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: 𝑢𝑡𝑡−𝜇Δ𝑢−(𝜆+𝜇)∇div𝑢+𝜇𝑔∗Δ𝑢+(𝜆+𝜇)𝑔∗∇div𝑢+𝛼∇𝜃𝑡𝜃=𝑓,(𝑥,𝑡)∈Ω×(0,∞),𝑡𝑡−Δ𝜃𝑡−Δ𝜃+𝛽div𝑢𝑡=ℎ,(𝑥,𝑡)∈Ω×(0,∞),(1.1) where the sign “∗” denotes the convolution product in time, which is defined by 𝑔∗𝑣(𝑡)=ğ‘¡âˆ’âˆžğ‘”(𝑡−𝑠)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),𝑥∈Ω,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 [1–3]. The following basic conditions on the relaxation function 𝑔(𝑡) are (H1)𝑔∈𝐶1[0,∞)∩𝐿1(0,∞); (H2)𝑔(𝑡)≥0,ğ‘”î…ž(𝑡)≤0,𝑡>0; (H3)∫𝜅=1−∞0𝑔(𝑡)𝑑𝑡>0.

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

When 𝑓=𝑔=ℎ=0, system (1.1)–(1.4) is reduced to the thermoelastic system: 𝑢𝑡𝑡−𝜇Δ𝑢−(𝜆+𝜇)∇div𝑢+𝛼∇𝜃𝑡𝜃=0,(𝑥,𝑡)∈Ω×(0,∞),𝑡𝑡−Δ𝜃𝑡−Δ𝜃+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),𝑥∈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) or𝑢∣𝑥=0=0,ğœŽâˆ£ğ‘¥=𝑙=0,𝜃𝑥∣𝑥=0,𝑙=0,(1.7)𝑢∣𝑥=0=0,ğœŽâˆ£ğ‘¥=𝑙=0,𝜃∣𝑥=0=0,𝜃𝑥∣𝑥=𝑙=0,(1.8) or𝑢∣𝑥=0=0,ğœŽâˆ£ğ‘¥=𝑙=0,𝜃∣𝑥=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:𝑢∣𝑥=0=0,ğœŽâˆ£ğ‘¥=𝑙=0,𝜃𝑥∣𝑥=0=0,𝜃∣𝑥=𝑙=0,(1.10) where ğœŽ=𝑢𝑥−𝛾𝜃 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=ℎ=0, system (1.1)-(1.4) is decoupled into the following viscoelastic system: 𝑢𝑡𝑡−𝜇Δ𝑢−(𝜆+𝜇)∇div𝑢+𝜇𝑔∗Δ𝑢+(𝜆+𝜇)𝑔∗∇div𝑢=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, 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≤+∞,𝑊𝑚,𝑝,𝑚∈𝑁,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),𝑘∈ℕ0, the space of k-times continuously differentiable functions from 𝐽⊆𝐼 into a Banach space 𝐵, and likewise by 𝐿𝑝(𝐼,𝐵),1≤𝑝≤+∞, 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 ‖𝑤‖2𝐿2𝐻(𝑔,(0,∞),10(Ω)𝑛)=∞0𝑔(𝑠)‖𝑤(𝑠)‖2𝐻10(Ω)𝑛𝑑𝑠<∞.