Abstract

We consider the global existence of strong solution 𝑢, corresponding to a class of fully nonlinear wave equations with strongly damped terms 𝑢𝑡𝑡−𝑘Δ𝑢𝑡=𝑓(𝑥,Δ𝑢)+𝑔(𝑥,𝑢,𝐷𝑢,𝐷2𝑢) in a bounded and smooth domain Ω in 𝑅𝑛, 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 ğ‘¢âˆˆğ¿âˆžloc((0,∞),𝑊2,𝑝(Ω)∩𝑊01,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 ğ¶âˆž domain Ω⊂𝑅𝑛: 𝑢𝑡𝑡−𝑘Δ𝑢𝑡=𝑓(𝑥,Δ𝑢)+𝑔𝑥,𝑢,𝐷𝑢,𝐷2𝑢[,in0,∞)×Ω,𝑢(0,𝑥)=𝜑,𝑢𝑡[(0,𝑥)=𝜓,inΩ,𝑢(𝑡,𝑥)=0,on0,∞)×𝜕Ω,(1.1) where 𝑢𝑡=𝜕𝑢𝜕𝑡,𝑢𝑡𝑡=𝜕2𝑢𝜕𝑡2,Δ=𝑛𝑖=1𝜕2𝜕𝑥2𝑖𝜕,𝐷=𝜕𝑥1𝜕,…,𝜕𝑥𝑛,𝐷2=𝜕2𝜕𝑥𝛼11⋯𝜕𝑥𝛼𝑛𝑛,𝛼1+⋯+𝛼𝑛𝑥=2,𝑥=1,…,𝑥𝑛,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: 𝑢𝑡𝑡+𝑢𝑡−𝑢𝑥𝑥[𝑢=𝑓(𝑢),in0,∞)×Ω,(0,𝑥)=𝑢0(𝑥),𝑢𝑡(0,𝑥)=𝑢1(𝑥),inΩ;(1.3) they discussed the global existence of weak solutions and regularity in 𝑅1 and 𝑅𝑛 [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|ğ‘ž and ğ‘ž>6: 𝑢𝑡𝑡(𝑡,𝑥)−Δ𝑢(𝑡,𝑥)−Δ𝑢𝑡||||(𝑡,𝑥)=𝑢(𝑡,𝑥)ğ‘ž[in0,∞)×Ω,𝑢(0,𝑥)=𝑢0(𝑥),𝑢𝑡(0,𝑥)=𝑢1[(𝑥)inΩ,𝑢(𝑡,𝑥)=0,on0,∞)×𝜕Ω.(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 𝑅𝑛. They assume the nonlinear term 𝑓 grows like |𝑢|ğ‘ž and ğ‘ž<(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: 𝑋⊂𝐻3⊂𝑋2⊂𝑋1𝑋⊂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⟶𝑋1beone-for-onedenselinearoperator,⟨𝐿𝑢,𝑣⟩𝐻=⟨𝑢,𝑣⟩𝐻1,∀𝑢,𝑣∈𝑋.(2.2)

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

We consider the following abstract wave equation model: 𝑑2𝑢𝑑𝑡2𝑑+𝑘𝑑𝑡ℒ𝑢=𝐺(𝑢),𝑘>0,𝑢(0)=𝜑,𝑢𝑡(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∈𝑊1,∞((0,𝑇),𝐻1)âˆ©ğ¿âˆž((0,𝑇),𝑋2) is a global weak solution of the (2.4) provided for (𝜑,𝜓)∈𝑋2×𝐻1⟨𝑢𝑡,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 𝑢𝑛, 𝑢0∈𝐿𝑝((0,𝑇),𝑋2). We say that 𝑢𝑛⇀𝑢0 in 𝐿𝑝((0,𝑇),𝑋2) is uniformly weakly convergent if {𝑢𝑛}âŠ‚ğ¿âˆž((0,𝑇),𝐻) is bounded, and 𝑢𝑛⇀𝑢0,in𝐿𝑝(0,𝑇),𝑋2,limğ‘›â†’âˆžî€œğ‘‡0||⟨𝑢𝑛−𝑢0,𝑣⟩𝐻||2𝑑𝑡=0,∀𝑣∈𝐻.(2.7)

