August  2021, 26(8): 4433-4458. doi: 10.3934/dcdsb.2020294

Pullback attractors for a weakly damped wave equation with delays and sup-cubic nonlinearity

1. 

Hunan Province Cooperative Innovation Center for the Construction and, Development of Dongting Lake Ecological Economic Zone & , College of Mathematics and Physics Science, Hunan University of Arts and Science, Changde 415000, China

2. 

School of Mathematics and Statistics, Changsha University of Science and Technology, Changsha 410114, China

3. 

School of Mathematical and Statistics, Shenzhen University, Shenzhen 518060, China

* Corresponding author: Kaixuan Zhu

Received  May 2020 Revised  August 2020 Published  August 2021 Early access  October 2020

Fund Project: This work is supported by the Natural Science Foundation of Hunan Province (Grants No. 2018JJ2416)

In this paper, we consider the weakly damped wave equations with hereditary effects and the nonlinearity $ f $ satisfying sup-cubic growth. Based on the recent extension of the Strichartz estimates to the case of bounded domains, we establish the global well-posedness of the Shatah-Struwe solutions for the non-autonomous case. Then, we prove the existence of the pullback $ \mathcal{D} $-attractors in $ C_{H_{0}^{1}(\Omega)}\times C_{L^{2}(\Omega)} $ for the solutions process $ \{U(t,\tau)\}_{t\geq\tau} $ by applying the idea of contractive functions.

Citation: Kaixuan Zhu, Yongqin Xie, Xinyu Mei. Pullback attractors for a weakly damped wave equation with delays and sup-cubic nonlinearity. Discrete and Continuous Dynamical Systems - B, 2021, 26 (8) : 4433-4458. doi: 10.3934/dcdsb.2020294
References:
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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]

A. V. Babin and M. I. Vishik, Attractors of Evolution Equations, North-Holland, Amsterdam, 1992.

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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.

[4]

M. D. BlairH. 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.

[5]

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.

[6]

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.

[7]

T. CaraballoX. Han and P. E. Kloeden, Nonautonomous chemostats with variable delays, SIAM J. Math. Anal., 47 (2015), 2178-2199.  doi: 10.1137/14099930X.

[8]

T. CaraballoP. E. Kloeden and P. Marín-Rubio, Numerical and finite delay approximations of attractors for logistic differential-integral equations with infinite delay, Discrete Contin. Dyn. Syst., 19 (2007), 177-196.  doi: 10.3934/dcds.2007.19.177.

[9]

T. CaraballoP. E. Kloeden and J. Real, Pullback and forward attractors for a damped wave equation with delays, Stoch. Dyn., 4 (2004), 405-423.  doi: 10.1142/S0219493704001139.

[10]

T. CaraballoG. Ƚukasiewicz and J. Real, Pullback attractors for asymptotically compact non-autonomous dynamical systems, Nonlinear Anal., 64 (2006), 484-498.  doi: 10.1016/j.na.2005.03.111.

[11]

T. CaraballoG. Ƚukaszewica and J. Real, Pullback attractors for non-autonomous 2D-Navier-Stokes equations in some unbounded domains, C. R. Math. Acad. Sci. Paris, 342 (2006), 263-268.  doi: 10.1016/j.crma.2005.12.015.

[12]

T. CaraballoP. Marín-Rubio and J. Valero, Autonomous and non-autonomous attractors for differential equations with delays, J. Differential Equations, 208 (2005), 9-41.  doi: 10.1016/j.jde.2003.09.008.

[13]

T. CaraballoP. Marín-Rubio and J. Valero, Attractors for differential equations with unbounded delays, J. Differential Equations, 239 (2007), 311-342.  doi: 10.1016/j.jde.2007.05.015.

[14]

T. Caraballo and J. Real, Attractors for 2D-Navier-Stokes models with delays, J. Differential Equations, 205 (2004), 271-297.  doi: 10.1016/j.jde.2004.04.012.

[15]

T. CaraballoJ. Real and A. M. Márquez, Three-dimensional system of globally modified Navier-Stokes equations with delay, Internat. J. Bifur. Chaos Appl. Sci. Engrg., 20 (2010), 2869-2883.  doi: 10.1142/S0218127410027428.

[16]

A. N. Carvalho, J. A. Langa and J. C. 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.

[17]

I. Chueshov and I. Lasiecka, Long-time dynamics of von Karman semi-flows with non-linear boundary/interior damping, J. Differential Equations, 233 (2007), 42-86.  doi: 10.1016/j.jde.2006.09.019.

