Journal of Applied Mathematics

Journal of Applied Mathematics / 2012 / Article

Research Article | Open Access

Volume 2012 |Article ID 805158 | 15 pages | https://doi.org/10.1155/2012/805158

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

Academic Editor: Kuppalapalle Vajravelu
Received20 Feb 2012
Revised08 May 2012
Accepted09 May 2012
Published12 Jul 2012

Abstract

We consider the global existence of strong solution 𝑒, corresponding to a class of fully nonlinear wave equations with strongly damped terms π‘’π‘‘π‘‘βˆ’π‘˜Ξ”π‘’π‘‘=𝑓(π‘₯,Δ𝑒)+𝑔(π‘₯,𝑒,𝐷𝑒,𝐷2𝑒) in a bounded and smooth domain Ξ© in 𝑅𝑛, where 𝑓(π‘₯,Δ𝑒) is a given monotone in Δ𝑒 nonlinearity satisfying some dissipativity and growth restrictions and 𝑔(π‘₯,𝑒,𝐷𝑒,𝐷2𝑒) is in a sense subordinated to 𝑓(π‘₯,Δ𝑒). By using spatial sequence techniques, the Galerkin approximation method, and some monotonicity arguments, we obtained the global existence of a solution π‘’βˆˆπΏβˆžloc((0,∞),π‘Š2,𝑝(Ξ©)βˆ©π‘Š01,𝑝(Ξ©)).

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 𝐢∞ domain Ξ©βŠ‚π‘…π‘›: π‘’π‘‘π‘‘βˆ’π‘˜Ξ”π‘’π‘‘ξ€·=𝑓(π‘₯,Δ𝑒)+𝑔π‘₯,𝑒,𝐷𝑒,𝐷2𝑒[,in0,∞)Γ—Ξ©,𝑒(0,π‘₯)=πœ‘,𝑒𝑑[(0,π‘₯)=πœ“,inΞ©,𝑒(𝑑,π‘₯)=0,on0,∞)Γ—πœ•Ξ©,(1.1) where 𝑒𝑑=πœ•π‘’πœ•π‘‘,𝑒𝑑𝑑=πœ•2π‘’πœ•π‘‘2,Ξ”=𝑛𝑖=1πœ•2πœ•π‘₯2π‘–ξ‚΅πœ•,𝐷=πœ•π‘₯1πœ•,…,πœ•π‘₯𝑛,𝐷2=πœ•2πœ•π‘₯𝛼11β‹―πœ•π‘₯𝛼𝑛𝑛,𝛼1+β‹―+𝛼𝑛π‘₯=2,π‘₯=1,…,π‘₯𝑛,π‘˜>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: 𝑒𝑑𝑑+π‘’π‘‘βˆ’π‘’π‘₯π‘₯[𝑒=𝑓(𝑒),in0,∞)Γ—Ξ©,(0,π‘₯)=𝑒0(π‘₯),𝑒𝑑(0,π‘₯)=𝑒1(π‘₯),inΞ©;(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 Ξ©βŠ‚π‘…2 with a compact smooth boundary πœ•Ξ© for a semilinear strongly damped wave equation with a power-type nonlinearity |𝑒|π‘ž and π‘ž>6: 𝑒𝑑𝑑(𝑑,π‘₯)βˆ’Ξ”π‘’(𝑑,π‘₯)βˆ’Ξ”π‘’π‘‘||||(𝑑,π‘₯)=𝑒(𝑑,π‘₯)π‘ž[in0,∞)Γ—Ξ©,𝑒(0,π‘₯)=𝑒0(π‘₯),𝑒𝑑(0,π‘₯)=𝑒1[(π‘₯)inΞ©,𝑒(𝑑,π‘₯)=0,on0,∞)Γ—πœ•Ξ©.(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 |𝑒|π‘ž and π‘ž<(𝑛+2)/(π‘›βˆ’2) if 𝑛β‰₯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 𝐷𝑒 and the nonlinear terms 𝑓 containing Δ𝑒. Here 𝑓(π‘₯,Δ𝑒) is a given monotone in Δ𝑒 nonlinearity satisfying some dissipativity and growth restrictions and 𝑔(π‘₯,𝑒,𝐷𝑒,𝐷2𝑒) is in a sense subordinated to 𝑓(π‘₯,Δ𝑒).

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: π‘‹βŠ‚π»3βŠ‚π‘‹2βŠ‚π‘‹1π‘‹βŠ‚π»,2βŠ‚π»2βŠ‚π»1βŠ‚π»,(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 πΏβˆΆπ‘‹βŸΆπ‘‹1beone-for-onedenselinearoperator,βŸ¨πΏπ‘’,π‘£βŸ©π»=βŸ¨π‘’,π‘£βŸ©π»1,βˆ€π‘’,π‘£βˆˆπ‘‹.(2.2)

Furthermore, 𝐿 has eigenvectors {π‘’π‘˜} satisfying πΏπ‘’π‘˜=πœ†π‘˜π‘’π‘˜,(π‘˜=1,2,…),(2.3) and {π‘’π‘˜} constitutes common orthogonal basis of 𝐻 and 𝐻3.

We consider the following abstract wave equation model: 𝑑2𝑒𝑑𝑑2𝑑+π‘˜π‘‘π‘‘β„’π‘’=𝐺(𝑒),π‘˜>0,𝑒(0)=πœ‘,𝑒𝑑(0)=πœ“,(2.4) where πΊβˆΆπ‘‹2×𝑅+β†’π‘‹βˆ—1 is a map, 𝑅+=[0,∞), and β„’βˆΆπ‘‹2→𝑋1 is a bounded linear operator, satisfying βŸ¨β„’π‘’,πΏπ‘£βŸ©π»=βŸ¨π‘’,π‘£βŸ©π»2,βˆ€π‘’,π‘£βˆˆπ‘‹2.(2.5)

Definition 2.1. We say that π‘’βˆˆπ‘Š1,∞((0,𝑇),𝐻1)∩𝐿∞((0,𝑇),𝑋2) is a global weak solution of the (2.4) provided for (πœ‘,πœ“)βˆˆπ‘‹2×𝐻1βŸ¨π‘’π‘‘,π‘£βŸ©π»+π‘˜βŸ¨β„’π‘’,π‘£βŸ©π»=ξ€œπ‘‘0⟨𝐺(𝑒),π‘£βŸ©π‘‘πœ+βŸ¨πœ“,π‘£βŸ©π»+π‘˜βŸ¨β„’πœ‘,π‘£βŸ©π»,(2.6) for each π‘£βˆˆπ‘‹1 and 0≀𝑑≀𝑇<∞.