Lemma 2.3 (see [16]). Let {𝑢𝑛}∈𝐿𝑝((0,𝑇),𝑊𝑚,𝑝(Ω))(𝑚≥1) be bounded sequences and {𝑢𝑛} uniformly weakly convergent to 𝑢0∈𝐿𝑝((0,𝑇),𝑊𝑚,𝑝(Ω)). Then, for each |𝛼|≤𝑚−1, it follows that 𝐷𝛼𝑢𝑛⟶𝐷𝛼𝑢0,in𝐿2((0,𝑇)×Ω).(2.8)

Lemma 2.4 (see [17]). Let Ω⊂𝑅𝑛 be an open set and 𝑓∶Ω×𝑅𝑁→𝑅1 satisfy Caratheodory condition and ||||𝑓(𝑥,𝜉)≤𝐶𝑁𝑖=1||𝜉𝑖||𝑝𝑖/𝑝+𝑏(𝑥).(2.9) If {𝑢𝑖𝑘}⊂𝐿𝑝𝑖(Ω)(1≤𝑖≤𝑁) is bounded and 𝑢𝑖𝑘 convergent to 𝑢𝑖 in Ω0 for all bounded Ω0⊂Ω, then for each 𝑣∈𝐿𝑝′(Ω), the following equality holds limğ‘˜â†’âˆžî€œÎ©ğ‘“î€·ğ‘¥,𝑢1𝑘,…,𝑢𝑁𝑘𝑣𝑑𝑥=Ω𝑓𝑥,𝑢1,…,𝑢𝑁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⟩≤𝐶1𝑘𝐹(𝑢)+2‖𝑣‖2𝐻1+𝐶2,∀𝑢,𝑣∈𝑋,(3.3)

for 𝑔∈𝐿1loc(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∈𝑊1,∞(0,∞),𝐻1∩𝑊1,2(0,∞),𝐻2î€¸âˆ©ğ¿âˆžî€·(0,∞),𝑋2.(3.4)(2)If 𝐺=𝐴+𝐵 satisfies (A1)–(A3), then (2.4) has a global weak solution 𝑢∈𝑊1,∞loc(0,∞),𝐻1∩𝑊1,2loc(0,∞),𝐻2î€¸âˆ©ğ¿âˆžloc(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⟩≤𝐶11𝐹(𝑢)+2‖𝑣‖2𝐻+𝐶2,(3.6)then 𝑢∈𝑊2,2loc((0,∞),𝐻).

Proof. Let {𝑒𝑘}⊂𝑋 be a common orthogonal basis of 𝐻 and 𝐻3, satisfying (2.3). Set 𝑋𝑛=𝑛𝑖=1𝛼𝑖𝑒𝑖∣𝛼𝑖∈𝑅1,𝑋𝑛=𝑛𝑗=1𝛽𝑗(𝑡)𝑒𝑗∣𝛽𝑗∈𝐶2[.0,∞)(3.7) Clearly, 𝐿𝑋𝑛=𝑋𝑛, 𝐿𝑋𝑛=𝑋𝑛.
By using Galerkin method, there exits 𝑢𝑛∈𝐶2([0,∞),𝑋𝑛) satisfying 𝑑𝑢𝑛𝑑𝑡,𝑣𝐻+𝑘⟨ℒ𝑢𝑛,𝑣⟩𝐻=𝑡0𝐺𝑢𝑛,𝑣𝑑𝜏+⟨𝜓𝑛,𝑣⟩𝐻+𝑘⟨ℒ𝜑𝑛,𝑣⟩𝐻,𝑢𝑛(0)=𝜑𝑛,ğ‘¢î…žğ‘›(0)=𝜓𝑛,(3.8) for ∀𝑣∈𝑋𝑛, and 𝑡0𝑑2𝑢𝑛𝑑𝑡2,𝑣𝐻ℒ+𝑘𝑑𝑢𝑛𝑑𝑡,𝑣𝐻𝑑𝜏=𝑡0⟨𝐺𝑢𝑛,𝑣⟩𝑑𝜏(3.9) for 𝑋∀𝑣∈𝑛.
