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August  2019, 24(8): 4117-4143. doi: 10.3934/dcdsb.2019053

Attractors for A sup-cubic weakly damped wave equation in $ \mathbb{R}^{3} $

Xinyu Mei 1, and  Chunyou Sun 1,,

1. 

School of Mathematical and Statistics, Lanzhou University, Lanzhou, Gansu, China

Corresponding author: sunchy@lzu.edu.cn

Received  May 2018 Revised  September 2018 Published  February 2019

Fund Project: This work was partly supported by the NSFC (Grants No. 11471148, 11522109) and lzujbky-2017-160.

In this paper, the dynamical behavior of weakly damped wave equations with a sup-cubic nonlinearity is considered in locally uniform spaces. We first prove the global well-posedness of the Shatah-Struwe solutions, then we obtain 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) $-global attractor for the Shatah-Struwe solutions semigroup of this equation. The results are crucially based on the recent extension of Strichartz estimates to the case of bounded domains.

Keywords: Wave equation, locally uniform spaces, global attractor, Shatah-Struwe solution, contractive function.
Mathematics Subject Classification: Primary: 35B41, 35Q55, 35Q56, 37L40, 76F20.
Citation: Xinyu Mei, Chunyou Sun. Attractors for A sup-cubic weakly damped wave equation in $ \mathbb{R}^{3} $. Discrete & Continuous Dynamical Systems - B, 2019, 24 (8) : 4117-4143. doi: 10.3934/dcdsb.2019053
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

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A. V. Babin and M. I. Vishik, Attractor of partial differential evolution equations in an unbounded domain, Proc. Roy. Soc. Edinburgh Sect. A, 116 (1990), 221-243.  doi: 10.1017/S0308210500031498.  Google Scholar

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

[6]

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

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M. D. Blair, H. F. Smith and C. D. Sogge, Strichartz estimates for the wave equation on manifolds with boundary, Ann. Inst. H. Poincar$\acute{e}$ Anal. Non Lin$\acute{e}$aire, 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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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

[12]

J. W. Cholewa and T. Dlotko, Cauchy problems in weighted Lebesgue spaces, Czechoslovak Math. J., 54 (2004), 991-1013.  doi: 10.1007/s10587-004-6447-z.  Google Scholar

[13]

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

[14]

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

[15]

I. Chueshov and I. Lasiecka, Long-Time Behavior of Second Order Evolution Equations with Nonlinear Damping, Mem. Amer. Math. Soc., vol. 195, (Providence, RI: American Mathematical Society), 2008. doi: 10.1090/memo/0912.  Google Scholar

[16]

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

[17]

E. Feireisl, P. Laurençot, F. Simondon and H. Tour$\acute{e}$, Compact attractors for reaction-diffusion equations in $\mathbb{R}^{N}$, C. R. Acad. Sci. Paris S$\acute{e}$r. I Math., 319 (1994), 147–151.  Google Scholar

[18]

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

[19]

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

[20]

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

[21]

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

[22]

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

[23]

J. L. Lions, Quelques M$\acute{e}$thodes de R$\acute{e}$solution des Probl$\grave{e}$mes aux Limites Nonlin$\acute{e}$aires, Dunod, Paris, 1969.  Google Scholar

[24]

S. Merino, On the existence of the compact global attractor for semilinear reaction diffusion systems on $\mathbb{R}^{N}$, J. Differential Equations, 132 (1996), 87-106.  doi: 10.1006/jdeq.1996.0172.  Google Scholar

[25]

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

[26]

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

[27]

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

[28]

A. Miranville and S. V. Zelik, Attractors for dissipative partial differential equations in bounded and unbounded domains, Handbook of Differential Equations: Evolutionary Equations, vol. Ⅳ, Elsevier, Amsterdam, (2008), 103–200. doi: 10.1016/S1874-5717(08)00003-0.  Google Scholar

[29]

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

[30]

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

[31]

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

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

[34]

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

[35]

M. I. Vishik and S. V. Zelik, A trajectory attractor of a nonlinear elliptic system in a cylindrical domain, Mat. Sb., 187 (1996), 21-56.  doi: 10.1070/SM1996v187n12ABEH000177.  Google Scholar

[36]

M. H. Yang and C. Y. Sun, Dynamics of strongly damped wave equations in locally uniform spaces: attractors and asymptotic regularity, Trans. Amer. Math. Soc., 361 (2009), 1069-1101.  doi: 10.1090/S0002-9947-08-04680-1.  Google Scholar

[37]

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, Attractor of partial differential evolution equations in an unbounded domain, Proc. Roy. Soc. Edinburgh Sect. A, 116 (1990), 221-243.  doi: 10.1017/S0308210500031498.  Google Scholar

[5]

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

[6]

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

[7]

M. D. Blair, H. F. Smith and C. D. Sogge, Strichartz estimates for the wave equation on manifolds with boundary, Ann. Inst. H. Poincar$\acute{e}$ Anal. Non Lin$\acute{e}$aire, 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]

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

[11]

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

[12]

J. W. Cholewa and T. Dlotko, Cauchy problems in weighted Lebesgue spaces, Czechoslovak Math. J., 54 (2004), 991-1013.  doi: 10.1007/s10587-004-6447-z.  Google Scholar

[13]

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

[14]

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

[15]

I. Chueshov and I. Lasiecka, Long-Time Behavior of Second Order Evolution Equations with Nonlinear Damping, Mem. Amer. Math. Soc., vol. 195, (Providence, RI: American Mathematical Society), 2008. doi: 10.1090/memo/0912.  Google Scholar

[16]

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

[17]

E. Feireisl, P. Laurençot, F. Simondon and H. Tour$\acute{e}$, Compact attractors for reaction-diffusion equations in $\mathbb{R}^{N}$, C. R. Acad. Sci. Paris S$\acute{e}$r. I Math., 319 (1994), 147–151.  Google Scholar

[18]

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

[19]

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

[20]

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

[21]

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

[22]

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

[23]

J. L. Lions, Quelques M$\acute{e}$thodes de R$\acute{e}$solution des Probl$\grave{e}$mes aux Limites Nonlin$\acute{e}$aires, Dunod, Paris, 1969.  Google Scholar

[24]

S. Merino, On the existence of the compact global attractor for semilinear reaction diffusion systems on $\mathbb{R}^{N}$, J. Differential Equations, 132 (1996), 87-106.  doi: 10.1006/jdeq.1996.0172.  Google Scholar

[25]

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

[26]

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

[27]

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

[28]

A. Miranville and S. V. Zelik, Attractors for dissipative partial differential equations in bounded and unbounded domains, Handbook of Differential Equations: Evolutionary Equations, vol. Ⅳ, Elsevier, Amsterdam, (2008), 103–200. doi: 10.1016/S1874-5717(08)00003-0.  Google Scholar

[29]

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

[30]

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

[31]

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

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

[34]

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

[35]

M. I. Vishik and S. V. Zelik, A trajectory attractor of a nonlinear elliptic system in a cylindrical domain, Mat. Sb., 187 (1996), 21-56.  doi: 10.1070/SM1996v187n12ABEH000177.  Google Scholar

[36]

M. H. Yang and C. Y. Sun, Dynamics of strongly damped wave equations in locally uniform spaces: attractors and asymptotic regularity, Trans. Amer. Math. Soc., 361 (2009), 1069-1101.  doi: 10.1090/S0002-9947-08-04680-1.  Google Scholar

[37]

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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