(2.1) Put 𝐻ℋ=10(Ω)𝑛×𝐿2(Ω)𝑛×𝐻10(Ω)×𝐿2(Ω)×𝐿2𝐻𝑔,(0,∞),10(Ω)𝑛(2.2) with the energy norm ‖‖𝑢,𝑣,𝜃,𝜃𝑡‖‖,𝑤ℋ=𝜅‖𝑢‖2𝐻10(Ω)𝑛+12‖𝑣‖2+𝛼𝛽‖‖𝜃𝑡‖‖2++â€–âˆ‡ğœƒâ€–âˆž0𝑔(𝑠)‖𝑤(𝑠)‖2𝐻10(Ω)𝑛𝑑𝑠1/2,(2.3) where 𝜅 denotes the positive constant in (𝐻3), that is, 𝜅=1−∞0𝑔(𝑡)𝑑𝑡>0.(2.4) Thus we consider the following thermoviscoelastic system:𝑢𝑡𝑡−𝜇Δ𝑢−(𝜆+𝜇)∇div𝑢+𝜇𝑔∗Δ𝑢+(𝜆+𝜇)𝑔∗∇div𝑢+𝛼∇𝜃𝑡𝜃=0,(𝑥,𝑡)∈Ω×(0,∞),𝑡𝑡−Δ𝜃𝑡−Δ𝜃+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),𝑥∈Ω,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,𝑡−𝑠).(2.6) Since ğœ•î€œğœ•ğœˆğ‘¡âˆ’âˆžğœ•ğ‘”(𝑡−𝑠)𝑢(𝑠)𝑑𝑠=î€œğœ•ğœˆâˆž0=𝑔(𝑠)𝑢(𝑡−𝑠)ğ‘‘ğ‘ âˆž0𝜕𝑔(𝑠)=𝜕𝜈(𝑢(𝑡)−𝑤(𝑡,𝑠))𝑑𝑠(1−𝜅)𝜕𝑢(𝑥,𝑡)âˆ’î€œğœ•ğœˆâˆž0𝑔(𝑠)𝜕𝑤(𝑡,𝑠)𝜕𝜈𝑑𝑠,(2.7) System (2.5) can be written as follows: 𝑢𝑡𝑡−𝜅𝜇Δ𝑢−𝜅(𝜆+𝜇)∇div𝑢+ğ›¼âˆ‡ğœƒğ‘¡î€œâˆ’ğœ‡âˆž0𝑔(𝑠)Δ𝑤(𝑡,𝑠)𝑑𝑠−(𝜆+𝜇)∞0𝜃𝑔(𝑠)∇div𝑤(𝑡,𝑠)𝑑𝑠=0,(𝑥,𝑡)∈Ω×(0,∞),𝑡𝑡−Δ𝜃𝑡−Δ𝜃+𝛽div𝑢𝑡𝑢=0,(𝑥,𝑡)∈Ω×(0,∞),𝑤(𝑥,𝑡,𝑠)=𝑢(𝑥,𝑡)−𝑢(𝑥,𝑡−𝑠),(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),𝑥∈Ω,𝑤(0,𝑠)=𝑤0(𝑠),(𝑥,𝑡)∈Ω×(0,∞).(2.8) We define a linear unbounded operator 𝐴 on ℋ by 𝐴𝑢,𝑣,𝜃,𝜃𝑡=,𝑤𝑣,𝐵(𝑢,𝑤)−𝛼∇𝜃𝑡,𝜃𝑡,Δ𝜃𝑡+Δ𝜃−𝛽div𝑣,𝑣−𝑤𝑠,(2.9) where 𝑤𝑠=𝜕𝑤/𝜕𝑠 and 𝐵(𝑢,𝑤)=𝜅𝜇Δ𝑢+𝜅(𝜆+𝜇)∇div𝑢+ğœ‡âˆž0𝑔(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,𝑡−𝑠),Φ=𝑢,𝑣,𝜃,𝜃𝑡,𝑤,𝐾=(0,𝑓,0,ℎ,0).(2.11) Then problem (2.8) can be formulated as an abstract Cauchy problem 𝑑Φ𝑑𝑡=𝐴Φ+𝐾,(2.12) on the Hilbert space ℋ for an initial condition Φ(0)=(𝑢0,𝑢1,𝜃0,𝜃1,𝑤0). The domain of A is given by 𝐷(𝐴)=(𝑢,𝑣,𝜃,𝑤)∈ℋ∶𝜃∈𝐻10(Ω),𝜃𝑡∈𝐻10(Ω),𝜃+𝜃𝑡∈𝐻2(Ω)∩𝐻10𝐻(Ω),𝑣∈10(Ω)𝑛,𝜅𝑢+∞0𝐻𝑔(𝑠)𝑤(𝑠)𝑑𝑠∈2(Ω)∩𝐻10(Ω)𝑛,𝑤(𝑠)∈𝐻1𝐻𝑔,(0,∞),10(Ω)𝑛,,𝑤(0)=0(2.13) where 𝐻1𝐻𝑔,(0,∞),10(Ω)𝑛=𝑤∶𝑤,𝑤𝑠∈𝐿2𝐻𝑔,(0,∞),10(Ω)𝑛.(2.14) It is clear that 𝐷(𝐴) is dense in ℋ.