Firstly, we consider 𝐺=𝐴. Let 𝑣=(𝑑/𝑑𝑡)𝐿𝑢𝑛 in (3.9). Taking into account (2.2) and (3.1), it follows that 0=𝑡0𝑑2𝑢𝑛𝑑𝑡2,𝑑𝑑𝑡𝐿𝑢𝑛ℒ+𝑘𝑑𝑢𝑛,𝑑𝑑𝑡𝑑𝑡𝐿𝑢𝑛𝐻𝑑𝜏−𝑡0𝐴𝑢𝑛,𝑑𝑑𝑡𝐿𝑢𝑛𝑑𝜏,0=𝑡012𝑑𝑑𝑡𝑑𝑢𝑛,𝑑𝑡𝑑𝑢𝑛𝑑𝑡𝐻1+𝑘𝑑𝑢𝑛,𝑑𝑡𝑑𝑢𝑛𝑑𝑡𝐻2+𝑢𝐷𝐹𝑛,𝑑𝑢𝑛=1𝑑𝑡𝑑𝜏2‖‖‖𝑑𝑢𝑛‖‖‖𝑑𝑡2𝐻1−12‖‖𝜓𝑛‖‖2𝐻1+𝑘𝑡0‖‖‖𝑑𝑢𝑛‖‖‖𝑑𝑡2𝐻2𝑢𝑑𝜏+𝐹𝑛𝜑−𝐹𝑛.(3.10)
We get 12‖‖‖𝑑𝑢𝑛‖‖‖𝑑𝑡2𝐻1+𝑘𝑡0‖‖‖𝑑𝑢𝑛‖‖‖𝑑𝑡2𝐻2𝑢𝑑𝜏+𝐹𝑛𝜑=𝐹𝑛+12‖‖𝜓𝑛‖‖2𝐻1.(3.11)
Let 𝜑∈𝐻3. From (2.1) and (2.2), it is known that {𝑒𝑛} are orthogonal basis of 𝐻1. We find that 𝜑𝑛→𝜑 in 𝐻3, and 𝜓𝑛→𝜓 in 𝐻1. Since 𝐻3⊂𝑋2 is imbedding, it follows that 𝜑𝑛⟶𝜑,in𝑋2,𝜓𝑛⟶𝜓,in𝐻1.(3.12)
From (3.2), (3.11), and (3.12), we obtain that, 𝑢𝑛⊂𝑊1,∞loc(0,∞),𝐻1∩𝑊1,2loc(0,∞),𝐻2î€¸âˆ©ğ¿âˆžloc(0,∞),𝑋2isbounded.(3.13)
Let 𝑢𝑛∗⇀𝑢0,in𝑊1,∞loc(0,∞),𝐻1î€¸âˆ©ğ¿âˆžloc(0,∞),𝑋2,𝑢𝑛⇀𝑢0,in𝑊1,2loc(0,∞),𝐻2,(3.14) which implies that 𝑢𝑛→𝑢0 in 𝑊1,2loc((0,∞),𝐻) is uniformly weakly convergent from that 𝐻2⊂𝐻 is compact imbedding.
If we have the following equality: limğ‘›â†’âˆžî‚¸âˆ’î€œğ‘¡0||⟨𝐺𝑢𝑛−𝐺𝑢0,𝐿𝑢𝑛−𝐿𝑢0⟩||𝑘𝑑𝜏+2‖‖𝑢𝑛−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𝑑𝑑𝑡ℒ𝑢𝑛−𝑑𝑑𝑡ℒ𝑢0,𝐿𝑢𝑛−𝐿𝑢0𝐻1𝑑𝜏=2𝑡0𝑑𝑑𝑡⟨𝑢𝑛−𝑢0,𝑢𝑛−𝑢0⟩𝐻2=1𝑑𝜏2‖‖𝑢𝑛(𝑡)−𝑢0(‖‖𝑡)2𝐻2−12‖‖𝜑𝑛‖‖−𝜑2𝐻2.(3.16)
