doi: 10.3934/dcdsb.2021310
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Multi-valued random dynamics of stochastic wave equations with infinite delays

1. 

College of Applied Mathematics, Shanxi University of Finance and Economics, Taiyuan 030006, China

2. 

School of Mathematics and Statistics, Gansu Key Laboratory of Applied Mathematics and Complex Systems, Lanzhou University, Lanzhou 730000, China

3. 

Departamento de Ecuaciones Diferenciales y Análisis Numérico, Facultad de Matemáticas, Universidad de Sevilla, c/ Tarfia s/n, 41012-Sevilla, Spain

* Corresponding author

Received  August 2021 Early access January 2022

Fund Project: This work was supported by NSF of China (Grants No. 11801335, 41875084), Scientific and Technologial Innovation Programs of Higher Education Institutions in Shanxi (Grant No. 2020L0249). The research of T. Caraballo has been partially supported by Ministerio de Ciencia Innovación y Universidades (Spain), FEDER (European Community) under grant PGC2018-096540-B-I00, and by FEDER and Junta de Andalucía (Consejería de Economía y Conocimiento) under projects US-1254251 and P18-FR-4509

This paper is devoted to the asymptotic behavior of solutions to a non-autonomous stochastic wave equation with infinite delays and additive white noise. The nonlinear terms of the equation are not expected to be Lipschitz continuous, but only satisfy continuity assumptions along with growth conditions, under which the uniqueness of the solutions may not hold. Using the theory of multi-valued non-autonomous random dynamical systems, we prove the existence and measurability of a compact global pullback attractor.

Citation: Jingyu Wang, Yejuan Wang, Tomás Caraballo. Multi-valued random dynamics of stochastic wave equations with infinite delays. Discrete and Continuous Dynamical Systems - B, doi: 10.3934/dcdsb.2021310
References:
[1] L. Arnold, Random Dynamical Systems, Springer-Verlag, New York, Berlin, 1998.  doi: 10.1007/978-3-662-12878-7.
[2]

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.

[3]

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

[4]

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.

[5]

S. Borini and V. Pata, Uniform attractors for a strongly damped wave equation with linear memory, Asymptot. Anal., 20 (1999), 263-277. 

[6]

T. CaraballoA. N. CarvalhoJ. A. Langa and F. Rivero, A non-autonomous strongly damped wave equation: Existence and continuity of the pullback attractor, Nonlinear Anal., 74 (2011), 2272-2283.  doi: 10.1016/j.na.2010.11.032.

[7]

T. CaraballoM. J. Garrido-AtienzaB. Schmalfuß and J. Valero, Nonautonomous and random attractors for delay random semilinear equations without uniqueness, Discrete Contin. Dyn. Syst., 21 (2008), 415-443.  doi: 10.3934/dcds.2008.21.415.

[8]

T. CaraballoM. J. Garrido-AtienzaB. Schmalfuß and J. Valero, Asymptotic behaviour of a stochastic semilinear dissipative functional equation without uniqueness of solutions, Discrete Contin. Dyn. Syst. Ser. B, 14 (2010), 439-455.  doi: 10.3934/dcdsb.2010.14.439.

[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. CaraballoE. Morillas and J. Valero, Random attractors for stochastic lattice systems with non-Lipschitz nonlinearity, J. Difference Equ. Appl., 17 (2011), 161-184.  doi: 10.1080/10236198.2010.549010.

[11]

V. V. Chepyzhov and M. I. Vishik, Attractors for Equations of Mathematical Physics, Amer. Math. Soc. Colloq. Publ., vol. 49, American Mathematical Society, Providence, RI, 2002. doi: 10.1051/cocv:2002056.

[12]

H. CrauelA. Debussche and F. Flandoli, Random attractors, J. Dynam. Differential Equations, 9 (1997), 307-341.  doi: 10.1007/BF02219225.

[13]

C. M. Dafermos, Asymptotic stability in viscoelasticity, Arch. Ration. Mech. Anal., 37 (1970), 297-308.  doi: 10.1007/BF00251609.

[14]

M. Fabrizio and A. Morro, Mathematical Problems in Linear Viscoelasticity, SIAM Studies in Applied Mathematics, vol. 12, SIAM, Philadelphia, 1992. doi: 10.1137/1.9781611970807.

