข such that ๐‘ขโˆˆ๐ถ1([[0,โˆž),๐ป)โˆฉ๐ถ(0,โˆž),๐ท(๐ต)).(3.8)

Lemma 3.3. Suppose that ๐พ=๐พ(๐‘ก), and ๐พ(๐‘ก)โˆˆ๐ถ1([0,โˆž),๐ป),๐‘ข0โˆˆ๐ท(๐ต).(3.9) Then problem (3.1) admits a unique global classical solution ๐‘ข such that ๐‘ขโˆˆ๐ถ1([[0,โˆž),๐ป)โˆฉ๐ถ(0,โˆž),๐ท(๐ต))(3.10) which can be expressed as ๐‘ข(๐‘ก)=๐‘†(๐‘ก)๐‘ข0+๎€œ๐‘ก0๐‘†(๐‘กโˆ’๐œ)๐พ(๐œ)๐‘‘๐œ.(3.11)

Proof. Since ๐‘†(๐‘ก)๐‘ข0 satisfies the homogeneous equation and nonhomogeneous initial condition, it suffices to verify that ๐‘ค(๐‘ก) given by ๎€œ๐‘ค(๐‘ก)=๐‘ก0๐‘†(๐‘กโˆ’๐œ)๐พ(๐œ)๐‘‘๐œ(3.12) belongs to ๐ถ1([0,โˆž),๐ป)โˆฉ๐ถ([0,โˆž),๐ท(๐ต)) and satisfies the nonhomogeneous equation. Consider the following quotient of difference ๐‘ค(๐‘ก+โ„Ž)โˆ’๐‘ค(๐‘ก)โ„Ž=1โ„Ž๎‚ต๎€œ0๐‘ก+โ„Ž๐‘†๎€œ(๐‘ก+โ„Žโˆ’๐œ)๐พ(๐œ)๐‘‘๐œโˆ’๐‘ก0๐‘†๎‚ถ=1(๐‘กโˆ’๐œ)๐พ(๐œ)๐‘‘๐œโ„Ž๎€œ๐‘ก๐‘ก+โ„Ž1๐‘†(๐‘ก+โ„Žโˆ’๐œ)๐พ(๐œ)๐‘‘๐œ+โ„Ž๎€œ๐‘ก0=1(๐‘†(๐‘ก+โ„Žโˆ’๐œ)โˆ’๐‘†(๐‘กโˆ’๐œ))๐พ(๐œ)๐‘‘๐œโ„Ž๎€œ๐‘ก๐‘ก+โ„Ž1๐‘†(๐‘ง)๐พ(๐‘ก+โ„Žโˆ’๐‘ง)๐‘‘๐‘ง+โ„Ž๎€œ๐‘ก0๐‘†(๐‘ง)(๐พ(๐‘ก+โ„Žโˆ’๐‘ง)โˆ’๐พ(๐‘กโˆ’๐‘ง))๐‘‘๐‘ง.(3.13) When โ„Žโ†’0, the terms in the last line of (3.13) have limits: ๎€œ๐‘†(๐‘ก)๐พ(0)+๐‘ก0[๐‘†(๐‘ง)๐พโ€ฒ(๐‘กโˆ’๐‘ง)๐‘‘๐‘งโˆˆ๐ถ(0,โˆž),๐ป).(3.14) It turns out that ๐‘คโˆˆ๐ถ1([0,โˆž),๐ป) and the terms in the third line of (3.13) have limits too, which should be ๐‘†(0)๐พ(๐‘ก)โˆ’๐ต๐‘ค(๐‘ก)=๐พ(๐‘ก)โˆ’๐ต๐‘ค(๐‘ก).(3.15) Thus the proof is complete.

Lemma 3.4. Suppose that ๐พ=๐พ(๐‘ก), and [๐พ(๐‘ก)โˆˆ๐ถ(0,โˆž),๐ท(๐ต)),u0โˆˆ๐ท(๐ต).(3.16) Then problem (3.1) admits a unique global classical solution.

