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Numerical analysis of two new finite difference methods for time-fractional telegraph equation
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 |
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.
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] |
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. 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. |
[5] |
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. |
[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. Caraballo, X. Han and P. E. Kloeden,
Nonautonomous chemostats with variable delays, SIAM J. Math. Anal., 47 (2015), 2178-2199.
doi: 10.1137/14099930X. |
[8] |
T. Caraballo, P. 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. Caraballo, P. 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. Caraballo, G. 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. Caraballo, G. 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. Caraballo, P. 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. Caraballo, P. 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. Caraballo, J. 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-Luengo, P. 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-Luengo, P. 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-Luengo, P. 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-Luengo, P. 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. Kalantarov, A. 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. Lu, H. 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-Rubio, A. 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-Rubio, A. 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. Meng, M. 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. Sun, D. 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. Xie, Q. 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. Zhou, C. 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. Zhu, Y. 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. 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] |
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. 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. |
[5] |
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. |
[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. Caraballo, X. Han and P. E. Kloeden,
Nonautonomous chemostats with variable delays, SIAM J. Math. Anal., 47 (2015), 2178-2199.
doi: 10.1137/14099930X. |
[8] |
T. Caraballo, P. 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. Caraballo, P. 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. Caraballo, G. 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. Caraballo, G. 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. Caraballo, P. 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. Caraballo, P. 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. Caraballo, J. 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-Luengo, P. 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-Luengo, P. 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-Luengo, P. 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-Luengo, P. 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. Kalantarov, A. 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. Lu, H. 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-Rubio, A. 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-Rubio, A. 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. Meng, M. 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. Sun, D. 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. Xie, Q. 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. Zhou, C. 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. Zhu, Y. 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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