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February  2021, 41(2): 569-600. doi: 10.3934/dcds.2020270

Pullback attractor for a weakly damped wave equation with sup-cubic nonlinearity

Xinyu Mei 1, , Yangmin Xiong 2, and  Chunyou Sun 2,,

1. 

School of Mathematics and Statistics, Shenzhen University, Shenzhen 518060, Guangdong, China, College of Physics and Optoelectronic Engineering, Shenzhen University, Shenzhen 518060, Guangdong, China
2. 

School of Mathematics and Statistics, Lanzhou University, Lanzhou 730000, Gansu, China

Corresponding author: Chunyou Sun, sunchy@lzu.edu.cn

Received  January 2020 Revised  May 2020 Published  July 2020

Fund Project: This work was partly supported by the NSFC Grants 11471148, 11522109 and 11871169

In this paper, the non-autonomous dynamical behavior of weakly damped wave equation with a sup-cubic nonlinearity is considered in locally uniform spaces. We first prove the global well-posedness of the Shatah-Struwe solutions, then establish the existence of the $ \big(H_{lu}^{1}(\mathbb{R}^{3})\times L_{lu}^{2}(\mathbb{R}^{3}),H_{\rho}^{1}(\mathbb{R}^{3})\times L_{\rho}^{2}(\mathbb{R}^{3})\big) $-pullback attractor for the Shatah-Struwe solutions process of this equation. The results are based on the recent extension of Strichartz estimates for the bounded domains.

Keywords: Wave equation, Locally uniform spaces, Shatah-Struwe solution, Pullback attractor.
Mathematics Subject Classification: Primary:35B41, 35Q55, 35Q56, 37L40, 76F20.
Citation: 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
References:
[1]

J. Arrieta, A. N. Carvalho and J. K. Hale, A damped hyperbolic equation with critical exponent, Comm. Partial Differential Equations, 17 (1992), 841-866.  doi: 10.1080/03605309208820866.  Google Scholar

[2]

J. Arrieta, J. W. Cholewa, T. Dlotko and A. Rodríguez-Bernal, Linear parabolic equations in locally spaces, Math. Models Methods Appl. Sci., 14 (2004), 253-293.  doi: 10.1142/S0218202504003234.  Google Scholar

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J. Arrieta, J. W. Cholewa, T. Dlotko and A. Rodríguez-Bernal, Dissipative parabolic equations in locally uniform spaces, Math. Nachr., 280 (2007), 1643-1663.  doi: 10.1002/mana.200510569.  Google Scholar

[4]

A. V. Babin and M. I. Vishik, Attractors of Evolutionary Equations, Stud. Math. Appl., vol. 25, North-Holland, Amsterdam, 1992.  Google Scholar

[5]

J. M. Ball, Global attractors for damped semilinear wave equations, Discrete Contin. Dyn. Syst., 10 (2004), 31-52.  doi: 10.3934/dcds.2004.10.31.  Google Scholar

[6]

P. W. Bates, K. N. Lu and B. X. Wang, Random attractors for stochastic reaction-diffusion equations on unbounded domains, J. Differential Equations, 246 (2009), 845-869.  doi: 10.1016/j.jde.2008.05.017.  Google Scholar

[7]

M. D. Blair, H. F. Smith and C. D. Sogge, Strichartz estimates for the wave equation on manifolds with boundary,, Ann. Inst. H. Poincaré Anal. Non Linéaire, 26 (2009), 1817-1829. doi: 10.1016/j.anihpc.2008.12.004.  Google Scholar

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N. Burq, G. Lebeau and F. Planchon, Global existence for energy critical waves in 3-D domains, J. Amer. Math. Soc., 21 (2008), 831-845.  doi: 10.1090/S0894-0347-08-00596-1.  Google Scholar

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N. Burq and F. Planchon, Global existence for energy critical waves in 3-D domains: Neumann boundary conditions, Amer. J. Math., 131 (2009), 1715-1742.  doi: 10.1353/ajm.0.0084.  Google Scholar

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T. Caraballo, G. Ƚukaszewicz and J. Real, Pullback attractors for asymptotically compact nonautonomous dynamical systems, Nonlinear Anal., 64 (2006), 484-498.  doi: 10.1016/j.na.2005.03.111.  Google Scholar