Proof. From the proof of Lemma 3.2, we can obtain ๐‘ค(๐‘ก+โ„Ž)โˆ’๐‘ค(๐‘ก)โ„Ž=1โ„Ž๎€œ๐‘ก๐‘ก+โ„Ž๐‘†1(๐‘ก+โ„Žโˆ’๐œ)๐พ(๐œ)๐‘‘๐œ+โ„Ž๎€œ๐‘ก0=1(๐‘†(๐‘ก+โ„Žโˆ’๐œ)โˆ’๐‘†(๐‘กโˆ’๐œ))๐พ(๐œ)๐‘‘๐œโ„Ž๎€œ๐‘ก๐‘ก+โ„Ž1๐‘†(๐‘ก+โ„Žโˆ’๐œ)๐พ(๐œ)๐‘‘๐œ+โ„Ž๎€œ๐‘ก0๎‚ต๐‘†(๐‘กโˆ’๐œ)๐‘†(โ„Ž)โˆ’๐ผโ„Ž๎‚ถ๐พ(๐œ)๐‘‘๐œ.(3.17) When โ„Žโ†’0, the last terms in the line of (3.17) have limits: ๎€œ๐‘†(0)๐พ(๐‘ก)โˆ’๐‘ก0๎€œ๐‘†(๐‘กโˆ’๐œ)๐ต๐พ(๐œ)๐‘‘๐œ=๐‘†(0)๐พ(๐‘ก)โˆ’๐ต๐‘ก0๐‘†(๐‘กโˆ’๐œ)๐พ(๐œ)๐‘‘๐œ=๐พ(๐‘ก)โˆ’๐ต๐‘ค(๐‘ก).(3.18) Combining the results of Lemma 3.3 proves the lemma.

Lemma 3.5. Suppose that ๐พ=๐พ(๐‘ก), and [๐พ(๐‘ก)โˆˆ๐ถ(0,โˆž),๐ป),๐‘ข0โˆˆ๐ท(๐ต),(3.19) and for any ๐‘‡>0, ๐พ๐‘กโˆˆ๐ฟ1([]0,๐‘‡,๐ป).(3.20) Then problem (3.1) admits a unique global classical solution.

Proof. We first prove that for any ๐พ1โˆˆ๐ฟ1([0,๐‘‡],๐ป), the function ๐‘ค given by the following integral: ๎€œ๐‘ค(๐‘ก)=๐‘ก0๐‘†(๐‘กโˆ’๐œ)๐พ1๐‘‘๐œ(3.21) belongs to ๐ถ([0,๐‘‡],๐ป). Indeed, we infer from the difference ๐‘ค๎€œ(๐‘ก+โ„Ž)โˆ’๐‘ค(๐‘ก)=0๐‘ก+โ„Ž๐‘†(๐‘ก+โ„Žโˆ’๐œ)๐พ1๎€œ(๐œ)๐‘‘๐œโˆ’๐‘ก0๐‘†(๐‘กโˆ’๐œ)๐พ1๎€œ(๐œ)๐‘‘๐œ=(๐‘†(โ„Ž)โˆ’๐ผ)๐‘ค(๐‘ก)+๐‘ก๐‘ก+โ„Ž๐‘†(๐‘ก+โ„Žโˆ’๐œ)๐พ1(๐œ)๐‘‘๐œ(3.22) that as โ„Žโ†’0, ๎€œโ€–๐‘ค(๐‘ก+โ„Ž)โˆ’๐‘ค(๐‘ก)โ€–โ‰คโ€–(๐‘†(โ„Ž)โˆ’๐ผ)๐‘ค(๐‘ก)โ€–+๐‘ก๐‘ก+โ„Žโ€–๐พ1(๐œ)โ€–๐‘‘๐œโŸถ0,(3.23) where we have used the strong continuity of ๐‘†(๐‘ก) and the absolute continuity of integral for โ€–๐พ1โ€–โˆˆ๐ฟ1[0,๐‘ก].
Now it can be seen from the last line of (3.13) that for almost every ๐‘กโˆˆ[0,๐‘‡],๐‘‘๐‘ค/๐‘‘๐‘ก exists, and it equals ๎€œ๐‘†(๐‘ก)๐พ(0)+๐‘ก0๎€œ๐‘†(๐‘ง)๐พโ€ฒ(๐‘กโˆ’๐‘ง)๐‘‘๐‘ง=๐‘†(๐‘ก)๐พ(0)+๐‘ก0[]๐‘†(๐‘กโˆ’๐œ)๐พโ€ฒ(๐œ)๐‘‘๐œโˆˆ๐ถ(0,๐‘‡,๐ป).(3.24) Thus, for almost every ๐‘ก, ๐‘‘๐‘ค๐‘‘๐‘ก=โˆ’๐ต๐‘ค+๐พ.(3.25) Since ๐‘ค and ๐พ both belong to ๐ถ([0,๐‘‡],๐ป), it follows from (3.25) that for almost every ๐‘ก, ๐ต๐‘ค equals a function belonging to ๐ถ([0,๐‘‡],๐ป). Since ๐ต is a closed operator, we conclude that []๐‘คโˆˆ๐ถ(0,๐‘‡,๐ท(๐ต))โˆฉ๐ถ1([]0,๐‘‡,๐ป)(3.26) and (3.25) holds for every ๐‘ก. Thus the proof is complete.

