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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

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.
Citation: Filippo Dell'Oro. Global attractors for strongly damped wave equations with subcritical-critical nonlinearities. Communications on Pure and 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.

[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.

[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.

[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.

[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.

[6]

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

[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.

[8]

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

[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.

[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.

[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.

[12]

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

[13]

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

[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.

[15]

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

[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.

[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.

[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.

[19]

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

show all references

References:
[1]

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

[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.

[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.

[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.

[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.

[6]

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

[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.

[8]

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

[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.

[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.

[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.

[12]

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

[13]

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

[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.

[15]

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

[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.

[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.

[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.

[19]

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

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