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A. Carvalho, J. Langa and J. Robinson, Attractors for Infinite-Dimensional Non-autonomous Dynamical Systems, Applied Mathematical Sciences, vol. 182, Springer, New York, 2013. doi: 10.1007/978-1-4614-4581-4.  Google Scholar

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V. V. Chepyzhov and M. I. Vishik, Attractors for Equations of Mathematical Physics, American Mathematical Society, Providence, Rhode Island, 2002.  Google Scholar

[13]

J. W. Cholewa and A. Rodríguez-Bernal, Linear higher order parabolic problems in locally uniform Lebesgue's spaces, J. Math. Anal. Appl., 449 (2017), 1-45.  doi: 10.1016/j.jmaa.2016.11.085.  Google Scholar

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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

[15]

A. N. Carvalho and T. Dlotko, Partly dissipative systems in uniformly local spaces, Colloq. Math., 100 (2004), 221-242.  doi: 10.4064/cm100-2-6.  Google Scholar

[16]

J. W. Cholewa and T. Dlotko, Hyperbolic equations in uniform spaces, Bull. Pol. Acad. Sci. Math., 52 (2004), 249-263.  doi: 10.4064/ba52-3-5.  Google Scholar

[17]

J. W. Cholewa and T. Dlotko, Strongly damped wave equation in uniform spaces, Nonlinear Anal., 64 (2006), 174-187.  doi: 10.1016/j.na.2005.06.021.  Google Scholar

[18]

M. A. Efendiev and S. V. Zelik, The attractor for a nonlinear reaction-diffusion system in an unbounded domain, Comm. Pure Appl. Math., 54 (2001), 625-688.  doi: 10.1002/cpa.1011.  Google Scholar

[19]

E. Feireisl, Asymptotic behaviour and attractors for a semilinear damped wave equation with supercritical exponent, Proc. Roy. Soc. Edinburgh Sect. A, 125 (1995), 1051-1062.  doi: 10.1017/S0308210500022630.  Google Scholar

[20]

E. Feireisl, Bounded, locally compact global attractors for semilinear damped wave equations on $\mathbb{R}^{N}$, Differential Integral Equations, 9 (1996), 1147-1156.   Google Scholar

[21]

M. Grillakis, Regularity and asymptotic behaviour of the wave equation with a critical nonlinearity, Ann. of Math., 132 (1990), 485-509.  doi: 10.2307/1971427.  Google Scholar

[22]

V. Kalantarov, A. Savostianov and S. V. Zelik, Attractors for damped quintic wave equations in bounded domains,, Ann. Henri Poincaré, 17 (2016), 2555-2584. doi: 10.1007/s00023-016-0480-y.  Google Scholar

[23]

L. Kapitanski, Minimal compact global attractor for a damped semilinear wave equation, Comm. Partial Differential Equations, 20 (1995), 1303-1323.  doi: 10.1080/03605309508821133.  Google Scholar

[24]

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

[25]

P. E. Kloeden, Pullback attractors in nonautonomous difference equations, J. Differ. Equations Appl., 6 (2000), 33-52.  doi: 10.1080/10236190008808212.  Google Scholar

[26]

J. L. Lions, Quelques Méthodes de Résolution des Problèmes aux Limites Nonlinéaires, Dunod, Paris, 1969.  Google Scholar

[27]

S. S. Lu, H. Q. Wu and C. K. Zhong, Attractors for nonautonomous 2D Navier-Stokes equations with normal external forces, Discrete Contin. Dyn. Syst., 13 (2005), 701-719.  doi: 10.3934/dcds.2005.13.701.  Google Scholar

[28]

X. Y. Mei and C. Y. Sun, Attractors for a sup-cubic weakly damped wave equation in $\mathbb{R}^{3}$, Discrete Contin. Dyn. Syst. Ser. B, 24 (2019), 4117-4143.  doi: 10.3934/dcdsb.2019053.  Google Scholar

[29]

M. Mich$\acute{a}$lek, D. Pra$\check{z}\acute{a}$k and J. Slavík, Semilinear damped wave equation in locally uniform spaces, Commun. Pure Appl. Anal., 16 (2017), 1673-1695.  doi: 10.3934/cpaa.2017080.  Google Scholar

[30]

A. Mielke and G. Schneider, Attractors for modulation equations unbounded domains-existence and comparison, Nonlinearity, 8 (1995), 743-768.  doi: 10.1088/0951-7715/8/5/006.  Google Scholar

[31]