To prove that the operator ๐ด defined by (2.14) is dissipative, we need the following lemma.

Lemma 3.6. If the function ๐‘“โˆถ[0,โˆž)โ†’๐‘… is uniformly continuous and is in ๐ฟ1(0,โˆž), then lim๐‘กโ†’โˆž๐‘“(๐‘ก)=0.(3.27)

Lemma 3.7. Suppose that the relaxation function ๐‘” satisfies (๐ป1) and (๐ป2). If ๐‘คโˆˆ๐ป1(๐‘”,(0,โˆž),(๐ป10(ฮฉ))๐‘›) and ๐‘ค(0)=0, then ๐‘”๎…žโ€–(๐‘ )โ€–๐‘ค(๐‘ )2(๐ป10(ฮฉ))๐‘›โˆˆ๐ฟ1(0,โˆž),lim๐‘ โ†’โˆž๐‘”(๐‘ )โ€–๐‘ค(๐‘ )โ€–2(๐ป10(ฮฉ))๐‘›=0.(3.28)

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

Lemma 3.8. Suppose relaxation function ๐‘” satisfies (H1)โ€“(H3). The operator ๐ด defined by (2.13) is dissipative; furthermore, 0โˆˆ๐œŒ(๐ด), where ๐œŒ(๐ด) is the resolvent of the operator ๐ด.

