Global Existence of Strong Solutions to a Class of Fully Nonlinear Wave Equations with Strongly Damped Terms

Abstract

We consider the global existence of strong solution 𝑢, corresponding to a class of fully nonlinear wave equations with strongly damped terms 𝑢_𝑡𝑡−𝑘Δ𝑢_𝑡=𝑓(𝑥,Δ𝑢)+𝑔(𝑥,𝑢,𝐷𝑢,𝐷²𝑢) in a bounded and smooth domain Ω in 𝑅^𝑛, where 𝑓(𝑥,Δ𝑢) is a given monotone in Δ𝑢 nonlinearity satisfying some dissipativity and growth restrictions and 𝑔(𝑥,𝑢,𝐷𝑢,𝐷²𝑢) is in a sense subordinated to 𝑓(𝑥,Δ𝑢). By using spatial sequence techniques, the Galerkin approximation method, and some monotonicity arguments, we obtained the global existence of a solution 𝑢∈𝐿^∞_loc((0,∞),𝑊^2,𝑝(Ω)∩𝑊^01,𝑝(Ω)).

1. Introduction

We are concerned with the following mixed problem for a class of fully nonlinear wave equations with strongly damped terms in a bounded and 𝐶^∞ domain Ω⊂𝑅^𝑛: 𝑢_𝑡𝑡−𝑘Δ𝑢_𝑡=𝑢𝑓(0(,𝑥𝑥,)Δ=𝑢)𝜑+,𝑔𝑢𝑥_𝑡,𝑢,𝐷𝑢,𝐷²𝑢,in[0,∞)×Ω,𝑢(𝑡,𝑥)=0,o(n0,𝑥[0),=∞𝜓)×,𝜕iΩn,Ω,(1.1) where 𝑢_𝑡=𝜕𝑢𝜕𝑡,𝑢^𝑡𝑡=𝜕²𝑢𝜕𝑡²,Δ=^𝑛_𝑖=1𝜕²𝜕𝑥^2𝑖,𝐷=𝜕𝜕𝑥₁,…,𝜕𝜕𝑥_𝑛,𝐷²=𝜕²𝜕𝑥^𝛼1₁⋯𝜕𝑥^𝛼𝑛_𝑛𝛼₁,+⋯+𝛼_𝑛=2,𝑥=𝑥₁,…,𝑥_𝑛,𝑘>0.(1.2) Equations of type (1.1) are a class essential nonlinear wave equations describing the speed of strain waves in a viscoelastic configuration (e.g., a bar if the space dimension 𝑁=1 and a plate if 𝑁=2) made up of the material of the rate type [1, 2]. They can also be seen as field equations governing the longitudinal motion of a viscoelastic bar obeying the nonlinear Voigt model [3]. Concerning damped cases, there is much to the global existence of solutions for the problem: 𝑢(0𝑢,_𝑡𝑡𝑥+)=𝑢_𝑡𝑢−₀𝑢_𝑥𝑥=𝑓(𝑢),in[0,∞)×Ω,(𝑥),𝑢_𝑡(0,𝑥)=𝑢₁(𝑥),inΩ;(1.3) they discussed the global existence of weak solutions and regularity in 𝑅¹ and 𝑅^𝑛 [4–8]. On the other hand, Ikehata and Inoue [9] considered the global existence of weak solutions for two-dimensional problem in an exterior domain Ω⊂𝑅² with a compact smooth boundary 𝜕Ω for a semilinear strongly damped wave equation with a power-type nonlinearity |𝑢|^𝑞 and 𝑞>6: 𝑢_𝑡𝑡(𝑡,𝑥)𝑢−(0Δ,𝑢𝑥()𝑡=,𝑥𝑢)₀−Δ𝑢_𝑡(𝑡,𝑥)=||𝑢(𝑡,𝑥)||^𝑞in[0,∞)×Ω,𝑢(𝑡,𝑥)(𝑥=)0,,𝑢o_𝑡(n0,[𝑥0),∞=)𝑢×₁(𝜕𝑥Ω).inΩ,(1.4) Cholewa and Dlotko [10] discussed the global solvability and asymptotic behavior of solutions to semilinear Cauchy problem for strongly damped wave equation in the whole of 𝑅^𝑛. They assume the nonlinear term 𝑓 grows like |𝑢|^𝑞 and 𝑞<(𝑛+2)/(𝑛−2) if 𝑛≥3. Similar problems attracted attention of the researchers for many years [11–13]. Especially, Yang [14] studied the global existence of weak solutions to the more general equation including (1.4), but he did not discuss the regularity of weak solution for the quasilinear wave equation. We are interested in discussing the global existence and regularity of weak solutions for strongly damped wave equation with the dissipative terms 𝑔 containing 𝐷𝑢 and the nonlinear terms 𝑓 containing Δ𝑢. Here 𝑓(𝑥,Δ𝑢) is a given monotone in Δ𝑢 nonlinearity satisfying some dissipativity and growth restrictions and 𝑔(𝑥,𝑢,𝐷𝑢,𝐷²𝑢) is in a sense subordinated to 𝑓(𝑥,Δ𝑢).

In [15], we have investigated the existence of global solutions to a class of nonlinear damped wave operator equations. In this paper, our first aim is to study the global existence of strong solutions to the more general equation including (1.4), which is the motivation that we establish our abstract strongly damped wave equation model with. The second aim is to deal with the global existence of strong solutions to a class of fully nonlinear wave equations with strongly damped terms under some weakly growing conditions.

This paper is organized as follows:(i)in Section 2, we recall some preliminary tools and definitions;(ii)in Section 3, we put forward our abstract strongly damped wave equation model and proof the global existence of strong solution of it;(iii)in Section 4, we provide the proof of the main results about the mixed problem (1.1).

