Existence of Global Attractors for the Nonlinear Plate Equation with Memory Term

Abstract

A two-dimensional nonlinear plate equation is revisited, which arises from the model of the viscoelastic thin rectangular plate with four edges supported. We establish that the system is exponentially decayed if the memory kernel satisfies the condition of the exponential decay. Furthermore, we show the existence of the global attractor by verifying the condition (C).

1. Introduction

In this paper, we investigate the nonlinear plate equation with memory type: 𝜌𝑢_𝑡𝑡+𝑟₁𝑟₂𝑢_𝑡+𝜙(0)▵²𝑢−𝑁₁+𝛽𝑢_𝑥²𝑢_𝑥𝑥−𝑁₂+𝛽𝑢_𝑦²𝑢_𝑦𝑦+^∞₀𝜙^′(𝑠)▵²𝑢(𝑡−𝑠)𝑑𝑠=𝑟₁𝑓𝑥,𝑦,inΩ×𝑅⁺, (1.1) verifying the initial conditions: 𝑢𝑥,𝑦,0=𝑢₀𝑥,𝑦,𝑡 (1.2) and the boundary conditions: 𝑢()()𝑢0,𝑦,𝑡=𝑢1,𝑦,𝑡=𝑢𝑥,0,𝑡=𝑢𝑥,1,𝑡=0,_𝑥𝑥0,𝑦,𝑡=𝑢_𝑥𝑥1,𝑦,𝑡=𝑢_𝑦𝑦(𝑥,0,𝑡)=𝑢_𝑦𝑦(𝑥,1,𝑡), (1.3) where 𝑥,𝑦∈[0,1],  𝑡∈𝑅,𝜂,𝑟₁,𝑟₂,𝑁₁,𝑁₂,𝛽 are nonnegative constants, Ω=[0,1]×[0,1] is a bounded domain with boundary 𝜕Ω,𝜙(0),𝜙(∞)>0 and 𝜙^′(𝑠)<0 for every 𝑠∈𝑅⁺.

The asymptotical behavior of solutions for the nonlinear plate equations had been studied by many authors [1–10]; of those, Santos and Junior [5] studied a kind of plate equation with memory type. Yang and Zhong [11] studies the plate equation: 𝑢_𝑡𝑡+𝛼(𝑥)𝑔𝑢_𝑡+Δ²𝑢+𝜆𝑢+𝑓(𝑢)=ℎ(𝑥),𝑥∈Ω, (1.4) where Ω⊂𝑅^𝑛 is a bounded domain and proves the existence of a global attractor in the space 𝐻¹₀(Ω)×𝐿²(Ω). After Yang and Zhong [11], Yue and Zhong [12] obtained the existence of a global attractor about some equations similar to (1.4). Xiao [13] discusses the long-time behavior of the plate equation: 𝜀𝑢_𝑡𝑡+Δ²𝑢+𝜆𝑢_𝑡+𝛽(𝑥)𝑢=𝑓(𝑥,𝑢),𝑥∈Ω=𝑅^𝑛,𝑡≥0 (1.5) on the unbounded domain 𝑅^𝑛 and show that there exists a compact global attractor for the above equation satisfying certain initial-boundary data. Wang and Zhang [14] prove that the two-dimensional nonlinear equation 𝜌𝑢_𝑡𝑡+𝐷Δ²𝑢+𝜀𝜇𝑢_𝑡−𝑁₁+𝑇2_Ω𝑢²_𝑥𝑑𝑥𝑑𝑦𝑢_𝑥𝑥−𝑁₂+𝑇2_Ω𝑢²_𝑦𝑑𝑥𝑑𝑦𝑢_𝑦𝑦=0 (1.6) has a global attractor in space (𝐻¹₀(Ω)⋂𝐿²(Ω)).

