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March  2013, 12(2): 1015-1027. doi: 10.3934/cpaa.2013.12.1015

Global attractors for strongly damped wave equations with subcritical-critical nonlinearities

Filippo Dell'Oro 1,

1. 

Dipartimento di Matematica "Francesco Brioschi", Politecnico di Milano, Via Bonardi 9, Milano 20133, Italy

Received  August 2011 Revised  March 2012 Published  September 2012

This paper is concerned with the nonlinear strongly damped wave equation \begin{eqnarray*} u_{t t}-\Delta u_t-\Delta u+f(u_t)+g(u)=h, \end{eqnarray*} with Dirichlet boundary conditions and a time-independent external force $h$. In the presence of nonlinearities $f$ and $g$ of subcritical and critical growth, respectively, the existence of a global attractor of optimal regularity is established.
Keywords: Strongly damped wave equation, regularity., critical growth, global attractor.
Mathematics Subject Classification: Primary: 35B33, 35B40, 35L05; Secondary: 35M1.
Citation: Filippo Dell'Oro. Global attractors for strongly damped wave equations with subcritical-critical nonlinearities. Communications on Pure & Applied Analysis, 2013, 12 (2) : 1015-1027. doi: 10.3934/cpaa.2013.12.1015
References:
[1]

A. V. Babin and M. I. Vishik, "Attractors of Evolution Equations," North-Holland, Amsterdam, 1992.  Google Scholar

[2]

A. N. Carvalho and J. W. Cholewa, Attractors for strongly damped wave equations with critical nonlinearities, Pacific J. Math., 207 (2002), 287-310. doi: 10.2140/pjm.2002.207.287.  Google Scholar

[3]

I. Chueshov and I. Lasiecka, Long-time behavior of second order evolution equations with nonlinear damping, Mem. Amer. Math. Soc., 195 (2008), viii+183 pp.  Google Scholar

[4]

M. Conti and V. Pata, On the regularity of global attractors, Discrete Cont. Dyn. Sys., 25 (2009), 1209-1217. doi: 10.3934/dcds.2009.25.1209.  Google Scholar

[5]

F. Dell'Oro and V. Pata, Long-term analysis of strongly damped nonlinear wave equations, Nonlinearity, 24 (2011), 3413-3435. doi: 10.1088/0951-7715/24/12/006.  Google Scholar

[6]

F. Dell'Oro and V. Pata, Strongly damped wave equations with critical nonlinearities, Nonlinear Anal., 75 (2012), 5723-5735.  Google Scholar

[7]

A. E. Green and P. M. Naghdi, On undamped heat waves in an elastic solid, J. Thermal Stresses, 15 (1992), 253-264. doi: 10.1080/01495739208946136.  Google Scholar

[8]

A. E. Green and P. M. Naghdi, Thermoelasticity without energy dissipation, J. Elasticity, 31 (1993), 189-208. doi: 10.1007/BF00044969.  Google Scholar

[9]

A. E. Green and P. M. Naghdi, A unified procedure for construction of theories of deformable media. I. Classical continuum physics, Proc. Roy. Soc. London A, 448 (1995), 335-356. doi: 10.1098/rspa.1995.0020.  Google Scholar

[10]

A. E. Green and P. M. Naghdi, A unified procedure for construction of theories of deformable media. II. Generalized continua, Proc. Roy. Soc. London A, 448 (1995), 357-377. doi: 10.1098/rspa.1995.0021.  Google Scholar

[11]

A. E. Green and P. M. Naghdi, A unified procedure for construction of theories of deformable media. III. Mixtures of interacting continua, Proc. Roy. Soc. London A, 448 (1995), 379-388. doi: 10.1098/rspa.1995.0022.  Google Scholar

[12]

J. K. Hale, "Asymptotic Behavior of Dissipative Systems," Amer. Math. Soc., Providence, 1988.  Google Scholar

[13]

A. Haraux, "Systèmes Dynamiques Dissipatifs et Applications," Masson, Paris, 1991.  Google Scholar

[14]

A. Kh. Khanmamedov, On the existence of a global attractor for the wave equation with nonlinear strong damping perturbed by nonmonotone term, Nonlinear Anal., 69 (2008), 3372-3385. doi: 10.1016/j.na.2007.09.028.  Google Scholar

[15]

V. Pata, Uniform estimates of Gronwall type, J. Math. Anal. Appl., 373 (2011), 264-270. doi: 10.1016/j.jmaa.2010.07.006.  Google Scholar

[16]

V. Pata and M. Squassina, On the strongly damped wave equation, Comm. Math. Phys., 253 (2005), 511-533. doi: 10.1007/s00220-004-1233-1.  Google Scholar