2. Preliminaries

We introduce two spatial sequences: 𝑋𝑋⊂𝐻₃⊂𝑋₂⊂𝑋₁⊂𝐻,₂⊂𝐻₂⊂𝐻₁⊂𝐻,(2.1) where 𝐻, 𝐻₁, 𝐻₂, and 𝐻₃ are Hilbert spaces, 𝑋 is a linear space, and 𝑋₁,𝑋₂ are Banach spaces.

All embeddings of (2.1) are dense. Let 𝐿∶𝑋⟶⟨𝐿𝑋𝑢₁,𝑣⟩b_𝐻eone-for-onedenselinearoperator,=⟨𝑢,𝑣⟩_𝐻1,∀𝑢,𝑣∈𝑋.(2.2)

Furthermore, 𝐿 has eigenvectors {𝑒_𝑘} satisfying 𝐿𝑒_𝑘=𝜆_𝑘𝑒_𝑘,(𝑘=1,2,…),(2.3) and {𝑒_𝑘} constitutes common orthogonal basis of 𝐻 and 𝐻₃.

We consider the following abstract wave equation model: 𝑑²𝑢𝑑𝑡²+𝑘𝑑𝑢(0)𝑑=𝑡ℒ𝜑,𝑢=𝐺𝑢_𝑡(𝑢),𝑘>0,(0)=𝜓,(2.4) where 𝐺∶𝑋₂×𝑅⁺→𝑋^∗1 is a map, 𝑅⁺=[0,∞), and ℒ∶𝑋₂→𝑋₁ is a bounded linear operator, satisfying ⟨ℒ𝑢,𝐿𝑣⟩_𝐻=⟨𝑢,𝑣⟩_𝐻2,∀𝑢,𝑣∈𝑋₂.(2.5)

Definition 2.1. We say that 𝑢∈𝑊^1,∞((0,𝑇),𝐻₁)∩𝐿^∞((0,𝑇),𝑋₂) is a global weak solution of the (2.4) provided for (𝜑,𝜓)∈𝑋₂×𝐻₁⟨𝑢_𝑡,𝑣⟩_𝐻+𝑘⟨ℒ𝑢,𝑣⟩_𝐻=^𝑡0⟨𝐺(𝑢),𝑣⟩𝑑𝜏+⟨𝜓,𝑣⟩_𝐻+𝑘⟨ℒ𝜑,𝑣⟩_𝐻,(2.6) for each 𝑣∈𝑋₁ and 0≤𝑡≤𝑇<∞.

Definition 2.2. Let 𝑢_𝑛, 𝑢₀∈𝐿^𝑝((0,𝑇),𝑋₂). We say that 𝑢_𝑛⇀𝑢₀ in 𝐿^𝑝((0,𝑇),𝑋₂) is uniformly weakly convergent if {𝑢_𝑛}⊂𝐿^∞((0,𝑇),𝐻) is bounded, and lim_𝑛→∞𝑢_𝑛⇀𝑢₀,in𝐿^𝑝(0,𝑇),𝑋₂,^𝑇0||⟨𝑢_𝑛−𝑢₀,𝑣⟩_𝐻||²𝑑𝑡=0,∀𝑣∈𝐻.(2.7)

Lemma 2.3 (see [16]). Let {𝑢_𝑛}∈𝐿^𝑝((0,𝑇),𝑊^𝑚,𝑝(Ω))(𝑚≥1) be bounded sequences and {𝑢_𝑛} uniformly weakly convergent to 𝑢₀∈𝐿^𝑝((0,𝑇),𝑊^𝑚,𝑝(Ω)). Then, for each |𝛼|≤𝑚−1, it follows that 𝐷^𝛼𝑢_𝑛⟶𝐷^𝛼𝑢₀,in𝐿²((0,𝑇)×Ω).(2.8)

Lemma 2.4 (see [17]). Let Ω⊂𝑅^𝑛 be an open set and 𝑓∶Ω×𝑅^𝑁→𝑅¹ satisfy Caratheodory condition and ||𝑓(𝑥,𝜉)||≤𝐶^𝑁_𝑖=1||𝜉_𝑖||^{𝑝𝑖/𝑝}+𝑏(𝑥).(2.9) If {𝑢_𝑖𝑘}⊂𝐿^𝑝𝑖(Ω)(1≤𝑖≤𝑁) is bounded and 𝑢_𝑖𝑘 convergent to 𝑢_𝑖 in Ω₀ for all bounded Ω₀⊂Ω, then for each 𝑣∈𝐿^𝑝′(Ω), the following equality holds lim_𝑘→∞_Ω𝑓𝑥,𝑢_1𝑘,…,𝑢_𝑁𝑘𝑣𝑑𝑥=_Ω𝑓𝑥,𝑢₁,…,𝑢_𝑁𝑣𝑑𝑥.(2.10)

3. Model Results

Let 𝐺=𝐴+𝐵∶𝑋₂×𝑅⁺→𝑋^∗1. Assume(A1) there is a 𝐶¹ functional 𝐹∶𝑋₂→𝑅¹ such that ⟨𝐴𝑢,𝐿𝑣⟩=⟨−𝐷𝐹(𝑢),𝑣⟩,∀𝑢,𝑣∈𝑋;(3.1)(A2) functional 𝐹∶𝑋₂→𝑅¹ is coercive, that is, 𝐹(𝑢)⟶∞,⟺‖𝑢‖_𝑋2⟶∞;(3.2)(A3)𝐵 satisfies ||⟨𝐵𝑢,𝐿𝑣⟩||≤𝐶₁𝐹(𝑢)+𝑘2‖𝑣‖^2𝐻1+𝐶₂,∀𝑢,𝑣∈𝑋,(3.3)

for 𝑔∈𝐿^1loc(0,∞).