Proof. By a straightforward calculation, it follows from Lemma 3.7 that ๎ซ๐ด๎€ท๐‘ข,๐‘ฃ,๐œƒ,๐œƒ๐‘ก๎€ธ,๎€ท,๐‘ค๐‘ข,๐‘ฃ,๐œƒ,๐œƒ๐‘ก,๐‘ค๎€ธ๎ฌโ„‹=๐œ…(๐‘ฃ,๐‘ข)(๐ป10(ฮฉ))๐‘›+12๎€ท๐ต(๐‘ข,๐‘ค)โˆ’๐›ผโˆ‡๐œƒ๐‘ก๎€ธ+๐›ผ,๐‘ฃ๎€ท2๐›ฝโˆ‡๐œƒ๐‘ก๎€ธ+๐›ผ,โˆ‡๐œƒ๎€ท2๐›ฝฮ”๐œƒ๐‘ก+ฮ”๐œƒโˆ’div๐‘ฃ,๐œƒ๐‘ก๎€ธ+๎€ท๐‘ฃโˆ’๐‘ค๐‘ ๎€ธ,๐‘ค๐ฟ2(๐‘”,(0,โˆž),(๐ป10(ฮฉ))๐‘›)๐›ผ=โˆ’โ€–โ€–2๐›ฝโˆ‡๐œƒ๐‘กโ€–โ€–2+๎€œโˆž0๐‘”๎…žโ€–(๐‘ )๐‘ค(๐‘ )โ€–2(๐ป10(ฮฉ))๐‘›๐‘‘๐‘ โ‰ค0.(3.29) Thus, ๐ด is dissipative.
To prove that 0โˆˆ๐œŒ(๐ด), for any ๐บ=(๐‘”1,๐‘”2,๐‘”3,๐‘”4,๐‘”5)โˆˆโ„‹, consider ๐ดฮฆ=๐บ,(3.30) that is, ๐‘ฃ=๐‘”1๎€ท๐ป,in10๎€ธ(ฮฉ)๐‘›,(3.31)๐ต(๐‘ข,๐‘ค)โˆ’๐›ผโˆ‡๐œƒ๐‘ก=๐‘”2๎€ท๐ฟ,in2(๎€ธฮฉ)๐‘›๐œƒ,(3.32)๐‘ก=๐‘”3,in๐ฟ2(ฮฉ),(3.33)ฮ”๐œƒ๐‘ก+ฮ”๐œƒโˆ’๐›ฝdiv๐‘ฃ=๐‘”4,in๐ฟ2(ฮฉ),(3.34)๐‘ฃโˆ’๐‘ค๐‘ =๐‘”5,in๐ฟ2๎€ท๎€ท๐ป๐‘”,(0,โˆž),10๎€ธ(ฮฉ)๐‘›๎€ธ.(3.35) Inserting ๐‘ฃ=๐‘”1 and ๐œƒ๐‘ก=๐‘”3 obtained from (3.31), (3.33) into (3.34), we obtain ฮ”๐œƒ=๐‘”4+๐›ฝdiv๐‘”1โˆ’ฮ”๐‘”3โˆˆ๐ฟ2(ฮฉ).(3.36) By the standard theory for the linear elliptic equations, we have a unique ๐œƒโˆˆ๐ป2(ฮฉ)โˆฉ๐ป10(ฮฉ) satisfying (3.36).
We plug ๐‘ฃ=๐‘”1 obtained from (3.31) into (3.35) to get ๐‘ค๐‘ =๐‘”1โˆ’๐‘”5โˆˆ๐ฟ2๎‚€๎‚€๐ป๐‘”,(0,โˆž),1ฮ“1๎‚(ฮฉ)๐‘›๎‚.(3.37) Applying the standard theory for the linear elliptic equations again, we have a unique ๐‘คโˆˆ๐ป1(๐‘”,(0,โˆž),(๐ป10(ฮฉ))๐‘›) satisfying (3.37). Then plugging ๐œƒ and ๐‘ค 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 ๐‘ขโˆˆ๐ท(๐ด) for (3.32) and such that โˆซ๐œ…๐‘ข+โˆž0๐‘”(๐‘ )๐‘ค(๐‘ )๐‘‘๐‘ โˆˆ(๐ป2(ฮฉ)โˆฉ๐ป10(ฮฉ))๐‘›. Thus the unique solvability of (3.30) follows. It is clear from the regularity theory for the linear elliptic equations that โ€–ฮฆโ€–โ„‹โ‰ค๐พโ€–๐บโ€–โ„‹ with ๐พ being a positive constant independent of ฮฆ. Thus the proof is completed.

Lemma 3.9. The operator ๐ด defined by (2.13) is closed.

