• Previous Article
    On global existence and bounds for blow-up time in nonlinear parabolic problems with time dependent coefficients
  • PROC Home
  • This Issue
  • Next Article
    Intricate bifurcation diagrams for a class of one-dimensional superlinear indefinite problems of interest in population dynamics
2013, 2013(special): 525-534. doi: 10.3934/proc.2013.2013.525

Attractors for weakly damped beam equations with $p$-Laplacian

1. 

Instituto de Ci^encias Matemáticas e de Computação, Universidade de São Paulo, 13566-590, São Carlos, SP, Brazil

2. 

Departamento de Ciências, Campus Regional de Goioerê, Universidade Estadual de Maringá, 87360-000, Goioerê, PR, Brazil

Received  September 2012 Revised  January 2013 Published  November 2013

This paper is concerned with a class of weakly damped one-dimensional beam equations with lower order perturbation of $p$-Laplacian type $$ u_{tt} + u_{xxxx} - (\sigma(u_x))_x + ku_t + f(u)= h \quad \hbox{in} \quad (0,L) \times \mathbb{R}^{+} , $$ where $\sigma(z)=|z|^{p-2}z$, $p \ge 2$, $k>0$ and $f(u)$ and $h(x)$ are forcing terms. Well-posedness, exponential stability and existence of a finite-dimensional attractor are proved.
Citation: T. F. Ma, M. L. Pelicer. Attractors for weakly damped beam equations with $p$-Laplacian. Conference Publications, 2013, 2013 (special) : 525-534. doi: 10.3934/proc.2013.2013.525
References:
[1]

L. An and A. Peirce, A weakly nonlinear analysis of elasto-plastic-microstructure models,, SIAM J. Appl. Math., 55 (1995), 136.   Google Scholar

[2]

D. Andrade, M. A. Jorge Silva and T. F. Ma, Exponential stability for a plate equation with $p$-Laplacian and memory terms,, Math. Meth. Appl. Sci., 35 (2012), 417.   Google Scholar

[3]

A. V. Babin and M. I. Vishik, "Attractors of Evolution Equations,'', Studies in Mathematics and its Application 25, (1992).   Google Scholar

[4]

I. Chueshov and I. Lasiecka, "Von Karman Evolution Equations, Well-Posedness and Long-Time Dynamics,'', Springer Monographs in Mathematics, (2010).   Google Scholar

[5]

I. Chueshov and I. Lasiecka, Existence, uniqueness of weakly solutions and global attractors for a class of nonlinear 2D Kirchhoff-Boussinesq models,, Discrete Contin. Dyn. Syst., 15 (2006), 777.   Google Scholar

[6]

I. Chueshov and I. Lasiecka, On global attractor for 2D Kirchhoff-Boussinesq model with supercritical nonlinearity,, Comm. Partial Differential Equations, 36 (2011), 67.   Google Scholar

[7]

J. K. Hale, "Asymptotic Behavior of Dissipative Systems,'', Mathematical Surveys and Monographs, (1988).   Google Scholar

[8]

J. U. Kim, A boundary thin obstacle problem for a wave equation,, Comm. Partial Differential Equations, 14 (1989), 1011.   Google Scholar

[9]

O. Ladyzhenskaya, "Attractors for Semigroups and Evolution Equations,'', Cambridge University Press, (1991).   Google Scholar

[10]

J. L. Lions, "Quelques Méthodes de Résolution des Problèmes aux Limites Non Linéaires,'', Dunod Gauthier-Villars, (1969).   Google Scholar

[11]

R. Temam, "Infinite-Dimensional Dynamical Systems in Mechanics and Physics,'', Applied Mathematical Sciences 68, (1988).   Google Scholar

[12]

Yang Zhijian, Global existence, asymptotic behavior and blowup of solutions for a class of nonlinear wave equations with dissipative term,, J. Differential Equations, 187 (2003), 520.   Google Scholar

[13]

Yang Zhijian, Longtime behavior for a nonlinear wave equation arising in elasto-plastic flow,, Math. Meth. Appl. Sci., 32 (2009), 1082.   Google Scholar

[14]

Yang Zhijian, Finite-dimensional attractors for the Kirchhoff models,, J. Math. Phys., 51 (2010).   Google Scholar