Definition 2.2. Let 𝑒𝑛, 𝑒0βˆˆπΏπ‘((0,𝑇),𝑋2). We say that 𝑒𝑛⇀𝑒0 in 𝐿𝑝((0,𝑇),𝑋2) is uniformly weakly convergent if {𝑒𝑛}βŠ‚πΏβˆž((0,𝑇),𝐻) is bounded, and 𝑒𝑛⇀𝑒0,in𝐿𝑝(0,𝑇),𝑋2ξ€Έ,limπ‘›β†’βˆžξ€œπ‘‡0||βŸ¨π‘’π‘›βˆ’π‘’0,π‘£βŸ©π»||2𝑑𝑑=0,βˆ€π‘£βˆˆπ».(2.7)

Lemma 2.3 (see [16]). Let {𝑒𝑛}βˆˆπΏπ‘((0,𝑇),π‘Šπ‘š,𝑝(Ξ©))(π‘šβ‰₯1) be bounded sequences and {𝑒𝑛} uniformly weakly convergent to 𝑒0βˆˆπΏπ‘((0,𝑇),π‘Šπ‘š,𝑝(Ξ©)). Then, for each |𝛼|β‰€π‘šβˆ’1, it follows that π·π›Όπ‘’π‘›βŸΆπ·π›Όπ‘’0,in𝐿2((0,𝑇)Γ—Ξ©).(2.8)

Lemma 2.4 (see [17]). Let Ξ©βŠ‚π‘…π‘› be an open set and π‘“βˆΆΞ©Γ—π‘…π‘β†’π‘…1 satisfy Caratheodory condition and ||||𝑓(π‘₯,πœ‰)≀𝐢𝑁𝑖=1||πœ‰π‘–||𝑝𝑖/𝑝+𝑏(π‘₯).(2.9) If {π‘’π‘–π‘˜}βŠ‚πΏπ‘π‘–(Ξ©)(1≀𝑖≀𝑁) is bounded and π‘’π‘–π‘˜ convergent to 𝑒𝑖 in Ξ©0 for all bounded Ξ©0βŠ‚Ξ©, then for each π‘£βˆˆπΏπ‘β€²(Ξ©), the following equality holds limπ‘˜β†’βˆžξ€œΞ©π‘“ξ€·π‘₯,𝑒1π‘˜,…,π‘’π‘π‘˜ξ€Έξ€œπ‘£π‘‘π‘₯=Ω𝑓π‘₯,𝑒1,…,𝑒𝑁𝑣𝑑π‘₯.(2.10)

3. Model Results

Let 𝐺=𝐴+π΅βˆΆπ‘‹2×𝑅+β†’π‘‹βˆ—1. Assume(A1) there is a 𝐢1 functional πΉβˆΆπ‘‹2→𝑅1 such that βŸ¨π΄π‘’,πΏπ‘£βŸ©=βŸ¨βˆ’π·πΉ(𝑒),π‘£βŸ©,βˆ€π‘’,π‘£βˆˆπ‘‹;(3.1)(A2) functional πΉβˆΆπ‘‹2→𝑅1 is coercive, that is, 𝐹(𝑒)⟢∞,βŸΊβ€–π‘’β€–π‘‹2⟢∞;(3.2)(A3)𝐡 satisfies ||||βŸ¨π΅π‘’,πΏπ‘£βŸ©β‰€πΆ1π‘˜πΉ(𝑒)+2‖𝑣‖2𝐻1+𝐢2,βˆ€π‘’,π‘£βˆˆπ‘‹,(3.3)

for π‘”βˆˆπΏ1loc(0,∞).

Theorem 3.1. Set πΊβˆΆπ‘‹2×𝑅+β†’π‘‹βˆ—1, for each (πœ‘,πœ“)βˆˆπ‘‹2×𝐻1, then the following assertions hold.(1)If 𝐺=𝐴 satisfies (A1) and (A2), then (2.4) has a globally weak solution π‘’βˆˆπ‘Š1,βˆžξ€·(0,∞),𝐻1ξ€Έβˆ©π‘Š1,2ξ€·(0,∞),𝐻2ξ€Έβˆ©πΏβˆžξ€·(0,∞),𝑋2ξ€Έ.(3.4)(2)If 𝐺=𝐴+𝐡 satisfies (A1)–(A3), then (2.4) has a global weak solution π‘’βˆˆπ‘Š1,∞locξ€·(0,∞),𝐻1ξ€Έβˆ©π‘Š1,2locξ€·(0,∞),𝐻2ξ€Έβˆ©πΏβˆžlocξ€·(0,∞),𝑋2ξ€Έ.(3.5)(3)Furthermore, if β„’βˆΆπ‘‹2→𝑋1 is symmetric sectorial operator, that is, βŸ¨β„’π‘’,π‘£βŸ©=βŸ¨π‘’,β„’π‘£βŸ©, and 𝐺=𝐴+𝐡 satisfies ||||βŸ¨πΊπ‘’,π‘£βŸ©β‰€πΆ11𝐹(𝑒)+2‖𝑣‖2𝐻+𝐢2,(3.6)then π‘’βˆˆπ‘Š2,2loc((0,∞),𝐻).