Theorem 3.1. Set 𝐺∶𝑋₂×𝑅⁺→𝑋^∗1, for each (𝜑,𝜓)∈𝑋₂×𝐻₁, then the following assertions hold.(1)If 𝐺=𝐴 satisfies (A1) and (A2), then (2.4) has a globally weak solution 𝑢∈𝑊^1,∞(0,∞),𝐻₁∩𝑊^1,2(0,∞),𝐻₂∩𝐿^∞(0,∞),𝑋₂.(3.4)(2)If 𝐺=𝐴+𝐵 satisfies (A1)–(A3), then (2.4) has a global weak solution 𝑢∈𝑊^1,∞loc(0,∞),𝐻₁∩𝑊^1,2loc(0,∞),𝐻₂∩𝐿^∞loc(0,∞),𝑋₂.(3.5)(3)Furthermore, if ℒ∶𝑋₂→𝑋₁ is symmetric sectorial operator, that is, ⟨ℒ𝑢,𝑣⟩=⟨𝑢,ℒ𝑣⟩, and 𝐺=𝐴+𝐵 satisfies ||⟨𝐺𝑢,𝑣⟩||≤𝐶₁𝐹(𝑢)+12‖𝑣‖^2𝐻+𝐶₂,(3.6)then 𝑢∈𝑊^2,2loc((0,∞),𝐻).

