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

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

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  February 2021 Early access  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.

Citation: Xinyu Mei, Yangmin Xiong, Chunyou Sun. Pullback attractor for a weakly damped wave equation with sup-cubic nonlinearity. Discrete and Continuous Dynamical Systems, 2021, 41 (2) : 569-600. doi: 10.3934/dcds.2020270
References:
[1]

J. ArrietaA. 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.

[2]

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

[3]

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

[4]

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

[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.

[6]

P. W. BatesK. 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.

[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.

[8]

N. BurqG. 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.

[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.

[10]

T. CaraballoG. 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.

[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.

[12]

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

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[20]

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

[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.

[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.

[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.

[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.

[25]

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

[26]

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

[27]

S. S. LuH. 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.

[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.

[29]

M. Mich$\acute{a}$lekD. 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.

[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.

[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.

[32]

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

[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. 

[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.

[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.

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

[38]

C. Y. SunD. 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.

[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.

[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.

[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.

[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.

[43]

X. H. WangK. 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.

[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.

show all references

References:
[1]

J. ArrietaA. 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.

[2]

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

[3]

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

[4]

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

[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.

[6]

P. W. BatesK. 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.

[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.

[8]

N. BurqG. 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.

[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.

[10]

T. CaraballoG. 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.

[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.

[12]

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

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[20]

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

[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.

[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.

[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.

[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.

[25]

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

[26]

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

[27]

S. S. LuH. 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.

[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.

[29]

M. Mich$\acute{a}$lekD. 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.

[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.

[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.

[32]

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

[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. 

[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.

[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.

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

[38]

C. Y. SunD. 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.

[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.

[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.

[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.

[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.

[43]

X. H. WangK. 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.

[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.

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