Proof. Let {π‘’π‘˜}βŠ‚π‘‹ be a common orthogonal basis of 𝐻 and 𝐻3, satisfying (2.3). Set 𝑋𝑛=𝑛𝑖=1π›Όπ‘–π‘’π‘–βˆ£π›Όπ‘–βˆˆπ‘…1ξƒ°,𝑋𝑛=𝑛𝑗=1𝛽𝑗(𝑑)π‘’π‘—βˆ£π›½π‘—βˆˆπΆ2[ξƒ°.0,∞)(3.7) Clearly, 𝐿𝑋𝑛=𝑋𝑛, 𝐿𝑋𝑛=𝑋𝑛.
By using Galerkin method, there exits π‘’π‘›βˆˆπΆ2([0,∞),𝑋𝑛) satisfying 𝑑𝑒𝑛𝑑𝑑,𝑣𝐻+π‘˜βŸ¨β„’π‘’π‘›,π‘£βŸ©π»=ξ€œπ‘‘0𝐺𝑒𝑛,π‘£π‘‘πœ+βŸ¨πœ“π‘›,π‘£βŸ©π»+π‘˜βŸ¨β„’πœ‘π‘›,π‘£βŸ©π»,𝑒𝑛(0)=πœ‘π‘›,π‘’ξ…žπ‘›(0)=πœ“π‘›,(3.8) for βˆ€π‘£βˆˆπ‘‹π‘›, and ξ€œπ‘‘0𝑑2𝑒𝑛𝑑𝑑2ξƒ’,𝑣𝐻ℒ+π‘˜π‘‘π‘’π‘›ξƒ’π‘‘π‘‘,π‘£π»ξ‚Ήξ€œπ‘‘πœ=𝑑0βŸ¨πΊπ‘’π‘›,π‘£βŸ©π‘‘πœ(3.9) for ξ‚π‘‹βˆ€π‘£βˆˆπ‘›.
Firstly, we consider 𝐺=𝐴. Let 𝑣=(𝑑/𝑑𝑑)𝐿𝑒𝑛 in (3.9). Taking into account (2.2) and (3.1), it follows that ξ€œ0=𝑑0𝑑2𝑒𝑛𝑑𝑑2,𝑑𝑑𝑑𝐿𝑒𝑛ℒ+π‘˜π‘‘π‘’π‘›,π‘‘π‘‘π‘‘π‘‘π‘‘πΏπ‘’π‘›ξƒ’π»ξ€œπ‘‘πœβˆ’π‘‘0𝐴𝑒𝑛,π‘‘π‘‘π‘‘πΏπ‘’π‘›ξ‚­ξ€œπ‘‘πœ,0=𝑑012𝑑𝑑𝑑𝑑𝑒𝑛,𝑑𝑑𝑑𝑒𝑛𝑑𝑑𝐻1+π‘˜π‘‘π‘’π‘›,𝑑𝑑𝑑𝑒𝑛𝑑𝑑𝐻2+𝑒𝐷𝐹𝑛,𝑑𝑒𝑛=1π‘‘π‘‘π‘‘πœ2‖‖‖𝑑𝑒𝑛‖‖‖𝑑𝑑2𝐻1βˆ’12β€–β€–πœ“π‘›β€–β€–2𝐻1ξ€œ+π‘˜π‘‘0‖‖‖𝑑𝑒𝑛‖‖‖𝑑𝑑2𝐻2ξ€·π‘’π‘‘πœ+πΉπ‘›ξ€Έξ€·πœ‘βˆ’πΉπ‘›ξ€Έ.(3.10)
We get 12‖‖‖𝑑𝑒𝑛‖‖‖𝑑𝑑2𝐻1ξ€œ+π‘˜π‘‘0‖‖‖𝑑𝑒𝑛‖‖‖𝑑𝑑2𝐻2ξ€·π‘’π‘‘πœ+πΉπ‘›ξ€Έξ€·πœ‘=𝐹𝑛+12β€–β€–πœ“π‘›β€–β€–2𝐻1.(3.11)
Let πœ‘βˆˆπ»3. From (2.1) and (2.2), it is known that {𝑒𝑛} are orthogonal basis of 𝐻1. We find that πœ‘π‘›β†’πœ‘ in 𝐻3, and πœ“π‘›β†’πœ“ in 𝐻1. Since 𝐻3βŠ‚π‘‹2 is imbedding, it follows that πœ‘π‘›βŸΆπœ‘,in𝑋2,πœ“π‘›βŸΆπœ“,in𝐻1.(3.12)
From (3.2), (3.11), and (3.12), we obtain that, ξ€½π‘’π‘›ξ€ΎβŠ‚π‘Š1,∞locξ€·(0,∞),𝐻1ξ€Έβˆ©π‘Š1,2locξ€·(0,∞),𝐻2ξ€Έβˆ©πΏβˆžlocξ€·(0,∞),𝑋2ξ€Έisbounded.(3.13)
Let π‘’π‘›βˆ—β‡€π‘’0,inπ‘Š1,∞locξ€·(0,∞),𝐻1ξ€Έβˆ©πΏβˆžlocξ€·(0,∞),𝑋2ξ€Έ,𝑒𝑛⇀𝑒0,inπ‘Š1,2locξ€·(0,∞),𝐻2ξ€Έ,(3.14) which implies that 𝑒𝑛→𝑒0 in π‘Š1,2loc((0,∞),𝐻) is uniformly weakly convergent from that 𝐻2βŠ‚π» is compact imbedding.
If we have the following equality: limπ‘›β†’βˆžξ‚Έβˆ’ξ€œπ‘‘0||βŸ¨πΊπ‘’π‘›βˆ’πΊπ‘’0,πΏπ‘’π‘›βˆ’πΏπ‘’0⟩||π‘˜π‘‘πœ+2β€–β€–π‘’π‘›βˆ’π‘’0β€–β€–2𝐻2ξ‚Ή=0,(3.15) then 𝑒0 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), ξ€œπ‘‘0ξ‚¬π‘‘π‘‘π‘‘β„’π‘’π‘›βˆ’π‘‘π‘‘π‘‘β„’π‘’0,πΏπ‘’π‘›βˆ’πΏπ‘’0𝐻1π‘‘πœ=2ξ€œπ‘‘0π‘‘π‘‘π‘‘βŸ¨π‘’π‘›βˆ’π‘’0,π‘’π‘›βˆ’π‘’0⟩𝐻2=1π‘‘πœ2‖‖𝑒𝑛(𝑑)βˆ’π‘’0(‖‖𝑑)2𝐻2βˆ’12β€–β€–πœ‘π‘›β€–β€–βˆ’πœ‘2𝐻2.(3.16)