[15]

X. Fan, Random attractors for damped stochastic wave equations with multiplicative noise, Internat. J. Math., 19 (2008), 421-437.  doi: 10.1142/S0129167X08004741.

[16]

E. Feireisl, Attractors for semilinear damped wave equations on $\mathbb{R}^3$, Nonlinear Anal., 23 (1994), 187-195.  doi: 10.1016/0362-546X(94)90041-8.

[17]

F. Flandoli and B. Schmalfuß, Random attractors for the $3$D stochastic Navier-Stokes equation with multiplicative noise, Stoch. Stoch. Rep., 59 (1996), 21-45.  doi: 10.1080/17442509608834083.

[18]

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

[19]

M. He and A. Liu, The oscillation of hyperbolic functional differential equations, Appl. Math. Comput., 142 (2003), 205-224.  doi: 10.1016/S0096-3003(02)00295-3.

[20]

R. Jones and B. Wang, Asymptotic behavior of a class of stochastic nonlinear wave equations with dispersive and dissipative terms, Nonlinear Anal. Real World Appl., 14 (2013), 1308-1322.  doi: 10.1016/j.nonrwa.2012.09.019.

[21]

A. Kh. Khanmamedov, Global attractors for wave equations with nonlinear interior damping and critical exponents, J. Differential Equations, 230 (2006), 702-719.  doi: 10.1016/j.jde.2006.06.001.

[22] V. Kolmanovskii and A. Myshkis, Introduction to the Theory and Applications of Functional Differential Equations, Kluwer Academic Publishers, Dordrecht, 1999.  doi: 10.1007/978-94-017-1965-0.
[23]

I. KucukI. Sadek and Y. Yilmaz, Active control of a smart beam with time delay by Legendre wavelets, Appl. Math. Comput., 218 (2012), 8968-8977.  doi: 10.1016/j.amc.2012.02.057.

[24]

H. LiY. You and J. Tu, Random attractors and averaging for non-autonomous stochastic wave equations with nonlinear damping, J. Differential Equations, 258 (2015), 148-190.  doi: 10.1016/j.jde.2014.09.007.

[25]

F. Morillas and J. Valero, Attractors for reaction-diffusion equations in $\mathbb{R}^N$ with continuous nonlinearity, Asymptot. Anal., 44 (2005), 111-130. 

[26]

S. Nicaise and C. Pignotti, Stability and instability results of the wave equation with a delay term in the boundary or internal feedbacks, SIAM J. Control Optim., 45 (2006), 1561-1585.  doi: 10.1137/060648891.

[27]

V. Pata, Attractors for a damped wave equation on $\mathbb{R}^3$ with linear memory, Math. Methods Appl. Sci., 23 (2000), 633-653.  doi: 10.1002/(SICI)1099-1476(20000510)23:7<633::AID-MMA135>3.0.CO;2-C.

[28]

N. Raskin and Y. Halevi, Control of flexible structures governed by the wave equation, American Control Conference, Arlington, VA, 2001, 2486–2491. doi: 10.1109/ACC.2001.946126.

[29] G. R. Sell and Y. You, Dynamics of Evolutionary Equations, Springer-Verlag, New York, 2002.  doi: 10.1007/978-1-4757-5037-9.
[30]

Z. ShenS. Zhou and W. Shen, One-dimensional random attractor and rotation number of the stochastic damped sine-Gordon equation, J. Differential Equations, 248 (2010), 1432-1457.  doi: 10.1016/j.jde.2009.10.007.

[31]

C. SunM. Yang and C. Zhong, Global attractors for the wave equation with nonlinear damping, J. Differential Equations, 227 (2006), 427-443.  doi: 10.1016/j.jde.2005.09.010.

[32] R. Temam, Infinite-Dimensional Dynamical Systems in Mechanics and Physics, Springer-Verlag, New York, 1997.  doi: 10.1007/978-1-4612-0645-3.
[33]

B. Wang, Asymptotic behavior of stochastic wave equations with critical exponents on $\mathbb{R}^3$, Trans. Amer. Math. Soc., 363 (2011), 3639-3663.  doi: 10.1090/S0002-9947-2011-05247-5.

[34]

B. 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.

[35]

B. Wang, Random attractors for non-autonomous stochastic wave equations with multiplicative noise, Discrete Contin. Dyn. Syst., 34 (2014), 269-300.  doi: 10.3934/dcds.2014.34.269.