Our hypotheses on 𝑓,ℎ can be stated as follows, which will be used in different theorems: (A1)𝑓=ℎ=0; (A2)𝑓=𝑓(𝑥,𝑡)∈𝐶1([0,∞),(𝐿2(Ω))𝑛),ℎ=ℎ(𝑥,𝑡)∈𝐶1([0,∞),𝐿2(Ω)); (A3)𝑓(𝑥,𝑡)∈𝐶([0,∞),(𝐻10(Ω))𝑛),ℎ(𝑥,𝑡)∈𝐶([0,∞),𝐻2(Ω)); (A4)𝑓(𝑥,𝑡)∈𝐶([0,∞),(𝐿2(Ω))𝑛),ℎ(𝑥,𝑡)∈𝐶([0,∞),𝐿2(Ω)), and for any 𝑇>0, 𝑓𝑡∈𝐿1((0,𝑇),(𝐿2(Ω))𝑛),â„Žğ‘¡âˆˆğ¿1((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,∞),ℋ)∩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,∞),ℋ)∩𝐶([0,∞),𝐷(𝐴)), that is, 𝑢∈𝐶1[𝐻0,∞),10(Ω)𝑛[𝐻∩𝐶0,∞),2(Ω)∩𝐻10(Ω)𝑛,𝑣∈𝐶1[𝐿0,∞),2(Ω)𝑛[𝐻∩𝐶0,∞),10(Ω)𝑛,𝜃∈𝐶1[0,∞),𝐻10[(Ω)∩𝐶0,∞),𝐻2(Ω)∩𝐻10,𝜃(Ω)𝑡∈𝐶1[0,∞),𝐿2[(Ω)∩𝐶0,∞),𝐻10,(Ω)𝑤∈𝐶1[0,∞),𝐿2𝐻𝑔,(0,∞),10(Ω)𝑛[∩𝐶0,∞),𝐻1𝐻𝑔,(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,∞),ℋ)∩𝐶([0,∞),𝐷(𝐴)) to system (2.8).

Corollary 2.4. If 𝑓(𝑥,𝑡) and ℎ(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,∞),ℋ)∩𝐶([0,∞),𝐷(𝐴)) to system (2.8).