Taking into account (2.2), (2.5), and (3.9), we get that βˆ’ξ€œπ‘‘0βŸ¨πΊπ‘’π‘›βˆ’πΊπ‘’0,πΏπ‘’π‘›βˆ’πΏπ‘’0π‘˜βŸ©π‘‘πœ+2β€–β€–π‘’π‘›βˆ’π‘’0β€–β€–2𝐻2=ξ€œπ‘‘0ξ‚ƒβŸ¨πΊπ‘’0βˆ’πΊπ‘’π‘›,πΏπ‘’π‘›βˆ’πΏπ‘’0ξ‚¬π‘‘βŸ©+π‘˜π‘‘π‘‘β„’π‘’π‘›βˆ’π‘‘π‘‘π‘‘β„’π‘’0,πΏπ‘’π‘›βˆ’πΏπ‘’0ξ‚­π»ξ‚„π‘˜π‘‘πœ+2β€–β€–πœ‘π‘›β€–β€–βˆ’πœ‘2𝐻2=ξ€œπ‘‘0ξƒ¬βŸ¨πΊπ‘’0,πΏπ‘’π‘›βˆ’πΏπ‘’0⟩+βŸ¨πΊπ‘’π‘›,𝐿𝑒0βŸ©βˆ’βŸ¨πΊπ‘’π‘›,πΏπ‘’π‘›ξƒ‘βŸ©βˆ’π‘˜π‘‘π‘’π‘›π‘‘π‘‘,𝑒0𝐻2ξƒ‘βˆ’π‘˜π‘‘π‘’0𝑑𝑑,π‘’π‘›βˆ’π‘’0𝐻2𝑑+π‘˜π‘‘π‘‘β„’π‘’π‘›,πΏπ‘’π‘›ξ‚­π»ξƒ­π‘˜π‘‘πœ+2β€–β€–πœ‘π‘›β€–β€–βˆ’πœ‘2𝐻2=ξ€œπ‘‘0ξƒ¬βŸ¨πΊπ‘’0,πΏπ‘’π‘›βˆ’πΏπ‘’0⟩+βŸ¨πΊπ‘’π‘›,𝐿𝑒0ξƒ‘βŸ©βˆ’π‘˜π‘‘π‘’π‘›π‘‘π‘‘,𝑒0𝐻2ξƒ‘βˆ’π‘˜π‘‘π‘’0𝑑𝑑,π‘’π‘›βˆ’π‘’0𝐻2βˆ’ξƒ‘π‘‘2𝑒𝑛𝑑𝑑2𝑑+π‘˜π‘‘π‘‘β„’π‘’π‘›,𝐿𝑒𝑛𝐻𝑑+π‘˜π‘‘π‘‘β„’π‘’π‘›,πΏπ‘’π‘›ξ‚­π»ξ‚Ήπ‘˜π‘‘πœ+2β€–β€–πœ‘π‘›β€–β€–βˆ’πœ‘2𝐻2=ξ€œπ‘‘0ξƒ¬βŸ¨πΊπ‘’0,πΏπ‘’π‘›βˆ’πΏπ‘’0⟩+βŸ¨πΊπ‘’π‘›,𝐿𝑒0βŸ©ξƒ‘βˆ’π‘˜π‘‘π‘’π‘›π‘‘π‘‘,𝑒0𝐻2ξ‚¬π‘‘βˆ’π‘˜π‘’π‘‘π‘‘0,π‘’π‘›βˆ’π‘’0𝐻2+𝑑𝑒𝑛,𝑑𝑑𝑑𝑒𝑛𝑑𝑑𝐻1ξƒ­βˆ’ξƒ‘π‘‘πœπ‘‘π‘’π‘›π‘‘π‘‘,𝑒𝑛𝐻1+βŸ¨πœ“π‘›,πœ‘π‘›βŸ©π»1+π‘˜2β€–β€–πœ‘π‘›β€–β€–βˆ’πœ‘2𝐻2.(3.17)
From (2.1) and (3.14), we have limπ‘›β†’βˆžβ€–β€–πœ‘π‘›β€–β€–βˆ’πœ‘π»2=0,limπ‘›β†’βˆžξ€œπ‘‘0βŸ¨πΊπ‘’0,πΏπ‘’π‘›βˆ’πΏπ‘’0βŸ©π‘‘πœ=0,limπ‘›β†’βˆžξ€œπ‘‘0𝑑𝑒𝑑𝑑0,π‘’π‘›βˆ’π‘’0𝐻2π‘‘πœ=0.(3.18)
Then, we get limπ‘›β†’βˆžβˆ’ξ€œπ‘‘0βŸ¨πΊπ‘’π‘›βˆ’πΊπ‘’0,πΏπ‘’π‘›βˆ’πΏπ‘’0π‘˜βŸ©π‘‘πœ+2limπ‘›β†’βˆžβ€–β€–π‘’π‘›βˆ’π‘’0β€–β€–2𝐻2=limπ‘›β†’βˆžβˆ«π‘‘0ξƒ¬βŸ¨πΊπ‘’π‘›,𝐿𝑒0ξƒ‘βŸ©βˆ’π‘˜π‘‘π‘’π‘›π‘‘π‘‘,𝑒0𝐻2+‖‖‖𝑑𝑒𝑛‖‖‖𝑑𝑑2𝐻1ξƒ­π‘‘πœβˆ’limπ‘›β†’βˆžξƒ‘π‘‘π‘’π‘›π‘‘π‘‘,𝑒𝑛𝐻1+βŸ¨πœ“,πœ‘βŸ©π»1.(3.19)
In view of (3.9), (3.14), we obtain for all π‘£βˆˆβˆͺβˆžπ‘›=1𝑋𝑛limπ‘›β†’βˆžξ€œπ‘‘0βŸ¨πΊπ‘’π‘›ξ€œ,πΏπ‘£βŸ©π‘‘πœ=𝑑0ξƒ¬π‘˜ξƒ‘π‘‘π‘’0𝑑𝑑,𝑣𝐻2βˆ’ξƒ‘π‘‘π‘’0,𝑑𝑑𝑑𝑣𝑑𝑑𝐻1ξƒ­+ξƒ‘π‘‘πœπ‘‘π‘’0𝑑𝑑,𝑣𝐻1βˆ’βŸ¨πœ“,𝑣(0)⟩𝐻1.(3.20)
Since βˆͺβˆžπ‘›=1𝑋𝑛 is dense in π‘Š1,2((0,𝑇),𝐻2)βˆ©πΏπ‘((0,𝑇),𝑋2), for all 𝑝<∞, (3.20) holds for all π‘£βˆˆπ‘Š1,2((0,𝑇),𝐻2)βˆ©πΏπ‘((0,𝑇),𝑋2). Thus, we have limπ‘›β†’βˆžξ€œπ‘‘0βŸ¨πΊπ‘’π‘›,𝐿𝑒0ξ€œβŸ©π‘‘πœ=𝑑0ξƒ¬π‘˜ξƒ‘π‘‘π‘’0𝑑𝑑,𝑒0𝐻2βˆ’β€–β€–β€–π‘‘π‘’0‖‖‖𝑑𝑑2𝐻1ξƒ­+ξƒ‘π‘‘πœπ‘‘π‘’0𝑑𝑑,𝑒0𝐻1βˆ’βŸ¨πœ“,πœ‘βŸ©π»1.(3.21)
From (3.14) and 𝐻2βŠ‚π»1 being compact imbedding, it follows that limπ‘›β†’βˆžξ€œπ‘‘0‖‖‖𝑑𝑒𝑛‖‖‖𝑑𝑑2𝐻1ξ€œπ‘‘πœ=𝑑0‖‖‖𝑑𝑒0‖‖‖𝑑𝑑2𝐻1π‘‘πœ,limπ‘›β†’βˆžξƒ‘π‘‘π‘’π‘›π‘‘π‘‘,𝑒𝑛𝐻1=𝑑𝑒0𝑑𝑑,𝑒0𝐻1,a.e.𝑑β‰₯0.(3.22)
Clearly, limπ‘›β†’βˆžξ€œπ‘‘0𝑑𝑒𝑛𝑑𝑑,𝑒𝑛𝐻1ξ€œπ‘‘πœ=𝑑0𝑑𝑒0𝑑𝑑,𝑒0𝐻1π‘‘πœ.(3.23)