[18]

J. García-Luengo and P. Marín-Rubio, Reaction-diffusion equations with non-autonomous force in $H^{-1}$ and delays under measurability conditions on the driving delay term, J. Math. Anal. Appl., 417 (2014), 80-95.  doi: 10.1016/j.jmaa.2014.03.026.

[19]

J. García-LuengoP. Marín-Rubio and J. Real, Pullback attractors for 2D Navier-Stokes equations with delays and their regularity, Adv. Nonlinear Stud., 13 (2013), 331-357.  doi: 10.1515/ans-2013-0205.

[20]

J. García-LuengoP. Marín-Rubio and J. Real, Some new regularity results of pullback attractors for 2D Navier-Stokes equations with delays, Commun. Pure Appl. Anal., 14 (2015), 1603-1621.  doi: 10.3934/cpaa.2015.14.1603.

[21]

J. García-LuengoP. Marín-Rubio and G. Planas, Attractors for a double time-delayed 2D-Navier-Stokes model, Discrete Contin. Dyn. Syst., 34 (2014), 4085-4105.  doi: 10.3934/dcds.2014.34.4085.

[22]

J. García-LuengoP. Marín-Rubio and J. Real, Regularity of pullback attractors and attraction in $H^{1}$ in arbitrarily large finite intervals for 2D Navier-Stokes equations with infinite delay, Discrete Contin. Dyn. Syst., 34 (2014), 181-201.  doi: 10.3934/dcds.2014.34.181.

[23]

J. K. Hale and S. M. Verduyn Lunel, Introduction to Functional Differential Equations, Springer-Verlag, New York, 1993. doi: 10.1007/978-1-4612-4342-7.

[24]

J. K. Hale, Asymptotic Behavior of Dissipative Systems, AMS, Providence, RI, 1988. doi: 10.1090/surv/025.

[25]

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

[26]

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.

[27]

P. E. Kloeden, Upper semi continuity of attractors of delay differential equations in the delay, Bull. Austral. Math. Soc., 73 (2006), 299-306.  doi: 10.1017/S0004972700038880.

[28]

P. E. Kloeden and T. Lorenz, Pullback incremental attraction, Nonauton. Dyn. Syst., 1 (2014), 53-60.  doi: 10.2478/msds-2013-0004.

[29]

P. E. Kloeden and P. Marín-Rubio, Equi-attraction and the continuous dependence of attractors on time delays, Discrete Contin. Dyn. Syst. Ser. B, 9 (2008), 581-593.  doi: 10.3934/dcdsb.2008.9.581.

[30]

Y. Li and C. Zhong, Pullback attractors for the norm-to-weak continuous process and application to the nonautonomous reaction-diffusion equations, Appl. Math. Comput., 190 (2007), 1020-1029.  doi: 10.1016/j.amc.2006.11.187.

[31]

S. LuH. Wu and C. 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.

[32]

P. Marín-RubioA. M. Márquez-Durán and J. Real, Three dimensional system of globally modified Navier-Stokes equations with infinite delays, Discrete Contin. Dyn. Syst. Ser. B, 14 (2010), 655-673.  doi: 10.3934/dcdsb.2010.14.655.

[33]

P. Marín-RubioA. M. Márquez-Durán and J. Real, Pullback attractors for globally modified Navier-Stokes equations with infinite delays, Discrete Contin. Dyn. Syst., 31 (2011), 779-796.  doi: 10.3934/dcds.2011.31.779.

[34]

F. MengM. Yang and C. Zhong, Attractors for wave equations with nonlinear damping on time-dependent space, Discrete Contin. Dyn. Syst. Ser. B, 21 (2016), 205-225.  doi: 10.3934/dcdsb.2016.21.205.

[35]

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. 

[36]

C. SunD. Cao and J. 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.

[37]

R. Temam, Infinite-Dimensional Dynamical Systems in Mechanics and Physics, Springer-Verlag, New York, 1997. doi: 10.1007/978-1-4612-0645-3.

[38]

Y. Wang, Pullback attractors for a damped wave equation with delays, Stoch. Dyn., 15 (2015), 1550003, 21 pp. doi: 10.1142/S0219493715500033.

[39]

Y. Wang and P. E. Kloeden, The uniform attractor of a multi-valued process generated by reaction-diffusion delay equations on an unbounded domain, Discrete Contin. Dyn. Syst., 34 (2014), 4343-4370.  doi: 10.3934/dcds.2014.34.4343.

[40]

F. Wu and P. E. Kloeden, Mean-square random attractors of stochastic delay differential equations with random delay, Discrete Contin. Dyn. Syst. Ser. B, 18 (2013), 1715-1734.  doi: 10.3934/dcdsb.2013.18.1715.

