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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), 136-155.  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), 417-426.  Google Scholar

[3]

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

[4]

I. Chueshov and I. Lasiecka, "Von Karman Evolution Equations, Well-Posedness and Long-Time Dynamics,'' Springer Monographs in Mathematics, Springer, New York, 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), 777-809.  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), 67-99.  Google Scholar

[7]

J. K. Hale, "Asymptotic Behavior of Dissipative Systems,'' Mathematical Surveys and Monographs, vol. 25, American Mathematical Society, Providence, 1988.  Google Scholar

[8]

J. U. Kim, A boundary thin obstacle problem for a wave equation, Comm. Partial Differential Equations, 14 (1989), 1011-1026.  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, Paris, 1969.  Google Scholar

[11]

R. Temam, "Infinite-Dimensional Dynamical Systems in Mechanics and Physics,'' Applied Mathematical Sciences 68, Springer-Verlag, New York, 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), 520-540.  Google Scholar

[13]

Yang Zhijian, Longtime behavior for a nonlinear wave equation arising in elasto-plastic flow, Math. Meth. Appl. Sci., 32 (2009), 1082-1104.  Google Scholar

[14]

Yang Zhijian, Finite-dimensional attractors for the Kirchhoff models, J. Math. Phys., 51 092703 (2010), 25 pp.  Google Scholar

[15]

Yang Zhijian and Jin Baoxia, Global attractor for a class of Kirchhoff models, J. Math. Phys., 50 032701 (2009), 29 pp.  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), 136-155.  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), 417-426.  Google Scholar

[3]

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

[4]

I. Chueshov and I. Lasiecka, "Von Karman Evolution Equations, Well-Posedness and Long-Time Dynamics,'' Springer Monographs in Mathematics, Springer, New York, 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), 777-809.  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), 67-99.  Google Scholar

[7]

J. K. Hale, "Asymptotic Behavior of Dissipative Systems,'' Mathematical Surveys and Monographs, vol. 25, American Mathematical Society, Providence, 1988.  Google Scholar

[8]

J. U. Kim, A boundary thin obstacle problem for a wave equation, Comm. Partial Differential Equations, 14 (1989), 1011-1026.  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, Paris, 1969.  Google Scholar

[11]

R. Temam, "Infinite-Dimensional Dynamical Systems in Mechanics and Physics,'' Applied Mathematical Sciences 68, Springer-Verlag, New York, 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), 520-540.  Google Scholar

[13]

Yang Zhijian, Longtime behavior for a nonlinear wave equation arising in elasto-plastic flow, Math. Meth. Appl. Sci., 32 (2009), 1082-1104.  Google Scholar

[14]

Yang Zhijian, Finite-dimensional attractors for the Kirchhoff models, J. Math. Phys., 51 092703 (2010), 25 pp.  Google Scholar

[15]

Yang Zhijian and Jin Baoxia, Global attractor for a class of Kirchhoff models, J. Math. Phys., 50 032701 (2009), 29 pp.  Google Scholar

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