Then, (3.14) follows from (3.19)–(3.21), which implies assertion (1).
Secondly, we consider 𝐺=𝐴+𝐡. Let 𝑣=(𝑑/𝑑𝑑)𝐿𝑒𝑛 in (3.9). In view of (2.2) and (2.8), it follows that ξ€œπ‘‘0𝑑2𝑒𝑛𝑑𝑑2,𝑑𝑑𝑑𝐿𝑒𝑛𝐻ℒ+π‘˜π‘‘π‘’π‘›,π‘‘π‘‘π‘‘π‘‘π‘‘πΏπ‘’π‘›ξƒ’π»ξ‚Ήξ€œπ‘‘πœ=𝑑0(𝐴+𝐡)𝑒𝑛,π‘‘π‘‘π‘‘πΏπ‘’π‘›ξ‚­π»ξ€œπ‘‘πœπ‘‘012𝑑𝑑𝑑𝑑𝑒𝑛,𝑑𝑑𝑑𝑒𝑛𝑑𝑑𝐻1+π‘˜π‘‘π‘’π‘›,𝑑𝑑𝑑𝑒𝑛𝑑𝑑𝐻2ξƒ­=ξ€œπ‘‘πœπ‘‘0ξ€·π‘’ξ‚Έξƒ‘βˆ’π·πΉπ‘›ξ€Έ,𝑑𝑒𝑛+𝐡𝑒𝑑𝑑𝑛,𝑑𝑑𝑑𝐿𝑒𝑛𝐻1π‘‘πœ,2‖‖‖𝑑𝑒𝑛‖‖‖𝑑𝑑2𝐻1βˆ’12β€–β€–πœ“π‘›β€–β€–2𝐻1ξ€œ+π‘˜π‘‘0‖‖‖𝑑𝑒𝑛‖‖‖𝑑𝑑2𝐻2ξ€·π‘’π‘‘πœ+πΉπ‘›ξ€Έξ€·πœ‘βˆ’πΉπ‘›ξ€Έ=ξ€œπ‘‘0𝐡𝑒𝑛,𝑑𝑑𝑑𝐿𝑒𝑛𝐻1π‘‘πœ,2‖‖‖𝑑𝑒𝑛‖‖‖𝑑𝑑2𝐻1ξ€œ+π‘˜π‘‘0‖‖‖𝑑𝑒𝑛‖‖‖𝑑𝑑2𝐻2𝑒𝑑𝑑+𝐹𝑛=ξ€œπ‘‘0𝐡𝑒𝑛,π‘‘π‘‘π‘‘πΏπ‘’π‘›ξ‚­ξ€·πœ‘π‘‘πœ+𝐹𝑛+12β€–β€–πœ“π‘›β€–β€–2𝐻1.(3.24)
From (3.3), we have 12‖‖‖𝑑𝑒𝑛‖‖‖𝑑𝑑2𝐻1𝑒+πΉπ‘›ξ€Έξ€œ+π‘˜π‘‘0‖‖‖𝑑𝑒𝑛‖‖‖𝑑𝑑2𝐻2ξ€œπ‘‘πœβ‰€πΆπ‘‘0𝐢1𝐹𝑒𝑛+π‘˜2‖‖‖𝑑𝑒𝑛‖‖‖𝑑𝑑2𝐻1+𝐢2ξ‚Ήξ€·πœ‘π‘‘πœ+𝐹𝑛+12β€–β€–πœ“π‘›β€–β€–2𝐻1ξ€œβ‰€πΆπ‘‘0𝐹𝑒𝑛+12‖‖‖𝑑𝑒𝑛‖‖‖𝑑𝑑2𝐻1ξ‚Ήπ‘‘πœ+𝑓(𝑑),(3.25) where 𝑓(𝑑)=(1/2)β€–πœ“β€–2𝐻1+sup𝑛𝐹(πœ‘π‘›).
By using Gronwall inequality, it follows that 12‖‖‖𝑑𝑒𝑛‖‖‖𝑑𝑑2𝐻1𝑒+𝐹𝑛≀𝑓(0)𝑒𝐢𝑑+ξ€œπ‘‘0𝑓(𝜏)𝑒𝐢(π‘‘βˆ’πœ)π‘‘πœ,(3.26) which implies that for all 0<𝑇<∞, ξ€½π‘’π‘›ξ€ΎβŠ‚π‘Š1,βˆžξ€·(0,𝑇),𝐻1ξ€Έβˆ©πΏβˆžξ€·(0,𝑇),𝑋2ξ€Έisbounded.(3.27)
From (3.25) and (3.21), it follows that ξ€½π‘’π‘›ξ€ΎβŠ‚π‘Š1,2ξ€·(0,𝑇),𝐻2ξ€Έisbounded.(3.28)
Let π‘’π‘›βˆ—β‡€π‘’0,inπ‘Š1,βˆžξ€·(0,𝑇),𝐻1ξ€Έβˆ©πΏβˆžξ€·(0,𝑇),𝑋2ξ€Έ,𝑒𝑛⇀𝑒0,inπ‘Š1,2ξ€·(0,𝑇),𝐻2ξ€Έ,(3.29) which implies that 𝑒𝑛→𝑒0 in π‘Š1,2((0,𝑇),𝐻) is uniformly weakly convergent from that 𝐻2βŠ‚π» is compact imbedding.
The left proof is same as assertion (1).
Lastly, assume (3.6) hold. Let 𝑣=𝑑2𝑒𝑛/𝑑𝑑2 in (3.9). It follows that ξ€œπ‘‘0𝑑2𝑒𝑛𝑑𝑑2,𝑑2𝑒𝑛𝑑𝑑2𝐻ℒ+π‘˜π‘‘π‘’π‘›,𝑑𝑑𝑑2𝑒𝑛𝑑𝑑2𝐻=ξ€œπ‘‘πœπ‘‘0𝑒(𝐴+𝐡)𝑛,𝑑2𝑒𝑛𝑑𝑑2ξƒ’β‰€ξ€œπ‘‘πœπ‘‘0𝑒𝐢𝐹𝑛+12‖‖‖𝑑2𝑒𝑛𝑑𝑑2β€–β€–β€–2π»ξƒ­ξ€œ+𝑔(𝑑)π‘‘πœ,𝑑0𝑑2𝑒𝑛𝑑𝑑2,𝑑2𝑒𝑛𝑑𝑑2𝐻+π‘˜2ξ€œπ‘‘0ξ€œΞ©π‘‘ξ€·π‘’π‘‘π‘‘ξ…žπ‘›ξ€Έ(𝑑)2ξ‚Ήβ‰€ξ€œπ‘‘π‘₯π‘‘πœπ‘‘0𝑒𝐢𝐹𝑛+12‖‖‖𝑑2𝑒𝑛𝑑𝑑2β€–β€–β€–2π»ξƒ­ξ€œ+𝑔(𝑑)π‘‘πœ,𝑑0𝑑2𝑒𝑛𝑑𝑑2,𝑑2𝑒𝑛𝑑𝑑2ξƒ’π»π‘˜π‘‘π‘‘+2‖‖‖𝑑𝑒𝑛‖‖‖𝑑𝑑2𝐻1β‰€π‘˜2β€–β€–πœ“π‘›β€–β€–2𝐻+ξ€œπ‘‘012‖‖‖𝑑2𝑒𝑛𝑑𝑑2β€–β€–β€–2𝐻𝑒+𝐢𝐹𝑛+𝑔(𝜏)π‘‘πœ.(3.30)
From (3.26), the above inequality implies ξ€œπ‘‘0‖‖‖𝑑2𝑒𝑛𝑑𝑑2β€–β€–β€–2π»π‘‘πœβ‰€πΆ,(𝐢>0isconstant).(3.31) We see that for all 0<𝑇<∞, {𝑒𝑛}βŠ‚π‘Š2,2((0,𝑇),𝐻) is bounded. Thus π‘’βˆˆπ‘Š2,2((0,𝑇),𝐻).

