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Strong global and exponential attractors for a nonlinear strongly damped hyperbolic equation

  • * Corresponding author: Zhijian Yang

    * Corresponding author: Zhijian Yang 
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  • In this paper, we investigate the global well-posedness and the existence of strong global and exponential attractors for a nonlinear strongly damped hyperbolic equation in $ \Omega\subset{\mathbb R}^N $:

    $ u_{tt}+\Delta ^2u+\Delta ^2u_t+\Delta \phi (\Delta u) = g(x), $

    with the hinged boundary condition. We show that (i) when the nonlinearity $ \phi $ is quasi-monotone and is of at most the critical growth: $ 1\leq p\leq p^{*}: = \frac{N+2}{(N-2)^{+}} \ (N\geq 2) $ and $ g = 0 $, the model has in phase space $ {\mathcal H} = V_3\times L^2 $ a trivial global and exponential attractor, respectively. (ii) In particular when $ N = 1 $, without any polynomial growth restriction for $ \phi $, the model has a strong global and a strong exponential attractor, respectively. These results deepen and extend the related researches on this topic in recent literature [16,22]. The method developed here allows us to establish the existence of the strong global and exponential attractor for this nonlinear model.

    Mathematics Subject Classification: Primary: 37L30, 35B33; Secondary: 35B40, 35B41, 35B65.