A. Mielke, The complex Ginzburg-Landau equation on large and unbounded domains: Sharper bounds and attractors, Nonlinearity, 10 (1997), 199-222.  doi: 10.1088/0951-7715/10/1/014.  Google Scholar

[32]

A. Savostianov and S. V. Zelik, Recent progress in attractors for quintic wave equations, Math. Bohem., 139 (2014), 657-665.   Google Scholar

[33]

A. Savostianov, Strichartz estimates and smooth attractors for a sub-quintic wave equation with fractional damping in bounded domains, Adv. Differential Equations, 20 (2015), 495-530.   Google Scholar

[34]

A. Savostianov and S. V. Zelik, Uniform attractors for measure-driven quintic wave equation, Uspekhi Mat. Nauk, 75 (2020), 61-132.  doi: 10.4213/rm9932.  Google Scholar

[35]

J. Shatah and M. Struwe, Well-posedness in the energy space for semilinear wave equations with critical growth, Internat. Math. Res. Notices, 1994 (1994), 303-309.  doi: 10.1155/S1073792894000346.  Google Scholar

[36] C. D. Sogge, Lectures on Non-linear Wave Equations, 2nd edition, International Press, Boston, 2008.   Google Scholar
[37]

R. Strichartz, Restrictions of Fourier transforms to quadratic surfaces and decay of solutions of wave equations, Duke Math. J., 44 (1977), 705-714.  doi: 10.1215/S0012-7094-77-04430-1.  Google Scholar

[38]

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

[39]

T. Tao, Nonlinear Dispersive Equations: Local and Global Analysis, CBMS Regi. Conf. Seri. Math., Providence, vol. 106. Washington, 2006. doi: 10.1090/cbms/106.  Google Scholar

[40]

R. Temam, Infinite-Dimensional Dynamical Systems in Mechanics and Physics, 2nd edition. App. Math. Sci., vol. 68. Springer-Verlag, New York, 1997. doi: 10.1007/978-1-4612-0645-3.  Google Scholar

[41]

B. X. Wang, Sufficient and necessary criteria for existence of pullback attractors for non-compact random dynamical systems, J. Differential Equations, 253 (2012), 1544-1583.  doi: 10.1016/j.jde.2012.05.015.  Google Scholar

[42]

B. X. Wang, Multivalued non-autonomous random dynamical systems for wave equations without uniqueness, Discrete Contin. Dyn. Syst. B, 22 (2017), 2011-2051.  doi: 10.3934/dcdsb.2017119.  Google Scholar

[43]

X. H. Wang, K. N. Lu and B. X. Wang, Wong-Zakai approximations and attractors for stochastic reaction-diffusion equations on unbounded domains, J. Differential Equations, 264 (2018), 378-424.  doi: 10.1016/j.jde.2017.09.006.  Google Scholar

[44]

S. V. Zelik, The attractor for a nonlinear hyperbolic equation in the unbounded domain, Discrete Contin. Dyn. Syst., 7 (2001), 593-641.  doi: 10.3934/dcds.2001.7.593.  Google Scholar

show all references

References:
[1]

J. Arrieta, A. N. Carvalho and J. K. Hale, A damped hyperbolic equation with critical exponent, Comm. Partial Differential Equations, 17 (1992), 841-866.  doi: 10.1080/03605309208820866.  Google Scholar

[2]

J. Arrieta, J. W. Cholewa, T. Dlotko and A. Rodríguez-Bernal, Linear parabolic equations in locally spaces, Math. Models Methods Appl. Sci., 14 (2004), 253-293.  doi: 10.1142/S0218202504003234.  Google Scholar

[3]

J. Arrieta, J. W. Cholewa, T. Dlotko and A. Rodríguez-Bernal, Dissipative parabolic equations in locally uniform spaces, Math. Nachr., 280 (2007), 1643-1663.  doi: 10.1002/mana.200510569.  Google Scholar

[4]

A. V. Babin and M. I. Vishik, Attractors of Evolutionary Equations, Stud. Math. Appl., vol. 25, North-Holland, Amsterdam, 1992.  Google Scholar

[5]

J. M. Ball, Global attractors for damped semilinear wave equations, Discrete Contin. Dyn. Syst., 10 (2004), 31-52.  doi: 10.3934/dcds.2004.10.31.  Google Scholar

[6]