Proof. To prove that ๐ด is closed, let (๐‘ข๐‘›,๐‘ฃ๐‘›,๐œƒ๐‘›,๐œƒ๐‘›๐‘ก,๐‘ค๐‘›)โˆˆ๐ท(๐ด) be such that ๎€ท๐‘ข๐‘›,๐‘ฃ๐‘›,๐œƒ๐‘›,๐œƒ๐‘›๐‘ก,๐‘ค๐‘›๎€ธโŸถ๎€ท๐‘ข,๐‘ฃ,๐œƒ,๐œƒ๐‘ก๎€ธ๐ด๎€ท๐‘ข,๐‘คinโ„‹,๐‘›,๐‘ฃ๐‘›,๐œƒ๐‘›,๐œƒ๐‘ก๐‘›,๐‘ค๐‘›๎€ธโŸถ(๐‘Ž,๐‘,๐‘,๐‘‘,๐‘’)inโ„‹.(3.38) Then we have ๐‘ข๐‘›๎€ท๐ปโŸถ๐‘ขin10๎€ธ(ฮฉ)๐‘›,๐‘ฃ(3.39)๐‘›๎€ท๐ฟโŸถ๐‘ฃin2๎€ธ(ฮฉ)๐‘›,๐œƒ(3.40)๐‘›โŸถ๐œƒin๐ป10๐œƒ(ฮฉ),(3.41)๐‘›๐‘กโŸถ๐œƒ๐‘กin๐ฟ2๐‘ค(ฮฉ),(3.42)๐‘›โŸถ๐‘คin๐ฟ2๎€ท๎€ท๐ป๐‘”,(0,โˆž),10๎€ธ(ฮฉ)๐‘›๎€ธ๐‘ฃ,(3.43)๐‘›๎€ท๐ปโŸถ๐‘Žin10๎€ธ(ฮฉ)๐‘›๐ต๎€ท๐‘ข,(3.44)๐‘›,๐‘ค๐‘›๎€ธโˆ’๐›ผโˆ‡๐œƒ๐‘›๐‘ก๎€ท๐ฟโŸถ๐‘in2๎€ธ(ฮฉ)๐‘›๐œƒ,(3.45)๐‘›๐‘กโŸถ๐‘in๐ป10(ฮฉ),(3.46)ฮ”๐œƒ๐‘›๐‘ก+ฮ”๐œƒ๐‘›โˆ’๐›ฝdiv๐‘ฃ๐‘›โŸถ๐‘‘in๐ฟ2๐‘ฃ(ฮฉ),(3.47)๐‘›โˆ’๐‘ค๐‘›๐‘ โŸถ๐‘’in๐ฟ2๎€ท๎€ท๐ป๐‘”,(0,โˆž),10๎€ธ(ฮฉ)๐‘›๎€ธ.(3.48) By (3.40) and (3.44), we deduce ๐‘ฃ๐‘›๎€ท๐ปโŸถ๐‘ฃin10๎€ธ(ฮฉ)๐‘›,๎€ท๐ป(3.49)๐‘ฃ=๐‘Žโˆˆ10๎€ธ(ฮฉ)๐‘›.(3.50) By (3.42) and (3.46), we deduce ๐œƒ๐‘›๐‘กโŸถ๐œƒ๐‘กin๐ป10๐œƒ(ฮฉ),(3.51)๐‘ก=๐‘โˆˆ๐ป10(ฮฉ).(3.52) By (3.47) and (3.49), we deduce ฮ”๐œƒ๐‘›๐‘ก+ฮ”๐œƒ๐‘›โŸถ๐‘‘+๐›ฝdiv๐‘ฃin๐ฟ2(ฮฉ),(3.53) and consequently, it follows from (3.41), that ๐œƒ๐‘›๐‘ก+๐œƒ๐‘›โŸถ๐œƒ๐‘ก+๐œƒin๐ป2(ฮฉ)โˆฉ๐ป10(ฮฉ),(3.54) since ฮ” is an isomorphism from ๐ป2(ฮฉ)โˆฉ๐ป10(ฮฉ) onto ๐ฟ2(ฮฉ). It therefore follows from (3.47) and (3.54) that ๐‘‘=ฮ”๐œƒ๐‘›๐‘ก+ฮ”๐œƒ๐‘›โˆ’๐›ฝdiv๐‘ฃ,๐œƒ๐‘ก+๐œƒโˆˆ๐ป2(ฮฉ)โˆฉ๐ป10(ฮฉ).(3.55) By (3.43), (3.48), and (3.49), we deduce ๐‘ค๐‘›โŸถ๐‘คin๐ป1๎‚€๎‚€๐ป๐‘”,(0,โˆž),1ฮ“1๎‚(ฮฉ)๐‘›๎‚,(3.56)๐‘’=๐‘ฃโˆ’๐‘ค๐‘ ,๐‘คโˆˆ๐ป1๎€ท๎€ท๐ป๐‘”,(0,โˆž),10๎€ธ(ฮฉ)๐‘›๎€ธ,๐‘ค(0)=0.(3.57) In addition, it follows from (3.39), (3.43), (3.51) that ๐ต๎€ท๐‘ข๐‘›,๐‘ค๐‘›๎€ธโˆ’๐›ผโˆ‡๐œƒ๐‘›๐‘กโŸถ๐ต(๐‘ข,๐‘ค)โˆ’๐›ผโˆ‡๐œƒ๐‘ก(3.58) in the distribution. It therefore follows from (3.45) and (3.58) that ๐‘=๐ต(๐‘ข,๐‘ค)โˆ’๐›ผโˆ‡๐œƒ๐‘ก๎€ท๐ฟ,๐ต(๐‘ข,๐‘ค)โˆˆ2๎€ธ(ฮฉ)๐‘›,(3.59) and consequently, ๎€œ๐œ…๐‘ข+โˆž0๎€ท๐ป๐‘”(๐‘ )๐‘ค(๐‘ )๐‘‘๐‘ โˆˆ2(ฮฉ)โˆฉ๐ป10๎€ธ(ฮฉ)๐‘›,(3.60) since ๐œ‡ฮ”+(๐œ†+๐œ‡)โˆ‡div is an isomorphism from ๐ป2(ฮฉ)โˆฉ๐ป10(ฮฉ) onto ๐ฟ2(ฮฉ). Thus, by (3.50), (3.52), (3.55), (3.57), (3.59), (3.60), we deduce ๐ด๎€ท๐‘ข,๐‘ฃ,๐œƒ,๐œƒ๐‘ก๎€ธ=๎€ท,๐‘ค(๐‘Ž,๐‘,๐‘,๐‘‘,๐‘’),๐‘ข,๐‘ฃ,๐œƒ,๐œƒ๐‘ก๎€ธ,๐‘คโˆˆ๐ท(๐ด).(3.61) Hence, ๐ด is closed.