4. Main Result

Now, we begin to consider the mixed problem (1.1). Set ξ€œπΉ(π‘₯,𝑦)=𝑦0𝑓(π‘₯,𝑧)𝑑𝑧.(4.1) We assume 𝐹(π‘₯,𝑦)β‰₯𝐢1||𝑦||π‘βˆ’πΆ2||||ξ‚€||𝑦||,𝑝β‰₯2,𝑓(π‘₯,𝑦)β‰€πΆπ‘βˆ’1,𝑓+1(4.2)π‘₯,𝑦1ξ€Έξ€·βˆ’π‘“π‘₯,𝑦2𝑦1βˆ’π‘¦2ξ€Έ||𝑦β‰₯πœ†1βˆ’π‘¦2||2||||ξ‚€|,πœ†>0,(4.3)𝑔(π‘₯,𝑧,πœ‰,πœ‚)≀𝐢𝑧|𝑝/2+||πœ‰||𝑝/2+||πœ‚||𝑝/2||𝑔+1,(4.4)π‘₯,𝑧,πœ‰,πœ‚1ξ€Έξ€·βˆ’π‘”π‘₯,𝑧,πœ‰,πœ‚2ξ€Έ||≀𝐾1||πœ‚1βˆ’πœ‚2||,(4.5) where 𝐢,𝐢1,𝐢2 are constant and 𝐾1<πœ†πΎ, 𝐾 is the best constant satisfying 𝐾2‖𝑒‖2𝐻2β‰€ξ€œΞ©||||Δ𝑒2𝑑π‘₯.(4.6)

Theorem 4.1. If the assumptions of (4.1)–(4.5) hold, for (πœ‘,πœ“)βˆˆπ‘Š2,𝑝(Ξ©)βˆ©π‘Š01,𝑝(Ξ©)×𝐻10(Ξ©), then (1.1) is a strong solution π‘’βˆˆπΏβˆžlocξ‚€(0,∞),π‘Š2,𝑝(Ξ©)βˆ©π‘Š01,𝑝,𝑒(Ξ©)π‘‘βˆˆπΏβˆžlocξ€·(0,∞),𝐻10ξ€Έ(Ξ©)∩𝐿2locξ€·(0,∞),𝐻2ξ€Έ,𝑒(Ξ©)π‘‘π‘‘βˆˆπΏπ‘β€²((0,𝑇)Γ—Ξ©),π‘ξ…ž=π‘π‘βˆ’1,βˆ€0<𝑇<∞.(4.7)

