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January  2022, 27(1): 583-600. doi: 10.3934/dcdsb.2021056

## Asymptotics of singularly perturbed damped wave equations with super-cubic exponent

 Center for Mathematical Sciences, School of Mathematics and Statistics, Huazhong University of Science and Technology, Wuhan, Hubei 430074, China

Received  August 2020 Published  January 2022 Early access  February 2021

This work is devoted to studying the relations between the asymptotic behavior for a class of hyperbolic equations with super-cubic nonlinearity and a class of heat equations, where the problem is considered in a smooth bounded three dimensional domain. Based on the extension of the Strichartz estimates to the case of bounded domain, we show the regularity of the pullback, uniform, and cocycle attractors for the non-autonomous dynamical system given by hyperbolic equation. Then we prove that all types of non-autonomous attractors converge, upper semicontiously, to the natural extension global attractor of the limit parabolic equations.

Citation: Dandan Li. Asymptotics of singularly perturbed damped wave equations with super-cubic exponent. Discrete and Continuous Dynamical Systems - B, 2022, 27 (1) : 583-600. doi: 10.3934/dcdsb.2021056
##### References:
 [1] J. Arrieta, A. N. Carvalho and J. K. Hale, A damped hyerbolic equation with critical exponent, Communications in Partial Differential Equations, 17 (1992), 841-866.  doi: 10.1080/03605309208820866. [2] A. V. Babin and M. I. Vishik, Regular attractors of semi-groups and evolution equations, J. Math. Pures Appl, 62 (1983), 441-491. [3] A. V. Babin, M. I. Vishik, Attractors of Evolution Equations, Studies in Mathematics and Its Applications, 25. North-Holland Publishing Co., Amsterdam, 1992. [4] J. M. Ball, On the asymptotic behavior of generalized processes, with applications to nonlinear evolution equations, Journal of Differential Equations, 27 (1978), 224-265.  doi: 10.1016/0022-0396(78)90032-3. [5] M. D. Blair, H. F. Smith and C. D. Sogge, Strichartz estimates for the wave equation on manifolds with boundary, Annales de l'Institut Henri Poincare (C) Non Linéaire, 26 (2009), 1817-1829.  doi: 10.1016/j.anihpc.2008.12.004. [6] M. C. Bortolan, A. N. Carvalho and J. A. Langa, Structure of attractors for skew product semiflows, Journal of Differential Equations, 257 (2014), 490-522.  doi: 10.1016/j.jde.2014.04.008. [7] N. Burq, G. Lebeau and F. Planchon, Global existence for energy critical waves in 3-D domains, Journal of the American Mathematical Society, 21 (2008), 831-845.  doi: 10.1090/S0894-0347-08-00596-1. [8] A. N. Carvalho, J. A. Langa, J. C. Robinson, Attractors for Infinite-Dimensional Non-Autonomous Dynamical Systems, Springer, 2013. doi: 10.1007/978-1-4614-4581-4. [9] A. N. Carvalho, J. W. Cholewa and T. Dlotko, Damped wave equations with fast growing dissipative nonlinearities, Discrete and Continuous Dynamical Systems-A, 24 (2009), 1147-1165.  doi: 10.3934/dcds.2009.24.1147. [10] C. I. Christov, P. M. Jordan, Heat conduction paradox involving second-sound propagation in moving media, Physical Review Letters, 94 (2005), 154301. doi: 10.1103/PhysRevLett.94.154301. [11] V. V. Chepyzhov, M. I. Vishik, Attractors for Equations of Mathematical Physics, American Mathematical Soc., 2002. [12] L. C. Evans, Partial Differential Equations, Second edition. Graduate Studies in Mathematics, 19. American Mathematical Society, Providence, RI, 2010. doi: 10.1090/gsm/019. [13] P. Fabrie, C. Galusinski, A. Miranville and S. Zelik, Uniform exponential attractors for a singularly perturbed damped wave equation, Discrete and Continuous Dynamical Systems, 10 (2004), 211-238.  