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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  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 & Continuous Dynamical Systems - B, doi: 10.3934/dcdsb.2020294
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.  Google Scholar

[2]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[24]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[28]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[37]

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

[38]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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.  Google Scholar

[2]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[24]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[28]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[37]

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

[38]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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