Proof. Let {𝑒_𝑘}⊂𝑋 be a common orthogonal basis of 𝐻 and 𝐻₃, satisfying (2.3). Set 𝑋_𝑛=^𝑛_𝑖=1𝛼_𝑖𝑒_𝑖∣𝛼_𝑖∈𝑅¹𝑋_𝑛,=^𝑛_𝑗=1𝛽_𝑗(𝑡)𝑒_𝑗∣𝛽_𝑗∈𝐶²[0,∞).(3.7) Clearly, 𝐿𝑋_𝑛=𝑋_𝑛, 𝐿𝑋_𝑛=𝑋_𝑛.
By using Galerkin method, there exits 𝑢_𝑛∈𝐶²([0,∞),𝑋_𝑛) satisfying 𝑑𝑑𝑢𝑡^𝑛,𝑣_𝐻+𝑘⟨ℒ𝑢_𝑛,𝑣𝑢⟩_𝐻𝑛=^𝑡0𝐺𝑢_𝑛,𝑣𝑑𝜏+⟨𝜓_𝑛,𝑣⟩_𝐻+𝑘⟨ℒ𝜑_𝑛,𝑣⟩_𝐻,(0)=𝜑_𝑛,𝑢^𝑛(0)=𝜓_𝑛,(3.8) for ∀𝑣∈𝑋_𝑛, and ^𝑡0𝑑²𝑢_𝑛𝑑𝑡²,𝑣_𝐻+𝑘ℒ𝑑𝑢^𝑛𝑑𝑡,𝑣_𝐻𝑑𝜏=^𝑡0⟨𝐺𝑢_𝑛,𝑣⟩𝑑𝜏(3.9) for ∀𝑣∈𝑋_𝑛.
Firstly, we consider 𝐺=𝐴. Let 𝑣=(𝑑/𝑑𝑡)𝐿𝑢_𝑛 in (3.9). Taking into account (2.2) and (3.1), it follows that 0=^𝑡0𝑑²𝑢_𝑛𝑑𝑡²,𝑑𝑑𝑡𝐿𝑢^𝑛+𝑘ℒ𝑑𝑑𝑢𝑡^𝑛,𝑑𝑑𝑡𝐿𝑢^𝑛_𝐻𝑑𝜏−^𝑡0𝐴𝑢^𝑛,𝑑0=_𝑡0𝑑𝑡𝐿𝑢^𝑛𝑑𝜏,12𝑑𝑑𝑡𝑑𝑑𝑢𝑡^𝑛,𝑑𝑑𝑢𝑡^𝑛_𝐻1+𝑘𝑑𝑢^𝑛𝑑𝑡,𝑑𝑑𝑢𝑡^𝑛_𝐻2+𝐷𝐹𝑢_𝑛,𝑑𝑢^𝑛=1𝑑𝑡𝑑𝜏2‖‖‖𝑑𝑑𝑢𝑡^𝑛‖‖‖^2𝐻₁−12‖‖𝜓^𝑛‖‖^2𝐻₁+𝑘^𝑡0‖‖‖𝑑𝑢_𝑛𝑑𝑡‖‖‖^2𝐻₂𝑑𝜏+𝐹𝑢_𝑛−𝐹𝜑_𝑛.(3.10)
We get 12‖‖‖𝑑𝑑𝑢𝑡^𝑛‖‖‖^2𝐻₁+𝑘^𝑡0‖‖‖𝑑𝑢_𝑛𝑑𝑡‖‖‖^2𝐻₂𝑑𝜏+𝐹𝑢_𝑛=𝐹𝜑_𝑛+12‖‖𝜓^𝑛‖‖^2𝐻₁.(3.11)
Let 𝜑∈𝐻₃. From (2.1) and (2.2), it is known that {𝑒_𝑛} are orthogonal basis of 𝐻₁. We find that 𝜑_𝑛→𝜑 in 𝐻₃, and 𝜓_𝑛→𝜓 in 𝐻₁. Since 𝐻₃⊂𝑋₂ is imbedding, it follows that 𝜓𝜑_𝑛𝑛⟶𝜑,in𝑋₂,⟶𝜓,in𝐻₁.(3.12)
From (3.2), (3.11), and (3.12), we obtain that, 𝑢_𝑛⊂𝑊^1,∞loc(0,∞),𝐻₁∩𝑊^1,2loc(0,∞),𝐻₂∩𝐿^∞loc(0,∞),𝑋₂isbounded.(3.13)
Let 𝑢_𝑛∗⇀𝑢₀,𝑢_𝑛in𝑊^1,∞loc(0,∞),𝐻₁∩𝐿^∞loc(0,∞),𝑋₂,⇀𝑢₀,in𝑊^1,2loc(0,∞),𝐻₂,(3.14) which implies that 𝑢_𝑛→𝑢₀ in 𝑊^1,2loc((0,∞),𝐻) is uniformly weakly convergent from that 𝐻₂⊂𝐻 is compact imbedding.
If we have the following equality: lim_𝑛→∞−^𝑡0||⟨𝐺𝑢_𝑛−𝐺𝑢₀,𝐿𝑢_𝑛−𝐿𝑢₀⟩||𝑑𝜏+𝑘2‖‖𝑢^𝑛−𝑢₀‖‖^2𝐻₂=0,(3.15) then 𝑢₀ is a weak solution of (2.4) in view of (3.8), (3.14).
We will show (3.15) as follows. It follows that from (2.5), ^𝑡0𝑑𝑑𝑡ℒ𝑢^𝑛−𝑑𝑑𝑡ℒ𝑢⁰,𝐿𝑢_𝑛−𝐿𝑢₀_𝐻𝑑𝜏=12^𝑡0𝑑𝑑𝑡⟨𝑢^𝑛=1−𝑢₀,𝑢_𝑛−𝑢₀⟩_𝐻2𝑑𝜏2‖‖𝑢^𝑛(𝑡)−𝑢₀(𝑡)‖‖^2𝐻₂−12‖‖𝜑^𝑛−𝜑‖‖^2𝐻₂.(3.16)