Lemma 3.10. Let ๐ด be a linear operator with dense domain ๐ท(๐ด) in a Hilbert space ๐ป. If ๐ด is dissipative and 0โˆˆ๐œŒ(๐ด), the resolvent set of ๐ด, then ๐ด is the infinitesimal generator of a ๐ถ0-semigroup of contractions on ๐ป.

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

Lemma 3.11. Let ๐ด be a densely defined linear operator on a Hilbert space ๐ป. Then ๐ด generates a ๐ถ0-semigroup of contractions on ๐ป if and only if ๐ด is dissipative and ๐‘…(๐ผโˆ’๐ด)=๐ป.

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

4. Proofs of Theorems 2.1โ€“2.5

Proof of Theorem 2.1. By (2.2), it is clear that โ„‹ is a Hilbert space. By Lemmas 3.8โ€“3.10, we can deduce that the operator ๐ด is the infinitesimal generator of a ๐ถ0-semigroup of contractions on Hilbert space โ„‹. Applying the result and Lemma 3.2, we can obtain our result.

Proof of Theorem 2.2. we have known that the operator ๐ด is the infinitesimal generator of a ๐ถ0-semigroup of contractions on Hilbert space โ„‹. Applying the result and Lemma 3.11, we can conclude that ๐‘…(๐ผโˆ’๐ด)=๐ป. If we choose operator ๐ต=โˆ’๐ด, we can obtain ๐ท(๐ต)=๐ท(๐ด) and ๐ท(๐ต) is dense in โ„‹. Noting that by (๐ด2), we know that ๐พ=(0,๐‘“,0,โ„Ž,0)โˆˆ๐ถ1([0,โˆž),โ„‹); therefore, applying Lemma 3.1, we can conclude the operator ๐ต 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 (๐ด3) or (๐ด4), we derive that ๐พ=(0,๐‘“,0,โ„Ž,0)โˆˆ๐ถ([0,โˆž),๐ท(๐ด)) or ๐พโˆˆ๐ถ([0,โˆž),โ„‹), and for any ๐‘‡>0,๐พ๐‘กโˆˆ๐ฟ1((0,๐‘‡),โ„‹). Noting that ๐ต=โˆ’๐ด 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 ๐พ(๐‘ฅ,๐‘ก)=(0,๐‘“,0,โ„Ž,0) are Lipschitz continuous functions from [0,๐‘‡] into โ„‹. Moreover, by (2.2), it is clear that โ„‹ is a reflexive Banach space. Therefore, ๐พ๐‘กโˆˆ๐ฟ1([0,๐‘‡],๐ป). 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 ๐ต=โˆ’๐ด is the maximal accretive operator of a ๐ถ0 semigroup ๐‘†(๐‘ก). On the other hand, ๐พ=(0,๐‘“,0,โ„Ž,0) satisfies the global Lipschitz condition on โ„‹. 