[41]

Y. XieQ. Li and K. Zhu, Attractors for nonclassical diffusion equations with arbitrary polynomial growth nonlinearity, Nonlinear Anal. Real World Appl., 31 (2016), 23-37.  doi: 10.1016/j.nonrwa.2016.01.004.

[42]

F. ZhouC. Sun and X. Li, Dynamics for the damped wave equations on time-dependent domains, Discrete Contin. Dyn. Syst. Ser. B, 23 (2018), 1645-1674.  doi: 10.3934/dcdsb.2018068.

[43]

K. ZhuY. Xie and F. Zhou, Pullback attractors for a damped semilinear wave equation with delays, Acta Math. Sin. (Engl. Ser.), 34 (2018), 1131-1150.  doi: 10.1007/s10114-018-7420-3.

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]

A. V. Babin and M. I. Vishik, Attractors of Evolution Equations, North-Holland, Amsterdam, 1992.

[3]

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.

[4]

M. D. BlairH. 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.

[5]

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.

[6]

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.

[7]

T. CaraballoX. Han and P. E. Kloeden, Nonautonomous chemostats with variable delays, SIAM J. Math. Anal., 47 (2015), 2178-2199.  doi: 10.1137/14099930X.

[8]

T. CaraballoP. E. Kloeden and P. Marín-Rubio, Numerical and finite delay approximations of attractors for logistic differential-integral equations with infinite delay, Discrete Contin. Dyn. Syst., 19 (2007), 177-196.  doi: 10.3934/dcds.2007.19.177.

[9]

T. CaraballoP. E. Kloeden and J. Real, Pullback and forward attractors for a damped wave equation with delays, Stoch. Dyn., 4 (2004), 405-423.  doi: 10.1142/S0219493704001139.

[10]

T. CaraballoG. Ƚukasiewicz and J. Real, Pullback attractors for asymptotically compact non-autonomous dynamical systems, Nonlinear Anal., 64 (2006), 484-498.  doi: 10.1016/j.na.2005.03.111.

[11]

T. CaraballoG. Ƚukaszewica and J. Real, Pullback attractors for non-autonomous 2D-Navier-Stokes equations in some unbounded domains, C. R. Math. Acad. Sci. Paris, 342 (2006), 263-268.  doi: 10.1016/j.crma.2005.12.015.

[12]

T. CaraballoP. Marín-Rubio and J. Valero, Autonomous and non-autonomous attractors for differential equations with delays, J. Differential Equations, 208 (2005), 9-41.  doi: 10.1016/j.jde.2003.09.008.

[13]

T. CaraballoP. Marín-Rubio and J. Valero, Attractors for differential equations with unbounded delays, J. Differential Equations, 239 (2007), 311-342.  doi: 10.1016/j.jde.2007.05.015.

[14]

T. Caraballo and J. Real, Attractors for 2D-Navier-Stokes models with delays, J. Differential Equations, 205 (2004), 271-297.  doi: 10.1016/j.jde.2004.04.012.

[15]

T. CaraballoJ. Real and A. M. Márquez, Three-dimensional system of globally modified Navier-Stokes equations with delay, Internat. J. Bifur. Chaos Appl. Sci. Engrg., 20 (2010), 2869-2883.  doi: 10.1142/S0218127410027428.

[16]

A. N. Carvalho, J. A. Langa and J. C. 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.

[17]

I. Chueshov and I. Lasiecka, Long-time dynamics of von Karman semi-flows with non-linear boundary/interior damping, J. Differential Equations, 233 (2007), 42-86.  doi: 10.1016/j.jde.2006.09.019.

[18]

J. García-Luengo and P. Marín-Rubio, Reaction-diffusion equations with non-autonomous force in $H^{-1}$ and delays under measurability conditions on the driving delay term, J. Math. Anal. Appl., 417 (2014), 80-95.  doi: 10.1016/j.jmaa.2014.03.026.

[19]

J. García-LuengoP. Marín-Rubio and J. Real, Pullback attractors for 2D Navier-Stokes equations with delays and their regularity, Adv. Nonlinear Stud., 13 (2013), 331-357.  doi: 10.1515/ans-2013-0205.

[20]

J. García-LuengoP. Marín-Rubio and J. Real, Some new regularity results of pullback attractors for 2D Navier-Stokes equations with delays, Commun. Pure Appl. Anal., 14 (2015), 1603-1621.  doi: 10.3934/cpaa.2015.14.1603.