Taking into account (2.2), (2.5), and (3.9), we get that −^𝑡0⟨𝐺𝑢_𝑛−𝐺𝑢₀,𝐿𝑢_𝑛−𝐿𝑢₀⟩𝑑𝜏+𝑘2‖‖𝑢^𝑛−𝑢₀‖‖^2𝐻₂=^𝑡0⟨𝐺𝑢⁰−𝐺𝑢_𝑛,𝐿𝑢_𝑛−𝐿𝑢₀⟩+𝑘𝑑𝑑𝑡ℒ𝑢^𝑛−𝑑𝑑𝑡ℒ𝑢⁰,𝐿𝑢_𝑛−𝐿𝑢₀_𝐻𝑑𝜏+𝑘2‖‖𝜑^𝑛−𝜑‖‖^2𝐻₂=^𝑡0⟨𝐺𝑢⁰,𝐿𝑢_𝑛−𝐿𝑢₀⟩+⟨𝐺𝑢_𝑛,𝐿𝑢₀⟩−⟨𝐺𝑢_𝑛,𝐿𝑢_𝑛⟩−𝑘𝑑𝑢^𝑛𝑑𝑡,𝑢⁰_𝐻2−𝑘𝑑𝑢⁰𝑑𝑡,𝑢^𝑛−𝑢₀_𝐻2+𝑘𝑑𝑑𝑡ℒ𝑢^𝑛,𝐿𝑢_𝑛_𝐻𝑑𝜏+𝑘2‖‖𝜑^𝑛−𝜑‖‖^2𝐻₂=^𝑡0⟨𝐺𝑢⁰,𝐿𝑢_𝑛−𝐿𝑢₀⟩+⟨𝐺𝑢_𝑛,𝐿𝑢₀⟩−𝑘𝑑𝑢^𝑛𝑑𝑡,𝑢⁰_𝐻2−𝑘𝑑𝑢⁰𝑑𝑡,𝑢^𝑛−𝑢₀_𝐻2−𝑑²𝑢_𝑛𝑑𝑡²+𝑘𝑑𝑑𝑡ℒ𝑢^𝑛,𝐿𝑢_𝑛_𝐻+𝑘𝑑𝑑𝑡ℒ𝑢^𝑛,𝐿𝑢_𝑛_𝐻𝑑𝜏+𝑘2‖‖𝜑^𝑛−𝜑‖‖^2𝐻₂=^𝑡0⟨𝐺𝑢⁰−𝑘𝑑,𝑢𝐿_𝑛𝑢_𝑛−𝐿𝑢₀⟩+⟨𝐺𝑢_𝑛,𝐿𝑢₀⟩𝑑𝑡,𝑢⁰_𝐻2−𝑘𝑑𝑑𝑡𝑢⁰,𝑢_𝑛−𝑢₀_𝐻2+𝑑𝑢^𝑛−𝑑𝑢_𝑛𝑑𝑡,𝑑𝑑𝑢𝑡^𝑛_𝐻1𝑑𝜏𝑑𝑡,𝑢^𝑛_𝐻1+⟨𝜓_𝑛,𝜑_𝑛⟩_𝐻1+𝑘2‖‖𝜑^𝑛−𝜑‖‖^2𝐻₂.(3.17)
From (2.1) and (3.14), we have lim_𝑛→∞lim_𝑛→∞‖‖𝜑_𝑛−𝜑‖‖_𝐻2=0,^𝑡0lim_𝑛→∞⟨𝐺𝑢₀,𝐿𝑢_𝑛−𝐿𝑢₀⟩𝑑𝜏=0,^𝑡0𝑑𝑑𝑡𝑢⁰,𝑢_𝑛−𝑢₀_𝐻2𝑑𝜏=0.(3.18)
Then, we get lim_𝑛→∞−^𝑡0⟨𝐺𝑢_𝑛−𝐺𝑢₀,𝐿𝑢_𝑛−𝐿𝑢₀⟩𝑑𝜏+𝑘2lim^𝑛→∞‖‖𝑢_𝑛−𝑢₀‖‖^2𝐻₂=lim_𝑛→∞∫^𝑡0⟨𝐺𝑢^𝑛,𝐿𝑢₀⟩−𝑘𝑑𝑢^𝑛𝑑𝑡,𝑢⁰_𝐻2+‖‖‖𝑑𝑢^𝑛−lim_𝑛→∞𝑑𝑡‖‖‖^2𝐻₁𝑑𝜏𝑑𝑑𝑢𝑡^𝑛,𝑢^𝑛_𝐻1+⟨𝜓,𝜑⟩_𝐻1.(3.19)
In view of (3.9), (3.14), we obtain for all 𝑣∈∪^∞𝑛=1𝑋_𝑛lim_𝑛→∞^𝑡0⟨𝐺𝑢_𝑛,𝐿𝑣⟩𝑑𝜏=^𝑡0𝑘𝑑𝑑𝑢𝑡⁰,𝑣_𝐻2−𝑑𝑢⁰+𝑑𝑢₀𝑑𝑡,𝑑𝑑𝑣𝑡_𝐻1𝑑𝜏𝑑𝑡,𝑣_𝐻1−⟨𝜓,𝑣(0)⟩_𝐻1.(3.20)
Since ∪^∞𝑛=1𝑋_𝑛 is dense in 𝑊^1,2((0,𝑇),𝐻₂)∩𝐿^𝑝((0,𝑇),𝑋₂), for all 𝑝<∞, (3.20) holds for all 𝑣∈𝑊^1,2((0,𝑇),𝐻₂)∩𝐿^𝑝((0,𝑇),𝑋₂). Thus, we have lim_𝑛→∞^𝑡0⟨𝐺𝑢_𝑛,𝐿𝑢₀⟩𝑑𝜏=^𝑡0𝑘𝑑𝑑𝑢𝑡⁰,𝑢⁰_𝐻2−‖‖‖𝑑𝑢⁰+𝑑𝑢₀𝑑𝑡‖‖‖^2𝐻₁𝑑𝜏𝑑𝑡,𝑢⁰_𝐻1−⟨𝜓,𝜑⟩_𝐻1.(3.21)
From (3.14) and 𝐻₂⊂𝐻₁ being compact imbedding, it follows that lim_𝑛→∞^𝑡0‖‖‖𝑑𝑢_𝑛𝑑𝑡‖‖‖^2𝐻₁𝑑𝜏=^𝑡0‖‖‖𝑑𝑢₀𝑑𝑡‖‖‖^2𝐻₁lim_𝑛→∞𝑑𝜏,𝑑𝑑𝑢𝑡^𝑛,𝑢^𝑛_𝐻1=𝑑𝑢⁰𝑑𝑡,𝑢⁰_𝐻1,a.e.𝑡≥0.(3.22)
Clearly, lim_𝑛→∞^𝑡0𝑑𝑑𝑢𝑡^𝑛,𝑢^𝑛_𝐻1𝑑𝜏=^𝑡0𝑑𝑑𝑢𝑡⁰,𝑢⁰_𝐻1𝑑𝜏.(3.23)
Then, (3.14) follows from (3.19)–(3.21), which implies assertion (1).