[36]

B. Wang, Multivalued non-autonomous random dynamical systems for wave equations without uniqueness, Discrete Contin. Dyn. Syst. Ser. B, 22 (2017), 2011-2051.  doi: 10.3934/dcdsb.2017119.

[37]

J. WangF. Meng and S. Liu, Integral average method for oscillation of second order partial differential equations with delays, Appl. Math. Comput., 187 (2007), 815-823.  doi: 10.1016/j.amc.2006.08.160.

[38]

J. WangY. Wang and D. Zhao, Pullback attractors for multi-valued non-compact random dynamical systems generated by semi-linear degenerate parabolic equations with unbounded delays, Stoch. Dyn., 17 (2016), 1750001, 49 pp.  doi: 10.1142/S0219493717500010.

[39]

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

[40]

Y. WangY. Qin and J. Wang, Pullback attractors for a strongly damped delay wave equation in $\mathbb{R}^n$, Stoch. Dyn., 18 (2018), 1850016, 24 pp.  doi: 10.1142/S0219493718500168.

[41]

Y. Wang and J. Wang, Pullback attractors for multi-valued non-compact random dynamical systems generated by reaction-diffusion equations on an unbounded domain, J. Differential Equations, 259 (2015), 728-776.  doi: 10.1016/j.jde.2015.02.026.

[42]

Y. Wang and S. Zhou, Kernel sections and uniform attractors of multi-valued semiprocesses, J. Differential Equations, 232 (2007), 573-622.  doi: 10.1016/j.jde.2006.07.005.

[43]

Z. WangS. Zhou and A. Gu, Random attractor for a stochastic damped wave equation with multiplicative noise on unbounded domains, Nonlinear Anal. Real World Appl., 12 (2011), 3468-3482.  doi: 10.1016/j.nonrwa.2011.06.008.

[44] J. Wu, Theory and Applications of Partial Functional Differential Equations, Springer-Verlag, New York, 1996.  doi: 10.1007/978-1-4612-4050-1.
[45]

M. YangJ. Duan and P. Kloeden, Asymptotic behavior of solutions for random wave equations with nonlinear damping and white noise, Nonlinear Anal. Real World Appl., 12 (2011), 464-478.  doi: 10.1016/j.nonrwa.2010.06.032.

[46]

Y. You, Global dynamics of nonlinear wave equations with cubic non-monotone damping, Dyn. Partial Differ. Equ., 1 (2004), 65-86.  doi: 10.4310/DPDE.2004.v1.n1.a3.

[47]

S. Zelik, Asymptotic regularity of solutions of a non-autonomous damped wave equation with a critical growth exponent, Commun. Pure Appl. Anal., 3 (2004), 921-934.  doi: 10.3934/cpaa.2004.3.921.

[48]

S. Zhou and M. Zhao, Random attractors for damped non-autonomous wave equations with memory and white noise, Nonlinear Anal., 120 (2015), 202-226.  doi: 10.1016/j.na.2015.03.009.

show all references

References:
[1] L. Arnold, Random Dynamical Systems, Springer-Verlag, New York, Berlin, 1998.  doi: 10.1007/978-3-662-12878-7.
[2]

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.

[3]

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

[4]

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.

[5]

S. Borini and V. Pata, Uniform attractors for a strongly damped wave equation with linear memory, Asymptot. Anal., 20 (1999), 263-277. 

[6]

T. CaraballoA. N. CarvalhoJ. A. Langa and F. Rivero, A non-autonomous strongly damped wave equation: Existence and continuity of the pullback attractor, Nonlinear Anal., 74 (2011), 2272-2283.  doi: 10.1016/j.na.2010.11.032.

[7]

T. CaraballoM. J. Garrido-AtienzaB. Schmalfuß and J. Valero, Nonautonomous and random attractors for delay random semilinear equations without uniqueness, Discrete Contin. Dyn. Syst., 21 (2008), 415-443.  doi: 10.3934/dcds.2008.21.415.

[8]

T. CaraballoM. J. Garrido-AtienzaB. Schmalfuß and J. Valero, Asymptotic behaviour of a stochastic semilinear dissipative functional equation without uniqueness of solutions, Discrete Contin. Dyn. Syst. Ser. B, 14 (2010), 439-455.  doi: 10.3934/dcdsb.2010.14.439.