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 ๐œ™(ฮฆ)=๐‘†(๐‘ก)ฮฆ0+๎€œ๐‘ก0๎‚ป[๐‘†(๐‘กโˆ’๐œ)๐พ(ฮฆ(๐œ))๐‘‘๐œ,(4.1)ฮฉ=ฮฆโˆˆ๐ถ(0,+โˆž),๐ป)โˆฃsup๐‘กโ‰ฅ0๎€ทโ€–ฮฆ(๐‘ก)โ€–๐‘’โˆ’๐‘˜๐‘ก๎€ธ๎‚ผ<โˆž,(4.2) where ๐‘˜ is a positive constant such that ๐‘˜>๐ฟ. In ฮฉ, we introduce the following norm: โ€–ฮฆโ€–ฮฉ=sup๐‘กโ‰ฅ0๎€ทโ€–ฮฆ(๐‘ก)โ€–๐‘’โˆ’๐‘˜๐‘ก๎€ธ.(4.3) Clearly, ฮฉ is a Banach space. We now show that the nonlinear operator ๐œ™ defined by (4.1) maps ฮฉ into itself, and the mapping is a contraction. Indeed, for ฮฆโˆˆฮฉ, we have โ€–๐œ™(ฮฆ)โ€–โ‰คโ€–๐‘†(๐‘ก)ฮฆ0๎€œโ€–+๐‘ก0โ€–๐‘†(๐‘กโˆ’๐œ)โ€–โ€–๐พ(ฮฆ)โ€–๐‘‘๐œโ‰คโ€–ฮฆ0๎€œโ€–+๐‘ก0โ€–๐พ(ฮฆ)โ€–๐‘‘๐œโ‰คโ€–ฮฆ0๎€œโ€–+๐‘ก0(๐ฟโ€–ฮฆ(๐œ)โ€–+โ€–๐พ(0)โ€–)๐‘‘๐œโ‰คโ€–ฮฆ0โ€–+๐ถ0๐‘ก+๐ฟsup๐‘กโ‰ฅ0โ€–ฮฆ(๐‘ก)โ€–๐‘’โˆ’๐‘˜๐‘ก๎€œ๐‘ก0๐‘’๐‘˜๐œ๐‘‘๐œโ‰คโ€–ฮฆ0โ€–+๐ถ0๐ฟ๐‘ก+๐‘˜๐‘’๐‘˜๐‘กโ€–ฮฆโ€–ฮฉ,(4.4) where ๐ถ0=โ€–๐พ(0)โ€–. Thus, โ€–โ€–๐œ™(ฮฆ)ฮฉโ‰คsup๐‘กโ‰ฅ0๎€บ๎€ทโ€–ฮฆ0โ€–+๐ถ0๐‘ก๎€ธ๐‘’โˆ’๐‘˜๐‘ก๎€ป+๐ฟ๐‘˜โ€–ฮฆโ€–ฮฉ<โˆž.(4.5) that is, ๐œ™(ฮฆ)โˆˆฮฉ.
For ฮฆ1,ฮฆ2โˆˆฮฉ, we have โ€–โ€–๐œ™๎€ทฮฆ1๎€ธ๎€ทฮฆโˆ’๐œ™2๎€ธโ€–โ€–ฮฉ=sup๐‘กโ‰ฅ0๐‘’โˆ’๐‘˜๐‘กโ€–โ€–โ€–๎€œ๐‘ก0๎€ท๐พ๎€ทฮฆ๐‘†(๐‘กโˆ’๐œ)1(๎€ธ๎€ทฮฆ๐œ)โˆ’๐พ2(โ€–โ€–โ€–๐œ)๎€ธ๎€ธ๐‘‘๐œโ‰คsup๐‘กโ‰ฅ0๐‘’โˆ’๐‘˜๐‘ก๐ฟ๎€œ๐‘ก0โ€–ฮฆ1โˆ’ฮฆ2โ€–๐‘‘๐œโ‰คsup๐‘กโ‰ฅ0๎‚€๐‘’โˆ’๐‘˜๐‘กโ‹…๐ฟ๐‘˜โ‹…๎€ท๐‘’๐‘˜๐‘ก๎€ธ๎‚โ€–โ€–ฮฆโˆ’11โˆ’ฮฆ2โ€–โ€–ฮฉโ‰ค๐ฟ๐‘˜โ€–ฮฆ1โˆ’ฮฆ2โ€–ฮฉ.(4.6) Therefore, by the contraction mapping theorem, the problem has a unique solution in ฮฉ. To show that the uniqueness also holds in ๐ถ([0,โˆž),๐ป), let ฮฆ1,ฮฆ2โˆˆ๐ถ([0,โˆž),๐ป) be two solutions of the problem and let ฮฆ=ฮฆ1โˆ’ฮฆ2. Then ๎€œฮฆ(๐‘ก)=๐‘ก0๎€ท๐พ๎€ทฮฆ๐‘†(๐‘กโˆ’๐œ)1๎€ธ๎€ทฮฆโˆ’๐พ2๎€œ๎€ธ๎€ธ๐‘‘๐œ,โ€–ฮฆ(๐‘ก)โ€–โ‰ค๐ฟ๐‘ก0โ€–ฮฆ(๐œ)โ€–๐‘‘๐œ.(4.7) By the Gronwall inequality, we immediately conclude that ฮฆ(๐‘ก)=0; that is, the uniqueness in ๐ถ([0,โˆž),๐ป) follows. Thus the proof is complete.