Secondly, we consider 𝐺=𝐴+𝐵. Let 𝑣=(𝑑/𝑑𝑡)𝐿𝑢_𝑛 in (3.9). In view of (2.2) and (2.8), it follows that ^𝑡0𝑑²𝑢_𝑛𝑑𝑡²,𝑑𝑑𝑡𝐿𝑢^𝑛_𝐻+𝑘ℒ𝑑𝑢^𝑛𝑑𝑡,𝑑𝑑𝑡𝐿𝑢^𝑛_𝐻𝑑𝜏=^𝑡0(𝐴+𝐵)𝑢^𝑛,𝑑𝑑𝑡𝐿𝑢^𝑛_𝐻𝑑𝜏_𝑡012𝑑𝑑𝑡𝑑𝑑𝑢𝑡^𝑛,𝑑𝑑𝑢𝑡^𝑛_𝐻1+𝑘𝑑𝑢^𝑛=𝑑𝑡,𝑑𝑑𝑢𝑡^𝑛_𝐻2𝑑𝜏_𝑡01−𝐷𝐹𝑢^𝑛,𝑑𝑑𝑢𝑡^𝑛+𝐵𝑢^𝑛,𝑑𝑑𝑡𝐿𝑢^𝑛_𝐻𝑑𝜏,2‖‖‖𝑑𝑑𝑢𝑡^𝑛‖‖‖^2𝐻₁−12‖‖𝜓^𝑛‖‖^2𝐻₁+𝑘^𝑡0‖‖‖𝑑𝑢_𝑛𝑑𝑡‖‖‖^2𝐻₂𝑑𝜏+𝐹𝑢_𝑛−𝐹𝜑_𝑛=^𝑡0𝐵𝑢^𝑛,𝑑𝑑𝑡𝐿𝑢^𝑛_𝐻1𝑑𝜏,2‖‖‖𝑑𝑑𝑢𝑡^𝑛‖‖‖^2𝐻₁+𝑘^𝑡0‖‖‖𝑑𝑢_𝑛𝑑𝑡‖‖‖^2𝐻₂𝑑𝑡+𝐹𝑢_𝑛=^𝑡0𝐵𝑢^𝑛,𝑑𝑑𝑡𝐿𝑢^𝑛𝑑𝜏+𝐹𝜑^𝑛+21‖‖𝜓^𝑛‖‖^2𝐻₁.(3.24)
From (3.3), we have 12‖‖‖𝑑𝑑𝑢𝑡^𝑛‖‖‖^2𝐻₁+𝐹𝑢_𝑛+𝑘^𝑡0‖‖‖𝑑𝑢_𝑛𝑑𝑡‖‖‖^2𝐻₂𝑑𝜏≤𝐶^𝑡0𝐶¹𝐹𝑢_𝑛+𝑘2‖‖‖𝑑𝑑𝑢𝑡^𝑛‖‖‖^2𝐻₁+𝐶₂+𝐹𝜑_𝑛𝑑𝜏+12‖‖𝜓^𝑛‖‖^2𝐻₁≤𝐶^𝑡0𝐹𝑢^𝑛+12‖‖‖𝑑𝑑𝑢𝑡^𝑛‖‖‖^2𝐻₁𝑑𝜏+𝑓(𝑡),(3.25) where 𝑓(𝑡)=(1/2)‖𝜓‖^2𝐻₁+sup_𝑛𝐹(𝜑_𝑛).
By using Gronwall inequality, it follows that 12‖‖‖𝑑𝑑𝑢𝑡^𝑛‖‖‖^2𝐻₁+𝐹𝑢_𝑛≤𝑓(0)𝑒^𝐶𝑡+^𝑡0𝑓(𝜏)𝑒^{𝐶(𝑡−𝜏)}𝑑𝜏,(3.26) which implies that for all 0<𝑇<∞, 𝑢_𝑛⊂𝑊^1,∞(0,𝑇),𝐻₁∩𝐿^∞(0,𝑇),𝑋₂isbounded.(3.27)
From (3.25) and (3.21), it follows that 𝑢_𝑛⊂𝑊^1,2(0,𝑇),𝐻₂isbounded.(3.28)
Let 𝑢_𝑛∗⇀𝑢₀,𝑢_𝑛in𝑊^1,∞(0,𝑇),𝐻₁∩𝐿^∞(0,𝑇),𝑋₂,⇀𝑢₀,in𝑊^1,2(0,𝑇),𝐻₂,(3.29) which implies that 𝑢_𝑛→𝑢₀ in 𝑊^1,2((0,𝑇),𝐻) is uniformly weakly convergent from that 𝐻₂⊂𝐻 is compact imbedding.
The left proof is same as assertion (1).
Lastly, assume (3.6) hold. Let 𝑣=𝑑²𝑢_𝑛/𝑑𝑡² in (3.9). It follows that ^𝑡0𝑑²𝑢_𝑛𝑑𝑡²,𝑑²𝑢_𝑛𝑑𝑡²_𝐻+𝑘ℒ𝑑𝑢^𝑛𝑑𝑡,𝑑²𝑢_𝑛=𝑑𝑡²_𝐻𝑑𝜏_𝑡0(𝐴+𝐵)𝑢^𝑛,𝑑²𝑢_𝑛≤𝑑𝑡²𝑑𝜏_𝑡0𝐶𝐹𝑢^𝑛+12‖‖‖𝑑²𝑢_𝑛𝑑𝑡²‖‖‖^2𝐻+𝑔(𝑡)𝑑𝜏,_𝑡0𝑑²𝑢_𝑛𝑑𝑡²,𝑑²𝑢_𝑛𝑑𝑡²_𝐻+𝑘2^𝑡0_Ω𝑑𝑑𝑡𝑢^𝑛(𝑡)²≤𝑑𝑥𝑑𝜏_𝑡0𝐶𝐹𝑢^𝑛+12‖‖‖𝑑²𝑢_𝑛𝑑𝑡²‖‖‖^2𝐻+𝑔(𝑡)𝑑𝜏,_𝑡0𝑑²𝑢_𝑛𝑑𝑡²,𝑑²𝑢_𝑛𝑑𝑡²_𝐻𝑑𝑡+𝑘2‖‖‖𝑑𝑑𝑢𝑡^𝑛‖‖‖^2𝐻₁≤𝑘2‖‖𝜓^𝑛‖‖^2𝐻+^𝑡012‖‖‖𝑑²𝑢_𝑛𝑑𝑡²‖‖‖^2𝐻+𝐶𝐹𝑢_𝑛+𝑔(𝜏)𝑑𝜏.(3.30)
From (3.26), the above inequality implies ^𝑡0‖‖‖𝑑²𝑢_𝑛𝑑𝑡²‖‖‖^2𝐻𝑑𝜏≤𝐶,(𝐶>0isconstant).(3.31) We see that for all 0<𝑇<∞, {𝑢_𝑛}⊂𝑊^2,2((0,𝑇),𝐻) is bounded. Thus 𝑢∈𝑊^2,2((0,𝑇),𝐻).

