Global Existence of the Higher-Dimensional Linear System of Thermoviscoelasticity

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: 𝑢_𝑡𝑡+−𝜇𝜇𝑔Δ∗𝑢Δ−𝑢(𝜆++(𝜆𝜇+)∇𝜇)d𝑔iv∗𝑢∇div𝑢+𝛼∇𝜃_𝑡𝜃=𝑓,(𝑥,𝑡)∈Ω×(0,∞),_𝑡𝑡−Δ𝜃_𝑡−Δ𝜃+𝛽div𝑢_𝑡=ℎ,(𝑥,𝑡)∈Ω×(0,∞),(1.1) where the sign “∗” denotes the convolution product in time, which is defined by 𝑔∗𝑣(𝑡)=^𝑡−∞𝑔(𝑡−𝑠)𝑣(𝑥,𝑠)𝑑𝑠(1.2) with the initial data 𝜃𝑢(𝑥,0)=𝑢₀(𝑥),𝑢_𝑡(𝑥,0)=𝑢₁(𝑥),𝜃(𝑥,0)=𝜃₀(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 𝑅^𝑛 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),…,𝑢_𝑛(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,𝑔^(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‖‖𝑣_𝑖‖‖²,for𝑣=𝑣₁,𝑣₂,…,𝑣_𝑛.(1.5)

When 𝑓=𝑔=ℎ=0, system (1.1)–(1.4) is reduced to the thermoelastic system: 𝑢_𝑡𝑡−𝜃𝜇Δ𝑢−(𝜆+𝜇)∇div𝑢+𝛼∇𝜃_𝑡=0,(𝑥,𝑡)∈Ω×(0,∞),_𝑡𝑡𝑢(𝑥,0)=𝑢−₀Δ𝜃𝑢_𝑡−=Δ0,𝜃+𝜃𝛽=d0iv,𝑢_𝑡(=𝑥,0𝑡),∈(Γ𝑥,×𝑡)(∈0,Ω∞×),(0,∞),(𝑥),𝑢_𝑡(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,𝜎∣_𝑥=𝑙=0,𝜃_𝑥∣_𝑥=0,𝑙=0,(1.7)𝑢∣_𝑥=0=0,𝜎∣_𝑥=𝑙=0,𝜃∣_𝑥=0=0,𝜃_𝑥∣_𝑥=𝑙=0,(1.8) or𝑢∣_𝑥=0=0,𝜎∣_𝑥=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,𝜎∣_𝑥=𝑙=0,𝜃_𝑥∣_𝑥=0=0,𝜃∣_𝑥=𝑙=0,(1.10) where 𝜎=𝑢_𝑥−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: 𝑢_𝑡𝑡−𝜇Δ𝑢−(𝜆+𝜇𝑢)(∇𝑥,d0iv)𝑢=+𝑢𝑢₀𝜇=𝑔∗0,Δ𝑢(𝑥+,(𝑡𝜆)∈+Γ𝜇)×𝑔(∗0∇,∞d)iv,𝑢=0,(𝑥,𝑡)∈Ω×(0,∞),𝑢(𝑥,0)−𝑢(𝑥(,𝑥−)𝑠,)=𝑢𝑤_𝑡(₀𝑥,0)=𝑢₁(𝑥),(𝑥,𝑡)∈Ω,(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. 