[15]

Yang Zhijian and Jin Baoxia, Global attractor for a class of Kirchhoff models,, J. Math. Phys., 50 (2009).   Google Scholar

show all references

References:
[1]

L. An and A. Peirce, A weakly nonlinear analysis of elasto-plastic-microstructure models,, SIAM J. Appl. Math., 55 (1995), 136.   Google Scholar

[2]

D. Andrade, M. A. Jorge Silva and T. F. Ma, Exponential stability for a plate equation with $p$-Laplacian and memory terms,, Math. Meth. Appl. Sci., 35 (2012), 417.   Google Scholar

[3]

A. V. Babin and M. I. Vishik, "Attractors of Evolution Equations,'', Studies in Mathematics and its Application 25, (1992).   Google Scholar

[4]

I. Chueshov and I. Lasiecka, "Von Karman Evolution Equations, Well-Posedness and Long-Time Dynamics,'', Springer Monographs in Mathematics, (2010).   Google Scholar

[5]

I. Chueshov and I. Lasiecka, Existence, uniqueness of weakly solutions and global attractors for a class of nonlinear 2D Kirchhoff-Boussinesq models,, Discrete Contin. Dyn. Syst., 15 (2006), 777.   Google Scholar

[6]

I. Chueshov and I. Lasiecka, On global attractor for 2D Kirchhoff-Boussinesq model with supercritical nonlinearity,, Comm. Partial Differential Equations, 36 (2011), 67.   Google Scholar

[7]

J. K. Hale, "Asymptotic Behavior of Dissipative Systems,'', Mathematical Surveys and Monographs, (1988).   Google Scholar

[8]

J. U. Kim, A boundary thin obstacle problem for a wave equation,, Comm. Partial Differential Equations, 14 (1989), 1011.   Google Scholar

[9]

O. Ladyzhenskaya, "Attractors for Semigroups and Evolution Equations,'', Cambridge University Press, (1991).   Google Scholar

[10]

J. L. Lions, "Quelques Méthodes de Résolution des Problèmes aux Limites Non Linéaires,'', Dunod Gauthier-Villars, (1969).   Google Scholar

[11]

R. Temam, "Infinite-Dimensional Dynamical Systems in Mechanics and Physics,'', Applied Mathematical Sciences 68, (1988).   Google Scholar

[12]

Yang Zhijian, Global existence, asymptotic behavior and blowup of solutions for a class of nonlinear wave equations with dissipative term,, J. Differential Equations, 187 (2003), 520.   Google Scholar

[13]

Yang Zhijian, Longtime behavior for a nonlinear wave equation arising in elasto-plastic flow,, Math. Meth. Appl. Sci., 32 (2009), 1082.   Google Scholar

[14]

Yang Zhijian, Finite-dimensional attractors for the Kirchhoff models,, J. Math. Phys., 51 (2010).   Google Scholar

[15]

Yang Zhijian and Jin Baoxia, Global attractor for a class of Kirchhoff models,, J. Math. Phys., 50 (2009).   Google Scholar

[1]

Cheng He, Changzheng Qu. Global weak solutions for the two-component Novikov equation. Electronic Research Archive, 2020, 28 (4) : 1545-1562. doi: 10.3934/era.2020081

[2]

Jianhua Huang, Yanbin Tang, Ming Wang. Singular support of the global attractor for a damped BBM equation. Discrete & Continuous Dynamical Systems - B, 2020  doi: 10.3934/dcdsb.2020345

[3]

Biyue Chen, Chunxiang Zhao, Chengkui Zhong. The global attractor for the wave equation with nonlocal strong damping. Discrete & Continuous Dynamical Systems - B, 2021  doi: 10.3934/dcdsb.2021015

[4]

Yichen Zhang, Meiqiang Feng. A coupled $ p $-Laplacian elliptic system: Existence, uniqueness and asymptotic behavior. Electronic Research Archive, 2020, 28 (4) : 1419-1438. doi: 10.3934/era.2020075

[5]

Stanislav Nikolaevich Antontsev, Serik Ersultanovich Aitzhanov, Guzel Rashitkhuzhakyzy Ashurova. An inverse problem for the pseudo-parabolic equation with p-Laplacian. Evolution Equations & Control Theory, 2021  doi: 10.3934/eect.2021005

