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

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

Fengjuan Meng 1, , Meihua Yang 2, and  Chengkui Zhong 3,

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.
Keywords: nonlinear damping, wave equation, Time-dependent attractor, critical exponent..
Mathematics Subject Classification: Primary: 35L05, 37L05, 35B40; Secondary: 35B41, 58J2.
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.  Google Scholar

[2]

V. V. Chepyzhov and M. I. Vishik, Attractors of nonautonomous dynamical systems and their dimension, J. Math. Pures Appl., 73 (1994), 279-333.  Google Scholar

[3]

V. V. Chepyzhov and M. I. Vishik, Attractors for Equations of Mathematical Physics, Colloquium Publications, vol. 49, American Mathematical Society, Providence, RI, 2002.  Google Scholar

[4]

V. V. Chepyzhov, M. Conti and V. Pata, A minimal approach to the theory of global attractor, Discrete Contin. Dyn. Syst., 32 (2012), 2079-2088. doi: 10.3934/dcds.2012.32.2079.  Google Scholar

[5]

I. D. Chueshov, Introduction to the Theory of Infinite-Dimensional Dissipative Systems, Acta Scientific Publishing House, Kharkiv, 2002. Google Scholar

[6]

I. Chueshov and I. Lasiecka, Attractors for second-order evolution equations with a nonlinear damping, J. Dynam. Differential Equations, 16 (2004), 469-512. doi: 10.1007/s10884-004-4289-x.  Google Scholar

[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, 153-192. Amer. Math. Soc., Providence, RI, 2007. doi: 10.1090/conm/426/08188.  Google Scholar

[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), 459-509.  Google Scholar

[9]

I. Chueshov and I. Lasiecka, Long-time behavior of second order evolution equations with nonlinear damping, Mem. Amer. Math. Soc., 195 (2008), viii+183 pp. doi: 10.1090/memo/0912.  Google Scholar

[10]

M. Conti, V. Pata and R. Temam, Attractors for processes on time-dependent space. Application to Wave equation, J. Differential Equations, 255 (2013), 1254-1277. doi: 10.1016/j.jde.2013.05.013.  Google Scholar

[11]

M. Conti and V. Pata, Asymptotic structure of the attractor for processes on time-dependent spaces, Nonlinear Analysis RWA, 19 (2014), 1-10. doi: 10.1016/j.nonrwa.2014.02.002.  Google Scholar

[12]

M. Conti and V. Pata, On the time-dependent cattaneo law in space dimension one, Applied Mathematic and Computation, 259 (2015), 32-44. doi: 10.1016/j.amc.2015.02.039.  Google Scholar

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H. Crauel, A. Debussche and F. Flandoli, Random attractors, J. Dynam. Differential Equations, 9 (1997), 307-341. doi: 10.1007/BF02219225.  Google Scholar

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E. Feireisl, Attractors for wave equations with nonlinear dissipation and critical exponent, C. R. Acad. Sci. Paris, 315 (1992), 551-555.  Google Scholar

[16]

E. Feireisl and E. Zuazua, Global attractors for semilinear wave equations with locally distributed nonlinear damping and critical exponent, Commun. PDE., 18 (1993), 1539-1555. doi: 10.1080/03605309308820985.  Google Scholar

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E. Feireisl, Global attractors for damped wave equations with supercritical exponent, J. Differential Equations, 116 (1995), 431-447. doi: 10.1006/jdeq.1995.1042.  Google Scholar

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J. K. Hale, Asymptotic Behavior of Dissipative Systems, AMS, Providence, RI, 1988.  Google Scholar

[19]

A. Kh. Khanmamedov, Global attractors for wave equations with nonlinear damping and critical exponents, J. Differential Equations, 230 (2006), 702-719. doi: 10.1016/j.jde.2006.06.001.  Google Scholar

[20]

A. Kh. Khanmamedov, Global attractors for von Karman equations with nonlinear interior dissipation, J. Math. Anal. Appl., 318 (2006), 92-101. doi: 10.1016/j.jmaa.2005.05.031.  Google Scholar

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A. Kh. Khanmamedov, Remark on the regularity of the global attractor for the wave equation with nonlinear damping, Nonlinear Anal., 72 (2010), 1993-1999. doi: 10.1016/j.na.2009.09.041.  Google Scholar

[22]