4. Main Result

Now, we begin to consider the mixed problem (1.1). Set 𝐹(𝑥,𝑦)=^𝑦0𝑓(𝑥,𝑧)𝑑𝑧.(4.1) We assume 𝐹||(𝑓𝑥,(𝑥𝑦),𝑦≥)||𝐶≤₁||𝐶𝑦||^𝑝−||𝑦||𝐶_2𝑝−1,𝑝≥2,𝑓𝑥,𝑦₁+1,(4.2)||𝑔(𝑥,−𝑧,𝑓𝜉,𝜂𝑥)||,𝑦≤₂𝐶𝑦|𝑧₁|−_𝑝/2𝑦₂≥𝜆||𝑦₁−𝑦₂||²,𝜆>0,(4.3)||𝑔𝑥,𝑧,𝜉,𝜂₁+||𝜉||^𝑝/2+||𝜂||^𝑝/2+1,(4.4)−𝑔𝑥,𝑧,𝜉,𝜂₂||≤𝐾₁||𝜂₁−𝜂₂||,(4.5) where 𝐶,𝐶₁,𝐶₂ are constant and 𝐾₁<𝜆𝐾, 𝐾 is the best constant satisfying 𝐾²‖𝑢‖^2𝐻₂≤_Ω||Δ𝑢||²𝑑𝑥.(4.6)