2. Preliminaries

We denote by 𝐻=𝐿²(Ω),𝑉=𝐻²₀(Ω) endowed with the scalar product and the norm on 𝐻 and 𝑉(⋅,⋅),|⋅|,((⋅,⋅)),‖⋅‖, respectively, where (𝑢,𝑣)=∫_Ω𝑢(𝑥,𝑦)𝑣(𝑥,𝑦)𝑑𝑥𝑑𝑦,  ((𝑢,𝑣))=∫_ΩΔ𝑢(𝑥,𝑦)Δ𝑣(𝑥,𝑦)𝑑𝑥𝑑𝑦. Define 𝐷(𝐴)={𝑣∈𝑉,𝐴𝑣∈𝐻}, where 𝐴=Δ². For the operator 𝐴, we assume that 𝐴∶𝐷(𝐴)→𝐻 are isomorphism, and there exists 𝛼>0 such that (𝐴𝑢,𝑢)≥𝛼‖𝑢‖², for all 𝑢∈𝑉. We also define the power 𝐴^𝑠 of 𝐴 for 𝑠∈𝑅 which operates on the spaces 𝐷(𝐴^𝑠), and we write 𝑉_2𝑠=𝐷(𝐴^𝑠),𝑠∈𝑅. This is a Hilbert space with the inner product and norm defined (𝑢,𝑣)_2𝑠=𝐴^𝑠𝑢,𝐴^𝑠𝑣,‖𝑢‖_2𝑠=(𝑢,𝑣)_2𝑠^1/2,∀𝑢,𝑣∈𝐷𝐴^𝑠, (2.1) and 𝐴^𝑟 is an isomorphism from 𝐷(𝐴^𝑠) onto 𝐷(𝐴^𝑠−𝑟), for all 𝑠,𝑟∈𝑅. In particular, 𝐷(𝐴⁰)=𝐻,𝐷(𝐴^1/2)=𝑉,𝐷(𝐴^−1/2)=𝑉^∗,𝐷(𝐴)⊂𝑉⊂𝐻=𝐻^∗⊂𝑉^∗, where  𝐻^∗,𝑉^∗ are the dual space, respectively, and each space is dense in the following one and the injections are continuous. Using the Poincáre inequality we have ‖𝑣‖≥𝜆₁|𝑣|,∀𝑣∈𝑉, (2.2) where 𝜆₁ denotes the first eigenvalue of 𝐴^1/2.

Let 𝜇(𝑠)=−𝜙^′(𝑠),𝜙(∞)=1 and assume that the memory kernel 𝜇 is required to satisfy the following assumptions:(ℎ₁)  𝜇∈𝐶¹(𝑅⁺)⋂𝐿¹(𝑅⁺),𝜇^′(𝑠)≤0, for all 𝑠∈𝑅⁺; (ℎ₂)  ∫^∞₀𝜇(𝑠)𝑑𝑠=𝑀>0; (ℎ₃)  𝜇^′(𝑠)+𝛼𝜇(𝑠)≤0,for all 𝑠∈𝑅⁺,𝛼>0.

In view of ℎ₁, let 𝐿²_𝜇(𝑅⁺,𝐻²₀) be the Hilbert space of 𝐻²₀-valued functions on 𝑅⁺, endowed with the following inner product and the norm: 𝜙,𝜓_𝜇,𝑉=^∞₀𝜇(𝑠)Δ𝜙(𝑠),Δ𝜓(𝑠)𝑑𝑠 (2.3) and |𝜙|²_𝜇,𝑉=(𝜙,𝜙)_𝜇,𝑉=∫^∞₀𝜇(𝑠)‖𝜙‖²𝑑𝑠. Finally, we introduce the following Hilbert spaces: 𝐻₀=𝑉×𝐻×𝐿²_𝜇𝑅⁺,𝑉,𝐻₁=𝐷(𝐴)×𝑉×𝐿²_𝜇𝑅⁺,𝐷(𝐴). (2.4)