[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. CaraballoE. Morillas and J. Valero, Random attractors for stochastic lattice systems with non-Lipschitz nonlinearity, J. Difference Equ. Appl., 17 (2011), 161-184.  doi: 10.1080/10236198.2010.549010.

[11]

V. V. Chepyzhov and M. I. Vishik, Attractors for Equations of Mathematical Physics, Amer. Math. Soc. Colloq. Publ., vol. 49, American Mathematical Society, Providence, RI, 2002. doi: 10.1051/cocv:2002056.

[12]

H. CrauelA. Debussche and F. Flandoli, Random attractors, J. Dynam. Differential Equations, 9 (1997), 307-341.  doi: 10.1007/BF02219225.

[13]

C. M. Dafermos, Asymptotic stability in viscoelasticity, Arch. Ration. Mech. Anal., 37 (1970), 297-308.  doi: 10.1007/BF00251609.

[14]

M. Fabrizio and A. Morro, Mathematical Problems in Linear Viscoelasticity, SIAM Studies in Applied Mathematics, vol. 12, SIAM, Philadelphia, 1992. doi: 10.1137/1.9781611970807.

[15]

X. Fan, Random attractors for damped stochastic wave equations with multiplicative noise, Internat. J. Math., 19 (2008), 421-437.  doi: 10.1142/S0129167X08004741.

[16]

E. Feireisl, Attractors for semilinear damped wave equations on $\mathbb{R}^3$, Nonlinear Anal., 23 (1994), 187-195.  doi: 10.1016/0362-546X(94)90041-8.

[17]

F. Flandoli and B. Schmalfuß, Random attractors for the $3$D stochastic Navier-Stokes equation with multiplicative noise, Stoch. Stoch. Rep., 59 (1996), 21-45.  doi: 10.1080/17442509608834083.

[18]

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

[19]

M. He and A. Liu, The oscillation of hyperbolic functional differential equations, Appl. Math. Comput., 142 (2003), 205-224.  doi: 10.1016/S0096-3003(02)00295-3.

[20]

R. Jones and B. Wang, Asymptotic behavior of a class of stochastic nonlinear wave equations with dispersive and dissipative terms, Nonlinear Anal. Real World Appl., 14 (2013), 1308-1322.  doi: 10.1016/j.nonrwa.2012.09.019.

[21]

A. Kh. Khanmamedov, Global attractors for wave equations with nonlinear interior damping and critical exponents, J. Differential Equations, 230 (2006), 702-719.  doi: 10.1016/j.jde.2006.06.001.

[22] V. Kolmanovskii and A. Myshkis, Introduction to the Theory and Applications of Functional Differential Equations, Kluwer Academic Publishers, Dordrecht, 1999.  doi: 10.1007/978-94-017-1965-0.
[23]

I. KucukI. Sadek and Y. Yilmaz, Active control of a smart beam with time delay by Legendre wavelets, Appl. Math. Comput., 218 (2012), 8968-8977.  doi: 10.1016/j.amc.2012.02.057.

[24]

H. LiY. You and J. Tu, Random attractors and averaging for non-autonomous stochastic wave equations with nonlinear damping, J. Differential Equations, 258 (2015), 148-190.  doi: 10.1016/j.jde.2014.09.007.

[25]

F. Morillas and J. Valero, Attractors for reaction-diffusion equations in $\mathbb{R}^N$ with continuous nonlinearity, Asymptot. Anal., 44 (2005), 111-130. 

[26]

S. Nicaise and C. Pignotti, Stability and instability results of the wave equation with a delay term in the boundary or internal feedbacks, SIAM J. Control Optim., 45 (2006), 1561-1585.  doi: 10.1137/060648891.

[27]

V. Pata, Attractors for a damped wave equation on $\mathbb{R}^3$ with linear memory, Math. Methods Appl. Sci., 23 (2000), 633-653.  doi: 10.1002/(SICI)1099-1476(20000510)23:7<633::AID-MMA135>3.0.CO;2-C.

[28]

N. Raskin and Y. Halevi, Control of flexible structures governed by the wave equation, American Control Conference, Arlington, VA, 2001, 2486–2491. doi: 10.1109/ACC.2001.946126.