[21]

J. García-LuengoP. Marín-Rubio and G. Planas, Attractors for a double time-delayed 2D-Navier-Stokes model, Discrete Contin. Dyn. Syst., 34 (2014), 4085-4105.  doi: 10.3934/dcds.2014.34.4085.

[22]

J. García-LuengoP. Marín-Rubio and J. Real, Regularity of pullback attractors and attraction in $H^{1}$ in arbitrarily large finite intervals for 2D Navier-Stokes equations with infinite delay, Discrete Contin. Dyn. Syst., 34 (2014), 181-201.  doi: 10.3934/dcds.2014.34.181.

[23]

J. K. Hale and S. M. Verduyn Lunel, Introduction to Functional Differential Equations, Springer-Verlag, New York, 1993. doi: 10.1007/978-1-4612-4342-7.

[24]

J. K. Hale, Asymptotic Behavior of Dissipative Systems, AMS, Providence, RI, 1988. doi: 10.1090/surv/025.

[25]

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

[26]

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.

[27]

P. E. Kloeden, Upper semi continuity of attractors of delay differential equations in the delay, Bull. Austral. Math. Soc., 73 (2006), 299-306.  doi: 10.1017/S0004972700038880.

[28]

P. E. Kloeden and T. Lorenz, Pullback incremental attraction, Nonauton. Dyn. Syst., 1 (2014), 53-60.  doi: 10.2478/msds-2013-0004.

[29]

P. E. Kloeden and P. Marín-Rubio, Equi-attraction and the continuous dependence of attractors on time delays, Discrete Contin. Dyn. Syst. Ser. B, 9 (2008), 581-593.  doi: 10.3934/dcdsb.2008.9.581.

[30]

Y. Li and C. Zhong, Pullback attractors for the norm-to-weak continuous process and application to the nonautonomous reaction-diffusion equations, Appl. Math. Comput., 190 (2007), 1020-1029.  doi: 10.1016/j.amc.2006.11.187.

[31]

S. LuH. Wu and C. 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.

[32]

P. Marín-RubioA. M. Márquez-Durán and J. Real, Three dimensional system of globally modified Navier-Stokes equations with infinite delays, Discrete Contin. Dyn. Syst. Ser. B, 14 (2010), 655-673.  doi: 10.3934/dcdsb.2010.14.655.

[33]

P. Marín-RubioA. M. Márquez-Durán and J. Real, Pullback attractors for globally modified Navier-Stokes equations with infinite delays, Discrete Contin. Dyn. Syst., 31 (2011), 779-796.  doi: 10.3934/dcds.2011.31.779.

[34]

F. MengM. Yang and C. Zhong, Attractors for wave equations with nonlinear damping on time-dependent space, Discrete Contin. Dyn. Syst. Ser. B, 21 (2016), 205-225.  doi: 10.3934/dcdsb.2016.21.205.

[35]

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. 

[36]

C. SunD. Cao and J. 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.

[37]

R. Temam, Infinite-Dimensional Dynamical Systems in Mechanics and Physics, Springer-Verlag, New York, 1997. doi: 10.1007/978-1-4612-0645-3.

[38]

Y. Wang, Pullback attractors for a damped wave equation with delays, Stoch. Dyn., 15 (2015), 1550003, 21 pp. doi: 10.1142/S0219493715500033.

[39]

Y. Wang and P. E. Kloeden, The uniform attractor of a multi-valued process generated by reaction-diffusion delay equations on an unbounded domain, Discrete Contin. Dyn. Syst., 34 (2014), 4343-4370.  doi: 10.3934/dcds.2014.34.4343.

[40]

F. Wu and P. E. Kloeden, Mean-square random attractors of stochastic delay differential equations with random delay, Discrete Contin. Dyn. Syst. Ser. B, 18 (2013), 1715-1734.  doi: 10.3934/dcdsb.2013.18.1715.

[41]

Y. XieQ. Li and K. Zhu, Attractors for nonclassical diffusion equations with arbitrary polynomial growth nonlinearity, Nonlinear Anal. Real World Appl., 31 (2016), 23-37.  doi: 10.1016/j.nonrwa.2016.01.004.

[42]

F. ZhouC. Sun and X. Li, Dynamics for the damped wave equations on time-dependent domains, Discrete Contin. Dyn. Syst. Ser. B, 23 (2018), 1645-1674.  doi: 10.3934/dcdsb.2018068.

[43]

K. ZhuY. Xie and F. Zhou, Pullback attractors for a damped semilinear wave equation with delays, Acta Math. Sin. (Engl. Ser.), 34 (2018), 1131-1150.  doi: 10.1007/s10114-018-7420-3.

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