doi: 10.3934/dcds.2004.10.211. [14] M. M. Freitas, P. Kalita and J. A. Langa, Continuity of non-autonomous attractors for hyperbolic perturbation of parabolic equations, Journal of Differential Equations, 264 (2018), 1886-1945.  doi: 10.1016/j.jde.2017.10.007. [15] T. Gallay and G. Raugel, Scaling variables and asymptotic expansions in damped wave equations, J. Differential Equations, 150 (1998), 42-97.  doi: 10.1006/jdeq.1998.3459. [16] J. K. Hale, Asymptotic Behavior of Dissipative Systems, American Mathematical Society, Providence, RI, 1988. doi: 10.1090/surv/025. [17] J. K. Hale and G. Raugel, Upper semicontinuity of the attractor for a singularly perturbed hyperbolic equation, Journal of Differential Equations, 73 (1988), 197-214.  doi: 10.1016/0022-0396(88)90104-0. [18] A. Haraux, Two remarks on hyperbolic dissipative problems, Nonlinear Partial Differential Equations and their Applications. College de France seminar, 7 198), 1983–1984. [19] L. T. Hoang, E. J. Olson and J. C. Robinson, Continuity of pullback and uniform attractors, Journal of Differential Equations, 264 (2018), 4067-4093.  doi: 10.1016/j.jde.2017.12.002. [20] V. Kalantarov, A. Savostianov and S. Zelik, Attractors for damped quintic wave equations in bounded domains, Annales Henri Poincaré, 17 (2016), 2555-2584.  doi: 10.1007/s00023-016-0480-y. [21] S. Konabe and T. Nikuni, Coarse-grained finite-temperature theory for the bose condensate in optical lattices, Journal of Low Temperature Physics, 150 (2008), 12-46.  doi: 10.1007/s10909-007-9517-4. [22] A. C. Lazer and P. J. McKenna, Large-amplitude periodic oscillations in suspension bridges: some new connections with nonlinear analysis, SIAM Rev., 32 (1990), 537-578.  doi: 10.1137/1032120. [23] D. Li, Q. Chang, C. Sun, Pullback attractors for a critical degenerate wave equation with time-dependent damping, To appear. [24] Y. Lv and W. Wang, Limiting dynamics for stochastic wave equations, Journal of Differential Equations, 244 (2008), 1-23.  doi: 10.1016/j.jde.2007.10.009. [25] C. Matheus, M. C. Bortolan, A. N. Carvalho, J. A. Langa, Attractors Under Autonomous and Non-autonomous Perturbations, American Mathematical Society, 2020. [26] A. Miranville, V. Pata and S. Zelik, Exponential attractors for singularly perturbed damped wave equations: A simple construction., Asymptotic Analysis, 53 (2007), 1-12. [27] J. C. Robinson, Infinite-Dimensional Dynamical Systems: An Introduction to Dissipative Parabolic PDEs and the Theory of Global Attractors, Cambridge University Press, 2001.  doi: 10.1007/978-94-010-0732-0. [28] G. Somieski, Shimmy analysis of a simple aircraft nose landing gear model using different mathematical methods, Aerospace Science and Technology, 1 (1997), 545-555.  doi: 10.1016/S1270-9638(97)90003-1. [29] Y. Wang and C. Zhong, Upper semicontinuity of global attractors for damped wave equations, Asymptotic Analysis, 91 (2015), 1-10.  doi: 10.3233/ASY-141253. [30] S. Zelik, Asymptotic regularity of solutions of a nonautonomous damped wave equation with a critical growth exponent, Communications on Pure and Applied Analysis, 3 (2004), 921-934.  doi: 10.3934/cpaa.2004.3.921. [31] S. Zelik, Asymptotic regularity of solutions of singularly perturbed damped wave equations with supercritical nonlinearities, Discrete and Continuous Dynamical Systems, 11 (2004), 351-392.  doi: 10.3934/dcds.2004.11.351.