𝐿^𝑝,1≤𝑝≤+∞,𝑊^𝑚,𝑝,𝑚∈𝑁,𝐻¹=𝑊^1,2,𝐻¹⁰=𝑊^01,2 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 𝐶^𝑘(𝐼,𝐵),𝑘∈ℕ₀, the space of k-times continuously differentiable functions from 𝐽⊆𝐼 into a Banach space 𝐵, and likewise by 𝐿^𝑝(𝐼,𝐵),1≤𝑝≤+∞, the corresponding Lebesgue spaces. 𝐶^𝛽([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‖^2𝐿_{2𝐻(𝑔,(0,∞),10(Ω)𝑛)}=^∞0𝑔(𝑠)‖𝑤(𝑠)‖^2𝐻_{10(Ω)𝑛}𝑑𝑠<∞.(2.1) Put ℋ=𝐻¹⁰(Ω)^𝑛×𝐿²(Ω)^𝑛×𝐻¹⁰(Ω)×𝐿²(Ω)×𝐿²𝑔,(0,∞),𝐻¹⁰(Ω)^𝑛(2.2) with the energy norm ‖‖𝑢,𝑣,𝜃,𝜃_𝑡,𝑤‖‖_ℋ=𝜅‖𝑢‖^2𝐻_{10(Ω)𝑛}+12‖𝑣‖²+𝛼𝛽‖‖𝜃^𝑡+‖‖²+‖∇𝜃‖_∞0𝑔(𝑠)‖𝑤(𝑠)‖^2𝐻_{10(Ω)𝑛}𝑑𝑠^1/2,(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:𝑢_𝑡𝑡−𝜇Δ𝑢−(𝜆+𝜇)∇𝜃div𝑢+𝜇𝑔∗Δ𝑢+(𝜆+𝜇)𝑔∗∇div𝑢+𝛼∇𝜃_𝑡=0,(𝑥,𝑡)∈Ω×(0,∞),_𝑡𝑡𝑢(𝑥,0)=𝑢₀−Δ𝜃𝑢_𝑡=−0Δ,𝜃+𝜃𝛽=di0v,𝑢_𝑡(=𝑥,0𝑡,)∈(𝑥Γ,×𝑡)∈(0,Ω∞×),(0,∞),(𝑥),𝑢(𝑥𝑢,_𝑡0(𝑥),−0)𝑢=(𝑥,𝑢−₁(𝑠𝑥))=,𝑤₀𝜃(𝑥,0)=𝜃₀(𝑥),𝜃_𝑡(𝑥,0)=𝜃₁(𝑥),𝑥∈Ω,(𝑥,𝑠),(𝑥,𝑡)∈Ω×(0,∞).(2.5) Let 𝑣(𝑥,𝑡)=𝑢_𝑡(𝑥,𝑡),𝑤(𝑥,𝑡,𝑠)=𝑢(𝑥,𝑡)−𝑢(𝑥,𝑡−𝑠).(2.6) Since 𝜕𝜕𝜈^𝑡−∞𝑔(𝑡−𝑠)𝑢(𝑠)𝑑𝑠=𝜕𝜕𝜈^∞0=𝑔(𝑠)𝑢(𝑡−𝑠)𝑑𝑠_∞0𝑔(𝑠)𝜕=(1−𝜅)𝜕𝜕𝑢𝜈(𝑥(𝑢,𝑡()𝑡)−𝑤(𝑡,𝑠))𝑑𝑠𝜕𝜈−^∞0𝑔(𝑠)𝜕𝑤(𝑡,𝑠)𝜕𝜈𝑑𝑠,(2.7) System (2.5) can be written as follows: 𝑢_𝑡𝑡+−𝛼𝜅∇𝜇𝜃Δ_𝑡𝑢−𝜅(𝜆+𝜇)∇div𝑢−(𝜆+𝜇−)𝜇_∞0^∞0𝑔(𝑠)Δ𝑤(𝑡,𝑠)𝑑𝑠𝜃𝑔(𝑠)∇div𝑤(𝑡,𝑠)𝑑𝑠=0,(𝑥,𝑡)∈Ω×(0,∞),_𝑡𝑡𝑢𝑤𝑢(=(𝑥−𝑥,0,0Δ,𝑡),𝜃𝑠=_𝑡)𝜃−𝑢=₀=Δ𝑢𝜃0(𝑥,+,𝑡𝛽)(𝑥−d,i𝑡v𝑢)𝑢(∈𝑥_𝑡,=Γ𝑡−×0,𝑠()0(,,𝑥∞,(𝑡)𝑥),,∈𝑡,Ω𝑠)×∈(Ω0,×∞()0,,∞)×(0,∞),𝑤(0,𝑠)=𝑤₀(𝑥),𝑢_𝑡(𝑥,0)=𝑢₁(𝑥),𝜃(𝑥,0)=𝜃₀(𝑥),𝜃_𝑡(𝑥,0)=𝜃₁(𝑥),𝑥∈Ω,(𝑠),(𝑥,𝑡)∈Ω×(0,∞).(2.8) We define a linear unbounded operator 𝐴 on ℋ by 𝐴𝑢,𝑣,𝜃,𝜃_𝑡,𝑤=𝑣,𝐵(𝑢,𝑤)−𝛼∇𝜃_𝑡,𝜃_𝑡,Δ𝜃_𝑡+Δ𝜃−𝛽div𝑣,𝑣−𝑤_𝑠,(2.9) where 𝑤_𝑠=𝜕𝑤/𝜕𝑠 and 𝐵(𝑢,𝑤)=+𝜅𝜇(𝜆Δ+𝑢+𝜇)𝜅_∞0(𝜆+𝜇)∇div𝑢+𝜇^∞0𝑔(𝑠)Δ𝑤(𝑠)𝑑𝑠𝑔(𝑠)∇div𝑤(𝑠)𝑑𝑠.