Pullback attractor for N-dimensional thermoelastic coupled structure equations
Abstract In this paper, proving the pullback asymptotic compactness of processes by the aid of a contractive function in space X 0 $X_{0}$ , we prove the existence of a pullback attractor for N-dimensional nonautonomous thermoelastic coupled structure equations u t t + α △ 2 u − [ β + σ ( ∫ Ω ( ∇ u...
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| Format: | Article |
| Language: | English |
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SpringerOpen
2018-01-01
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| Series: | Boundary Value Problems |
| Subjects: | |
| Online Access: | http://link.springer.com/article/10.1186/s13661-017-0921-7 |
| _version_ | 1818029466009468928 |
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| author | Danxia Wang Yinzhu Wang |
| author_facet | Danxia Wang Yinzhu Wang |
| author_sort | Danxia Wang |
| collection | DOAJ |
| description | Abstract In this paper, proving the pullback asymptotic compactness of processes by the aid of a contractive function in space X 0 $X_{0}$ , we prove the existence of a pullback attractor for N-dimensional nonautonomous thermoelastic coupled structure equations u t t + α △ 2 u − [ β + σ ( ∫ Ω ( ∇ u ) 2 d x ) ] △ u + γ △ θ + g ( u ) + η u t = h ( x , t ) , in Ω × [ τ , ∞ ) , θ t − △ θ − γ △ u t = q ( x , t ) in Ω × [ τ , ∞ ) , $$\begin{aligned} &u_{tt}+\alpha\triangle^{2}u-\biggl[\beta+\sigma\biggl( \int_{\Omega}(\nabla u)^{2}\,dx\biggr)\biggr]\triangle u+ \gamma\triangle\theta+g(u)+\eta u_{t}=h(x,t), \\ &\quad \mbox{in } \Omega \times [\tau,\infty), \\ &\theta_{t}-\triangle\theta-\gamma\triangle u_{t}=q(x,t) \quad \mbox{in } \Omega\times [\tau,\infty), \end{aligned}$$ with the lateral load distribution function h ( x , t ) $h(x,t)$ and the external heat supply function q ( x , t ) $q(x,t)$ unnecessarily bounded. The nonlinear source term g ( u ) $g(u)$ is essentially k 1 ( u + | u | ρ − 1 u ρ + 1 ) $k_{1}(u+\frac{|u|^{\rho-1}u}{\rho+1})$ ( k 1 > 0 ) $(k_{1}>0)$ with 1 < ρ ≤ N N − 2 $1<\rho\leq\frac{N}{N-2}$ if N ≥ 3 $N\geq 3$ and 1 < ρ < ∞ $1<\rho<\infty$ if N = 1 , 2 $N=1,2$ . |
| first_indexed | 2024-12-10T05:20:08Z |
| format | Article |
| id | doaj.art-2376dfc4c9e746a6a0669489f8d1faf8 |
| institution | Directory Open Access Journal |
| issn | 1687-2770 |
| language | English |
| last_indexed | 2024-12-10T05:20:08Z |
| publishDate | 2018-01-01 |
| publisher | SpringerOpen |
| record_format | Article |
| series | Boundary Value Problems |
| spelling | doaj.art-2376dfc4c9e746a6a0669489f8d1faf82022-12-22T02:00:49ZengSpringerOpenBoundary Value Problems1687-27702018-01-012018112110.1186/s13661-017-0921-7Pullback attractor for N-dimensional thermoelastic coupled structure equationsDanxia Wang0Yinzhu Wang1Department of Mathematics, Taiyuan University of TechnologyDepartment of Mathematics, Taiyuan University of Science and TechnologyAbstract In this paper, proving the pullback asymptotic compactness of processes by the aid of a contractive function in space X 0 $X_{0}$ , we prove the existence of a pullback attractor for N-dimensional nonautonomous thermoelastic coupled structure equations u t t + α △ 2 u − [ β + σ ( ∫ Ω ( ∇ u ) 2 d x ) ] △ u + γ △ θ + g ( u ) + η u t = h ( x , t ) , in Ω × [ τ , ∞ ) , θ t − △ θ − γ △ u t = q ( x , t ) in Ω × [ τ , ∞ ) , $$\begin{aligned} &u_{tt}+\alpha\triangle^{2}u-\biggl[\beta+\sigma\biggl( \int_{\Omega}(\nabla u)^{2}\,dx\biggr)\biggr]\triangle u+ \gamma\triangle\theta+g(u)+\eta u_{t}=h(x,t), \\ &\quad \mbox{in } \Omega \times [\tau,\infty), \\ &\theta_{t}-\triangle\theta-\gamma\triangle u_{t}=q(x,t) \quad \mbox{in } \Omega\times [\tau,\infty), \end{aligned}$$ with the lateral load distribution function h ( x , t ) $h(x,t)$ and the external heat supply function q ( x , t ) $q(x,t)$ unnecessarily bounded. The nonlinear source term g ( u ) $g(u)$ is essentially k 1 ( u + | u | ρ − 1 u ρ + 1 ) $k_{1}(u+\frac{|u|^{\rho-1}u}{\rho+1})$ ( k 1 > 0 ) $(k_{1}>0)$ with 1 < ρ ≤ N N − 2 $1<\rho\leq\frac{N}{N-2}$ if N ≥ 3 $N\geq 3$ and 1 < ρ < ∞ $1<\rho<\infty$ if N = 1 , 2 $N=1,2$ .http://link.springer.com/article/10.1186/s13661-017-0921-7pullback attractorpullback asymptotically compactcontractive functionthermoelastic coupled structure equations |
| spellingShingle | Danxia Wang Yinzhu Wang Pullback attractor for N-dimensional thermoelastic coupled structure equations Boundary Value Problems pullback attractor pullback asymptotically compact contractive function thermoelastic coupled structure equations |
| title | Pullback attractor for N-dimensional thermoelastic coupled structure equations |
| title_full | Pullback attractor for N-dimensional thermoelastic coupled structure equations |
| title_fullStr | Pullback attractor for N-dimensional thermoelastic coupled structure equations |
| title_full_unstemmed | Pullback attractor for N-dimensional thermoelastic coupled structure equations |
| title_short | Pullback attractor for N-dimensional thermoelastic coupled structure equations |
| title_sort | pullback attractor for n dimensional thermoelastic coupled structure equations |
| topic | pullback attractor pullback asymptotically compact contractive function thermoelastic coupled structure equations |
| url | http://link.springer.com/article/10.1186/s13661-017-0921-7 |
| work_keys_str_mv | AT danxiawang pullbackattractorforndimensionalthermoelasticcoupledstructureequations AT yinzhuwang pullbackattractorforndimensionalthermoelasticcoupledstructureequations |