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doi: 10.3934/dcdsb.2021223
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Asymptotic behavior of supercritical wave equations driven by colored noise on unbounded domains

Department of Mathematics, New Mexico Institute of Mining and Technology, Socorro, NM 87801, USA

Received  May 2021 Early access September 2021

This paper deals with the asymptotic behavior of the non-autonomous random dynamical systems generated by the wave equations with supercritical nonlinearity driven by colored noise defined on $ \mathbb{R}^n $ with $ n\le 6 $. Based on the uniform Strichartz estimates, we prove the well-posedness of the equation in the natural energy space and define a continuous cocycle associated with the solution operator. We also establish the existence and uniqueness of tempered random attractors of the equation by showing the uniform smallness of the tails of the solutions outside a bounded domain in order to overcome the non-compactness of Sobolev embeddings on unbounded domains.

Citation: Bixiang Wang. Asymptotic behavior of supercritical wave equations driven by colored noise on unbounded domains. Discrete & Continuous Dynamical Systems - B, doi: 10.3934/dcdsb.2021223
References:
[1]

J. M. 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, Studies in Mathematics and its Applications, 25. North-Holland Publishing Co., Amsterdam, 1992.  Google Scholar

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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.  Google Scholar

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P. W. BatesK. Lu and B. Wang, Random attractors for stochastic reaction-diffusion equations on unbounded domains, J. Differential Equations, 246 (2009), 845-869.  doi: 10.1016/j.jde.2008.05.017.  Google Scholar

[5]

T. CaraballoM. J. Garrido-AtienzaB. Schmalfuss and J. Valero, Non-autonomous 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.  Google Scholar

[6]

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.  Google Scholar

[7]

T. CaraballoJ. A. LangaV. S. Melnik and J. Valero, Pullback attractors for nonautonomous and stochastic multivalued dynamical systems, Set-Valued Anal., 11 (2003), 153-201.  doi: 10.1023/A:1022902802385.  Google Scholar

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

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H. Crauel and F. Flandoli, Attractors for random dynamical systems, Probab. Theory Related Fields, 100 (1994), 365-393.  doi: 10.1007/BF01193705.  Google Scholar

[10]

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

[11]

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.  Google Scholar

[12]

E. Feireisl, Asymptotic behavior 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

[13]

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

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J. Ginibre and G. Velo, Generalized Strichartz inequalities for the wave equation, J. Funct. Anal., 133 (1995), 50-68.  doi: 10.1006/jfan.1995.1119.  Google Scholar

[15]

A. GuB. Guo and B. Wang, Long term behavior of random Navier-Stokes equations driven by colored noise, Discrete Contin. Dyn. Syst. Ser. B, 25 (2020), 2495-2532.  doi: 10.3934/dcdsb.2020020.  Google Scholar

[16]

A. Gu and B. Wang, Asymptotic behavior of random FitzHugh-Nagumo systems driven by colored noise, Discrete Contin. Dyn. Syst. Ser. B, 23 (2018), 1689-1720.  doi: 10.3934/dcdsb.2018072.  Google Scholar

[17]

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

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L. V. Kapitanskii, Cauchy problem for a semilinear wave equation, Ⅱ, J. Soviet Math., 62 (1992), 2746-2777.  doi: 10.1007/BF01671000.  Google Scholar

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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

[20]

P. E. Kloeden and J. A. Langa, Flattening, squeezing and the existence of random attractors, Proc. R. Soc. Lond. Ser. A. Math. Phys. Eng. Sci., 463 (2007), 163-181.  doi: 10.1098/rspa.2006.1753.  Google Scholar

[21]

O. A. Ladyzhenskaya, The Boundary Value Problems of Mathematical Physics, Applied Mathematical Sciences Vol. 49, Springer, New York, 1985. doi: 10.1007/978-1-4757-4317-3.  Google Scholar

[22]

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.  Google Scholar

[23]