Theorem 2.5. Suppose relaxation function 𝑔 satisfies (H1)–(H3), 𝑓=𝑓(Φ), and ℎ=ℎ(Φ),Φ=(𝑢,𝑣,𝜃,𝜃𝑡,𝑤), and 𝐾=(0,𝑓,0,ℎ,0) satisfies the global Lipschitz condition on ℋ; that is, there is a positive constant 𝐿 such that for all Φ1,Φ2∈ℋ, ‖‖𝐾Φ1Φ−𝐾2‖‖ℋ‖‖Φ≤𝐿1−Φ2‖‖ℋ.(2.16) Then for any Φ(0)=(𝑢0,𝑢1,𝜃0,𝜃1,𝑤0)∈ℋ, there exists a global mild solution Φ to system (2.8) such that Φ∈𝐶([0,∞),ℋ), that is, [𝐻𝑢∈𝐶0,∞),1Γ1(Ω)𝑛[,𝜃∈𝐶0,∞),𝐻10(Ω),𝜃𝑡[∈𝐶0,∞),𝐿2,[𝐿(Ω)𝑣∈𝐶0,∞),2(Ω)𝑛[,𝑤∈𝐶(0,∞),𝐿2𝐻𝑔,(0,∞),10(Ω)𝑛.(2.17)

Theorem 2.6. Suppose 𝑓=𝑓(Φ) and ℎ=ℎ(Φ),Φ=(𝑢,𝑣,𝜃,𝜃𝑡,𝑤), and 𝐾=(0,𝑓,0,ℎ,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,Φ2∈𝐷(𝐴), ‖‖𝐾Φ1Φ−𝐾2‖‖𝐷(𝐴)‖‖Φ≤𝐿1−Φ2‖‖𝐷(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,∞),ℋ)∩𝐶([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, 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,𝐵∶𝐷(𝐵)⊂𝐻↦𝐻. Then the necessary and sufficient conditions for 𝐵 being maximal accretive are (i)Re(𝐵𝑥,𝑥)≥0,forall𝑥∈𝐷(𝐵), (ii)𝑅(𝐼+𝐵)=𝐻.

Proof. We first prove the necessity. 𝐵 is an accretive operator, so we have (𝑥,𝑥)=‖𝑥‖2≤‖𝑥+𝜆𝐵𝑥‖2=(𝑥,𝑥)+2𝜆Re(𝐵𝑥,𝑥)+𝜆2‖𝐵𝑥‖2.(3.2) Thus, for all 𝜆>0, 𝜆Re(𝐵𝑥,𝑥)≥−2‖𝐵𝑥‖2.(3.3) Letting 𝜆→0, 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, ‖𝑥−𝑦‖2≤Re(𝑥−𝑦,𝑥−𝑦+𝜆𝐵(𝑥−𝑦))≤‖𝑥−𝑦‖‖𝑥−𝑦+𝜆(𝐵𝑥−𝐵𝑦)‖.(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 𝑥∈𝐷(𝐵), 𝑥,𝑥0=0.(3.5) It follows from (ii) that there is 𝑥∗∈𝐷(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 𝑢0∈𝐷(𝐵). Then problem (3.1) has a unique classical solution 𝑢 such that 𝑢∈𝐶1([[0,∞),𝐻)∩𝐶(0,∞),𝐷(𝐵)).(3.8)

Lemma 3.3. Suppose that 𝐾=𝐾(𝑡), and 𝐾(𝑡)∈𝐶1([0,∞),𝐻),𝑢0∈𝐷(𝐵).(3.9) Then problem (3.1) admits a unique global classical solution 𝑢 such that 𝑢∈𝐶1([[0,∞),𝐻)∩𝐶(0,∞),𝐷(𝐵))(3.10) which can be expressed as 𝑢(𝑡)=𝑆(𝑡)𝑢0+𝑡0𝑆(𝑡−𝜏)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(𝑡−𝜏)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+ℎ)−𝑤(𝑡)ℎ=1ℎ0𝑡+â„Žğ‘†î€œ(𝑡+â„Žâˆ’ğœ)𝐾(𝜏)𝑑𝜏−𝑡0𝑆=1(𝑡−𝜏)𝐾(𝜏)ğ‘‘ğœâ„Žî€œğ‘¡ğ‘¡+ℎ1𝑆(𝑡+â„Žâˆ’ğœ)𝐾(𝜏)𝑑𝜏+â„Žî€œğ‘¡0=1(𝑆(𝑡+â„Žâˆ’ğœ)−𝑆(𝑡−𝜏))𝐾(𝜏)ğ‘‘ğœâ„Žî€œğ‘¡ğ‘¡+ℎ1𝑆(𝑧)𝐾(𝑡+â„Žâˆ’ğ‘§)𝑑𝑧+â„Žî€œğ‘¡0𝑆(𝑧)(𝐾(𝑡+â„Žâˆ’ğ‘§)−𝐾(𝑡−𝑧))𝑑𝑧.