P. W. Bates, K. N. Lu and B. X. Wang, Random attractors for stochastic reaction-diffusion equations on unbounded domains, J. Differential Equations, 246 (2009), 845-869.  doi: 10.1016/j.jde.2008.05.017.  Google Scholar

[7]

M. D. Blair, H. F. Smith and C. D. Sogge, Strichartz estimates for the wave equation on manifolds with boundary,, Ann. Inst. H. Poincaré Anal. Non Linéaire, 26 (2009), 1817-1829. doi: 10.1016/j.anihpc.2008.12.004.  Google Scholar

[8]

N. Burq, G. Lebeau and F. Planchon, Global existence for energy critical waves in 3-D domains, J. Amer. Math. Soc., 21 (2008), 831-845.  doi: 10.1090/S0894-0347-08-00596-1.  Google Scholar

[9]

N. Burq and F. Planchon, Global existence for energy critical waves in 3-D domains: Neumann boundary conditions, Amer. J. Math., 131 (2009), 1715-1742.  doi: 10.1353/ajm.0.0084.  Google Scholar

[10]

T. Caraballo, G. Ƚukaszewicz and J. Real, Pullback attractors for asymptotically compact nonautonomous dynamical systems, Nonlinear Anal., 64 (2006), 484-498.  doi: 10.1016/j.na.2005.03.111.  Google Scholar

[11]

A. Carvalho, J. Langa and J. Robinson, Attractors for Infinite-Dimensional Non-autonomous Dynamical Systems, Applied Mathematical Sciences, vol. 182, Springer, New York, 2013. doi: 10.1007/978-1-4614-4581-4.  Google Scholar

[12]

V. V. Chepyzhov and M. I. Vishik, Attractors for Equations of Mathematical Physics, American Mathematical Society, Providence, Rhode Island, 2002.  Google Scholar

[13]

J. W. Cholewa and A. Rodríguez-Bernal, Linear higher order parabolic problems in locally uniform Lebesgue's spaces, J. Math. Anal. Appl., 449 (2017), 1-45.  doi: 10.1016/j.jmaa.2016.11.085.  Google Scholar

[14]

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

[15]

A. N. Carvalho and T. Dlotko, Partly dissipative systems in uniformly local spaces, Colloq. Math., 100 (2004), 221-242.  doi: 10.4064/cm100-2-6.  Google Scholar

[16]

J. W. Cholewa and T. Dlotko, Hyperbolic equations in uniform spaces, Bull. Pol. Acad. Sci. Math., 52 (2004), 249-263.  doi: 10.4064/ba52-3-5.  Google Scholar

[17]

J. W. Cholewa and T. Dlotko, Strongly damped wave equation in uniform spaces, Nonlinear Anal., 64 (2006), 174-187.  doi: 10.1016/j.na.2005.06.021.  Google Scholar

[18]

M. A. Efendiev and S. V. Zelik, The attractor for a nonlinear reaction-diffusion system in an unbounded domain, Comm. Pure Appl. Math., 54 (2001), 625-688.  doi: 10.1002/cpa.1011.  Google Scholar

[19]

E. Feireisl, Asymptotic behaviour and attractors for a semilinear damped wave equation with supercritical exponent, Proc. Roy. Soc. Edinburgh Sect. A, 125 (1995), 1051-1062.  doi: 10.1017/S0308210500022630.  Google Scholar

[20]

E. Feireisl, Bounded, locally compact global attractors for semilinear damped wave equations on $\mathbb{R}^{N}$, Differential Integral Equations, 9 (1996), 1147-1156.   Google Scholar

[21]

M. Grillakis, Regularity and asymptotic behaviour of the wave equation with a critical nonlinearity, Ann. of Math., 132 (1990), 485-509.  doi: 10.2307/1971427.  Google Scholar

[22]

V. Kalantarov, A. Savostianov and S. V. Zelik, Attractors for damped quintic wave equations in bounded domains,, Ann. Henri Poincaré, 17 (2016), 2555-2584. doi: 10.1007/s00023-016-0480-y.  Google Scholar

[23]

L. Kapitanski, Minimal compact global attractor for a damped semilinear wave equation, Comm. Partial Differential Equations, 20 (1995), 1303-1323.  doi: 10.1080/03605309508821133.  Google Scholar

[24]

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

[25]

P. E. Kloeden, Pullback attractors in nonautonomous difference equations, J. Differ. Equations Appl., 6 (2000), 33-52.  doi: 10.1080/10236190008808212.  Google Scholar

