# American Institute of Mathematical Sciences

2016, 21(1): 205-225. doi: 10.3934/dcdsb.2016.21.205

## Attractors for wave equations with nonlinear damping on time-dependent space

 1 School of Mathematics and Physics, Jiangsu University of Technology, Changzhou 213001, China 2 School of Mathematics and Statistics, Huazhong University of Science and Technology, Wuhan 430074, China 3 Department of Mathematics, Nanjing University, Nanjing 210093

Received  January 2015 Revised  July 2015 Published  November 2015

In this paper, we consider the long time behavior of the solution for the following nonlinear damped wave equation \begin{eqnarray*} \varepsilon(t) u_{tt}+g(u_{t})-\Delta u+\varphi (u)=f \end{eqnarray*} with Dirichlet boundary condition, in which, the coefficient $\varepsilon$ depends explicitly on time, the damping $g$ is nonlinear and the nonlinearity $\varphi$ has a critical growth. Spirited by this concrete problem, we establish a sufficient and necessary condition for the existence of attractors on time-dependent spaces, which is equivalent to that provided by M. Conti et al.[10]. Furthermore, we give a technical method for verifying compactness of the process via contractive functions. Finally, by the new framework, we obtain the existence of the time-dependent attractors for the wave equations with nonlinear damping.
Citation: Fengjuan Meng, Meihua Yang, Chengkui Zhong. Attractors for wave equations with nonlinear damping on time-dependent space. Discrete & Continuous Dynamical Systems - B, 2016, 21 (1) : 205-225. doi: 10.3934/dcdsb.2016.21.205
##### References:
 [1] A. N. Carvaho, J. A. Langa and J. C. Robinson, Attractors for Infinite-Dimensional Non-Autonomous Dynamical Systems,, Springer, (2013). doi: 10.1007/978-1-4614-4581-4. [2] V. V. Chepyzhov and M. I. Vishik, Attractors of nonautonomous dynamical systems and their dimension,, J. Math. Pures Appl., 73 (1994), 279. [3] V. V. Chepyzhov and M. I. Vishik, Attractors for Equations of Mathematical Physics,, Colloquium Publications, (2002). [4] V. V. Chepyzhov, M. Conti and V. Pata, A minimal approach to the theory of global attractor,, Discrete Contin. Dyn. Syst., 32 (2012), 2079. doi: 10.3934/dcds.2012.32.2079. [5] I. D. Chueshov, Introduction to the Theory of Infinite-Dimensional Dissipative Systems,, Acta Scientific Publishing House, (2002). [6] I. Chueshov and I. Lasiecka, Attractors for second-order evolution equations with a nonlinear damping,, J. Dynam. Differential Equations, 16 (2004), 469. doi: 10.1007/s10884-004-4289-x. [7] I. Chueshov and I. Lasiecka, Long-time dynamics of semilinear wave equation with nonlinear interior-boundary damping and sources of critical exponents,, Contemp. Math., 426 (2007), 153. doi: 10.1090/conm/426/08188. [8] I. Chueshov, I. Lasiecka and D. Toundykov, Long-term dynamics of semilinear wave equations with nonlinear localized interior damping and a source term of critical exponent,, Discrete. Contin. Dyn. Syst., 20 (2008), 459. [9] I. Chueshov and I. Lasiecka, Long-time behavior of second order evolution equations with nonlinear damping,, Mem. Amer. Math. Soc., 195 (2008). doi: 10.1090/memo/0912. [10] M. Conti, V. Pata and R. Temam, Attractors for processes on time-dependent space. Application to Wave equation,, J. Differential Equations, 255 (2013), 1254. doi: 10.1016/j.jde.2013.05.013. [11] M. Conti and V. Pata, Asymptotic structure of the attractor for processes on time-dependent spaces,, Nonlinear Analysis RWA, 19 (2014), 1. doi: 10.1016/j.nonrwa.2014.02.002. [12] M. Conti and V. Pata, On the time-dependent cattaneo law in space dimension one,, Applied Mathematic and Computation, 259 (2015), 32. doi: 10.1016/j.amc.2015.02.039. [13] H. Crauel and F. Flandoli, Attractors for random dynamical systems,, Probab. Theory