(2.10) Set 𝑣(𝑥,𝑡)Φ==𝑢_𝑡(𝑢𝑥,,𝑣𝑡,)𝜃,,𝜃_𝑡𝑤(𝑥,𝑡,𝑠)=𝑢(𝑥,𝑡)−𝑢(𝑥,𝑡−𝑠),,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)=𝑣(𝑢∈,𝑣,𝐻𝜃,₁₀𝑤)∈ℋ∶𝜃∈𝐻¹⁰(Ω),𝜃_𝑡∈𝐻¹⁰(Ω),𝜃+𝜃_𝑡∈𝐻²(Ω)∩𝐻¹⁰(Ω),𝑤(𝑠)∈𝐻(Ω₁)^𝑛,𝜅𝑢+^∞0𝑔(𝑠)𝑤(𝑠)𝑑𝑠∈𝐻²(Ω)∩𝐻¹⁰(Ω)^𝑛,𝑔,(0,∞),𝐻¹⁰(Ω)^𝑛,𝑤(0)=0,(2.13) where 𝐻¹𝑔,(0,∞),𝐻¹⁰(Ω)^𝑛=𝑤∶𝑤,𝑤_𝑠∈𝐿²𝑔,(0,∞),𝐻¹⁰(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, 𝑓_𝑡∈𝐿¹((0,𝑇),(𝐿²(Ω))^𝑛),ℎ_𝑡∈𝐿¹((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,𝜃_𝑡,𝑤) to system (2.8) satisfying Φ=(𝑢,𝑣,𝜃,𝜃_𝑡,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,𝜃_𝑡,𝑤) to system (2.8) satisfying Φ=(𝑢,𝑣,𝜃,𝜃_𝑡,𝑤)∈𝐶¹([0,∞),ℋ)∩𝐶([0,∞),𝐷(𝐴)), that is, 𝑣𝑢∈∈𝐶𝐶¹₁[0,∞),𝐻¹⁰(Ω)^𝑛∩𝐶[0,∞),𝐻²(Ω)∩𝐻¹⁰(Ω)^𝑛,𝜃∈𝐶₁[0,∞),𝐿²(Ω)^𝑛∩𝐶[0,∞),𝐻¹⁰(Ω)^𝑛,𝜃[0,∞),𝐻¹⁰(Ω)∩𝐶[0,∞),𝐻²(Ω)∩𝐻¹⁰(Ω)_𝑡,𝑤∈∈𝐶𝐶¹₁[0,∞),𝐿²(Ω)∩𝐶[0,∞),𝐻¹⁰(Ω),[0,∞),𝐿²𝑔,(0,∞),𝐻¹⁰(Ω)^𝑛∩𝐶[0,∞),𝐻¹𝑔,(0,∞),𝐻¹⁰(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,𝜃_𝑡,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,𝜃_𝑡,𝑤)∈𝐷(𝐴), there exists a unique global classical solution Φ=(𝑢,𝑣,𝜃,𝜃_𝑡,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,𝜃_𝑡,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−𝐾Φ₂‖‖_ℋ≤𝐿‖‖Φ₁−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, 𝑢∈𝐶𝑣∈[0,𝐶∞)[0,,∞𝐻)^1Γ₁,(Ω𝐿)₂^𝑛,𝜃∈𝐶[0,∞),𝐻¹⁰(Ω),𝜃_𝑡∈𝐶[0,∞),𝐿²(Ω),(Ω)^𝑛,𝑤∈𝐶([0,∞),𝐿²𝑔,(0,∞),𝐻¹⁰(Ω)^𝑛.(2.17)