[26]

J. L. Lions, Quelques Méthodes de Résolution des Problèmes aux Limites Nonlinéaires, Dunod, Paris, 1969.  Google Scholar

[27]

S. S. Lu, H. Q. Wu and C. K. Zhong, Attractors for nonautonomous 2D Navier-Stokes equations with normal external forces, Discrete Contin. Dyn. Syst., 13 (2005), 701-719.  doi: 10.3934/dcds.2005.13.701.  Google Scholar

[28]

X. Y. Mei and C. Y. Sun, Attractors for a sup-cubic weakly damped wave equation in $\mathbb{R}^{3}$, Discrete Contin. Dyn. Syst. Ser. B, 24 (2019), 4117-4143.  doi: 10.3934/dcdsb.2019053.  Google Scholar

[29]

M. Mich$\acute{a}$lek, D. Pra$\check{z}\acute{a}$k and J. Slavík, Semilinear damped wave equation in locally uniform spaces, Commun. Pure Appl. Anal., 16 (2017), 1673-1695.  doi: 10.3934/cpaa.2017080.  Google Scholar

[30]

A. Mielke and G. Schneider, Attractors for modulation equations unbounded domains-existence and comparison, Nonlinearity, 8 (1995), 743-768.  doi: 10.1088/0951-7715/8/5/006.  Google Scholar

[31]

A. Mielke, The complex Ginzburg-Landau equation on large and unbounded domains: Sharper bounds and attractors, Nonlinearity, 10 (1997), 199-222.  doi: 10.1088/0951-7715/10/1/014.  Google Scholar

[32]

A. Savostianov and S. V. Zelik, Recent progress in attractors for quintic wave equations, Math. Bohem., 139 (2014), 657-665.   Google Scholar

[33]

A. Savostianov, Strichartz estimates and smooth attractors for a sub-quintic wave equation with fractional damping in bounded domains, Adv. Differential Equations, 20 (2015), 495-530.   Google Scholar

[34]

A. Savostianov and S. V. Zelik, Uniform attractors for measure-driven quintic wave equation, Uspekhi Mat. Nauk, 75 (2020), 61-132.  doi: 10.4213/rm9932.  Google Scholar

[35]

J. Shatah and M. Struwe, Well-posedness in the energy space for semilinear wave equations with critical growth, Internat. Math. Res. Notices, 1994 (1994), 303-309.  doi: 10.1155/S1073792894000346.  Google Scholar

[36] C. D. Sogge, Lectures on Non-linear Wave Equations, 2nd edition, International Press, Boston, 2008.   Google Scholar
[37]

R. Strichartz, Restrictions of Fourier transforms to quadratic surfaces and decay of solutions of wave equations, Duke Math. J., 44 (1977), 705-714.  doi: 10.1215/S0012-7094-77-04430-1.  Google Scholar

[38]

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

[39]

T. Tao, Nonlinear Dispersive Equations: Local and Global Analysis, CBMS Regi. Conf. Seri. Math., Providence, vol. 106. Washington, 2006. doi: 10.1090/cbms/106.  Google Scholar

[40]

R. Temam, Infinite-Dimensional Dynamical Systems in Mechanics and Physics, 2nd edition. App. Math. Sci., vol. 68. Springer-Verlag, New York, 1997. doi: 10.1007/978-1-4612-0645-3.  Google Scholar

[41]

B. X. Wang, Sufficient and necessary criteria for existence of pullback attractors for non-compact random dynamical systems, J. Differential Equations, 253 (2012), 1544-1583.  doi: 10.1016/j.jde.2012.05.015.  Google Scholar

[42]

B. X. Wang, Multivalued non-autonomous random dynamical systems for wave equations without uniqueness, Discrete Contin. Dyn. Syst. B, 22 (2017), 2011-2051.  doi: 10.3934/dcdsb.2017119.  Google Scholar

[43]

X. H. Wang, K. N. Lu and B. X. Wang, Wong-Zakai approximations and attractors for stochastic reaction-diffusion equations on unbounded domains, J. Differential Equations, 264 (2018), 378-424.  doi: 10.1016/j.jde.2017.09.006.  Google Scholar

[44]

S. V. Zelik, The attractor for a nonlinear hyperbolic equation in the unbounded domain, Discrete Contin. Dyn. Syst., 7 (2001), 593-641.  doi: 10.3934/dcds.2001.7.593.  Google Scholar

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