Related Fields, 100 (1994), 365. doi: 10.1007/BF01193705. [14] H. Crauel, A. Debussche and F. Flandoli, Random attractors,, J. Dynam. Differential Equations, 9 (1997), 307. doi: 10.1007/BF02219225. [15] E. Feireisl, Attractors for wave equations with nonlinear dissipation and critical exponent,, C. R. Acad. Sci. Paris, 315 (1992), 551. [16] E. Feireisl and E. Zuazua, Global attractors for semilinear wave equations with locally distributed nonlinear damping and critical exponent,, Commun. PDE., 18 (1993), 1539. doi: 10.1080/03605309308820985. [17] E. Feireisl, Global attractors for damped wave equations with supercritical exponent,, J. Differential Equations, 116 (1995), 431. doi: 10.1006/jdeq.1995.1042. [18] J. K. Hale, Asymptotic Behavior of Dissipative Systems,, AMS, (1988). [19] A. Kh. Khanmamedov, Global attractors for wave equations with nonlinear damping and critical exponents,, J. Differential Equations, 230 (2006), 702. doi: 10.1016/j.jde.2006.06.001. [20] A. Kh. Khanmamedov, Global attractors for von Karman equations with nonlinear interior dissipation,, J. Math. Anal. Appl., 318 (2006), 92. doi: 10.1016/j.jmaa.2005.05.031. [21] A. Kh. Khanmamedov, Remark on the regularity of the global attractor for the wave equation with nonlinear damping,, Nonlinear Anal., 72 (2010), 1993. doi: 10.1016/j.na.2009.09.041. [22] P. S. Landahl, O. H. Soerensen and P. L. Christiansen, Soliton excitations in Josephson tunnel junctions, Phys. Rev.B, 25 (1982), 5737. [23] I. Lasiecka and A. R. Ruzmaikina, Finite dimensionality and regularity of attractors for 2-D semilinear wave equation with nonlinear dissipation,, J. Math. Anal. Appl., 270 (2002), 16. doi: 10.1016/S0022-247X(02)00006-9. [24] O. A. Ladyzhenskaya, Attractors for Semigroups and Evolution Equations,, Cambridge University Prss, (1991). doi: 10.1017/CBO9780511569418. [25] J. L. Lions, Quelques Méthodes de Résolution des Problèmes Aux Limites N-linéaires,, Dunod, (1969). [26] I. Moise, R. Rosa and X. Wang, Attractors for noncompact nonautonomous systems via energy equations,, Discrete Contin. Dyn. Syst., 10 (2004), 473. doi: 10.3934/dcds.2004.10.473. [27] M. Nakao, Global attractors for nonlinear wave equations with nonlinear dissipative terms,, J. Differential Equations, 227 (2006), 204. doi: 10.1016/j.jde.2005.09.013. [28] F. Di Plinio, G. S.Duane and R. Temam, Time dependent attractor for the oscillon equation,, Discrete Contin. Dyn. Syst., 29 (2011), 141. doi: 10.3934/dcds.2011.29.141. [29] V. Pata and S. Zelik, Smooth attractors for strongly damped wave equations,, Nonlinearity, 19 (2006), 1495. doi: 10.1088/0951-7715/19/7/001. [30] G. Raugel, Une equation des ondes avec amortissment non lineaire dans le cas critique en dimensions trois,, C. R. Acad. Sci. Paris, 314 (1992), 177. [31] B. Schmalfuß, Backward cocycles and attractors of stochastic differential equations, International Seminar on Applied Mathematics and Nonlinear Dynamics: Attractor Approximation and Global Behaviour (eds. V. Reitmann, T. Riedrich and N. Koksch),, Dresden, 73 (1992), 185. [32] C. Y. Sun, D. M. Cao and J. Q. Duan, Non-autonomous dynamics of wave equations with nonlinear damping and critical nonlinearity,, Nonlinearity, 19 (2006), 2645. doi: 10.1088/0951-7715/19/11/008. [33] C. Y. Sun, M. H. Yang and C. K. Zhong, Global attractors for the wave equation with nonlinear damping,, J. Differential Equations, 227 (2006), 427. doi: 10.1016/j.jde.2005.09.010. [34] C. Y. Sun, D. M. Cao and J. Q. Duan, Uniform attractors for nonautonomous wave equations with nonlinear damping,, SIAM J. Appl. Dyn. Syst., 6 (2007), 293. doi: 10.1137/060663805. [35] R. Temam, Infinite-dimensional Dynamical Systems in Mechanics and Physics,, Second edition, 68 (1997). doi: 10.1007/978-1-4612-0645-3.