Proof of Theorem 2.6. Since ๐ต is the maximal accretive operator, ๐พ=(0,๐‘“,0,โ„Ž,0) satisfies the global Lipschitz condition on ๐ท(๐ด). Let ๐ด1=๐ท(๐ต),๐ต1=๐ต2๎€ท๐ตโˆถ๐ท1๎€ธ๎€ท๐ต=๐ท2๎€ธโŸผ๐ด1.(4.8) Then ๐ด1 is a Banach space, and ๐ต1=๐ต2 is a densely defined operator from ๐ท(๐ต2) into ๐ด1. In what follows we prove that ๐ต1 is ๐‘š-accretive in ๐ด1=๐ท(B).
Indeed, for any ๐‘ฅ,๐‘ฆโˆˆ๐ท(๐ต2), since ๐ต is accretive in ๐ป, we have โ€–โ€–๐‘ฅโˆ’๐‘ฆ+๐œ†(๐ต๐‘ฅโˆ’๐ต๐‘ฆ)๐ท(๐ต)=๎‚€โ€–โ€–๐‘ฅโˆ’๐‘ฆ+๐œ†(๐ต๐‘ฅโˆ’๐ต๐‘ฆ)2+โ€–โ€–๎€ท๐ต๐ต๐‘ฅโˆ’๐ต๐‘ฆ+๐œ†2๐‘ฅโˆ’๐ต2๐‘ฆ๎€ธโ€–โ€–2๎‚1/2โ‰ฅ๎€ทโ€–๐‘ฅโˆ’๐‘ฆโ€–2+โ€–๐ต๐‘ฅโˆ’๐ต๐‘ฆโ€–2๎€ธ1/2=โ€–๐‘ฅโˆ’๐‘ฆโ€–๐ท(๐ต).(4.9) that is, ๐ต1 is accretive in ๐ด1. Furthermore, since ๐ต is ๐‘š-accretive in ๐ป, for any ๐‘ฆโˆˆ๐ป, there is a unique ๐‘ฅโˆˆ๐ท(๐ต) such that ๐‘ฅ+๐ต๐‘ฅ=๐‘ฆ.(4.10) Now for any ๐‘ฆโˆˆ๐ด1=๐ท(๐ต), (4.10) admits a unique solution ๐‘ฅโˆˆ๐ท(๐ต). It turns out that ๐ต๐‘ฅ=๐‘ฆโˆ’๐‘ฅโˆˆ๐ท(๐ต).(4.11) Thus ๐‘ฅโˆˆ๐ท(๐ต2); that is, ๐ต1 is m-accretive in ๐ด1. Let ๐‘†1(๐‘ก) be the semigroup generated by ๐ต1. If ฮฆ0โˆˆ๐ท(๐ต2)=๐ท(๐ต1), then ฮฆ(๐‘ก)=๐‘†1(๐‘ก)ฮฆ0๎€ท[๎€ท๐ตโˆˆ๐ถ0,+โˆž),๐ท2๎€ธ๎€ธโˆฉ๐ถ1([0,+โˆž),๐ท(๐ต))(4.12) is unique classical solution of the problem. On the other hand, ฮฆ(๐‘ก)=๐‘†1(๐‘ก)ฮฆ0 is also a classical solution in [๐ถ(0,+โˆž),๐ท(๐ต))โˆฉ๐ถ1([0,+โˆž),๐ป).(4.13) This implies that ๐‘†1(๐‘ก) is a restriction of ๐‘†(๐‘ก) on ๐ด1. By virtue of the proof of Theorem 2.5, there exists a unique mild solution ฮฆโˆˆ๐ถ([0,+โˆž),๐ด1). Since ๐‘†1(๐‘ก) is a restriction of ๐‘†(๐‘ก) on ๐ท(๐ต), and moreover, we infer from ๐พ(ฮฆ) being an operator from ๐ท(๐ต) to ๐ท(๐ต) and Lemma 3.4 that ฮฆ 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

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Copyright ยฉ 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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