Proof. We introduce spatial sequences 𝑋=π‘’βˆˆπΆβˆž(||ΔΩ)π‘˜π‘’||πœ•Ξ©ξ€Ύ,𝑋=0,π‘˜=0,1,2,…1=𝐿𝑝𝑋(Ξ©),2=π‘Š2,𝑝(Ξ©)βˆ©π‘Š01,𝑝(Ξ©),𝐻=𝐿2𝐻(Ξ©),1=𝐻10𝐻(Ξ©),2=𝐻2(Ξ©)∩𝐻10𝐻(Ξ©),3=ξ€½π‘’βˆˆπ»2π‘š(Ξ©)βˆΆπ‘’|πœ•Ξ©=β‹―=Ξ”π‘šβˆ’1𝑒|πœ•Ξ©ξ€Ύ,=0(4.8) where the inner products of 𝐻2 and 𝐻3 are defined by βŸ¨π‘’,π‘£βŸ©π»2=ξ€œΞ©Ξ”π‘’Ξ”π‘£π‘‘π‘₯,βŸ¨π‘’,π‘£βŸ©π»2=ξ€œΞ©Ξ”π‘šπ‘’Ξ”π‘šπ‘£π‘‘π‘₯,(4.9) where π‘šβ‰₯1 such that 𝐻3βŠ‚π‘‹2 is an embedding.
Linear operators β„’βˆΆπ‘‹β†’π‘‹1 and πΏβˆΆπ‘‹β†’π‘‹1 are defined by ℒ𝑒=𝐿𝑒=βˆ’Ξ”π‘’.(4.10) It is known that β„’ and 𝐿 satisfy (2.2), (2.3), and (2.5). Define 𝐺=𝐴+π΅βˆΆπ‘‹2β†’π‘‹βˆ—1 by ξ€œβŸ¨π΄π‘’,π‘£βŸ©=Ξ©ξ€œπ‘“(π‘₯,Δ𝑒)𝑣𝑑π‘₯,βŸ¨π΅π‘’,π‘£βŸ©=Ω𝑔π‘₯,𝑒,𝐷𝑒,𝐷2𝑒𝑣𝑑π‘₯,forπ‘£βˆˆπ‘‹1.(4.11)
We show that 𝐺=𝐴+π΅βˆΆπ‘‹2β†’π‘‹βˆ—1 is 𝑇-coercively weakly continuous. Let {𝑒𝑛}βŠ‚πΏβˆž((0,𝑇),π‘Š2,𝑝(Ξ©)βˆ©π‘Š01,𝑝(Ξ©)) satisfying (2.7) and limπ‘›β†’βˆžξ€œπ‘‡0||βŸ¨πΊπ‘’π‘›βˆ’πΊπ‘’0,πΏπ‘’π‘›βˆ’πΏπ‘’0⟩||𝑑𝑑=limπ‘›β†’βˆžξ€œπ‘‡0ξ€œΞ©π‘“ξ€·ξ€Ίξ€·π‘₯,Ξ”π‘’π‘›ξ€Έξ€·βˆ’π‘“π‘₯,Δ𝑒0π‘’ξ€Έξ€Έξ€·π‘›βˆ’π‘’0ξ€Έ+𝑔π‘₯,𝑒𝑛,𝐷𝑒𝑛,𝐷2π‘’π‘›ξ€Έξ€·βˆ’π‘”π‘₯,𝑒0,𝐷𝑒0,𝐷2𝑒0π‘’ξ€Έξ€Έξ€·π‘›βˆ’π‘’0𝑑π‘₯𝑑𝑑=0.(4.12)
We need to prove that limπ‘›β†’βˆžξ€œπ‘‡0ξ€œΞ©ξ€Ίπ‘“ξ€·π‘₯,Δ𝑒𝑛+𝑔π‘₯,𝑒𝑛,𝐷𝑒𝑛,𝐷2𝑒𝑛=ξ€œξ€Έξ€»π‘£π‘‘π‘₯𝑑𝑑𝑇0ξ€œΞ©ξ€Ίπ‘“ξ€·π‘₯,Δ𝑒0ξ€Έξ€·+𝑔π‘₯,𝑒0,𝐷𝑒0,𝐷2𝑒0𝑣𝑑π‘₯𝑑𝑑.(4.13)
From (2.7) and Lemma 2.3, we obtain π‘’π‘›βŸΆπ‘’0,π·π‘’π‘›βŸΆπ·π‘’0in𝐿2((0,𝑇)Γ—Ξ©).(4.14)
From (4.3), we get ξ€œπ‘‡0ξ€œΞ©ξ€Ίπ‘“ξ€·π‘₯,Ξ”π‘’π‘›ξ€Έξ€·βˆ’π‘“π‘₯,Δ𝑒0ξ€Έξ€»ξ€·Ξ”π‘’π‘›βˆ’Ξ”π‘’0ξ€Έξ€œπ‘‘π‘₯𝑑𝑑β‰₯πœ†π‘‡0ξ€œΞ©||Ξ”π‘’π‘›βˆ’Ξ”π‘’0||2𝑑π‘₯𝑑𝑑.(4.15)
We have the deformation ξ€œπ‘‡0ξ€œΞ©ξ€Ίπ‘”ξ€·π‘₯,𝑒𝑛,𝐷𝑒𝑛,𝐷2π‘’π‘›ξ€Έξ€·βˆ’π‘”π‘₯,𝑒0,𝐷𝑒0,𝐷2𝑒0ξ€Έξ€»ξ€·Ξ”π‘’π‘›βˆ’Ξ”π‘’0ξ€Έ=ξ€œπ‘‘π‘₯𝑑𝑑𝑇0ξ€œΞ©ξ€Ίπ‘”ξ€·π‘₯,𝑒𝑛,𝐷𝑒𝑛,𝐷2𝑒0ξ€Έξ€·βˆ’π‘”π‘₯,𝑒0,𝐷𝑒0,𝐷2𝑒0ξ€Έξ€»ξ€·Ξ”π‘’π‘›βˆ’Ξ”π‘’0ξ€Έ+ξ€œπ‘‘π‘₯𝑑𝑑𝑇0ξ€œΞ©ξ€Ίπ‘”ξ€·π‘₯,𝑒𝑛,𝐷𝑒𝑛,𝐷2π‘’π‘›ξ€Έξ€·βˆ’π‘”π‘₯,𝑒𝑛,𝐷𝑒𝑛,𝐷2𝑒0ξ€Έξ€»ξ€·Ξ”π‘’π‘›βˆ’Ξ”π‘’0𝑑π‘₯𝑑𝑑.(4.16)
From (4.14) and Lemma 2.4, we have limπ‘›β†’βˆžξ€œπ‘‡0ξ€œΞ©ξ€Ίπ‘”ξ€·π‘₯,𝑒𝑛,𝐷𝑒𝑛,𝐷2𝑒0ξ€Έξ€·βˆ’π‘”π‘₯,𝑒0,𝐷𝑒0,𝐷2𝑒0ξ€Έξ€»ξ€·Ξ”π‘’π‘›βˆ’Ξ”π‘’0𝑑π‘₯𝑑𝑑=0.(4.17)
From (4.12), (4.15)–(4.17), it follows that ξ€œ0β‰₯πœ†π‘‡0ξ€œΞ©||Ξ”π‘’π‘›βˆ’Ξ”π‘’0||2ξ€œπ‘‘π‘₯𝑑𝑑+𝑇0ξ€œΞ©ξ€Ίπ‘”ξ€·π‘₯,𝑒𝑛,𝐷𝑒𝑛,𝐷2π‘’π‘›ξ€Έξ€·βˆ’π‘”π‘₯,𝑒𝑛,𝐷𝑒𝑛,𝐷2𝑒0Γ—ξ€·ξ€Έξ€»Ξ”π‘’π‘›βˆ’Ξ”π‘’0ξ€Έξ€œπ‘‘π‘₯𝑑𝑑β‰₯πœ†π‘‡0ξ€œΞ©||Ξ”π‘’π‘›βˆ’Ξ”π‘’0||2𝑑π‘₯π‘‘π‘‘βˆ’πΎ1ξ€œπ‘‡0ξ€œΞ©||𝐷2π‘’π‘›βˆ’π·2𝑒0||||Ξ”π‘’π‘›βˆ’Ξ”π‘’0||β‰₯πœ†π‘‘π‘₯𝑑𝑑2ξ€œπ‘‡0ξ€œΞ©||Ξ”π‘’π‘›βˆ’Ξ”π‘’0||2𝐾𝑑π‘₯π‘‘π‘‘βˆ’21ξ€œ2πœ†π‘‡0ξ€œΞ©||𝐷2π‘’π‘›βˆ’π·2𝑒0||2β‰₯πœ†π‘‘π‘₯𝑑𝑑2𝐾2βˆ’πΎ21ξ€œ2πœ†π‘‡0ξ€œΞ©||𝐷2π‘’π‘›βˆ’π·2𝑒0||2𝑑π‘₯𝑑𝑑.(4.18)
Since πœ†πΎ>𝐾1, we have limπ‘›β†’βˆžξ€œπ‘‡0ξ€œΞ©||𝐷2π‘’π‘›βˆ’π·2𝑒0||2𝑑π‘₯𝑑𝑑=0.(4.19)
From (4.14), (4.19), (4.1), (4.4), and Lemma 2.3, we get (4.13).
Let 𝐹1∫(𝑒)=Ω𝐹(π‘₯,Δ𝑒)𝑑π‘₯, where 𝐹 is same as (4.1). We get βŸ¨π΄π‘’,πΏπ‘’βŸ©=βˆ’βŸ¨π·πΉ1(𝑒),π‘£βŸ©,𝐹(𝑒)βŸΆβˆžβŸΊβ€–π‘’β€–π‘‹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 ||||=ξ€œβŸ¨π΅π‘’,πΏπ‘£βŸ©Ξ©||𝑔π‘₯,𝑒,𝐷𝑒,𝐷2𝑒||||||β‰€π‘˜Ξ”π‘£π‘‘π‘₯2ξ€œΞ©||||Δ𝑣22𝑑π‘₯+π‘˜ξ€œΞ©||𝑔π‘₯,𝑒,𝐷𝑒,𝐷2𝑒||2β‰€π‘˜π‘‘π‘₯2‖𝑣‖2𝐻2ξ€œ+𝐢Ω||𝐷2𝑒||𝑝+||||βˆ‡π‘’π‘+|𝑒|π‘ξ€»β‰€π‘˜+1𝑑π‘₯2‖𝑣‖2𝐻2+𝐢𝐹1(𝑒)+𝐢,(4.21) which imply Conditions (A3) of Theorem 3.1. From Theorem 3.1, (1.1) has a solution π‘’βˆˆπΏβˆžlocξ‚€(0,∞),π‘Š2,𝑝(Ξ©)βˆ©π‘Š01,𝑝,𝑒(Ξ©)π‘‘βˆˆπΏβˆžlocξ€·(0,∞),𝐻10ξ€Έ(Ξ©)∩𝐿2locξ€·(0,∞),𝐻2ξ€Έ,𝑒(Ξ©)π‘‘π‘‘βˆˆπΏπ‘β€²((0,𝑇)Γ—Ξ©),π‘ξ…ž=π‘π‘βˆ’1,βˆ€0<𝑇<∞,(4.22) satisfying ξ€œΞ©πœ•π‘’ξ€œπœ•π‘‘π‘£π‘‘π‘₯βˆ’π‘˜Ξ©ξ€œΞ”π‘’π‘£π‘‘π‘₯=𝑑0ξ€œΞ©ξ€œπ‘“(π‘₯,Δ𝑒)𝑣𝑑π‘₯π‘‘πœ+𝑑0ξ€œΞ©π‘”ξ€·π‘₯,𝑒,𝐷𝑒,𝐷2𝑒+ξ€œπ‘£π‘‘π‘₯π‘‘πœΞ©ξ€œπœ“π‘£π‘‘π‘₯βˆ’π‘˜Ξ©Ξ”πœ‘π‘£π‘‘π‘₯.(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).