[6]

Lihong Zhang, Wenwen Hou, Bashir Ahmad, Guotao Wang. Radial symmetry for logarithmic Choquard equation involving a generalized tempered fractional $ p $-Laplacian. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020445

[7]

Fuensanta Andrés, Julio Muñoz, Jesús Rosado. Optimal design problems governed by the nonlocal $ p $-Laplacian equation. Mathematical Control & Related Fields, 2021, 11 (1) : 119-141. doi: 10.3934/mcrf.2020030

[8]

Zaizheng Li, Qidi Zhang. Sub-solutions and a point-wise Hopf's lemma for fractional $ p $-Laplacian. Communications on Pure & Applied Analysis, , () : -. doi: 10.3934/cpaa.2020293

[9]

Raffaele Folino, Ramón G. Plaza, Marta Strani. Long time dynamics of solutions to $ p $-Laplacian diffusion problems with bistable reaction terms. Discrete & Continuous Dynamical Systems - A, 2020  doi: 10.3934/dcds.2020403

[10]

Bo Chen, Youde Wang. Global weak solutions for Landau-Lifshitz flows and heat flows associated to micromagnetic energy functional. Communications on Pure & Applied Analysis, 2021, 20 (1) : 319-338. doi: 10.3934/cpaa.2020268

[11]

Yongxiu Shi, Haitao Wan. Refined asymptotic behavior and uniqueness of large solutions to a quasilinear elliptic equation in a borderline case. Electronic Research Archive, , () : -. doi: 10.3934/era.2020119

[12]

Wenjun Liu, Hefeng Zhuang. Global attractor for a suspension bridge problem with a nonlinear delay term in the internal feedback. Discrete & Continuous Dynamical Systems - B, 2021, 26 (2) : 907-942. doi: 10.3934/dcdsb.2020147

[13]

Stefano Bianchini, Paolo Bonicatto. Forward untangling and applications to the uniqueness problem for the continuity equation. Discrete & Continuous Dynamical Systems - A, 2020  doi: 10.3934/dcds.2020384

[14]

Xinyu Mei, Yangmin Xiong, Chunyou Sun. Pullback attractor for a weakly damped wave equation with sup-cubic nonlinearity. Discrete & Continuous Dynamical Systems - A, 2021, 41 (2) : 569-600. doi: 10.3934/dcds.2020270

[15]

Fang Li, Bo You. On the dimension of global attractor for the Cahn-Hilliard-Brinkman system with dynamic boundary conditions. Discrete & Continuous Dynamical Systems - B, 2021  doi: 10.3934/dcdsb.2021024

[16]

José Luiz Boldrini, Jonathan Bravo-Olivares, Eduardo Notte-Cuello, Marko A. Rojas-Medar. Asymptotic behavior of weak and strong solutions of the magnetohydrodynamic equations. Electronic Research Archive, 2021, 29 (1) : 1783-1801. doi: 10.3934/era.2020091

[17]

Christian Beck, Lukas Gonon, Martin Hutzenthaler, Arnulf Jentzen. On existence and uniqueness properties for solutions of stochastic fixed point equations. Discrete & Continuous Dynamical Systems - B, 2020  doi: 10.3934/dcdsb.2020320

[18]

Mokhtar Bouloudene, Manar A. Alqudah, Fahd Jarad, Yassine Adjabi, Thabet Abdeljawad. Nonlinear singular $ p $ -Laplacian boundary value problems in the frame of conformable derivative. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020442

[19]

Karoline Disser. Global existence and uniqueness for a volume-surface reaction-nonlinear-diffusion system. Discrete & Continuous Dynamical Systems - S, 2021, 14 (1) : 321-330. doi: 10.3934/dcdss.2020326

[20]

Martin Kalousek, Joshua Kortum, Anja Schlömerkemper. Mathematical analysis of weak and strong solutions to an evolutionary model for magnetoviscoelasticity. Discrete & Continuous Dynamical Systems - S, 2021, 14 (1) : 17-39. doi: 10.3934/dcdss.2020331

 Impact Factor: 

Metrics

  • PDF downloads (59)
  • HTML views (0)
  • Cited by (0)

Other articles
by authors

[Back to Top]