(3.13) When ℎ→0, the terms in the last line of (3.13) have limits: 𝑆(𝑡)𝐾(0)+𝑡0[𝑆(𝑧)𝐾′(𝑡−𝑧)𝑑𝑧∈𝐶(0,∞),𝐻).(3.14) It turns out that 𝑤∈𝐶1([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)),u0∈𝐷(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+ℎ)−𝑤(𝑡)ℎ=1â„Žî€œğ‘¡ğ‘¡+â„Žğ‘†1(𝑡+â„Žâˆ’ğœ)𝐾(𝜏)𝑑𝜏+â„Žî€œğ‘¡0=1(𝑆(𝑡+â„Žâˆ’ğœ)−𝑆(𝑡−𝜏))𝐾(𝜏)ğ‘‘ğœâ„Žî€œğ‘¡ğ‘¡+ℎ1𝑆(𝑡+â„Žâˆ’ğœ)𝐾(𝜏)𝑑𝜏+â„Žî€œğ‘¡0𝑆(𝑡−𝜏)𝑆(ℎ)âˆ’ğ¼â„Žî‚¶ğ¾(𝜏)𝑑𝜏.(3.17) When ℎ→0, the last terms in the line of (3.17) have limits: 𝑆(0)𝐾(𝑡)−𝑡0𝑆(𝑡−𝜏)𝐵𝐾(𝜏)𝑑𝜏=𝑆(0)𝐾(𝑡)−𝐵𝑡0𝑆(𝑡−𝜏)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),𝑢0∈𝐷(𝐵),(3.19) and for any 𝑇>0, 𝐾𝑡∈𝐿1([]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 𝑤 given by the following integral: 𝑤(𝑡)=𝑡0𝑆(𝑡−𝜏)𝐾1𝑑𝜏(3.21) belongs to 𝐶([0,𝑇],𝐻). Indeed, we infer from the difference 𝑤(𝑡+ℎ)−𝑤(𝑡)=0𝑡+â„Žğ‘†(𝑡+â„Žâˆ’ğœ)𝐾1(𝜏)𝑑𝜏−𝑡0𝑆(𝑡−𝜏)𝐾1(𝜏)𝑑𝜏=(𝑆(ℎ)−𝐼)𝑤(𝑡)+𝑡𝑡+â„Žğ‘†(𝑡+â„Žâˆ’ğœ)𝐾1(𝜏)𝑑𝜏(3.22) that as ℎ→0, ‖𝑤(𝑡+ℎ)−𝑤(𝑡)‖≤‖(𝑆(ℎ)−𝐼)𝑤(𝑡)‖+𝑡𝑡+â„Žâ€–ğ¾1(𝜏)‖𝑑𝜏⟶0,(3.23) where we have used the strong continuity of 𝑆(𝑡) and the absolute continuity of integral for ‖𝐾1‖∈𝐿1[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)+𝑡0𝑆(𝑧)𝐾′(𝑡−𝑧)𝑑𝑧=𝑆(𝑡)𝐾(0)+𝑡0[]𝑆(𝑡−𝜏)𝐾′(𝜏)𝑑𝜏∈𝐶(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 []𝑤∈𝐶(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 limğ‘¡â†’âˆžğ‘“(𝑡)=0.(3.27)

Lemma 3.7. Suppose that the relaxation function 𝑔 satisfies (𝐻1) and (𝐻2). If 𝑤∈𝐻1(𝑔,(0,∞),(𝐻10(Ω))𝑛) and 𝑤(0)=0, then ğ‘”î…žâ€–(𝑠)‖𝑤(𝑠)2(𝐻10(Ω))𝑛∈𝐿1(0,∞),limğ‘ â†’âˆžğ‘”(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, 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𝑢,𝑣,𝜃,𝜃𝑡,,𝑤𝑢,𝑣,𝜃,𝜃𝑡,𝑤ℋ=𝜅(𝑣,𝑢)(𝐻10(Ω))𝑛+12𝐵(𝑢,𝑤)−𝛼∇𝜃𝑡+𝛼,𝑣2𝛽∇𝜃𝑡+𝛼,∇𝜃2𝛽Δ𝜃𝑡+Δ𝜃−div𝑣,𝜃𝑡+𝑣−𝑤𝑠,𝑤𝐿2(𝑔,(0,∞),(𝐻10(Ω))𝑛)𝛼=−‖‖2𝛽∇𝜃𝑡‖‖2+∞0ğ‘”î…žâ€–(𝑠)𝑤(𝑠)‖2(𝐻10(Ω))𝑛𝑑𝑠≤0.(3.29) Thus, 𝐴 is dissipative.