show all references

##### References:
 [1] A. N. Carvaho, J. A. Langa and J. C. Robinson, Attractors for Infinite-Dimensional Non-Autonomous Dynamical Systems,, Springer, (2013). doi: 10.1007/978-1-4614-4581-4. [2] V. V. Chepyzhov and M. I. Vishik, Attractors of nonautonomous dynamical systems and their dimension,, J. Math. Pures Appl., 73 (1994), 279. [3] V. V. Chepyzhov and M. I. Vishik, Attractors for Equations of Mathematical Physics,, Colloquium Publications, (2002). [4] V. V. Chepyzhov, M. Conti and V. Pata, A minimal approach to the theory of global attractor,, Discrete Contin. Dyn. Syst., 32 (2012), 2079. doi: 10.3934/dcds.2012.32.2079. [5] I. D. Chueshov, Introduction to the Theory of Infinite-Dimensional Dissipative Systems,, Acta Scientific Publishing House, (2002). [6] I. Chueshov and I. Lasiecka, Attractors for second-order evolution equations with a nonlinear damping,, J. Dynam. Differential Equations, 16 (2004), 469. doi: 10.1007/s10884-004-4289-x. [7] I. Chueshov and I. Lasiecka, Long-time dynamics of semilinear wave equation with nonlinear interior-boundary damping and sources of critical exponents,, Contemp. Math., 426 (2007), 153. doi: 10.1090/conm/426/08188. [8] I. Chueshov, I. Lasiecka and D. Toundykov, Long-term dynamics of semilinear wave equations with nonlinear localized interior damping and a source term of critical exponent,, Discrete. Contin. Dyn. Syst., 20 (2008), 459. [9] I. Chueshov and I. Lasiecka, Long-time behavior of second order evolution equations with nonlinear damping,, Mem. Amer. Math. Soc., 195 (2008). doi: 10.1090/memo/0912. [10] M. Conti, V. Pata and R. Temam, Attractors for processes on time-dependent space. Application to Wave equation,, J. Differential Equations, 255 (2013), 1254. doi: 10.1016/j.jde.2013.05.013. [11] M. Conti and V. Pata, Asymptotic structure of the attractor for processes on time-dependent spaces,, Nonlinear Analysis RWA, 19 (2014), 1. doi: 10.1016/j.nonrwa.2014.02.002. [12] M. Conti and V. Pata, On the time-dependent cattaneo law in space dimension one,, Applied Mathematic and Computation, 259 (2015), 32. doi: 10.1016/j.amc.2015.02.039. [13] H. Crauel and F. Flandoli, Attractors for random dynamical systems,, Probab. Theory Related Fields, 100 (1994), 365. doi: 10.1007/BF01193705. [14] H. Crauel, A. Debussche and F. Flandoli, Random attractors,, J. Dynam. Differential Equations, 9 (1997), 307. doi: 10.1007/BF02219225. [15] E. Feireisl, Attractors for wave equations with nonlinear dissipation and critical exponent,, C. R. Acad. Sci. Paris, 315 (1992), 551. [16] E. Feireisl and E. Zuazua, Global attractors for semilinear wave equations with locally distributed nonlinear damping and critical exponent,, Commun. PDE., 18 (1993), 1539. doi: 10.1080/03605309308820985. [17] E. Feireisl, Global attractors for damped wave equations with supercritical exponent,, J. Differential Equations, 116 (1995), 431. doi: 10.1006/jdeq.1995.1042. [18] J. K. Hale, Asymptotic Behavior of Dissipative Systems,, AMS, (1988). [19] A. Kh. Khanmamedov, Global