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

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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

Abstract
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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, (1992).
[2]	A. N. Carvalho and J. W. Cholewa, Attractors for strongly damped wave equations with critical nonlinearities,, Pacific J. Math., 207 (2002), 287. 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).
[4]	M. Conti and V. Pata, On the regularity of global attractors,, Discrete Cont. Dyn. Sys., 25 (2009), 1209. 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. 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.
[7]	A. E. Green and P. M. Naghdi, On undamped heat waves in an elastic solid,, J. Thermal Stresses, 15 (1992), 253. doi: 10.1080/01495739208946136.
[8]	A. E. Green and P. M. Naghdi, Thermoelasticity without energy dissipation,, J. Elasticity, 31 (1993), 189. 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. 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. 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. doi: 10.1098/rspa.1995.0022.
[12]	J. K. Hale, "Asymptotic Behavior of Dissipative Systems,", Amer. Math. Soc., (1988).
[13]	A. Haraux, "Systèmes Dynamiques Dissipatifs et Applications,", Masson, (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. doi: 10.1016/j.na.2007.09.028.
[15]	V. Pata, Uniform estimates of Gronwall type,, J. Math. Anal. Appl., 373 (2011), 264. 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. 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. doi: 10.3934/cpaa.2007.6.481.
[18]	V. Pata and S. Zelik, Smooth attractors for strongly damped wave equations,, Nonlinearity, 19 (2006), 1495. doi: 10.1088/0951-7715/19/7/001.
[19]	R. Temam, "Infinite-Dimensional Dynamical Systems in Mechanics and Physics,", Springer, (1997).

show all references

References:

[1]	A. V. Babin and M. I. Vishik, "Attractors of Evolution Equations,", North-Holland, (1992).
[2]	A. N. Carvalho and J. W. Cholewa, Attractors for strongly damped wave equations with critical nonlinearities,, Pacific J. Math., 207 (2002), 287. 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).
[4]	M. Conti and V. Pata, On the regularity of global attractors,, Discrete Cont. Dyn. Sys., 25 (2009), 1209. 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. 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.
[7]	A. E. Green and P. M. Naghdi, On undamped heat waves in an elastic solid,, J. Thermal Stresses, 15 (1992), 253. doi: 10.1080/01495739208946136.
[8]	A. E. Green and P. M. Naghdi, Thermoelasticity without energy dissipation,, J. Elasticity, 31 (1993), 189. 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. 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. 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. doi: 10.1098/rspa.1995.0022.
[12]	J. K. Hale, "Asymptotic Behavior of Dissipative Systems,", Amer. Math. Soc., (1988).
[13]	A. Haraux, "Systèmes Dynamiques Dissipatifs et Applications,", Masson, (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. doi: 10.1016/j.na.2007.09.028.
[15]	V. Pata, Uniform estimates of Gronwall type,, J. Math. Anal. Appl., 373 (2011), 264. 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. 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. doi: 10.3934/cpaa.2007.6.481.
[18]	V. Pata and S. Zelik, Smooth attractors for strongly damped wave equations,, Nonlinearity, 19 (2006), 1495. doi: 10.1088/0951-7715/19/7/001.
[19]	R. Temam, "Infinite-Dimensional Dynamical Systems in Mechanics and Physics,", Springer, (1997).

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