Theorem 2.6. Suppose 𝑓=𝑓(Φ) and ℎ=ℎ(Φ),Φ=(𝑢,𝑣,𝜃,𝜃_𝑡,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−𝐾Φ₂‖‖_𝐷(𝐴)≤𝐿‖‖Φ₁−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,𝜃_𝑡,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, 𝑑𝑢𝑑𝑡𝑢+(0)𝐵=𝑢=𝑢₀𝐾,,(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, Re(𝐵𝑥,𝑥)≥−𝜆2‖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≤≤R‖𝑥e−(𝑥𝑦−‖𝑦‖,𝑥𝑥−−𝑦𝑦++𝜆𝜆(𝐵𝐵(𝑥𝑥−−𝐵𝑦𝑦)))‖.(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=ℎ1^0𝑡+ℎ=1𝑆(𝑡+ℎ−𝜏)𝐾(𝜏)𝑑𝜏−^𝑡0𝑆(𝑡−𝜏)𝐾(𝜏)𝑑𝜏ℎ^𝑡𝑡+ℎ𝑆(𝑡+ℎ−𝜏)𝐾(𝜏)𝑑𝜏+1ℎ^𝑡0=1(𝑆(𝑡+ℎ−𝜏)−𝑆(𝑡−𝜏))𝐾(𝜏)𝑑𝜏ℎ^𝑡𝑡+ℎ𝑆(𝑧)𝐾(𝑡+ℎ−𝑧)𝑑𝑧+1ℎ^𝑡0𝑆(𝑧)(𝐾(𝑡+ℎ−𝑧)−𝐾(𝑡−𝑧))𝑑𝑧.(3.13) When ℎ→0, the terms in the last line of (3.13) have limits: 𝑆(𝑡)𝐾(0)+^𝑡0𝑆(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=ℎ1^𝑡𝑡+ℎ𝑆(𝑡+ℎ−𝜏)𝐾(𝜏)𝑑𝜏+1ℎ^𝑡0=1(𝑆(𝑡+ℎ−𝜏)−𝑆(𝑡−𝜏))𝐾(𝜏)𝑑𝜏ℎ^𝑡𝑡+ℎ𝑆(𝑡+ℎ−𝜏)𝐾(𝜏)𝑑𝜏+1ℎ^𝑡0𝑆(𝑡−𝜏)𝑆(ℎ)−𝐼ℎ𝐾(𝜏)𝑑𝜏.(3.17) When ℎ→0, the last terms in the line of (3.17) have limits: 𝑆(0)=𝐾𝑆(𝑡()0−)𝐾^𝑡0(𝑆𝑡)(−𝑡−𝐵𝜏_𝑡0)𝐵𝐾(𝜏)𝑑𝜏𝑆(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, 𝐾_𝑡∈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)==(𝑆(^0𝑡+ℎℎ𝑆)−(𝑡𝐼+)ℎ𝑤−(𝑡𝜏))+𝐾₁(𝜏_𝑡𝑡+ℎ)𝑑𝜏−^𝑡0𝑆(𝑡−𝜏)𝐾₁(𝜏)𝑑𝜏𝑆(𝑡+ℎ−𝜏)𝐾₁(𝜏)𝑑𝜏(3.22) that as ℎ→0, ‖𝑤(𝑡+ℎ)−𝑤(𝑡)‖≤‖(𝑆(ℎ)−𝐼)𝑤(𝑡)‖+^𝑡𝑡+ℎ‖𝐾₁(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)=𝐾𝑆(0()𝑡)+𝐾^𝑡0(0𝑆)(+𝑧)𝐾_𝑡0′(𝑡−𝑧)𝑑𝑧𝑆(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 lim_𝑡→∞𝑓(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 𝑔l^im(𝑠_𝑠→∞)‖𝑤(𝑠)‖^2(𝐻_10(Ω))𝑛∈𝐿¹(0,∞),𝑔(𝑠)‖𝑤(𝑠)‖^2(𝐻_10(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,𝜃_𝑡,𝑤,𝑢,𝑣,𝜃,𝜃_𝑡,𝑤_ℋ=𝜅(𝑣,𝑢)_(𝐻10(Ω))^𝑛+12𝐵(𝑢,𝑤)−𝛼∇𝜃^𝑡,𝑣+𝛼2𝛽∇𝜃^𝑡+𝛼,∇𝜃2𝛽Δ𝜃^𝑡+Δ𝜃−div𝑣,𝜃_𝑡+𝑣−𝑤_𝑠,𝑤_{𝐿2(𝑔,(0,∞),(𝐻10(Ω))}^𝑛₎=−𝛼2𝛽‖‖∇𝜃^𝑡≤0.