attractors for wave equations with nonlinear damping and critical exponents,, J. Differential Equations, 230 (2006), 702. doi: 10.1016/j.jde.2006.06.001. [20] A. Kh. Khanmamedov, Global attractors for von Karman equations with nonlinear interior dissipation,, J. Math. Anal. Appl., 318 (2006), 92. doi: 10.1016/j.jmaa.2005.05.031. [21] A. Kh. Khanmamedov, Remark on the regularity of the global attractor for the wave equation with nonlinear damping,, Nonlinear Anal., 72 (2010), 1993. doi: 10.1016/j.na.2009.09.041. [22] P. S. Landahl, O. H. Soerensen and P. L. Christiansen, Soliton excitations in Josephson tunnel junctions, Phys. Rev.B, 25 (1982), 5737. [23] I. Lasiecka and A. R. Ruzmaikina, Finite dimensionality and regularity of attractors for 2-D semilinear wave equation with nonlinear dissipation,, J. Math. Anal. Appl., 270 (2002), 16. doi: 10.1016/S0022-247X(02)00006-9. [24] O. A. Ladyzhenskaya, Attractors for Semigroups and Evolution Equations,, Cambridge University Prss, (1991). doi: 10.1017/CBO9780511569418. [25] J. L. Lions, Quelques Méthodes de Résolution des Problèmes Aux Limites N-linéaires,, Dunod, (1969). [26] I. Moise, R. Rosa and X. Wang, Attractors for noncompact nonautonomous systems via energy equations,, Discrete Contin. Dyn. Syst., 10 (2004), 473. doi: 10.3934/dcds.2004.10.473. [27] M. Nakao, Global attractors for nonlinear wave equations with nonlinear dissipative terms,, J. Differential Equations, 227 (2006), 204. doi: 10.1016/j.jde.2005.09.013. [28] F. Di Plinio, G. S.Duane and R. Temam, Time dependent attractor for the oscillon equation,, Discrete Contin. Dyn. Syst., 29 (2011), 141. doi: 10.3934/dcds.2011.29.141. [29] V. Pata and S. Zelik, Smooth attractors for strongly damped wave equations,, Nonlinearity, 19 (2006), 1495. doi: 10.1088/0951-7715/19/7/001. [30] G. Raugel, Une equation des ondes avec amortissment non lineaire dans le cas critique en dimensions trois,, C. R. Acad. Sci. Paris, 314 (1992), 177. [31] B. Schmalfuß, Backward cocycles and attractors of stochastic differential equations, International Seminar on Applied Mathematics and Nonlinear Dynamics: Attractor Approximation and Global Behaviour (eds. V. Reitmann, T. Riedrich and N. Koksch),, Dresden, 73 (1992), 185. [32] C. Y. Sun, D. M. Cao and J. Q. Duan, Non-autonomous dynamics of wave equations with nonlinear damping and critical nonlinearity,, Nonlinearity, 19 (2006), 2645. doi: 10.1088/0951-7715/19/11/008. [33] C. Y. Sun, M. H. Yang and C. K. Zhong, Global attractors for the wave equation with nonlinear damping,, J. Differential Equations, 227 (2006), 427. doi: 10.1016/j.jde.2005.09.010. [34] C. Y. Sun, D. M. Cao and J. Q. Duan, Uniform attractors for nonautonomous wave equations with nonlinear damping,, SIAM J. Appl. Dyn. Syst., 6 (2007), 293. doi: 10.1137/060663805. [35] R. Temam, Infinite-dimensional Dynamical Systems in Mechanics and Physics,, Second edition, 68 (1997). doi: 10.1007/978-1-4612-0645-3.
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