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.   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]

Tomás Caraballo, Marta Herrera-Cobos, Pedro Marín-Rubio. Global attractor for a nonlocal p-Laplacian equation without uniqueness of solution. Discrete & 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 & Continuous Dynamical Systems - A, 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 & 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 & 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 & Continuous Dynamical Systems - A, 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 & 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 & 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 & 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 & Continuous Dynamical Systems - A, 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 & 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 & Continuous Dynamical Systems - A, 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 & 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 & 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 & Continuous Dynamical Systems - A, 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 & Continuous Dynamical Systems - A, 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 & 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, 0 (0) : 0-0. doi: 10.3934/era.2020081
[19]

Hugo Beirão da Veiga, Francesca Crispo. On the global regularity for nonlinear systems of the $p$-Laplacian type. Discrete & Continuous Dynamical Systems - S, 2013, 6 (5) : 1173-1191. doi: 10.3934/dcdss.2013.6.1173
[20]

Ludovic Dan Lemle. $L^1(R^d,dx)$-uniqueness of weak solutions for the Fokker-Planck equation associated with a class of Dirichlet operators. Electronic Research Announcements, 2008, 15: 65-70. doi: 10.3934/era.2008.15.65

 Impact Factor: 

Tools

Metrics

  • PDF downloads (50)
  • HTML views (0)
  • Cited by (0)

Other articles
by authors

[Back to Top]

Copyright © 2020 American Institute of Mathematical Sciences