We define 𝜂^𝑡𝑥,𝑦,𝑠=𝑢𝑥,𝑦,𝑡−𝑢𝑥,𝑦,𝑡−𝑠=𝜂𝑥,𝑦,𝑠, (2.5) and (1.1) is transformed into the system 𝜌𝑢_𝑡𝑡+𝑟₁𝑟₂𝑢_𝑡+▵²𝑢−𝑁₁+𝛽𝑢_𝑥²𝑢_𝑥𝑥−𝑁₂+𝛽𝑢_𝑦²𝑢_𝑦𝑦+^∞₀𝜇(𝑠)▵²𝜂𝑑𝑠=𝑟₁𝑓,𝜂𝑥,𝑦_𝑡𝑡+𝜂^𝑡_𝑠=𝑢_𝑡, (2.6) where the second equation is obtained by differentiating (2.5). The corresponding initial-boundary value conditions are then given by 𝑢()()𝑢0,𝑦,𝑡=𝑢1,𝑦,𝑡=𝑢𝑥,0,𝑡=𝑢𝑥,1,𝑡=0,_𝑥𝑥0,𝑦,𝑡=𝑢_𝑥𝑥1,𝑦,𝑡=𝑢_𝑦𝑦(𝑥,0,𝑡)=𝑢_𝑦𝑦(),𝜂𝑥,1,𝑡_𝑡0,𝑦,𝑠=𝜂^𝑡1,𝑦,𝑠=𝜂^𝑡(𝑥,0,𝑠)=𝜂^𝑡()𝜂𝑥,1,𝑠=0,_𝑡𝑥𝑥0,𝑦,𝑠=𝜂^𝑡_𝑥𝑥1,𝑦,𝑠=𝜂^𝑡_𝑦𝑦(𝑥,0,𝑠)=𝜂^𝑡_𝑦𝑦()𝑢𝑥,1,𝑠=0,𝑥,𝑦,0=𝑢₁𝑥,𝑦,𝑢_𝑡𝑥,𝑦,0=𝑢₂,𝜂𝑥,𝑦₀𝑥,𝑦,𝑠=𝑢₀𝑥,𝑦,0−𝑢₀𝑥,𝑦,−𝑠, (2.7) where 𝑢₁(𝑥,𝑦)=𝑢₀(𝑥,𝑦,𝑡),𝑢₂(𝑥,𝑦)=𝜕_𝑡𝑢₀(𝑥,𝑦,𝑡)|_𝑡=0,𝑥,𝑦∈[0,1],𝑡≥0,𝑠∈𝑅⁺.

According to the classical Faedo-Galerkin method it is easy to obtain the existence and uniqueness of solutions and the continuous dependence to the initial value, so we omit it and only give the following theorem.

Theorem 2.1 (see [15, 16]). Let (ℎ₁) hold and 𝑓(𝑥,𝑦)∈𝐿²(Ω). Then given any time interval 𝐼=[0,1], problems (2.6)-(2.7) have a unique solution (𝑢,𝑢_𝑡,𝜂) in 𝐼 with initial data (𝑢₁,𝑢₂,𝜂₀)∈𝐻₀, and the mapping (𝑢₁,𝑢₂,𝜂₀)→(𝑢(𝑡),𝑢_𝑡(𝑡),𝜂^𝑡_𝑠) is continuous in 𝐻₀.

Thus, it admits to define a 𝐶⁰ semigroup 𝑆(𝑡)∶𝑢₁,𝑢₂,𝜂₀→𝑢(𝑡),𝑢_𝑡(𝑡),𝜂^𝑡_𝑠,𝑡∈𝑅⁺, (2.8) and they map 𝐻₀ into themselves.

In addition, the following abstract results will be used in our consideration.

Theorem 2.2 (see [17]). A 𝐶⁰ semigroup {𝑆(𝑡)}_𝑡≥0 in a Banach space 𝑋 is said to satisfy condition (𝐶) which arised by [17] if for any 𝜀>0 and for any bounded set 𝐵 of 𝑋, there exists 𝑡(𝐵)>0 and a finite dimensional subspace 𝑋₁ of 𝑋, such that {‖𝑃𝑆(𝑡)𝑥‖_𝑋,𝑥∈𝐵,𝑡≥𝑡(𝐵)} is bounded and ‖(𝐼−𝑃)𝑆(𝑡)𝑥‖_𝑋<_𝑋𝜀, (2.9) where 𝑃∶𝑋→𝑋₁ is a bounded projector.

Lemma 2.3 (see [17]). Let {𝑆(𝑡)}_𝑡≥0 be a 𝐶⁰ semigroup in a Hilbert space 𝑀. Then {𝑆(𝑡)}_𝑡≥0 has a global attractor if and only if (1){𝑆(𝑡)}_𝑡≥0 satisfies the condition 𝐶; (2)there exists a bounded absorbing subset 𝐵 of 𝑀.

3. Global Attractor Λ in 𝐻₀

Theorem 3.1. Assume (ℎ₁)–(ℎ₃) hold. Then the ball of 𝐻₀,𝐵₀=𝐵_𝐻0(0,𝑅₁), centered at 0 with 𝑅₁=𝑐/𝑐₀𝜌₀, is a bounded absorbing set in 𝐻₀ for the semigroup {𝑆(𝑡)}_𝑡≥0.