J. L. Lions, Quelques Methodes de Resolution des Problemes aux Limites Non Lineaires, Dunod, Gauthier-Villars, Paris, 1969.  Google Scholar

[24]

C. LiuF. Meng and C. Sun, Well-posedness and attractors for a super-cubic weakly damped wave equation with $H^{-1} $ source term, J. Differential Equations, 263 (2017), 8718-8748.  doi: 10.1016/j.jde.2017.08.047.  Google Scholar

[25]

Y. Lv and W. Wang, Limiting dynamics for stochastic wave equations, J. Differential Equations, 244 (2008), 1-23.  doi: 10.1016/j.jde.2007.10.009.  Google Scholar

[26]

M. Prizzi and K. P. Rybakowski, Attractors for semilinear damped wave equations on arbitrary unbounded domains, Topol. Methods Nonlinear Anal., 31 (2008), 49-82.   Google Scholar

[27]

M. Prizzi and K. P. Rybakowski, Attractors for singularly perturbed damped wave equations on unbounded domains, Topol. Methods Nonlinear Anal., 32 (2008), 1-20.   Google Scholar

[28] L. RidolfiP. D'Odorico and F. Laio, Noise-Induced Phenomena in the Environmental Sciences, Cambridge University Press, Cambridge, 2011.  doi: 10.1017/CBO9780511984730.  Google Scholar
[29]

B. Schmalfuss, Backward cocycles and attractors of stochastic differential equations, International Seminar on Applied Mathematics-Nonlinear Dynamics: Attractor Approximation and Global Behavior, Dresden, (1992), 185–192. Google Scholar

[30]

R. Sell and Y. You, Dynamics of Evolutionary Equations, Applied Mathematical Sciences, Springer-Verlag, New York, 2002. doi: 10.1007/978-1-4757-5037-9.  Google Scholar

[31]

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

[32]

R. Temam, Infinite-Dimensional Dynamical Systems in Mechanics and Physics, 2nd edition, Applied Mathematical Sciences, Springer-Verlag, New York, 1997. doi: 10.1007/978-1-4612-0645-3.  Google Scholar

[33]

G. Uhlenbeck and L. Ornstein, On the theory of Brownian motion, Phys. Rev., 36 (1930), 823-841.  doi: 10.1103/PhysRev.36.823.  Google Scholar

[34]

B. Wang, Attractors for reaction-diffusion equations in unbounded domains, Physica D, 128 (1999), 41-52.  doi: 10.1016/S0167-2789(98)00304-2.  Google Scholar

[35]

B. Wang, Random attractors for the stochastic benjamin-bona-mahony equation on unbounded domains, J. Differential Equations, 246 (2009), 2506-2537.  doi: 10.1016/j.jde.2008.10.012.  Google Scholar

[36]

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.  Google Scholar

[37]

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.  Google Scholar

[38]

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.  Google Scholar

[39]

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.  Google Scholar

[40]

B. Wang, Random attractors of supercritical stochastic wave equations, Pure and Applied Functional Analysis, (in press). Google Scholar

[41]

R. WangL. Shi and B. Wang, Asymptotic behavior of fractional nonclassical diffusion equations driven by nonlinear colored noise on $\mathbb{R}^N$, Nonlinearity, 32 (2019), 4524-4556.  doi: 10.1088/1361-6544/ab32d7.  Google Scholar

[42]

X. WangK. Lu and B. Wang, Wong-Zakai approximations and attractors for stochastic reaction-diffusion equations on unbounded domains, J. Differential Equations, 246 (2018), 378-424.  doi: 10.1016/j.jde.2017.09.006.  Google Scholar

[43]

M. Wang and G. Uhlenbeck, On the theory of Brownian motion. Ⅱ, Rev. Modern Phys., 17 (1945), 323-342.  doi: 10.1103/RevModPhys.17.323.  Google Scholar

[44]

Z. WangS. Zhou and A. Gu, Random attractors 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.  Google Scholar

[45]