P. S. Landahl, O. H. Soerensen and P. L. Christiansen, Soliton excitations in Josephson tunnel junctions Phys. Rev.B, 25 (1982), 5737-5348. Google Scholar

[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), 16-50. doi: 10.1016/S0022-247X(02)00006-9.  Google Scholar

[24]

O. A. Ladyzhenskaya, Attractors for Semigroups and Evolution Equations, Cambridge University Prss, Cambridge, 1991. doi: 10.1017/CBO9780511569418.  Google Scholar

[25]

J. L. Lions, Quelques Méthodes de Résolution des Problèmes Aux Limites N-linéaires, Dunod, Paris, 1969.  Google Scholar

[26]

I. Moise, R. Rosa and X. Wang, Attractors for noncompact nonautonomous systems via energy equations, Discrete Contin. Dyn. Syst., 10 (2004), 473-496. doi: 10.3934/dcds.2004.10.473.  Google Scholar

[27]

M. Nakao, Global attractors for nonlinear wave equations with nonlinear dissipative terms, J. Differential Equations, 227 (2006), 204-229. doi: 10.1016/j.jde.2005.09.013.  Google Scholar

[28]

F. Di Plinio, G. S.Duane and R. Temam, Time dependent attractor for the oscillon equation, Discrete Contin. Dyn. Syst., 29 (2011), 141-167. doi: 10.3934/dcds.2011.29.141.  Google Scholar

[29]

V. Pata and S. Zelik, Smooth attractors for strongly damped wave equations, Nonlinearity, 19 (2006), 1495-1506. doi: 10.1088/0951-7715/19/7/001.  Google Scholar

[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), 177-182.  Google Scholar

[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), 185-192. Google Scholar

[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), 2645-2665. doi: 10.1088/0951-7715/19/11/008.  Google Scholar

[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), 427-443. doi: 10.1016/j.jde.2005.09.010.  Google Scholar

[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), 293-318. doi: 10.1137/060663805.  Google Scholar

[35]

R. Temam, Infinite-dimensional Dynamical Systems in Mechanics and Physics, Second edition, Applied Mathematical Sciences, 68, Springer-Verlag, New York, 1997. doi: 10.1007/978-1-4612-0645-3.  Google Scholar

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.  Google Scholar

[2]

V. V. Chepyzhov and M. I. Vishik, Attractors of nonautonomous dynamical systems and their dimension, J. Math. Pures Appl., 73 (1994), 279-333.  Google Scholar

[3]

V. V. Chepyzhov and M. I. Vishik, Attractors for Equations of Mathematical Physics, Colloquium Publications, vol. 49, American Mathematical Society, Providence, RI, 2002.  Google Scholar

[4]

V. V. Chepyzhov, M. Conti and V. Pata, A minimal approach to the theory of global attractor, Discrete Contin. Dyn. Syst., 32 (2012), 2079-2088. doi: 10.3934/dcds.2012.32.2079.  Google Scholar

[5]

I. D. Chueshov, Introduction to the Theory of Infinite-Dimensional Dissipative Systems, Acta Scientific Publishing House, Kharkiv, 2002. Google Scholar

[6]

I. Chueshov and I. Lasiecka, Attractors for second-order evolution equations with a nonlinear damping, J. Dynam. Differential Equations, 16 (2004), 469-512. doi: 10.1007/s10884-004-4289-x.  Google Scholar

[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, 153-192. Amer. Math. Soc., Providence, RI, 2007. doi: 10.1090/conm/426/08188.  Google Scholar

[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), 459-509.  Google Scholar

[9]

I. Chueshov and I. Lasiecka, Long-time behavior of second order evolution equations with nonlinear damping, Mem. Amer. Math. Soc., 195 (2008), viii+183 pp. doi: 10.1090/memo/0912.  Google Scholar

[10]

M. Conti, V. Pata and R. Temam, Attractors for processes on time-dependent space. Application to Wave equation, J. Differential Equations, 255 (2013), 1254-1277. doi: 10.1016/j.jde.2013.05.013.  Google Scholar

[11]

M. Conti and V. Pata, Asymptotic structure of the attractor for processes on time-dependent spaces, Nonlinear Analysis RWA, 19 (2014), 1-10. doi: 10.1016/j.nonrwa.2014.02.002.  Google Scholar

[12]