References

  1. G. Andrews, β€œOn the existence of solutions to the equation utt=uxxt+σ(ux)x,” Journal of Differential Equations, vol. 35, no. 2, pp. 200–231, 1980. View at: Publisher Site | utt=uxxt+Οƒ(ux)x&author=G. Andrews&publication_year=1980" target="_blank">Google Scholar
  2. S. Kawashima and Y. Shibata, β€œGlobal existence and exponential stability of small solutions to nonlinear viscoelasticity,” Communications in Mathematical Physics, vol. 148, no. 1, pp. 189–208, 1992. View at: Google Scholar
  3. D. D. Ang and A. P. N. Dinh, β€œStrong solutions of a quasilinear wave equation with nonlinear damping,” SIAM Journal on Mathematical Analysis, vol. 19, no. 2, pp. 337–347, 1988. View at: Publisher Site | Google Scholar
  4. K. Nishihara and H. Zhao, β€œExistence and nonexistence of time-global solutions to damped wave equation on half-line,” Nonlinear Analysis: Theory, Methods & Applications, vol. 61, no. 6, pp. 931–960, 2005. View at: Publisher Site | Google Scholar
  5. N. Hayashi, E. I. Kaikina, and P. I. Naumkin, β€œDamped wave equation in the subcritical case,” Journal of Differential Equations, vol. 207, no. 1, pp. 161–194, 2004. View at: Publisher Site | Google Scholar
  6. M. Prizzi, β€œRegularity of invariant sets in semilinear damped wave equations,” Journal of Differential Equations, vol. 247, no. 12, pp. 3315–3337, 2009. View at: Publisher Site | Google Scholar
  7. S. Zelik, β€œAsymptotic regularity of solutions of a nonautonomous damped wave equation with a critical growth exponent,” Communications on Pure and Applied Analysis, vol. 3, no. 4, pp. 921–934, 2004. View at: Publisher Site | Google Scholar
  8. T. Ogawa and H. Takeda, β€œNon-existence of weak solutions to nonlinear damped wave equations in exterior domains,” Nonlinear Analysis: Theory, Methods & Applications, vol. 70, no. 10, pp. 3696–3701, 2009. View at: Publisher Site | Google Scholar
  9. R. Ikehata and Y.-k. Inoue, β€œGlobal existence of weak solutions for two-dimensional semilinear wave equations with strong damping in an exterior domain,” Nonlinear Analysis: Theory, Methods & Applications, vol. 68, no. 1, pp. 154–169, 2008. View at: Publisher Site | Google Scholar
  10. J. W. Cholewa and T. Dlotko, β€œStrongly damped wave equation in uniform spaces,” Nonlinear Analysis: Theory, Methods & Applications, vol. 64, no. 1, pp. 174–187, 2006. View at: Publisher Site | Google Scholar
  11. G. F. Webb, β€œExistence and asymptotic behavior for a strongly damped nonlinear wave equation,” Canadian Journal of Mathematics, vol. 32, no. 3, pp. 631–643, 1980. View at: Publisher Site | Google Scholar
  12. E. Feireisl, β€œBounded, locally compact global attractors for semilinear damped wave equations on RN,” Differential and Integral Equations, vol. 9, no. 5, pp. 1147–1156, 1996. View at: RN&author=E. Feireisl&publication_year=1996" target="_blank">Google Scholar
  13. R. Temam, Infinite-Dimensional Dynamical Systems in Mechanics and Physics, vol. 68 of Applied Mathematical Sciences, Springer, New York, NY, USA, 2nd edition, 1997.
  14. Z. Yang, β€œCauchy problem for quasi-linear wave equations with nonlinear damping and source terms,” Journal of Mathematical Analysis and Applications, vol. 300, no. 1, pp. 218–243, 2004. View at: Publisher Site | Google Scholar
  15. Z. G. Pan, Z. L. Pu, and T. Ma, β€œGlobal solutions to a class of nonlinear damped wave operator equations,” Boundary Value Problems, vol. 2012, article 42, 2012. View at: Publisher Site | Google Scholar
  16. T. Ma, Theories and Methods in Partial Differential Equations, Science Press, Beijing, China, 2011.
  17. E. Zeidler, Applied Functional Analysis: Applications to Mathematical Physics, vol. 108 of Applied Mathematical Sciences, Springer, New York, NY, USA, 1995.

Copyright © 2012 Zhigang Pan et al. 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.

477Β Views | 292Β Downloads | 2Β Citations
 PDF  Download Citation  Citation
 Download other formatsMore
 Order printed copiesOrder