Z. Wang and S. Zhou, Random attractor for stochastic non-autonomous damped wave equation with critical exponent, Discrete Contin. Dyn. Syst., 37 (2017), 545-573.  doi: 10.3934/dcds.2017022.  Google Scholar

[46]

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.  Google Scholar

[47]

S. Zhou and M. Zhao, Fractal dimension of random attractor for stochastic non-autonomous damped wave equation with linear multiplicative white noise, Discrete Contin. Dyn. Syst., 36 (2016), 2887-2914.  doi: 10.3934/dcds.2016.36.2887.  Google Scholar

show all references

References:
[1]

J. M. 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, Studies in Mathematics and its Applications, 25. North-Holland Publishing Co., 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]

P. W. BatesK. Lu and B. Wang, Random attractors for stochastic reaction-diffusion equations on unbounded domains, J. Differential Equations, 246 (2009), 845-869.  doi: 10.1016/j.jde.2008.05.017.  Google Scholar

[5]

T. CaraballoM. J. Garrido-AtienzaB. Schmalfuss and J. Valero, Non-autonomous 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.  Google Scholar

[6]

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.  Google Scholar

[7]

T. CaraballoJ. A. LangaV. S. Melnik and J. Valero, Pullback attractors for nonautonomous and stochastic multivalued dynamical systems, Set-Valued Anal., 11 (2003), 153-201.  doi: 10.1023/A:1022902802385.  Google Scholar

[8]

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

[9]

H. Crauel and F. Flandoli, Attractors for random dynamical systems, Probab. Theory Related Fields, 100 (1994), 365-393.  doi: 10.1007/BF01193705.  Google Scholar

[10]

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

[11]

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.  Google Scholar

[12]

E. Feireisl, Asymptotic behavior 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

[13]

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

[14]

J. Ginibre and G. Velo, Generalized Strichartz inequalities for the wave equation, J. Funct. Anal., 133 (1995), 50-68.  doi: 10.1006/jfan.1995.1119.  Google Scholar

[15]

A. GuB. Guo and B. Wang, Long term behavior of random Navier-Stokes equations driven by colored noise, Discrete Contin. Dyn. Syst. Ser. B, 25 (2020), 2495-2532.  doi: 10.3934/dcdsb.2020020.  Google Scholar

[16]

A. Gu and B. Wang, Asymptotic behavior of random FitzHugh-Nagumo systems driven by colored noise, Discrete Contin. Dyn. Syst. Ser. B, 23 (2018), 1689-1720.  doi: 10.3934/dcdsb.2018072.  Google Scholar

[17]

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

[18]

L. V. Kapitanskii, Cauchy problem for a semilinear wave equation, Ⅱ, J. Soviet Math., 62 (1992), 2746-2777.  doi: 10.1007/BF01671000.  Google Scholar

[19]

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

[20]

P. E. Kloeden and J. A. Langa, Flattening, squeezing and the existence of random attractors, Proc. R. Soc. Lond. Ser. A. Math. Phys. Eng. Sci., 463 (2007), 163-181.  doi: 10.1098/rspa.2006.1753.  Google Scholar

[21]

O. A. Ladyzhenskaya, The Boundary Value Problems of Mathematical Physics, Applied Mathematical Sciences Vol. 49, Springer, New York, 1985. doi: 10.1007/978-1-4757-4317-3.  Google Scholar

[22]

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.  Google Scholar

[23]

J. L. Lions, Quelques Methodes de Resolution des Problemes aux Limites Non Lineaires, Dunod, Gauthier-Villars, Paris, 1969.  Google Scholar

[24]

C. LiuF. Meng and C. Sun, Well-posedness and attractors for a super-cubic weakly damped wave equation with $H^{-1} $ source term, J. Differential Equations, 263 (2017), 8718-8748.  doi: 10.1016/j.jde.2017.08.047.  Google Scholar

[25]

Y. Lv and W. Wang, Limiting dynamics for stochastic wave equations, J. Differential Equations, 244 (2008), 1-23.  doi: 10.1016/j.jde.2007.10.009.  Google Scholar