Proof. We fixed 𝛿 and take 𝛿∈(0,𝛿₀), where 𝛿₀=min(𝑟₁𝑟₂/4𝜌,𝜆²₁/2𝑟₁𝑟₂).
First, taking the inner product of the first equation of (2.6) with 𝑣=𝑢_𝑡+𝛿𝑢, after computation we conclude 12𝑑𝜌|𝑣|𝑑𝑡²+|Δ𝑢|²+𝑁₁𝑢_𝑥²+𝑁₂𝑢_𝑦²+𝛽2𝑢_𝑥⁴+𝛽2𝑢_𝑦⁴+𝐼₁+𝑁₁𝛿𝑢_𝑥²+𝑁₂𝛿𝑢_𝑦²+𝛽𝛿𝑢_𝑥⁴+𝛽𝛿𝑢_𝑦⁴+𝜂,𝑣_𝜇,𝑉=𝑟₁𝑓,𝑣, (3.1) where 𝐼₁=−𝛿𝜌|𝑣|²+𝑟₁𝑟₂|𝑣|²+𝛿|Δ𝑢|²−(𝑟₁𝑟₂𝛿−𝜌𝛿²)(𝑢,𝑣), and we can easily obtain 𝐼₁≥𝛿2|Δ𝑢|²+12𝑟₁𝑟₂|𝑣|². (3.2) Combining with the second equation of (2.6) we have 𝜂,𝑣_𝜇,𝑉=12𝑑𝑑𝑡|𝜂|²_𝜇,𝑉+𝜂,𝜂_{𝑠𝜇,𝑉}+𝛿𝜂,𝑣_𝜇,𝑉. (3.3) According to (ℎ₂)–(ℎ₃), we conclude 𝜂,𝜂_{𝑠𝜇,𝑉}=12^∞₀𝜇(𝑠)𝑑𝑑𝑠Δ𝜂^𝑡(𝑠)²1𝑑𝑠=−2^∞₀𝜇^′(𝑠)𝑑𝑑𝑠Δ𝜂^𝑡(𝑠)²𝛼𝑑𝑠≥2𝜂²_𝜇,𝑉,𝛿𝜂,𝑣_𝜇,𝑉=𝛿^∞₀𝜇(𝑠)Δ𝜂^𝑡(𝑠),Δ𝑢𝑑𝑠≥−𝛿^∞₀𝜇(𝑠)Δ𝜂^𝑡(𝑠)²𝑑𝑠^1/2⋅^∞₀𝜇(𝑠)|Δ𝑢|²𝑑𝑠^1/2𝛼≥−4^∞₀𝜇(𝑠)𝛿𝑑𝑠−²𝛼^∞₀𝜇(𝑠)|Δ𝑢|²𝛼𝑑𝑠≥−4𝜂²_𝜇,𝑣−𝑀𝛿²𝛼|Δ𝑢|². (3.4) Integrating with (3.4), from (3.3), entails 𝜂,𝑣_𝜇,𝑉≥12𝑑𝜂𝑑𝑡^2𝜇,𝑣+𝛼4𝜂²_𝜇,𝑣−𝑀𝛿²𝛼|Δ𝑢|². (3.5) Write 𝐸(𝑡)={𝜌|𝑣|²+|Δ𝑢|²+𝑁₁|𝑢_𝑥|²+𝑁₂|𝑢_𝑦|²+(𝛽/2)|𝑢_𝑥|⁴+(𝛽/2)|𝑢_𝑦|⁴+|𝜂|²_𝜇,𝑉}, from (2.7), (3.2), and (3.5), we obtain 𝑑𝐸(𝑡)𝑑𝑡+𝛿1−2𝑀𝛿𝛼|Δ𝑢|²+𝑟₁𝑟₂2|𝑣|²+𝛼2𝜂²_𝜇,𝑉+2𝛿𝑁₁𝑢_𝑥²+2𝛿𝑁₂𝑢_𝑦²+𝛿𝛽𝑢_𝑥⁴+𝛿𝛽𝑢_𝑦⁴≤2𝑟₁𝑟₂𝑓². (3.6) Take 𝛿 small enough, such that 1−(2𝑀𝛿/𝛼)>1/2. Write 𝑐=(2𝑟₁/𝑟₂)|𝑓|²,  𝑐₀=min{𝛿/2,𝑟₁𝑟₂/2𝜌,𝛼/2}, thus in line with (3.6), we have 𝑑𝐸(𝑡)𝑑𝑡+𝑐₀𝐸(𝑡)≤𝐶. (3.7) By the Gronwall Lemma, we conclude 𝐸(𝑡)≤𝐸(0)exp−𝑐₀𝑡+𝑐𝑐₀1−exp𝑐₀,∀𝑡≥0. (3.8) Due to (3.8), write 𝜌₀=min(𝜌,1), we have |Δ𝑢|²∣+|𝑣|²∣+𝜂²_𝜇,𝑉≤1𝜌₀𝐸(0)exp−𝑐₀𝑡+𝑐𝑐₀𝜌₀1−exp𝑐₀,∀𝑡≥0. (3.9) Write 𝑅₁=𝑐/𝑐₀𝜌₀, we end up with limsup‖𝑢‖²∣+|𝑣|²∣+𝜂²_𝜇,𝑉≤𝑅²₁,∀𝑡≥0. (3.10)