‖‖²+^∞0𝑔^(𝑠)‖𝑤(𝑠)‖^2(𝐻_10(Ω))𝑛𝑑𝑠(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, 𝐵(𝑢,𝑤)𝑣−=𝛼𝑔∇₁,𝜃_𝑡in𝐻¹⁰(Ω)^𝑛,(3.31)𝜃=𝑔₂,in𝐿²(Ω)^𝑛,(3.32)_𝑡Δ𝜃_𝑡=𝑔₃,in𝐿²(Ω),(3.33)𝑣−𝑤_𝑠+Δ𝜃−𝛽div𝑣=𝑔₄,in𝐿²(Ω),(3.34)=𝑔₅,in𝐿²𝑔,(0,∞),𝐻¹⁰(Ω)^𝑛.(3.35) Inserting 𝑣=𝑔₁ and 𝜃_𝑡=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 𝑤_𝑠=𝑔₁−𝑔₅∈𝐿²𝑔,(0,∞),𝐻^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 (𝑢_𝑛,𝑣_𝑛,𝜃_𝑛,𝜃_𝑛𝑡,𝑤_𝑛)∈𝐷(𝐴) be such that 𝐴𝑢_𝑛,𝑢𝑣_𝑛,𝜃_𝑛,𝜃_𝑛𝑡,𝑤_𝑛⟶𝑢,𝑣,𝜃,𝜃_𝑡,𝑤inℋ,_𝑛,𝑣_𝑛,𝜃_𝑛,𝜃_𝑡𝑛,𝑤_𝑛⟶(𝑎,𝑏,𝑐,𝑑,𝑒)inℋ.(3.38) Then we have 𝑢𝑣_𝑛⟶𝑢in𝐻¹⁰(Ω)^𝑛,(3.39)_𝑛𝜃⟶𝑣in𝐿²(Ω)^𝑛,(3.40)_𝑛𝜃⟶𝜃in𝐻¹⁰(Ω),(3.41)_𝑛𝑡𝑤⟶𝜃_𝑡in𝐿²(Ω),(3.42)_𝑛⟶𝑤𝑣in𝐿²𝑔,(0,∞),𝐻¹⁰(Ω)^𝑛,(3.43)_𝑛𝐵𝑢⟶𝑎in𝐻¹⁰(Ω)^𝑛,(3.44)_𝑛,𝑤_𝑛𝜃−𝛼∇𝜃_𝑛𝑡⟶𝑏in𝐿²(Ω)^𝑛,(3.45)_𝑛𝑡Δ𝜃_𝑛𝑡⟶𝑐in𝐻¹⁰(Ω),(3.46)𝑣+Δ𝜃_𝑛−𝛽div𝑣_𝑛⟶𝑑in𝐿²(Ω),(3.47)_𝑛−𝑤_𝑛𝑠⟶𝑒in𝐿²𝑔,(0,∞),𝐻¹⁰(Ω)^𝑛.(3.48) By (3.40) and (3.44), we deduce 𝑣_𝑛𝑣⟶=𝑎𝑣∈in𝐻₁₀𝐻¹⁰(Ω)^𝑛,(3.49)(Ω)^𝑛.(3.50) By (3.42) and (3.46), we deduce 𝜃_𝑛𝑡𝜃⟶𝜃_𝑡in𝐻¹⁰(Ω),(3.51)_𝑡=𝑐∈𝐻¹⁰(Ω).(3.52) By (3.47) and (3.49), we deduce Δ𝜃_𝑛𝑡+Δ𝜃_𝑛⟶𝑑+𝛽div𝑣in𝐿²(Ω),(3.53) and consequently, it follows from (3.41), that 𝜃_𝑛𝑡+𝜃_𝑛⟶𝜃_𝑡+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=Δ𝜃_𝑛𝑡+Δ𝜃_𝑛−𝛽div𝑣,𝜃_𝑡+𝜃∈𝐻²(Ω)∩𝐻¹⁰(Ω).