[26]

M. Prizzi and K. P. Rybakowski, Attractors for semilinear damped wave equations on arbitrary unbounded domains, Topol. Methods Nonlinear Anal., 31 (2008), 49-82.   Google Scholar

[27]

M. Prizzi and K. P. Rybakowski, Attractors for singularly perturbed damped wave equations on unbounded domains, Topol. Methods Nonlinear Anal., 32 (2008), 1-20.   Google Scholar

[28] L. RidolfiP. D'Odorico and F. Laio, Noise-Induced Phenomena in the Environmental Sciences, Cambridge University Press, Cambridge, 2011.  doi: 10.1017/CBO9780511984730.  Google Scholar
[29]

B. Schmalfuss, Backward cocycles and attractors of stochastic differential equations, International Seminar on Applied Mathematics-Nonlinear Dynamics: Attractor Approximation and Global Behavior, Dresden, (1992), 185–192. Google Scholar

[30]

R. Sell and Y. You, Dynamics of Evolutionary Equations, Applied Mathematical Sciences, Springer-Verlag, New York, 2002. doi: 10.1007/978-1-4757-5037-9.  Google Scholar

[31]

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

[32]

R. Temam, Infinite-Dimensional Dynamical Systems in Mechanics and Physics, 2nd edition, Applied Mathematical Sciences, Springer-Verlag, New York, 1997. doi: 10.1007/978-1-4612-0645-3.  Google Scholar

[33]

G. Uhlenbeck and L. Ornstein, On the theory of Brownian motion, Phys. Rev., 36 (1930), 823-841.  doi: 10.1103/PhysRev.36.823.  Google Scholar

[34]

B. Wang, Attractors for reaction-diffusion equations in unbounded domains, Physica D, 128 (1999), 41-52.  doi: 10.1016/S0167-2789(98)00304-2.  Google Scholar

[35]

B. Wang, Random attractors for the stochastic benjamin-bona-mahony equation on unbounded domains, J. Differential Equations, 246 (2009), 2506-2537.  doi: 10.1016/j.jde.2008.10.012.  Google Scholar

[36]

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.  Google Scholar

[37]

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.  Google Scholar

[38]

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.  Google Scholar

[39]

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.  Google Scholar

[40]

B. Wang, Random attractors of supercritical stochastic wave equations, Pure and Applied Functional Analysis, (in press). Google Scholar

[41]

R. WangL. Shi and B. Wang, Asymptotic behavior of fractional nonclassical diffusion equations driven by nonlinear colored noise on $\mathbb{R}^N$, Nonlinearity, 32 (2019), 4524-4556.  doi: 10.1088/1361-6544/ab32d7.  Google Scholar

[42]

X. WangK. Lu and B. Wang, Wong-Zakai approximations and attractors for stochastic reaction-diffusion equations on unbounded domains, J. Differential Equations, 246 (2018), 378-424.  doi: 10.1016/j.jde.2017.09.006.  Google Scholar

[43]

M. Wang and G. Uhlenbeck, On the theory of Brownian motion. Ⅱ, Rev. Modern Phys., 17 (1945), 323-342.  doi: 10.1103/RevModPhys.17.323.  Google Scholar

[44]

Z. WangS. Zhou and A. Gu, Random attractors 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.  Google Scholar

[45]

Z. Wang and S. Zhou, Random attractor for stochastic non-autonomous damped wave equation with critical exponent, Discrete Contin. Dyn. Syst., 37 (2017), 545-573.  doi: 10.3934/dcds.2017022.  Google Scholar

[46]

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.  Google Scholar

[47]

S. Zhou and M. Zhao, Fractal dimension of random attractor for stochastic non-autonomous damped wave equation with linear multiplicative white noise, Discrete Contin. Dyn. Syst., 36 (2016), 2887-2914.  doi: 10.3934/dcds.2016.36.2887.  Google Scholar

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