Taking into account (2.2), (2.5), and (3.9), we get that −𝑡0⟨𝐺𝑢𝑛−𝐺𝑢0,𝐿𝑢𝑛−𝐿𝑢0𝑘⟩𝑑𝜏+2‖‖𝑢𝑛−𝑢0‖‖2𝐻2=𝑡0⟨𝐺𝑢0−𝐺𝑢𝑛,𝐿𝑢𝑛−𝐿𝑢0𝑑⟩+𝑘𝑑𝑡ℒ𝑢𝑛−𝑑𝑑𝑡ℒ𝑢0,𝐿𝑢𝑛−𝐿𝑢0𝐻𝑘𝑑𝜏+2‖‖𝜑𝑛‖‖−𝜑2𝐻2=𝑡0⟨𝐺𝑢0,𝐿𝑢𝑛−𝐿𝑢0⟩+⟨𝐺𝑢𝑛,𝐿𝑢0⟩−⟨𝐺𝑢𝑛,𝐿𝑢𝑛⟩−𝑘𝑑𝑢𝑛𝑑𝑡,𝑢0𝐻2−𝑘𝑑𝑢0𝑑𝑡,𝑢𝑛−𝑢0𝐻2𝑑+𝑘𝑑𝑡ℒ𝑢𝑛,𝐿𝑢𝑛𝐻𝑘𝑑𝜏+2‖‖𝜑𝑛‖‖−𝜑2𝐻2=𝑡0⟨𝐺𝑢0,𝐿𝑢𝑛−𝐿𝑢0⟩+⟨𝐺𝑢𝑛,𝐿𝑢0⟩−𝑘𝑑𝑢𝑛𝑑𝑡,𝑢0𝐻2−𝑘𝑑𝑢0𝑑𝑡,𝑢𝑛−𝑢0𝐻2−𝑑2𝑢𝑛𝑑𝑡2𝑑+𝑘𝑑𝑡ℒ𝑢𝑛,𝐿𝑢𝑛𝐻𝑑+𝑘𝑑𝑡ℒ𝑢𝑛,𝐿𝑢𝑛𝐻𝑘𝑑𝜏+2‖‖𝜑𝑛‖‖−𝜑2𝐻2=𝑡0⟨𝐺𝑢0,𝐿𝑢𝑛−𝐿𝑢0⟩+⟨𝐺𝑢𝑛,𝐿𝑢0⟩−𝑘𝑑𝑢𝑛𝑑𝑡,𝑢0𝐻2𝑑−𝑘𝑢𝑑𝑡0,𝑢𝑛−𝑢0𝐻2+𝑑𝑢𝑛,𝑑𝑡𝑑𝑢𝑛𝑑𝑡𝐻1−𝑑𝜏𝑑𝑢𝑛𝑑𝑡,𝑢𝑛𝐻1+⟨𝜓𝑛,𝜑𝑛⟩𝐻1+𝑘2‖‖𝜑𝑛‖‖−𝜑2𝐻2.(3.17)
From (2.1) and (3.14), we have limğ‘›â†’âˆžâ€–â€–ğœ‘ğ‘›â€–â€–âˆ’ğœ‘ğ»2=0,limğ‘›â†’âˆžî€œğ‘¡0⟨𝐺𝑢0,𝐿𝑢𝑛−𝐿𝑢0⟩𝑑𝜏=0,limğ‘›â†’âˆžî€œğ‘¡0𝑑𝑢𝑑𝑡0,𝑢𝑛−𝑢0𝐻2𝑑𝜏=0.(3.18)
Then, we get limğ‘›â†’âˆžâˆ’î€œğ‘¡0⟨𝐺𝑢𝑛−𝐺𝑢0,𝐿𝑢𝑛−𝐿𝑢0𝑘⟩𝑑𝜏+2limğ‘›â†’âˆžâ€–â€–ğ‘¢ğ‘›âˆ’ğ‘¢0‖‖2𝐻2=limğ‘›â†’âˆžâˆ«ğ‘¡0⟨𝐺𝑢𝑛,𝐿𝑢0⟩−𝑘𝑑𝑢𝑛𝑑𝑡,𝑢0𝐻2+‖‖‖𝑑𝑢𝑛‖‖‖𝑑𝑡2𝐻1𝑑𝜏−limğ‘›â†’âˆžîƒ¡ğ‘‘ğ‘¢ğ‘›ğ‘‘ğ‘¡,𝑢𝑛𝐻1+⟨𝜓,𝜑⟩𝐻1.(3.19)
In view of (3.9), (3.14), we obtain for all ğ‘£âˆˆâˆªâˆžğ‘›=1𝑋𝑛limğ‘›â†’âˆžî€œğ‘¡0⟨𝐺𝑢𝑛,𝐿𝑣⟩𝑑𝜏=𝑡0𝑘𝑑𝑢0𝑑𝑡,𝑣𝐻2−𝑑𝑢0,𝑑𝑡𝑑𝑣𝑑𝑡𝐻1+𝑑𝜏𝑑𝑢0𝑑𝑡,𝑣𝐻1−⟨𝜓,𝑣(0)⟩𝐻1.(3.20)
