• Previous Article
    The explicit nonlinear wave solutions of the generalized $b$-equation
  • CPAA Home
  • This Issue
  • Next Article
    Stability of rarefaction wave and boundary layer for outflow problem on the two-fluid Navier-Stokes-Poisson equations
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 & 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).   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.  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).   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[6]

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

[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.  Google Scholar

[8]

A. E. Green and P. M. Naghdi, Thermoelasticity without energy dissipation,, J. Elasticity, 31 (1993), 189.  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.  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.  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.  doi: 10.1098/rspa.1995.0022.  Google Scholar

[12]

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

[13]

A. Haraux, "Systèmes Dynamiques Dissipatifs et Applications,", Masson, (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.  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.  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.  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.  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.  doi: 10.1088/0951-7715/19/7/001.  Google Scholar

[19]

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

show all references

References:
[1]

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

[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.  Google Scholar

[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.  Google Scholar

[6]

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

[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.  Google Scholar

[8]

A. E. Green and P. M. Naghdi, Thermoelasticity without energy dissipation,, J. Elasticity, 31 (1993), 189.  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.  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.  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.  doi: 10.1098/rspa.1995.0022.  Google Scholar

[12]

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

[13]

A. Haraux, "Systèmes Dynamiques Dissipatifs et Applications,", Masson, (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.  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.  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.  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.  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.  doi: 10.1088/0951-7715/19/7/001.  Google Scholar

[19]

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

[1]

Ahmad Z. Fino, Wenhui Chen. A global existence result for two-dimensional semilinear strongly damped wave equation with mixed nonlinearity in an exterior domain. Communications on Pure & Applied Analysis, 2020, 19 (12) : 5387-5411. doi: 10.3934/cpaa.2020243

[2]

Jianhua Huang, Yanbin Tang, Ming Wang. Singular support of the global attractor for a damped BBM equation. Discrete & Continuous Dynamical Systems - B, 2020  doi: 10.3934/dcdsb.2020345

[3]

Xinyu Mei, Yangmin Xiong, Chunyou Sun. Pullback attractor for a weakly damped wave equation with sup-cubic nonlinearity. Discrete & Continuous Dynamical Systems - A, 2021, 41 (2) : 569-600. doi: 10.3934/dcds.2020270

[4]

Mengni Li. Global regularity for a class of Monge-Ampère type equations with nonzero boundary conditions. Communications on Pure & Applied Analysis, 2021, 20 (1) : 301-317. doi: 10.3934/cpaa.2020267

[5]

Kihoon Seong. Low regularity a priori estimates for the fourth order cubic nonlinear Schrödinger equation. Communications on Pure & Applied Analysis, 2020, 19 (12) : 5437-5473. doi: 10.3934/cpaa.2020247

[6]

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

[7]

Oussama Landoulsi. Construction of a solitary wave solution of the nonlinear focusing schrödinger equation outside a strictly convex obstacle in the $ L^2 $-supercritical case. Discrete & Continuous Dynamical Systems - A, 2021, 41 (2) : 701-746. doi: 10.3934/dcds.2020298

[8]

Thomas Bartsch, Tian Xu. Strongly localized semiclassical states for nonlinear Dirac equations. Discrete & Continuous Dynamical Systems - A, 2021, 41 (1) : 29-60. doi: 10.3934/dcds.2020297

[9]

Qingfang Wang, Hua Yang. Solutions of nonlocal problem with critical exponent. Communications on Pure & Applied Analysis, 2020, 19 (12) : 5591-5608. doi: 10.3934/cpaa.2020253

[10]

Laurence Cherfils, Stefania Gatti, Alain Miranville, Rémy Guillevin. Analysis of a model for tumor growth and lactate exchanges in a glioma. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020457

[11]

Emre Esentürk, Juan Velazquez. Large time behavior of exchange-driven growth. Discrete & Continuous Dynamical Systems - A, 2021, 41 (2) : 747-775. doi: 10.3934/dcds.2020299

[12]

João Marcos do Ó, Bruno Ribeiro, Bernhard Ruf. Hamiltonian elliptic systems in dimension two with arbitrary and double exponential growth conditions. Discrete & Continuous Dynamical Systems - A, 2021, 41 (1) : 277-296. doi: 10.3934/dcds.2020138

[13]

Ebraheem O. Alzahrani, Muhammad Altaf Khan. Androgen driven evolutionary population dynamics in prostate cancer growth. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020426

[14]

Haiyu Liu, Rongmin Zhu, Yuxian Geng. Gorenstein global dimensions relative to balanced pairs. Electronic Research Archive, 2020, 28 (4) : 1563-1571. doi: 10.3934/era.2020082

[15]

Bernold Fiedler. Global Hopf bifurcation in networks with fast feedback cycles. Discrete & Continuous Dynamical Systems - S, 2021, 14 (1) : 177-203. doi: 10.3934/dcdss.2020344

[16]

Adel M. Al-Mahdi, Mohammad M. Al-Gharabli, Salim A. Messaoudi. New general decay result for a system of viscoelastic wave equations with past history. Communications on Pure & Applied Analysis, 2021, 20 (1) : 389-404. doi: 10.3934/cpaa.2020273

[17]

Jun Zhou. Lifespan of solutions to a fourth order parabolic PDE involving the Hessian modeling epitaxial growth. Communications on Pure & Applied Analysis, 2020, 19 (12) : 5581-5590. doi: 10.3934/cpaa.2020252

[18]

Touria Karite, Ali Boutoulout. Global and regional constrained controllability for distributed parabolic linear systems: RHUM approach. Numerical Algebra, Control & Optimization, 2020  doi: 10.3934/naco.2020055

[19]

Youshan Tao, Michael Winkler. Critical mass for infinite-time blow-up in a haptotaxis system with nonlinear zero-order interaction. Discrete & Continuous Dynamical Systems - A, 2021, 41 (1) : 439-454. doi: 10.3934/dcds.2020216

[20]

Shengbing Deng, Tingxi Hu, Chun-Lei Tang. $ N- $Laplacian problems with critical double exponential nonlinearities. Discrete & Continuous Dynamical Systems - A, 2021, 41 (2) : 987-1003. doi: 10.3934/dcds.2020306

2019 Impact Factor: 1.105

Metrics

  • PDF downloads (40)
  • HTML views (0)
  • Cited by (3)

Other articles
by authors

[Back to Top]