American Institute of Mathematical Sciences

Advanced
  • 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

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.

[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.

[3]

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

[4]

I. Chueshov and I. Lasiecka, "Von Karman Evolution Equations, Well-Posedness and Long-Time Dynamics,'' Springer Monographs in Mathematics, Springer, New York, 2010.

[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.

[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.

[7]

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

[8]

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

[9]

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

[10]

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

[11]

R. Temam, "Infinite-Dimensional Dynamical Systems in Mechanics and Physics,'' Applied Mathematical Sciences 68, Springer-Verlag, New York, 1988.

[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.

[13]

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

[14]

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

[15]

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

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.

[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.

[3]

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

[4]

I. Chueshov and I. Lasiecka, "Von Karman Evolution Equations, Well-Posedness and Long-Time Dynamics,'' Springer Monographs in Mathematics, Springer, New York, 2010.

[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.

[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.

[7]

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

[8]

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

[9]

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

[10]

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

[11]

R. Temam, "Infinite-Dimensional Dynamical Systems in Mechanics and Physics,'' Applied Mathematical Sciences 68, Springer-Verlag, New York, 1988.

[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.

[13]

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

[14]

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

[15]

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

[1]

Tomás Caraballo, Marta Herrera-Cobos, Pedro Marín-Rubio. Global attractor for a nonlocal p-Laplacian equation without uniqueness of solution. Discrete and Continuous Dynamical Systems - B, 2017, 22 (5) : 1801-1816. doi: 10.3934/dcdsb.2017107
[2]

Shihui Zhu. Existence and uniqueness of global weak solutions of the Camassa-Holm equation with a forcing. Discrete and Continuous Dynamical Systems, 2016, 36 (9) : 5201-5221. doi: 10.3934/dcds.2016026
[3]

Ronghua Jiang, Jun Zhou. Blow-up and global existence of solutions to a parabolic equation associated with the fraction p-Laplacian. Communications on Pure and Applied Analysis, 2019, 18 (3) : 1205-1226. doi: 10.3934/cpaa.2019058
[4]

Nicholas J. Kass, Mohammad A. Rammaha. Local and global existence of solutions to a strongly damped wave equation of the $ p $-Laplacian type. Communications on Pure and Applied Analysis, 2018, 17 (4) : 1449-1478. doi: 10.3934/cpaa.2018070
[5]

Mohammad A. Rammaha, Daniel Toundykov, Zahava Wilstein. Global existence and decay of energy for a nonlinear wave equation with $p$-Laplacian damping. Discrete and Continuous Dynamical Systems, 2012, 32 (12) : 4361-4390. doi: 10.3934/dcds.2012.32.4361
[6]

Wen Tan. The regularity of pullback attractor for a non-autonomous p-Laplacian equation with dynamical boundary condition. Discrete and Continuous Dynamical Systems - B, 2019, 24 (2) : 529-546. doi: 10.3934/dcdsb.2018194
[7]

C. Cortázar, Marta García-Huidobro. On the uniqueness of ground state solutions of a semilinear equation containing a weighted Laplacian. Communications on Pure and Applied Analysis, 2006, 5 (4) : 813-826. doi: 10.3934/cpaa.2006.5.813
[8]

C. Cortázar, Marta García-Huidobro. On the uniqueness of ground state solutions of a semilinear equation containing a weighted Laplacian. Communications on Pure and Applied Analysis, 2006, 5 (1) : 71-84. doi: 10.3934/cpaa.2006.5.71
[9]

Zhenhua Guo, Mina Jiang, Zhian Wang, Gao-Feng Zheng. Global weak solutions to the Camassa-Holm equation. Discrete and Continuous Dynamical Systems, 2008, 21 (3) : 883-906. doi: 10.3934/dcds.2008.21.883
[10]

Robert Stegliński. On homoclinic solutions for a second order difference equation with p-Laplacian. Discrete and Continuous Dynamical Systems - B, 2018, 23 (1) : 487-492. doi: 10.3934/dcdsb.2018033
[11]

Michael Filippakis, Alexandru Kristály, Nikolaos S. Papageorgiou. Existence of five nonzero solutions with exact sign for a $p$-Laplacian equation. Discrete and Continuous Dynamical Systems, 2009, 24 (2) : 405-440. doi: 10.3934/dcds.2009.24.405
[12]

John R. Graef, Lingju Kong. Uniqueness and parameter dependence of positive solutions of third order boundary value problems with $p$-laplacian. Conference Publications, 2011, 2011 (Special) : 515-522. doi: 10.3934/proc.2011.2011.515
[13]

Dung Le. On the regular set of BMO weak solutions to $p$-Laplacian strongly coupled nonregular elliptic systems. Discrete and Continuous Dynamical Systems - B, 2014, 19 (10) : 3245-3265. doi: 10.3934/dcdsb.2014.19.3245
[14]

Zhiming Liu, Zhijian Yang. Global attractor of multi-valued operators with applications to a strongly damped nonlinear wave equation without uniqueness. Discrete and Continuous Dynamical Systems - B, 2020, 25 (1) : 223-240. doi: 10.3934/dcdsb.2019179
[15]

Igor Chueshov, Irena Lasiecka. Existence, uniqueness of weak solutions and global attractors for a class of nonlinear 2D Kirchhoff-Boussinesq models. Discrete and Continuous Dynamical Systems, 2006, 15 (3) : 777-809. doi: 10.3934/dcds.2006.15.777
[16]

Jonatan Lenells. Weak geodesic flow and global solutions of the Hunter-Saxton equation. Discrete and Continuous Dynamical Systems, 2007, 18 (4) : 643-656. doi: 10.3934/dcds.2007.18.643
[17]

Chien-Hong Cho, Marcus Wunsch. Global weak solutions to the generalized Proudman-Johnson equation. Communications on Pure and Applied Analysis, 2012, 11 (4) : 1387-1396. doi: 10.3934/cpaa.2012.11.1387
[18]

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
[19]

Fanqin Zeng, Yu Gao, Xiaoping Xue. Global weak solutions to the generalized mCH equation via characteristics. Discrete and Continuous Dynamical Systems - B, 2022, 27 (8) : 4317-4329. doi: 10.3934/dcdsb.2021229
[20]

Hugo Beirão da Veiga, Francesca Crispo. On the global regularity for nonlinear systems of the $p$-Laplacian type. Discrete and Continuous Dynamical Systems - S, 2013, 6 (5) : 1173-1191. doi: 10.3934/dcdss.2013.6.1173

 Impact Factor: 

Tools

Metrics

  • PDF downloads (114)
  • HTML views (0)
  • Cited by (0)

Other articles
by authors

[Back to Top]

Copyright © 2022 American Institute of Mathematical Sciences | Privacy Policy