Since âˆªâˆžğ‘›=1𝑋𝑛 is dense in 𝑊1,2((0,𝑇),𝐻2)∩𝐿𝑝((0,𝑇),𝑋2), for all 𝑝<∞, (3.20) holds for all 𝑣∈𝑊1,2((0,𝑇),𝐻2)∩𝐿𝑝((0,𝑇),𝑋2). Thus, we have limğ‘›â†’âˆžî€œğ‘¡0⟨𝐺𝑢𝑛,𝐿𝑢0⟩𝑑𝜏=𝑡0𝑘𝑑𝑢0𝑑𝑡,𝑢0𝐻2−‖‖‖𝑑𝑢0‖‖‖𝑑𝑡2𝐻1+𝑑𝜏𝑑𝑢0𝑑𝑡,𝑢0𝐻1−⟨𝜓,𝜑⟩𝐻1.(3.21)
From (3.14) and 𝐻2⊂𝐻1 being compact imbedding, it follows that limğ‘›â†’âˆžî€œğ‘¡0‖‖‖𝑑𝑢𝑛‖‖‖𝑑𝑡2𝐻1𝑑𝜏=𝑡0‖‖‖𝑑𝑢0‖‖‖𝑑𝑡2𝐻1𝑑𝜏,limğ‘›â†’âˆžîƒ¡ğ‘‘ğ‘¢ğ‘›ğ‘‘ğ‘¡,𝑢𝑛𝐻1=𝑑𝑢0𝑑𝑡,𝑢0𝐻1,a.e.𝑡≥0.(3.22)
Clearly, limğ‘›â†’âˆžî€œğ‘¡0𝑑𝑢𝑛𝑑𝑡,𝑢𝑛𝐻1𝑑𝜏=𝑡0𝑑𝑢0𝑑𝑡,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)𝐿𝑢𝑛 in (3.9). In view of (2.2) and (2.8), it follows that 𝑡0𝑑2𝑢𝑛𝑑𝑡2,𝑑𝑑𝑡𝐿𝑢𝑛𝐻ℒ+𝑘𝑑𝑢𝑛,𝑑𝑑𝑡𝑑𝑡𝐿𝑢𝑛𝐻𝑑𝜏=𝑡0(𝐴+𝐵)𝑢𝑛,𝑑𝑑𝑡𝐿𝑢𝑛𝐻𝑑𝜏𝑡012𝑑𝑑𝑡𝑑𝑢𝑛,𝑑𝑡𝑑𝑢𝑛𝑑𝑡𝐻1+𝑘𝑑𝑢𝑛,𝑑𝑡𝑑𝑢𝑛𝑑𝑡𝐻2=𝑑𝜏𝑡0𝑢−𝐷𝐹𝑛,𝑑𝑢𝑛+𝐵𝑢𝑑𝑡𝑛,𝑑𝑑𝑡𝐿𝑢𝑛𝐻1𝑑𝜏,2‖‖‖𝑑𝑢𝑛‖‖‖𝑑𝑡2𝐻1−12‖‖𝜓𝑛‖‖2𝐻1+𝑘𝑡0‖‖‖𝑑𝑢𝑛‖‖‖𝑑𝑡2𝐻2𝑢𝑑𝜏+𝐹𝑛𝜑−𝐹𝑛=𝑡0𝐵𝑢𝑛,𝑑𝑑𝑡𝐿𝑢𝑛𝐻1𝑑𝜏,2‖‖‖𝑑𝑢𝑛‖‖‖𝑑𝑡2𝐻1+𝑘𝑡0‖‖‖𝑑𝑢𝑛‖‖‖𝑑𝑡2𝐻2𝑢𝑑𝑡+𝐹𝑛=𝑡0𝐵𝑢𝑛,𝑑𝑑𝑡𝐿𝑢𝑛𝜑𝑑𝜏+𝐹𝑛+12‖‖𝜓𝑛‖‖2𝐻1.(3.24)
From (3.3), we have 12‖‖‖𝑑𝑢𝑛‖‖‖𝑑𝑡2𝐻1𝑢+𝐹𝑛+𝑘𝑡0‖‖‖𝑑𝑢𝑛‖‖‖𝑑𝑡2𝐻2𝑑𝜏≤𝐶𝑡0𝐶1𝐹𝑢𝑛+𝑘2‖‖‖𝑑𝑢𝑛‖‖‖𝑑𝑡2𝐻1+𝐶2𝜑𝑑𝜏+𝐹𝑛+12‖‖𝜓𝑛‖‖2𝐻1≤𝐶𝑡0𝐹𝑢𝑛+12‖‖‖𝑑𝑢𝑛‖‖‖𝑑𝑡2𝐻1𝑑𝜏+𝑓(𝑡),(3.25) where 𝑓(𝑡)=(1/2)‖𝜓‖2𝐻1+sup𝑛𝐹(𝜑𝑛).