[29] G. R. Sell and Y. You, Dynamics of Evolutionary Equations, Springer-Verlag, New York, 2002.  doi: 10.1007/978-1-4757-5037-9.
[30]

Z. ShenS. Zhou and W. Shen, One-dimensional random attractor and rotation number of the stochastic damped sine-Gordon equation, J. Differential Equations, 248 (2010), 1432-1457.  doi: 10.1016/j.jde.2009.10.007.

[31]

C. SunM. Yang and C. Zhong, Global attractors for the wave equation with nonlinear damping, J. Differential Equations, 227 (2006), 427-443.  doi: 10.1016/j.jde.2005.09.010.

[32] R. Temam, Infinite-Dimensional Dynamical Systems in Mechanics and Physics, Springer-Verlag, New York, 1997.  doi: 10.1007/978-1-4612-0645-3.
[33]

B. Wang, Asymptotic behavior of stochastic wave equations with critical exponents on $\mathbb{R}^3$, Trans. Amer. Math. Soc., 363 (2011), 3639-3663.  doi: 10.1090/S0002-9947-2011-05247-5.

[34]

B. 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.

[35]

B. Wang, Random attractors for non-autonomous stochastic wave equations with multiplicative noise, Discrete Contin. Dyn. Syst., 34 (2014), 269-300.  doi: 10.3934/dcds.2014.34.269.

[36]

B. Wang, Multivalued non-autonomous random dynamical systems for wave equations without uniqueness, Discrete Contin. Dyn. Syst. Ser. B, 22 (2017), 2011-2051.  doi: 10.3934/dcdsb.2017119.

[37]

J. WangF. Meng and S. Liu, Integral average method for oscillation of second order partial differential equations with delays, Appl. Math. Comput., 187 (2007), 815-823.  doi: 10.1016/j.amc.2006.08.160.

[38]

J. WangY. Wang and D. Zhao, Pullback attractors for multi-valued non-compact random dynamical systems generated by semi-linear degenerate parabolic equations with unbounded delays, Stoch. Dyn., 17 (2016), 1750001, 49 pp.  doi: 10.1142/S0219493717500010.

[39]

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

[40]

Y. WangY. Qin and J. Wang, Pullback attractors for a strongly damped delay wave equation in $\mathbb{R}^n$, Stoch. Dyn., 18 (2018), 1850016, 24 pp.  doi: 10.1142/S0219493718500168.

[41]

Y. Wang and J. Wang, Pullback attractors for multi-valued non-compact random dynamical systems generated by reaction-diffusion equations on an unbounded domain, J. Differential Equations, 259 (2015), 728-776.  doi: 10.1016/j.jde.2015.02.026.

[42]

Y. Wang and S. Zhou, Kernel sections and uniform attractors of multi-valued semiprocesses, J. Differential Equations, 232 (2007), 573-622.  doi: 10.1016/j.jde.2006.07.005.

[43]

Z. WangS. Zhou and A. Gu, Random attractor for a stochastic damped wave equation with multiplicative noise on unbounded domains, Nonlinear Anal. Real World Appl., 12 (2011), 3468-3482.  doi: 10.1016/j.nonrwa.2011.06.008.

[44] J. Wu, Theory and Applications of Partial Functional Differential Equations, Springer-Verlag, New York, 1996.  doi: 10.1007/978-1-4612-4050-1.
[45]

M. YangJ. Duan and P. Kloeden, Asymptotic behavior of solutions for random wave equations with nonlinear damping and white noise, Nonlinear Anal. Real World Appl., 12 (2011), 464-478.  doi: 10.1016/j.nonrwa.2010.06.032.

[46]

Y. You, Global dynamics of nonlinear wave equations with cubic non-monotone damping, Dyn. Partial Differ. Equ., 1 (2004), 65-86.  doi: 10.4310/DPDE.2004.v1.n1.a3.

[47]

S. Zelik, Asymptotic regularity of solutions of a non-autonomous damped wave equation with a critical growth exponent, Commun. Pure Appl. Anal., 3 (2004), 921-934.  doi: 10.3934/cpaa.2004.3.921.

[48]

S. Zhou and M. Zhao, Random attractors for damped non-autonomous wave equations with memory and white noise, Nonlinear Anal., 120 (2015), 202-226.  doi: 10.1016/j.na.2015.03.009.

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