To prove that 0∈𝜌(𝐴), for any 𝐺=(𝑔1,𝑔2,𝑔3,𝑔4,𝑔5)∈ℋ, consider 𝐴Φ=𝐺,(3.30) that is, 𝑣=𝑔1𝐻,in10(Ω)𝑛,(3.31)𝐵(𝑢,𝑤)−𝛼∇𝜃𝑡=𝑔2𝐿,in2(Ω)𝑛𝜃,(3.32)𝑡=𝑔3,in𝐿2(Ω),(3.33)Δ𝜃𝑡+Δ𝜃−𝛽div𝑣=𝑔4,in𝐿2(Ω),(3.34)𝑣−𝑤𝑠=𝑔5,in𝐿2𝐻𝑔,(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+𝛽div𝑔1−Δ𝑔3∈𝐿2(Ω).(3.36) By the standard theory for the linear elliptic equations, we have a unique 𝜃∈𝐻2(Ω)∩𝐻10(Ω) satisfying (3.36).
We plug 𝑣=𝑔1 obtained from (3.31) into (3.35) to get 𝑤𝑠=𝑔1−𝑔5∈𝐿2𝐻𝑔,(0,∞),1Γ1(Ω)𝑛.(3.37) Applying the standard theory for the linear elliptic equations again, we have a unique 𝑤∈𝐻1(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 𝑢∈𝐷(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 ‖Φ‖ℋ≤𝐾‖𝐺‖ℋ 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,𝑤𝑛⟶𝑢,𝑣,𝜃,𝜃𝑡𝐴𝑢,𝑤inℋ,𝑛,𝑣𝑛,𝜃𝑛,𝜃𝑡𝑛,𝑤𝑛⟶(ğ‘Ž,𝑏,𝑐,𝑑,𝑒)inℋ.(3.38) Then we have 𝑢𝑛𝐻⟶𝑢in10(Ω)𝑛,𝑣(3.39)𝑛𝐿⟶𝑣in2(Ω)𝑛,𝜃(3.40)𝑛⟶𝜃in𝐻10𝜃(Ω),(3.41)𝑛𝑡⟶𝜃𝑡in𝐿2𝑤(Ω),(3.42)𝑛⟶𝑤in𝐿2𝐻𝑔,(0,∞),10(Ω)𝑛𝑣,(3.43)ğ‘›î€·ğ»âŸ¶ğ‘Žin10(Ω)𝑛𝐵𝑢,(3.44)𝑛,𝑤𝑛−𝛼∇𝜃𝑛𝑡𝐿⟶𝑏in2(Ω)𝑛𝜃,(3.45)𝑛𝑡⟶𝑐in𝐻10(Ω),(3.46)Δ𝜃𝑛𝑡+Δ𝜃𝑛−𝛽div𝑣𝑛⟶𝑑in𝐿2𝑣(Ω),(3.47)𝑛−𝑤𝑛𝑠⟶𝑒in𝐿2𝐻𝑔,(0,∞),10(Ω)𝑛.(3.48) By (3.40) and (3.44), we deduce 𝑣𝑛𝐻⟶𝑣in10(Ω)𝑛,𝐻(3.49)𝑣=ğ‘Žâˆˆ10(Ω)𝑛.(3.50) By (3.42) and (3.46), we deduce 𝜃𝑛𝑡⟶𝜃𝑡in𝐻10𝜃(Ω),(3.51)𝑡=𝑐∈𝐻10(Ω).(3.52) By (3.47) and (3.49), we deduce Δ𝜃𝑛𝑡+Δ𝜃𝑛⟶𝑑+𝛽div𝑣in𝐿2(Ω),(3.53) and consequently, it follows from (3.41), that 𝜃𝑛𝑡+𝜃𝑛⟶𝜃𝑡+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+Δ𝜃𝑛−𝛽div𝑣,𝜃𝑡+𝜃∈𝐻2(Ω)∩𝐻10(Ω).(3.55) By (3.43), (3.48), and (3.49), we deduce 𝑤𝑛⟶𝑤in𝐻1𝐻𝑔,(0,∞),1Γ1(Ω)𝑛,(3.56)𝑒=𝑣−𝑤𝑠,𝑤∈𝐻1𝐻𝑔,(0,∞),10(Ω)𝑛,𝑤(0)=0.