(3.55) By (3.43), (3.48), and (3.49), we deduce 𝑒=𝑣−𝑤𝑤^𝑛_𝑠⟶𝑤in𝐻¹𝑔,(0,∞),𝐻^1Γ₁(Ω)^𝑛,(3.56),𝑤∈𝐻¹𝑔,(0,∞),𝐻¹⁰(Ω)^𝑛,𝑤(0)=0.(3.57) In addition, it follows from (3.39), (3.43), (3.51) that 𝐵𝑢_𝑛,𝑤_𝑛−𝛼∇𝜃_𝑛𝑡⟶𝐵(𝑢,𝑤)−𝛼∇𝜃_𝑡(3.58) in the distribution. It therefore follows from (3.45) and (3.58) that 𝑏=𝐵(𝑢,𝑤)−𝛼∇𝜃_𝑡,𝐵(𝑢,𝑤)∈𝐿²(Ω)^𝑛,(3.59) and consequently, 𝜅𝑢+^∞0𝑔(𝑠)𝑤(𝑠)𝑑𝑠∈𝐻²(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,𝜃_𝑡,𝑤=(𝑎,𝑏,𝑐,𝑑,𝑒),𝑢,𝑣,𝜃,𝜃_𝑡,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,𝐾_𝑡∈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, 𝐾_𝑡∈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(ΦΩ)==𝑆(Φ𝑡)∈Φ₀𝐶+([0^𝑡0,𝑆+∞(𝑡)−,𝐻𝜏))𝐾∣(sΦup_𝑡≥0(𝜏))𝑑𝜏,(4.1)‖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=sup_𝑡≥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)‖≤≤‖‖𝑆Φ₀(𝑡)Φ₀‖+^𝑡0‖𝑆(𝑡−𝜏)‖‖𝐾(Φ)‖𝑑𝜏≤‖Φ₀‖+^𝑡0‖𝐾(Φ)‖𝑑𝜏≤‖Φ₀‖+^𝑡0(𝐿‖Φ(𝜏)‖+‖𝐾(0)‖)𝑑𝜏‖+𝐶₀𝑡+𝐿sup_𝑡≥0‖Φ(𝑡)‖𝑒^−𝑘𝑡^𝑡0≤‖Φ₀𝑒^𝑘𝜏𝑑𝜏‖+𝐶₀𝑡+𝐿𝑘𝑒^𝑘𝑡‖Φ‖_Ω,(4.4) where 𝐶₀=‖𝐾(0)‖. Thus, ‖𝜙(Φ)‖_Ω≤sup_𝑡≥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=sup_𝑡≥0𝑒^−𝑘𝑡‖‖‖^𝑡0≤sup_𝑡≥0𝑆(𝑡−𝜏)𝐾Φ₁(𝜏)−𝐾Φ₂(𝜏)𝑑𝜏‖‖‖𝑒^−𝑘𝑡𝐿^𝑡0‖Φ₁−Φ₂‖𝑑𝜏≤sup_𝑡≥0𝑒^−𝑘𝑡⋅𝐿𝑘⋅𝑒^𝑘𝑡−1‖‖Φ₁−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)=‖^𝑡0Φ𝑆((𝑡𝑡)−‖≤𝜏)𝐿𝐾_𝑡0Φ₁−𝐾Φ₂𝑑𝜏,‖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∶𝐷𝐵₁=𝐷𝐵²⟼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+‖‖𝐵𝑥−𝐵𝑦+𝜆𝐵²𝑥−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∈𝐶[0,+∞),𝐷𝐵²∩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).