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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 |
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.
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. |
[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. |
[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. |
[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. 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. |
[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. 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. |
[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. 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. |
[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. 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. |
[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$\acute{a}$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. |
[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. 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. |
[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. 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. |
[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. 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. |
[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. |
[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. |
[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. 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. |
[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. 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. |
[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. 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. |
[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. 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. |
[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$\acute{a}$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. |
[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. 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. |
[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. 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. |
[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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