(3.57) In addition, it follows from (3.39), (3.43), (3.51) that 𝐵𝑢𝑛,𝑤𝑛−𝛼∇𝜃𝑛𝑡⟶𝐵(𝑢,𝑤)−𝛼∇𝜃𝑡(3.58) in the distribution. It therefore follows from (3.45) and (3.58) that 𝑏=𝐵(𝑢,𝑤)−𝛼∇𝜃𝑡𝐿,𝐵(𝑢,𝑤)∈2(Ω)𝑛,(3.59) and consequently, 𝜅𝑢+∞0𝐻𝑔(𝑠)𝑤(𝑠)𝑑𝑠∈2(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 𝐴𝑢,𝑣,𝜃,𝜃𝑡=,𝑤(ğ‘Ž,𝑏,𝑐,𝑑,𝑒),𝑢,𝑣,𝜃,𝜃𝑡,𝑤∈𝐷(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 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(𝐼−𝐴)=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 ℋ is a Hilbert space. By Lemmas 3.8–3.10, we can deduce that the operator 𝐴 is the infinitesimal generator of a 𝐶0-semigroup of contractions on Hilbert space ℋ. 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 ℋ. Applying the result and Lemma 3.11, we can conclude that 𝑅(𝐼−𝐴)=𝐻. If we choose operator 𝐵=−𝐴, we can obtain 𝐷(𝐵)=𝐷(𝐴) and 𝐷(𝐵) is dense in ℋ. Noting that by (𝐴2), we know that 𝐾=(0,𝑓,0,ℎ,0)∈𝐶1([0,∞),ℋ); 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,ℎ,0)∈𝐶([0,∞),𝐷(𝐴)) or 𝐾∈𝐶([0,∞),ℋ), and for any 𝑇>0,𝐾𝑡∈𝐿1((0,𝑇),ℋ). 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,ℎ,0) are Lipschitz continuous functions from [0,𝑇] into ℋ. Moreover, by (2.2), it is clear that ℋ is a reflexive Banach space. Therefore, 𝐾𝑡∈𝐿1([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,ℎ,0) satisfies the global Lipschitz condition on ℋ. 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(𝑡−𝜏)𝐾(Φ(𝜏))𝑑𝜏,(4.1)Ω=Φ∈𝐶(0,+∞),𝐻)∣sup𝑡≥0‖Φ(𝑡)‖𝑒−𝑘𝑡<∞,(4.2) where 𝑘 is a positive constant such that 𝑘>𝐿. In Ω, we introduce the following norm: ‖Φ‖Ω=sup𝑡≥0‖Φ(𝑡)‖𝑒−𝑘𝑡.(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 Φ∈Ω, we have ‖𝜙(Φ)‖≤‖𝑆(𝑡)Φ0‖+𝑡0‖𝑆(𝑡−𝜏)‖‖𝐾(Φ)‖𝑑𝜏≤‖Φ0‖+𝑡0‖𝐾(Φ)‖𝑑𝜏≤‖Φ0‖+𝑡0(𝐿‖Φ(𝜏)‖+‖𝐾(0)‖)𝑑𝜏≤‖Φ0‖+𝐶0𝑡+𝐿sup𝑡≥0‖Φ(𝑡)‖𝑒−𝑘𝑡𝑡0𝑒𝑘𝜏𝑑𝜏≤‖Φ0‖+𝐶0𝐿𝑡+𝑘𝑒𝑘𝑡‖Φ‖Ω,(4.4) where 𝐶0=‖𝐾(0)‖. Thus, ‖‖𝜙(Φ)Ω≤sup𝑡≥0‖Φ0‖+𝐶0𝑡𝑒−𝑘𝑡+𝐿𝑘‖Φ‖Ω<∞.(4.5) that is, 𝜙(Φ)∈Ω.