[17]

V. Pata and S. Zelik, A result on the existence of global attractors for semigroups of closed operators, Commun. Pure Appl. Anal., 6 (2007), 481-486. doi: 10.3934/cpaa.2007.6.481.  Google Scholar

[18]

V. Pata and S. Zelik, Smooth attractors for strongly damped wave equations, Nonlinearity, 19 (2006), 1495-1506. doi: 10.1088/0951-7715/19/7/001.  Google Scholar

[19]

R. Temam, "Infinite-Dimensional Dynamical Systems in Mechanics and Physics," Springer, New York, 1997.  Google Scholar

show all references

References:
[1]

A. V. Babin and M. I. Vishik, "Attractors of Evolution Equations," North-Holland, Amsterdam, 1992.  Google Scholar

[2]

A. N. Carvalho and J. W. Cholewa, Attractors for strongly damped wave equations with critical nonlinearities, Pacific J. Math., 207 (2002), 287-310. doi: 10.2140/pjm.2002.207.287.  Google Scholar

[3]

I. Chueshov and I. Lasiecka, Long-time behavior of second order evolution equations with nonlinear damping, Mem. Amer. Math. Soc., 195 (2008), viii+183 pp.  Google Scholar

[4]

M. Conti and V. Pata, On the regularity of global attractors, Discrete Cont. Dyn. Sys., 25 (2009), 1209-1217. doi: 10.3934/dcds.2009.25.1209.  Google Scholar

[5]

F. Dell'Oro and V. Pata, Long-term analysis of strongly damped nonlinear wave equations, Nonlinearity, 24 (2011), 3413-3435. doi: 10.1088/0951-7715/24/12/006.  Google Scholar

[6]

F. Dell'Oro and V. Pata, Strongly damped wave equations with critical nonlinearities, Nonlinear Anal., 75 (2012), 5723-5735.  Google Scholar

[7]

A. E. Green and P. M. Naghdi, On undamped heat waves in an elastic solid, J. Thermal Stresses, 15 (1992), 253-264. doi: 10.1080/01495739208946136.  Google Scholar

[8]

A. E. Green and P. M. Naghdi, Thermoelasticity without energy dissipation, J. Elasticity, 31 (1993), 189-208. doi: 10.1007/BF00044969.  Google Scholar

[9]

A. E. Green and P. M. Naghdi, A unified procedure for construction of theories of deformable media. I. Classical continuum physics, Proc. Roy. Soc. London A, 448 (1995), 335-356. doi: 10.1098/rspa.1995.0020.  Google Scholar

[10]

A. E. Green and P. M. Naghdi, A unified procedure for construction of theories of deformable media. II. Generalized continua, Proc. Roy. Soc. London A, 448 (1995), 357-377. doi: 10.1098/rspa.1995.0021.  Google Scholar

[11]

A. E. Green and P. M. Naghdi, A unified procedure for construction of theories of deformable media. III. Mixtures of interacting continua, Proc. Roy. Soc. London A, 448 (1995), 379-388. doi: 10.1098/rspa.1995.0022.  Google Scholar

[12]

J. K. Hale, "Asymptotic Behavior of Dissipative Systems," Amer. Math. Soc., Providence, 1988.  Google Scholar

[13]

A. Haraux, "Systèmes Dynamiques Dissipatifs et Applications," Masson, Paris, 1991.  Google Scholar

[14]

A. Kh. Khanmamedov, On the existence of a global attractor for the wave equation with nonlinear strong damping perturbed by nonmonotone term, Nonlinear Anal., 69 (2008), 3372-3385. doi: 10.1016/j.na.2007.09.028.  Google Scholar

[15]

V. Pata, Uniform estimates of Gronwall type, J. Math. Anal. Appl., 373 (2011), 264-270. doi: 10.1016/j.jmaa.2010.07.006.  Google Scholar

[16]

V. Pata and M. Squassina, On the strongly damped wave equation, Comm. Math. Phys., 253 (2005), 511-533. doi: 10.1007/s00220-004-1233-1.  Google Scholar

[17]

V. Pata and S. Zelik, A result on the existence of global attractors for semigroups of closed operators, Commun. Pure Appl. Anal., 6 (2007), 481-486. doi: 10.3934/cpaa.2007.6.481.  Google Scholar

[18]

V. Pata and S. Zelik, Smooth attractors for strongly damped wave equations, Nonlinearity, 19 (2006), 1495-1506. doi: 10.1088/0951-7715/19/7/001.  Google Scholar

[19]

R. Temam, "Infinite-Dimensional Dynamical Systems in Mechanics and Physics," Springer, New York, 1997.  Google Scholar

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