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##### References:
 [1] J. Arrieta, A. N. Carvalho and J. K. Hale, A damped hyerbolic equation with critical exponent, Communications in Partial Differential Equations, 17 (1992), 841-866.  doi: 10.1080/03605309208820866. [2] A. V. Babin and M. I. Vishik, Regular attractors of semi-groups and evolution equations, J. Math. Pures Appl, 62 (1983), 441-491. [3] A. V. Babin, M. I. Vishik, Attractors of Evolution Equations, Studies in Mathematics and Its Applications, 25. North-Holland Publishing Co., Amsterdam, 1992. [4] J. M. Ball, On the asymptotic behavior of generalized processes, with applications to nonlinear evolution equations, Journal of Differential Equations, 27 (1978), 224-265.  doi: 10.1016/0022-0396(78)90032-3. [5] M. D. Blair, H. F. Smith and C. D. Sogge, Strichartz estimates for the wave equation on manifolds with boundary, Annales de l'Institut Henri Poincare (C) Non Linéaire, 26 (2009), 1817-1829.  doi: 10.1016/j.anihpc.2008.12.004. [6] M. C. Bortolan, A. N. Carvalho and J. A. Langa, Structure of attractors for skew product semiflows, Journal of Differential Equations, 257 (2014), 490-522.  doi: 10.1016/j.jde.2014.04.008. [7] N. Burq, G. Lebeau and F. Planchon, Global existence for energy critical waves in 3-D domains, Journal of the American Mathematical Society, 21 (2008), 831-845.  doi: 10.1090/S0894-0347-08-00596-1. [8] A. N. Carvalho, J. A. Langa, J. C. Robinson, Attractors for Infinite-Dimensional Non-Autonomous Dynamical Systems, Springer, 2013. doi: 10.1007/978-1-4614-4581-4. [9] A. N. Carvalho, J. W. Cholewa and T. Dlotko, Damped wave equations with fast growing dissipative nonlinearities, Discrete and Continuous Dynamical Systems-A, 24 (2009), 1147-1165.  doi: 10.3934/dcds.2009.24.1147. [10] C. I. Christov, P. M. Jordan, Heat conduction paradox involving second-sound propagation in moving media, Physical Review Letters, 94 (2005), 154301. doi: 10.1103/PhysRevLett.94.154301. [11] V. V. Chepyzhov, M. I. Vishik, Attractors for Equations of Mathematical Physics, American Mathematical Soc., 2002. [12] L. C. Evans, Partial Differential Equations, Second edition. Graduate Studies in Mathematics, 19. American Mathematical Society, Providence, RI, 2010. doi: 10.1090/gsm/019. [13] P. Fabrie, C. Galusinski, A. Miranville and S. Zelik, Uniform exponential attractors for a singularly perturbed damped wave equation, Discrete and Continuous Dynamical Systems, 10 (2004), 211-238.  doi: 10.3934/dcds.2004.10.211. [14] M. M. Freitas, P. Kalita and J. A. Langa, Continuity of non-autonomous attractors for hyperbolic perturbation of parabolic equations, Journal of Differential Equations, 264 (2018), 1886-1945.  doi: 10.1016/j.jde.2017.10.007. [15] T. Gallay and G. Raugel, Scaling variables and asymptotic expansions in damped wave equations, J. Differential Equations, 150 (1998), 42-97.  doi: 10.1006/jdeq.1998.3459. [16] J. K. Hale, Asymptotic Behavior of Dissipative Systems, American Mathematical Society, Providence, RI, 1988. doi: 10.1090/surv/025. [17] J. K. Hale and G. Raugel, Upper semicontinuity of the attractor for a singularly perturbed hyperbolic equation, Journal of Differential Equations, 73 (1988), 197-214.  doi: 10.1016/0022-0396(88)90104-0. [18] A. Haraux, Two remarks on hyperbolic dissipative problems, Nonlinear Partial Differential Equations and their Applications. College de France seminar, 7 198), 1983–1984. [19] L. T. Hoang, E. J. Olson and J. C. Robinson, Continuity of pullback and uniform attractors, Journal of Differential Equations, 264 (2018), 4067-4093.  doi: 10.1016/j.jde.2017.12.002. [20] V. Kalantarov, A. Savostianov and S. Zelik, Attractors for damped quintic wave equations in bounded domains, Annales Henri Poincaré, 17 (2016), 2555-2584.  doi: 10.1007/s00023-016-0480-y. [21] S. Konabe and T. Nikuni, Coarse-grained finite-temperature theory for the bose condensate in optical lattices, Journal of Low Temperature Physics, 150 (2008), 12-46.  doi: 10.1007/s10909-007-9517-4. [22] A. C. Lazer and P. J. McKenna, Large-amplitude periodic oscillations in suspension bridges: some new connections with nonlinear analysis, SIAM Rev., 32 (1990), 537-578.  doi: 10.1137/1032120. [23] D. Li, Q. Chang, C. Sun, Pullback attractors for a critical degenerate wave equation with time-dependent damping, To appear. [24] Y. Lv and W. Wang, Limiting dynamics for stochastic wave equations, Journal of Differential Equations, 244 (2008), 1-23.  doi: 10.1016/j.jde.2007.10.009. [25] C. Matheus, M. C. Bortolan, A. N. Carvalho, J. A. Langa, Attractors Under Autonomous and Non-autonomous Perturbations, American Mathematical Society, 2020. [26] A. Miranville, V. Pata and S. Zelik, Exponential attractors for singularly perturbed damped wave equations: A simple construction., Asymptotic Analysis, 53 (2007), 1-12. [27] J. C. Robinson, Infinite-Dimensional Dynamical Systems: An Introduction to Dissipative Parabolic PDEs and the Theory of Global Attractors, Cambridge University Press, 2001.  doi: 10.1007/978-94-010-0732-0. [28] G. Somieski, Shimmy analysis of a simple aircraft nose landing gear model using different mathematical methods, Aerospace Science and Technology, 1 (1997), 545-555.  doi: 10.1016/S1270-9638(97)90003-1. [29] Y. Wang and C. Zhong, Upper semicontinuity of global attractors for damped wave equations, Asymptotic Analysis, 91 (2015), 1-10.  doi: 10.3233/ASY-141253. [30] S. Zelik, Asymptotic regularity of solutions of a nonautonomous damped wave equation with a critical growth exponent, Communications on Pure and Applied Analysis, 3 (2004), 921-934.  doi: 10.3934/cpaa.2004.3.921. [31] S. Zelik, Asymptotic regularity of solutions of singularly perturbed damped wave equations with supercritical nonlinearities, Discrete and Continuous Dynamical Systems, 11 (2004), 351-392.  doi: 10.3934/dcds.2004.11.351.
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