Theorem 4.1. If the assumptions of (4.1)–(4.5) hold, for (𝜑,𝜓)∈𝑊^2,𝑝(Ω)∩𝑊^01,𝑝(Ω)×𝐻¹⁰(Ω), then (1.1) is a strong solution 𝑢𝑢∈𝐿^∞loc(0,∞),𝑊^2,𝑝(Ω)∩𝑊^01,𝑝(Ω)_𝑡,𝑢∈𝐿^∞loc(0,∞),𝐻¹⁰(Ω)∩𝐿^2loc(0,∞),𝐻²(Ω)_𝑡𝑡,∈𝐿^𝑝′((0,𝑇)×Ω),𝑝^=𝑝𝑝−1,∀0<𝑇<∞.(4.7)

Proof. We introduce spatial sequences 𝑋𝑋=𝑢∈𝐶^∞(Ω)||Δ^𝑘𝑢||_𝜕Ω=0,𝑘=0,1,2,…₁,𝑋=𝐿^𝑝(Ω),₂𝐻==𝑊𝐿^2,𝑝₂(Ω)∩𝑊^01,𝑝(Ω),𝐻(Ω),₁𝐻=𝐻¹⁰(Ω),₂𝐻=𝐻²(Ω)∩𝐻¹⁰(Ω),₃=𝑢∈𝐻^2𝑚(Ω)∶𝑢|_𝜕Ω=⋯=Δ^𝑚−1𝑢|_𝜕Ω=0,(4.8) where the inner products of 𝐻₂ and 𝐻₃ are defined by ⟨𝑢,𝑣⟩_𝐻2=_ΩΔ𝑢Δ𝑣𝑑𝑥,⟨𝑢,𝑣⟩_𝐻2=_ΩΔ^𝑚𝑢Δ^𝑚𝑣𝑑𝑥,(4.9) where 𝑚≥1 such that 𝐻₃⊂𝑋₂ is an embedding.
Linear operators ℒ∶𝑋→𝑋₁ and 𝐿∶𝑋→𝑋₁ are defined by ℒ𝑢=𝐿𝑢=−Δ𝑢.(4.10) It is known that ℒ and 𝐿 satisfy (2.2), (2.3), and (2.5). Define 𝐺=𝐴+𝐵∶𝑋₂→𝑋^∗1 by ⟨𝐴𝑢,𝑣⟩=_Ω𝑓(𝑥,Δ𝑢)𝑣𝑑𝑥,⟨𝐵𝑢,𝑣⟩=_Ω𝑔𝑥,𝑢,𝐷𝑢,𝐷²𝑢𝑣𝑑𝑥,for𝑣∈𝑋₁.(4.11)
We show that 𝐺=𝐴+𝐵∶𝑋₂→𝑋^∗1 is 𝑇-coercively weakly continuous. Let {𝑢_𝑛}⊂𝐿^∞((0,𝑇),𝑊^2,𝑝(Ω)∩𝑊^01,𝑝(Ω)) satisfying (2.7) and lim_𝑛→∞^𝑇0=li||⟨m𝐺_𝑛→∞𝑢_𝑛−𝐺𝑢₀,𝐿𝑢_𝑛−𝐿𝑢₀⟩||𝑑𝑡^𝑇0_Ω𝑓𝑥,Δ𝑢_𝑛−𝑓𝑥,Δ𝑢₀𝑢_𝑛+𝑔𝑥,𝑢_𝑛−𝑢₀,𝐷𝑢_𝑛,𝐷²𝑢_𝑛−𝑔𝑥,𝑢₀,𝐷𝑢₀,𝐷²𝑢₀𝑢_𝑛−𝑢₀𝑑𝑥𝑑𝑡=0.(4.12)
We need to prove that lim_𝑛→∞^𝑇0_Ω𝑓𝑥,Δ𝑢_𝑛+𝑔𝑥,𝑢_𝑛=,𝐷𝑢_𝑛,𝐷²𝑢_𝑛𝑣𝑑𝑥𝑑𝑡_𝑇0_Ω𝑓𝑥,Δ𝑢₀+𝑔𝑥,𝑢₀,𝐷𝑢₀,𝐷²𝑢₀𝑣𝑑𝑥𝑑𝑡.(4.13)
From (2.7) and Lemma 2.3, we obtain 𝑢_𝑛⟶𝑢₀,𝐷𝑢_𝑛⟶𝐷𝑢₀in𝐿²((0,𝑇)×Ω).(4.14)
From (4.3), we get ^𝑇0_Ω𝑓𝑥,Δ𝑢_𝑛−𝑓𝑥,Δ𝑢₀Δ𝑢_𝑛−Δ𝑢₀𝑑𝑥𝑑𝑡≥𝜆^𝑇0_Ω||Δ𝑢_𝑛−Δ𝑢₀||²𝑑𝑥𝑑𝑡.(4.15)
We have the deformation ^𝑇0_Ω𝑔𝑥,𝑢_𝑛=,𝐷𝑢_𝑛,𝐷²𝑢_𝑛−𝑔𝑥,𝑢₀,𝐷𝑢₀,𝐷²𝑢₀Δ𝑢_𝑛−Δ𝑢₀𝑑𝑥𝑑𝑡_𝑇0_Ω𝑔𝑥,𝑢_𝑛+,𝐷𝑢_𝑛,𝐷²𝑢₀−𝑔𝑥,𝑢₀,𝐷𝑢₀,𝐷²𝑢₀Δ𝑢_𝑛−Δ𝑢₀𝑑𝑥𝑑𝑡_𝑇0_Ω𝑔𝑥,𝑢_𝑛,𝐷𝑢_𝑛,𝐷²𝑢_𝑛−𝑔𝑥,𝑢_𝑛,𝐷𝑢_𝑛,𝐷²𝑢₀Δ𝑢_𝑛−Δ𝑢₀𝑑𝑥𝑑𝑡.(4.16)
From (4.14) and Lemma 2.4, we have lim_𝑛→∞^𝑇0_Ω𝑔𝑥,𝑢_𝑛,𝐷𝑢_𝑛,𝐷²𝑢₀−𝑔𝑥,𝑢₀,𝐷𝑢₀,𝐷²𝑢₀Δ𝑢_𝑛−Δ𝑢₀𝑑𝑥𝑑𝑡=0.(4.17)
From (4.12), (4.15)–(4.17), it follows that 0≥𝜆^𝑇0_Ω||Δ𝑢_𝑛−Δ𝑢₀||²𝑑𝑥𝑑𝑡+^𝑇0_Ω𝑔𝑥,𝑢_𝑛×Δ𝑢_𝑛,𝐷𝑢_𝑛,𝐷²𝑢_𝑛−𝑔𝑥,𝑢_𝑛,𝐷𝑢_𝑛,𝐷²𝑢₀≥𝜆_𝑇0−Δ𝑢₀𝑑𝑥𝑑𝑡_Ω||Δ𝑢_𝑛−Δ𝑢₀||²𝑑𝑥𝑑𝑡−𝐾₁^𝑇0_Ω≥𝜆||𝐷²𝑢_𝑛−𝐷²𝑢₀||||Δ𝑢_𝑛−Δ𝑢₀||𝑑𝑥𝑑𝑡2^𝑇0_Ω||Δ𝑢_𝑛−Δ𝑢₀||²𝑑𝑥𝑑𝑡−𝐾²¹2𝜆^𝑇0_Ω≥𝜆||𝐷²𝑢_𝑛−𝐷²𝑢₀||²𝑑𝑥𝑑𝑡₂𝐾²−𝐾²¹2𝜆^𝑇0_Ω||𝐷²𝑢_𝑛−𝐷²𝑢₀||²𝑑𝑥𝑑𝑡.(4.18)
Since 𝜆𝐾>𝐾₁, we have lim_𝑛→∞^𝑇0_Ω||𝐷²𝑢_𝑛−𝐷²𝑢₀||²𝑑𝑥𝑑𝑡=0.(4.19)
From (4.14), (4.19), (4.1), (4.4), and Lemma 2.3, we get (4.13).
Let 𝐹₁(𝑢)=∫_Ω𝐹(𝑥,Δ𝑢)𝑑𝑥, where 𝐹 is same as (4.1). We get 𝐹⟨(𝐴𝑢)𝑢⟶,𝐿𝑢⟩∞=⟺−⟨𝐷‖𝑢𝐹‖₁(_𝑋2𝑢),𝑣⟩,⟶∞,(4.20) which implies Conditions (A1), (A2) of model results in Theorem 3.1.
We will show (3.3) as follows. It follows that ||⟨𝐵𝑢,𝐿𝑣⟩||≤=𝑘_Ω||𝑔𝑥,𝑢,𝐷𝑢,𝐷²𝑢||||Δ𝑣||𝑑𝑥2_Ω||Δ𝑣||²𝑑𝑥+2𝑘_Ω≤𝑘||𝑔𝑥,𝑢,𝐷𝑢,𝐷²𝑢||²𝑑𝑥2‖𝑣‖^2𝐻2≤𝑘+𝐶_Ω||𝐷²𝑢||^𝑝+||∇𝑢||^𝑝+|𝑢|^𝑝+1𝑑𝑥2‖𝑣‖^2𝐻2+𝐶𝐹₁(𝑢)+𝐶,(4.21) which imply Conditions (A3) of Theorem 3.1. From Theorem 3.1, (1.1) has a solution 𝑢𝑢∈𝐿^∞loc(0,∞),𝑊^2,𝑝(Ω)∩𝑊^01,𝑝(Ω)_𝑡,𝑢∈𝐿^∞loc(0,∞),𝐻¹⁰(Ω)∩𝐿^2loc(0,∞),𝐻²(Ω)_𝑡𝑡,∈𝐿^𝑝′((0,𝑇)×Ω),𝑝^=𝑝𝑝−1,∀0<𝑇<∞,(4.22) satisfying _Ω𝜕𝑢𝜕𝑡𝑣𝑑𝑥−𝑘_ΩΔ𝑢𝑣𝑑𝑥=^𝑡0_Ω𝑓(𝑥,Δ𝑢)𝑣𝑑𝑥𝑑𝜏+^𝑡0_Ω+𝑔𝑥,𝑢,𝐷𝑢,𝐷²𝑢𝑣𝑑𝑥𝑑𝜏_Ω𝜓𝑣𝑑𝑥−𝑘_ΩΔ𝜑𝑣𝑑𝑥.(4.23)

Acknowledgments

The authors are grateful to the anonymous reviewers for their careful reading and useful suggestions, which greatly improved the presentation of the paper. This paper was funded by the National Natural Science Foundation of China (no. 11071177) and the NSF of Sichuan Science and Technology Department of China (no. 2010JY0057).