By using Gronwall inequality, it follows that 12‖‖‖𝑑𝑢𝑛‖‖‖𝑑𝑡2𝐻1𝑢+𝐹𝑛≤𝑓(0)𝑒𝐶𝑡+𝑡0𝑓(𝜏)𝑒𝐶(𝑡−𝜏)𝑑𝜏,(3.26) which implies that for all 0<𝑇<∞, 𝑢𝑛⊂𝑊1,∞(0,𝑇),𝐻1î€¸âˆ©ğ¿âˆžî€·(0,𝑇),𝑋2isbounded.(3.27)
From (3.25) and (3.21), it follows that 𝑢𝑛⊂𝑊1,2(0,𝑇),𝐻2isbounded.(3.28)
Let 𝑢𝑛∗⇀𝑢0,in𝑊1,∞(0,𝑇),𝐻1î€¸âˆ©ğ¿âˆžî€·(0,𝑇),𝑋2,𝑢𝑛⇀𝑢0,in𝑊1,2(0,𝑇),𝐻2,(3.29) which implies that 𝑢𝑛→𝑢0 in 𝑊1,2((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=𝑑2𝑢𝑛/𝑑𝑡2 in (3.9). It follows that 𝑡0𝑑2𝑢𝑛𝑑𝑡2,𝑑2𝑢𝑛𝑑𝑡2𝐻ℒ+𝑘𝑑𝑢𝑛,𝑑𝑑𝑡2𝑢𝑛𝑑𝑡2𝐻=𝑑𝜏𝑡0𝑢(𝐴+𝐵)𝑛,𝑑2𝑢𝑛𝑑𝑡2≤𝑑𝜏𝑡0𝑢𝐶𝐹𝑛+12‖‖‖𝑑2𝑢𝑛𝑑𝑡2‖‖‖2𝐻+𝑔(𝑡)𝑑𝜏,𝑡0𝑑2𝑢𝑛𝑑𝑡2,𝑑2𝑢𝑛𝑑𝑡2𝐻+𝑘2𝑡0î€œÎ©ğ‘‘î€·ğ‘¢ğ‘‘ğ‘¡î…žğ‘›î€¸(𝑡)2≤𝑑𝑥𝑑𝜏𝑡0𝑢𝐶𝐹𝑛+12‖‖‖𝑑2𝑢𝑛𝑑𝑡2‖‖‖2𝐻+𝑔(𝑡)𝑑𝜏,𝑡0𝑑2𝑢𝑛𝑑𝑡2,𝑑2𝑢𝑛𝑑𝑡2𝐻𝑘𝑑𝑡+2‖‖‖𝑑𝑢𝑛‖‖‖𝑑𝑡2𝐻1≤𝑘2‖‖𝜓𝑛‖‖2𝐻+𝑡012‖‖‖𝑑2𝑢𝑛𝑑𝑡2‖‖‖2𝐻𝑢+𝐶𝐹𝑛+𝑔(𝜏)𝑑𝜏.(3.30)
From (3.26), the above inequality implies 𝑡0‖‖‖𝑑2𝑢𝑛𝑑𝑡2‖‖‖2𝐻𝑑𝜏≤𝐶,(𝐶>0isconstant).(3.31) We see that for all 0<𝑇<∞, {𝑢𝑛}⊂𝑊2,2((0,𝑇),𝐻) is bounded. Thus 𝑢∈𝑊2,2((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(𝑥,𝑦)≥𝐶1||𝑦||𝑝−𝐶2||||||𝑦||,𝑝≥2,𝑓(𝑥,𝑦)≤𝐶𝑝−1,𝑓+1(4.2)𝑥,𝑦1−𝑓𝑥,𝑦2𝑦1−𝑦2||𝑦≥𝜆1−𝑦2||2|||||,𝜆>0,(4.3)𝑔(𝑥,𝑧,𝜉,𝜂)≤𝐶𝑧|𝑝/2+||𝜉||𝑝/2+||𝜂||𝑝/2||𝑔+1,(4.4)𝑥,𝑧,𝜉,𝜂1−𝑔𝑥,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||||Δ𝑢2𝑑𝑥.(4.6)