Theorem 3.2. Suppose 𝑓∈𝐿²(Ω) and conditions (ℎ₁)–(ℎ₃) are hold. Then the solution semigroup {𝑆(𝑡)}_𝑡≥0 associated with system (2.6) and (2.7) has a global attractor Λ in 𝐻₀, and it attracts all bounded subsets of 𝐻₀, in the norm of 𝐻₀.

Proof. Applying Lemma 2.3, we only to prove that the condition (𝐶) holds in 𝐻₀.
Let 𝑤_𝑖 be an orthonormal basis of 𝐷(𝐴) which consists of eigenvectors of 𝐴. It is also an orthonormal basis of 𝐻,𝑉, respectively. The corresponding eigenvalues are denoted by 0<𝜆₁<𝜆₂<𝜆₃<⋯,𝜆_𝑖→∞,as𝑖→∞ (3.11) with 𝐴𝑤_𝑖=𝜆_𝑖𝑤_𝑖, for all 𝑖∈𝑁. We write 𝐻_𝑚=span{𝑤₁,𝑤₂,…,𝑤_𝑚}. For any (𝑢,𝑢_𝑡,𝜂)∈𝐻₀, we decompose that (𝑢,𝑢_𝑡,𝜂)=(𝑢₁,𝑢_1𝑡,𝜂₁)+(𝑢₂,𝑢_2𝑡,𝜂₂), where (𝑢₁,𝑢_1𝑡,𝜂₁)=(𝑃_𝑚𝑢,𝑃_𝑚𝑢_𝑡,𝑃_𝑚𝜂), and 𝑃_𝑚∶𝐻→𝐻_𝑚 is the orthogonal projector. Since 𝑓∈𝐻, for any 𝜀>0, there exists some 𝑚, such that 𝐼−𝑃_𝑚𝑓<𝜀4. (3.12) Taking the scalar product of the first equation of (2.6) in 𝐻 with 𝑣₂=𝑢_2𝑡+𝐴𝑢₂, combining with the second equation and using the same way with Theorem 3.1, we find 𝑑𝑑𝑡𝐸(𝑡)+𝐶₀𝐸(𝑡)≤2𝑟₁𝑟₂𝑓²≤𝑟₁𝜀²8𝑟₂, (3.13) where 𝐸(𝑡)={𝜌|𝑣₂|²+|Δ𝑢₂|²+𝑁₁|𝑢_2𝑥|²+𝑁₂|𝑢_2𝑦|²+(𝛽/2)|𝑢_2𝑥|⁴+(𝛽/2)|𝑢_2𝑦|⁴+|𝜂₂|²_𝜇,𝑉. According to Theorem 3.1, some 𝑀₀ and 𝑡₀ exist, for any 𝑡≥𝑡₀, such that 𝐸(𝑡₀)≤𝑀₀. By the Gronwall Lemma, we conclude 𝐸(𝑡)≤𝐸𝑡₀exp−𝑐₀𝑡−𝑡₀+𝑟₁𝜀²8𝑐₀𝑟₂≤𝑀₀exp−𝑐₀𝑡−𝑡₀+𝑟₁𝜀²8𝑐₀𝑟₂,∀𝑡≥𝑡₀. (3.14) Take 𝑡₁ large enough, such that 𝑡₁−𝑡₀≥(1/𝑐₀)ln𝑀₀/𝜀², so we conclude 𝑣₂²+Δ𝑢₂²+𝜂₂²_𝜇,𝑉≤1𝜌₀𝑟1+¹8𝑐₀𝑟₂𝜀²,for𝑡≥𝑡₀. (3.15) Thus 𝑆{𝑡}_𝑡≥0 satisfies the condition (𝐶).

Acknowledgments

Project supported by the Nature Science Foundation of Shanxi province (Grant no. 200611005) and Natural Science Foundation of China (Grant no. 10772131).