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  • [1] M. Aassila and A. Guesmia, Energy decay for a damped nonlinear hyperbolic equation, Appl. Math. Lett., 12 (1999), 49-52.  doi: 10.1016/S0893-9659(98)00171-2.
    [2] A. S. AcklehH. T. Banks and G. A. Pinter, A nonlinear beam equation, Appl. Math. Lett., 15 (2002), 381-387.  doi: 10.1016/S0893-9659(01)00147-1.
    [3] H. Amann, Nonhomogeneous linear and quasilinear elliptic and parabolic boundary value problem, in: Schmeisser/Triebel: Function Spaces, Differential Operators and Nonlinear Analysis, Teubner Texte zur Mathematik, vol. 133, Teubner, 1993, 9-126. doi: 10.1007/978-3-663-11336-2_1.
    [4] J. M. Arrieta and A. N. Carvalho, Abstract parabolic problems with critical nonlinearities and applications to Navier-Stokes and heat equations, Trans. Amer. Math. Soc., 352 (1999), 285-310.  doi: 10.1090/S0002-9947-99-02528-3.
    [5] A. V. Babin and M. I. Vishik, Attractors of Evolutionary Equations, North-Holland, Amsterdam, 1992.
    [6] H. T. BanksD. S. Gilliam and V. I. Shubov, Global solvability for damped abstractnonlinear hyperbolic systems, Differ. Integral Equ., 100 (1997), 309-332. 
    [7] H. T. Banks, D. S. Gilliam and V. I. Shubov, Well-Posedness for A One Dimensional Nonlinear Beam, In Computation and Control IV, Progress in Systems and Control Theory, Volume 20, Birkh user, Boston, MA, 1995.
    [8] V. Belleri and V. Pata, Attractors for semilinear strongly damped wave equations on $\mathbb R^3$, Discrete Contin. Dyn. Syst., 7 (2001), 719-735.  doi: 10.3934/dcds.2001.7.719.
    [9] G. Chen and F. Da, Blow-up of solution of Cauchy problem for three-dimensional damped nonlinear hyperbolic equation, Nonlinear Anal., 71 (2009), 358-372.  doi: 10.1016/j.na.2008.10.132.
    [10] G. ChenY. Wang and Z. Zhao, Blow-up of solution of an initial boundary value problem for a damped nonlinear hyperbolic equation, Appl. Math. Lett., 17 (2004), 491-497.  doi: 10.1016/S0893-9659(04)90116-4.
    [11] I. Chueshov, Dynamics of Quasi-Stable Dissipative Systems, Springer, 2015. doi: 10.1007/978-3-319-22903-4.
    [12] I. Chueshov and I. Lasiecka, Attractors for second order evolution equations with a nonlinear damping, J. Dynam. Diff. Eqs., 16 (2004), 469-512.  doi: 10.1007/s10884-004-4289-x.
    [13] P. DingZ. Yang and Y. Zhao, Attractors and their upper semicontinuity for the nonlinear membrane equation with structural damping (in Chinese), Sci. Sin. Math., 51 (2021), 315-332.  doi: 10.1360/SCM-2018-0901.
    [14] M. EfendievY. Yamamoto and A. Yagi, Exponential attractors for non-autonomous dissipative systems, Journal of the Mathematical Society of Japan, 63 (2011), 647-673.  doi: 10.2969/jmsj/06320647.
    [15] D. Fujiwara, Concrete characterization of the domains of fractional powers of some elliptic differential operators of the second order, Proc. Japan Acad., 43 (1967), 82-86.  doi: 10.3792/pja/1195521686.
    [16] V. Kalantarov and S. Zelik, Finite-dimensional attractors for the nonlinear strongly damped wave equation, J. Differential Equations, 247 (2009), 1120-1155.  doi: 10.1016/j.jde.2009.04.010.
    [17] I. Lasiecka and R. Triggiani, Regularity theory for a class of non-homogeneous Euler-Bernoulli equations: A cosine operator approach, Series in Pure Mathematics, Topics in Mathematical Analysis, World Scientific Publ. Co., 1989,623-657. doi: 10.1142/9789814434201_0028.
    [18] Y. Li and Z. Yang, Optimal attractors of the Kirchhoff wave model with structural nonlinear damping, J. Differential Equations, 268 (2020), 7741-7773.  doi: 10.1016/j.jde.2019.11.084.
    [19] Y. Li and Z. Yang, Strong attractors and their continuity for the semilinear wave equations with fractional damping, Advances in Differential Equations, 26 (2021), 45-82. 
    [20] Y. LiZ. Yang and P. Ding, Regular solutions and strong attractors for the Kirchhoff wave model with structural nonlinear damping, Appl. Math. Lett., 104 (2020), 106258.  doi: 10.1016/j.aml.2020.106258.
    [21] H. Ma and C. Zhong, Attractors for the Kirchhoff equations with strong nonlinear damping, Appl. Math. Lett., 74 (2017), 127-133.  doi: 10.1016/j.aml.2017.06.002.
    [22] T. Pang and J. Shen, Global existence and asymptotic behaviour of solution for a damped nonlinear hyperbolic equation, Nonlinear Anal., 198 (2020), 111885.  doi: 10.1016/j.na.2020.111885.
    [23] J. Simon, Compact sets in the space $L^{p}(0, T;B)$, Ann. Mat. Pura. Appl., 146 (1986), 65-96.  doi: 10.1007/BF01762360.
    [24] C. Song and Z. Yang, Existence and nonexistence of global solutions to the Cauchy problem for a nonlinear beam equation, Math. Meth. Appl. Sci., 33 (2010), 563-575.  doi: 10.1002/mma.1175.
    [25] Y. Sun and Z. Yang, Strong attractors and their robustness for an extensible beam model with energy damping, Discrete Contin. Dyn. Syst. -B, 27 (2022), 3101-3129.  doi: 10.3934/dcdsb.2021175.
    [26] R. Temam, Infinite-Dimensional Dynamical Systems in Mechanics and Physics, Springer, New York, 1988. doi: 10.1007/978-1-4684-0313-8.
    [27] C.-K. ZhongM.-H. Yang and C.-Y. Sun, The existence of global attractors for the norm-to-weak continuous semigroup and application to the nonlinear reaction-diffusion equations, J. Differential Equations, 223 (2006), 367-399.  doi: 10.1016/j.jde.2005.06.008.
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