Theorem 4.1. If the assumptions of (4.1)–(4.5) hold, for (𝜑,𝜓)∈𝑊2,𝑝(Ω)∩𝑊01,𝑝(Ω)×𝐻10(Ω), then (1.1) is a strong solution ğ‘¢âˆˆğ¿âˆžloc(0,∞),𝑊2,𝑝(Ω)∩𝑊01,𝑝,𝑢(Ω)ğ‘¡âˆˆğ¿âˆžloc(0,∞),𝐻10(Ω)∩𝐿2loc(0,∞),𝐻2,𝑢(Ω)𝑡𝑡∈𝐿𝑝′((0,𝑇)×Ω),ğ‘î…ž=𝑝𝑝−1,∀0<𝑇<∞.(4.7)

Proof. We introduce spatial sequences 𝑋=ğ‘¢âˆˆğ¶âˆž(||ΔΩ)𝑘𝑢||𝜕Ω,𝑋=0,𝑘=0,1,2,…1=𝐿𝑝𝑋(Ω),2=𝑊2,𝑝(Ω)∩𝑊01,𝑝(Ω),𝐻=𝐿2𝐻(Ω),1=𝐻10𝐻(Ω),2=𝐻2(Ω)∩𝐻10𝐻(Ω),3=𝑢∈𝐻2𝑚(Ω)∶𝑢|𝜕Ω=⋯=Δ𝑚−1𝑢|𝜕Ω,=0(4.8) where the inner products of 𝐻2 and 𝐻3 are defined by ⟨𝑢,𝑣⟩𝐻2=ΩΔ𝑢Δ𝑣𝑑𝑥,⟨𝑢,𝑣⟩𝐻2=ΩΔ𝑚𝑢Δ𝑚𝑣𝑑𝑥,(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⟩=Ω𝑓(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 {𝑢𝑛}âŠ‚ğ¿âˆž((0,𝑇),𝑊2,𝑝(Ω)∩𝑊01,𝑝(Ω)) satisfying (2.7) and limğ‘›â†’âˆžî€œğ‘‡0||⟨𝐺𝑢𝑛−𝐺𝑢0,𝐿𝑢𝑛−𝐿𝑢0⟩||𝑑𝑡=limğ‘›â†’âˆžî€œğ‘‡0Ω𝑓𝑥,Δ𝑢𝑛−𝑓𝑥,Δ𝑢0𝑢𝑛−𝑢0+𝑔𝑥,𝑢𝑛,𝐷𝑢𝑛,𝐷2𝑢𝑛−𝑔𝑥,𝑢0,𝐷𝑢0,𝐷2𝑢0𝑢𝑛−𝑢0𝑑𝑥𝑑𝑡=0.(4.12)
We need to prove that limğ‘›â†’âˆžî€œğ‘‡0Ω𝑓𝑥,Δ𝑢𝑛+𝑔𝑥,𝑢𝑛,𝐷𝑢𝑛,𝐷2𝑢𝑛=𝑣𝑑𝑥𝑑𝑡𝑇0Ω𝑓𝑥,Δ𝑢0+𝑔𝑥,𝑢0,𝐷𝑢0,𝐷2𝑢0𝑣𝑑𝑥𝑑𝑡.(4.13)
From (2.7) and Lemma 2.3, we obtain 𝑢𝑛⟶𝑢0,𝐷𝑢𝑛⟶𝐷𝑢0in𝐿2((0,𝑇)×Ω).(4.14)
From (4.3), we get 𝑇0Ω𝑓𝑥,Δ𝑢𝑛−𝑓𝑥,Δ𝑢0Δ𝑢𝑛−Δ𝑢0𝑑𝑥𝑑𝑡≥𝜆𝑇0Ω||Δ𝑢𝑛−Δ𝑢0||2𝑑𝑥𝑑𝑡.(4.15)