M. Conti and V. Pata, On the time-dependent cattaneo law in space dimension one, Applied Mathematic and Computation, 259 (2015), 32-44. doi: 10.1016/j.amc.2015.02.039.  Google Scholar

[13]

H. Crauel and F. Flandoli, Attractors for random dynamical systems, Probab. Theory Related Fields, 100 (1994), 365-393. doi: 10.1007/BF01193705.  Google Scholar

[14]

H. Crauel, A. Debussche and F. Flandoli, Random attractors, J. Dynam. Differential Equations, 9 (1997), 307-341. doi: 10.1007/BF02219225.  Google Scholar

[15]

E. Feireisl, Attractors for wave equations with nonlinear dissipation and critical exponent, C. R. Acad. Sci. Paris, 315 (1992), 551-555.  Google Scholar

[16]

E. Feireisl and E. Zuazua, Global attractors for semilinear wave equations with locally distributed nonlinear damping and critical exponent, Commun. PDE., 18 (1993), 1539-1555. doi: 10.1080/03605309308820985.  Google Scholar

[17]

E. Feireisl, Global attractors for damped wave equations with supercritical exponent, J. Differential Equations, 116 (1995), 431-447. doi: 10.1006/jdeq.1995.1042.  Google Scholar

[18]

J. K. Hale, Asymptotic Behavior of Dissipative Systems, AMS, Providence, RI, 1988.  Google Scholar

[19]

A. Kh. Khanmamedov, Global attractors for wave equations with nonlinear damping and critical exponents, J. Differential Equations, 230 (2006), 702-719. doi: 10.1016/j.jde.2006.06.001.  Google Scholar

[20]

A. Kh. Khanmamedov, Global attractors for von Karman equations with nonlinear interior dissipation, J. Math. Anal. Appl., 318 (2006), 92-101. doi: 10.1016/j.jmaa.2005.05.031.  Google Scholar

[21]

A. Kh. Khanmamedov, Remark on the regularity of the global attractor for the wave equation with nonlinear damping, Nonlinear Anal., 72 (2010), 1993-1999. doi: 10.1016/j.na.2009.09.041.  Google Scholar

[22]

P. S. Landahl, O. H. Soerensen and P. L. Christiansen, Soliton excitations in Josephson tunnel junctions Phys. Rev.B, 25 (1982), 5737-5348. Google Scholar

[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), 16-50. doi: 10.1016/S0022-247X(02)00006-9.  Google Scholar

[24]

O. A. Ladyzhenskaya, Attractors for Semigroups and Evolution Equations, Cambridge University Prss, Cambridge, 1991. doi: 10.1017/CBO9780511569418.  Google Scholar

[25]

J. L. Lions, Quelques Méthodes de Résolution des Problèmes Aux Limites N-linéaires, Dunod, Paris, 1969.  Google Scholar

[26]

I. Moise, R. Rosa and X. Wang, Attractors for noncompact nonautonomous systems via energy equations, Discrete Contin. Dyn. Syst., 10 (2004), 473-496. doi: 10.3934/dcds.2004.10.473.  Google Scholar

[27]

M. Nakao, Global attractors for nonlinear wave equations with nonlinear dissipative terms, J. Differential Equations, 227 (2006), 204-229. doi: 10.1016/j.jde.2005.09.013.  Google Scholar

[28]

F. Di Plinio, G. S.Duane and R. Temam, Time dependent attractor for the oscillon equation, Discrete Contin. Dyn. Syst., 29 (2011), 141-167. doi: 10.3934/dcds.2011.29.141.  Google Scholar

[29]

V. Pata and S. Zelik, Smooth attractors for strongly damped wave equations, Nonlinearity, 19 (2006), 1495-1506. doi: 10.1088/0951-7715/19/7/001.  Google Scholar

[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), 177-182.  Google Scholar

[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), 185-192. Google Scholar

[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), 2645-2665. doi: 10.1088/0951-7715/19/11/008.  Google Scholar

[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), 427-443. doi: 10.1016/j.jde.2005.09.010.  Google Scholar

[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), 293-318. doi: 10.1137/060663805.  Google Scholar

[35]

R. Temam, Infinite-dimensional Dynamical Systems in Mechanics and Physics, Second edition, Applied Mathematical Sciences, 68, Springer-Verlag, New York, 1997. doi: 10.1007/978-1-4612-0645-3.  Google Scholar

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