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2013, 2013(special): 525-534. doi: 10.3934/proc.2013.2013.525

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

T. F. Ma 1, and  M. L. Pelicer 2,

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.
Keywords: uniqueness of weak solutions., Beam equation, global attractor, $p$-Laplacian.
Mathematics Subject Classification: Primary: 35L75, 35B40; Secondary: 74J3.
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

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