We have the deformation 𝑇0Ω𝑔𝑥,𝑢𝑛,𝐷𝑢𝑛,𝐷2𝑢𝑛−𝑔𝑥,𝑢0,𝐷𝑢0,𝐷2𝑢0Δ𝑢𝑛−Δ𝑢0=𝑑𝑥𝑑𝑡𝑇0Ω𝑔𝑥,𝑢𝑛,𝐷𝑢𝑛,𝐷2𝑢0−𝑔𝑥,𝑢0,𝐷𝑢0,𝐷2𝑢0Δ𝑢𝑛−Δ𝑢0+𝑑𝑥𝑑𝑡𝑇0Ω𝑔𝑥,𝑢𝑛,𝐷𝑢𝑛,𝐷2𝑢𝑛−𝑔𝑥,𝑢𝑛,𝐷𝑢𝑛,𝐷2𝑢0Δ𝑢𝑛−Δ𝑢0𝑑𝑥𝑑𝑡.(4.16)
From (4.14) and Lemma 2.4, we have limğ‘›â†’âˆžî€œğ‘‡0Ω𝑔𝑥,𝑢𝑛,𝐷𝑢𝑛,𝐷2𝑢0−𝑔𝑥,𝑢0,𝐷𝑢0,𝐷2𝑢0Δ𝑢𝑛−Δ𝑢0𝑑𝑥𝑑𝑡=0.(4.17)
From (4.12), (4.15)–(4.17), it follows that 0≥𝜆𝑇0Ω||Δ𝑢𝑛−Δ𝑢0||2𝑑𝑥𝑑𝑡+𝑇0Ω𝑔𝑥,𝑢𝑛,𝐷𝑢𝑛,𝐷2𝑢𝑛−𝑔𝑥,𝑢𝑛,𝐷𝑢𝑛,𝐷2𝑢0×Δ𝑢𝑛−Δ𝑢0𝑑𝑥𝑑𝑡≥𝜆𝑇0Ω||Δ𝑢𝑛−Δ𝑢0||2𝑑𝑥𝑑𝑡−𝐾1𝑇0Ω||𝐷2𝑢𝑛−𝐷2𝑢0||||Δ𝑢𝑛−Δ𝑢0||≥𝜆𝑑𝑥𝑑𝑡2𝑇0Ω||Δ𝑢𝑛−Δ𝑢0||2𝐾𝑑𝑥𝑑𝑡−212𝜆𝑇0Ω||𝐷2𝑢𝑛−𝐷2𝑢0||2≥𝜆𝑑𝑥𝑑𝑡2𝐾2−𝐾212𝜆𝑇0Ω||𝐷2𝑢𝑛−𝐷2𝑢0||2𝑑𝑥𝑑𝑡.(4.18)
Since 𝜆𝐾>𝐾1, we have limğ‘›â†’âˆžî€œğ‘‡0Ω||𝐷2𝑢𝑛−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 ⟨𝐴𝑢,𝐿𝑢⟩=−⟨𝐷𝐹1(𝑢),𝑣⟩,𝐹(𝑢)âŸ¶âˆžâŸºâ€–ğ‘¢â€–ğ‘‹2⟶∞,(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⟩Ω||𝑔𝑥,𝑢,𝐷𝑢,𝐷2𝑢||||||≤𝑘Δ𝑣𝑑𝑥2Ω||||Δ𝑣22𝑑𝑥+𝑘Ω||𝑔𝑥,𝑢,𝐷𝑢,𝐷2𝑢||2≤𝑘𝑑𝑥2‖𝑣‖2𝐻2+𝐶Ω||𝐷2𝑢||𝑝+||||∇𝑢𝑝+|𝑢|𝑝≤𝑘+1𝑑𝑥2‖𝑣‖2𝐻2+𝐶𝐹1(𝑢)+𝐶,(4.21) which imply Conditions (A3) of Theorem 3.1. From Theorem 3.1, (1.1) has a solution ğ‘¢âˆˆğ¿âˆžloc(0,∞),𝑊2,𝑝(Ω)∩𝑊01,𝑝,𝑢(Ω)ğ‘¡âˆˆğ¿âˆžloc(0,∞),𝐻10(Ω)∩𝐿2loc(0,∞),𝐻2,𝑢(Ω)𝑡𝑡∈𝐿𝑝′((0,𝑇)×Ω),ğ‘î…ž=𝑝𝑝−1,∀0<𝑇<∞,(4.22) satisfying Ω𝜕𝑢𝜕𝑡𝑣𝑑𝑥−𝑘ΩΔ𝑢𝑣𝑑𝑥=𝑡0Ω𝑓(𝑥,Δ𝑢)𝑣𝑑𝑥𝑑𝜏+𝑡0Ω𝑔𝑥,𝑢,𝐷𝑢,𝐷2𝑢+𝑣𝑑𝑥𝑑𝜏Ω𝜓𝑣𝑑𝑥−𝑘ΩΔ𝜑𝑣𝑑𝑥.(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).