For Φ1,Φ2∈Ω, we have ‖‖𝜙Φ1Φ−𝜙2‖‖Ω=sup𝑡≥0𝑒−𝑘𝑡‖‖‖𝑡0𝐾Φ𝑆(𝑡−𝜏)1(Φ𝜏)−𝐾2(‖‖‖𝜏)𝑑𝜏≤sup𝑡≥0𝑒−𝑘𝑡𝐿𝑡0‖Φ1−Φ2‖𝑑𝜏≤sup𝑡≥0𝑒−𝑘𝑡⋅𝐿𝑘⋅𝑒𝑘𝑡‖‖Φ−11−Φ2‖‖Ω≤𝐿𝑘‖Φ1−Φ2‖Ω.(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,Φ2∈𝐶([0,∞),𝐻) be two solutions of the problem and let Φ=Φ1−Φ2. Then Φ(𝑡)=𝑡0𝐾Φ𝑆(𝑡−𝜏)1Φ−𝐾2𝑑𝜏,‖Φ(𝑡)‖≤𝐿𝑡0‖Φ(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,ℎ,0) satisfies the global Lipschitz condition on 𝐷(𝐴). Let 𝐴1=𝐷(𝐵),𝐵1=𝐵2𝐵∶𝐷1𝐵=𝐷2⟼𝐴1.(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,𝑦∈𝐷(𝐵2), since 𝐵 is accretive in 𝐻, we have ‖‖𝑥−𝑦+𝜆(𝐵𝑥−𝐵𝑦)𝐷(𝐵)=‖‖𝑥−𝑦+𝜆(𝐵𝑥−𝐵𝑦)2+‖‖𝐵𝐵𝑥−𝐵𝑦+𝜆2𝑥−𝐵2𝑦‖‖21/2≥‖𝑥−𝑦‖2+‖𝐵𝑥−𝐵𝑦‖21/2=‖𝑥−𝑦‖𝐷(𝐵).(4.9) that is, 𝐵1 is accretive in 𝐴1. Furthermore, since 𝐵 is 𝑚-accretive in 𝐻, for any 𝑦∈𝐻, there is a unique 𝑥∈𝐷(𝐵) such that 𝑥+𝐵𝑥=𝑦.(4.10) Now for any 𝑦∈𝐴1=𝐷(𝐵), (4.10) admits a unique solution 𝑥∈𝐷(𝐵). It turns out that 𝐵𝑥=𝑦−𝑥∈𝐷(𝐵).(4.11) Thus 𝑥∈𝐷(𝐵2); that is, 𝐵1 is m-accretive in 𝐴1. Let 𝑆1(𝑡) be the semigroup generated by 𝐵1. If Φ0∈𝐷(𝐵2)=𝐷(𝐵1), then Φ(𝑡)=𝑆1(𝑡)Φ0[𝐵∈𝐶0,+∞),𝐷2∩𝐶1([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 Φ∈𝐶([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).