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Upper semi-continuity of attractors for non-autonomous fractional stochastic parabolic equations with delay
Department of Mathematics, Northwest Normal University, Lanzhou 730070, China |
This paper is concerned with the asymptotic behavior of the solutions to a class of non-autonomous nonlocal fractional stochastic parabolic equations with delay defined on bounded domain. We first prove the existence of a continuous non-autonomous random dynamical system for the equations as well as the uniform estimates of solutions with respect to the delay time and noise intensity. We then show pullback asymptotical compactness of solutions as well as the existence and uniqueness of tempered random attractors by utilizing the Arzela-Ascoli theorem and the uniform estimates of solutions in fractional Sobolev space $ H^\alpha(\mathbb{R}^n) $ with $ \alpha\in (0,1) $ as well as their time derivatives in $ L^2(\mathbb{R}^n) $. Finally, we establish the upper semi-continuity of the random attractors when noise intensity and time delay approaches zero, respectively.
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Random attractors for stochastic FitzHugh-Nagumo systems driven by deterministic non-autonomous forcing, Discrete Contin. Dyn. Syst. Ser. B, 18 (2013), 643-666.
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P. W. Bates, K. Lu and B. Wang,
Attractors of non-autonomous stochastic lattice systems in weighted spaces, Phys. D, 289 (2014), 32-50.
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L. A. Caffarelli, J.-M. Roquejoffre and Y. Sire,
Variational problems for free boundaries for the fractional Laplacian, J. Eur. Math. Soc., 12 (2010), 1151-1179.
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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. |
[8] |
T. Caraballo, M. J. Garrido-Atienza, B. 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. Caraballo, M. J. Garrido-Atienza and T. Taniguchi,
The existence and exponential behavior of solutions to stochastic delay evolution equations with a fractional Brownian motion, Nonlinear Anal., 74 (2011), 3671-3684.
doi: 10.1016/j.na.2011.02.047. |
[10] |
T. Caraballo and A. M. Márquez-Durán,
Existence, uniqueness and asymptotic behavior of solutions for a nonclassical diffusion equation with delay, Dyn. Partial Differ. Equ., 10 (2013), 267-281.
doi: 10.4310/DPDE.2013.v10.n3.a3. |
[11] |
T. Caraballo, M. J. Garrido-Atienza, B. Schmalfuss and J. Valero,
Attractors for a random evolution equation with infinite memory: Theoretical results, Discrete Contin. Dyn. Syst. Ser. B, 22 (2017), 1779-1800.
doi: 10.3934/dcdsb.2017106. |
[12] |
P. Chen, Y. Li and X. Zhang, Cauchy problem for stochastic non-autonomous evolution equations governed by noncompact evolution families, Discrete Contin. Dyn. Syst. Ser. B, published online, 2020.
doi: 10.3934/dcdsb.2020171. |
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Existence and approximate controllability of fractional evolution equations with nonlocal conditions via resolvent operators, Fract. Calcu. Appl. Anal., 23 (2020), 268-291.
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Random attractors, J. Dynam. Differential Equations, 9 (1997), 307-341.
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Ergodicity of the infinite dimensional fractional Brownian motion, J. Dynam. Differential Equations, 23 (2011), 671-681.
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B. Gess,
Random attractors for singular stochastic evolution equations, J. Differential Equations, 255 (2013), 524-559.
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A. Gu, D. Li, B. Wang and H. Yang,
Regularity of random attractors for fractional stochastic reaction-diffusion equations on $\mathbb{R}^n$, J. Differential Equations, 264 (2018), 7094-7137.
doi: 10.1016/j.jde.2018.02.011. |
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X. Han, P. E. Kloden and B. Usman,
Upper semi-continuous convergence of attractors for a Hopfield-type lattice model, Nonlinearity, 33 (2020), 1881-1906.
doi: 10.1088/1361-6544/ab6813. |
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J. Huang, T. Shen and Y. Li,
Dynamics of stochastic fractional Boussinesq equations, Discrete Contin. Dyn. Syst. Ser. B, 20 (2015), 2051-2067.
doi: 10.3934/dcdsb.2015.20.2051. |
[25] |
P. E. Kloeden,
Upper semicontinuity of attractors of delay differential equations in the delay, Bull. Austral. Math. Soc., 73 (2006), 299-306.
doi: 10.1017/S0004972700038880. |
[26] |
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. |
[27] |
P. E. Kloeden and T. Lorenz,
Pullback attractors of reaction-diffusion inclusions with space-dependent delay, Discrete Contin. Dyn. Syst. Ser. B, 22 (2017), 1909-1964.
doi: 10.3934/dcdsb.2017114. |
[28] |
D. Li, K. Lu, B. Wang and X. Wang,
Limiting dynamics for non-autonomous stochastic retarded reaction-diffusion equations on thin domains, Discrete Contin. Dyn. Syst., 39 (2019), 3717-3747.
doi: 10.3934/dcds.2019151. |
[29] |
D. Li, L. Shi and X. Wang,
Long term behavior of stochastic discrete complex Ginzburg-Landau equations with time delays in weighted spaces, Discrete Contin. Dyn. Syst. Ser. B, 24 (2019), 5121-5148.
doi: 10.3934/dcdsb.2019046. |
[30] |
Y. Li and Y. Wang,
The existence and asymptotic behavior of solutions to fractional stochastic evolution equations with infinite delay, J. Differential Equations, 266 (2019), 3514-3558.
doi: 10.1016/j.jde.2018.09.009. |
[31] |
D. Li, B. Wang and X. Wang, Random dynamics of fractional stochastic reaction-diffusion equations on $\mathbb{R}^{n}$ without uniqueness, J. Math. Phys., 60 (2019), 072704, 21 pp.
doi: 10.1063/1.5063840. |
[32] |
H. Lu, P. W. Bates, S. Lü and M. Zhang,
Dynamics of 3-D fractional complex Ginzburg-Landau equation, J. Differential Equations, 259 (2015), 5276-5301.
doi: 10.1016/j.jde.2015.06.028. |
[33] |
H. Lu, P. W. Bates, S. Lü and M. Zhang,
Dynamics of the 3D fractional Ginzburg-Landau equation with multiplicative noise on an unbounded domain, Commun. Math. Sci., 14 (2016), 273-295.
|
[34] |
H. Lu, P. W. Bates, J. Xin and M. Zhang,
Asymptotic behavior of stochastic fractional power dissipative equations on $\mathbb{R}^{n}$, Nonlinear Anal., 128 (2015), 176-198.
doi: 10.1016/j.na.2015.06.033. |
[35] |
H. Lu, J. Qi, B. Wang and M. Zhang,
Random attractors for non-autonomous fractional stochastic parabolic equations on unbounded domains, Discrete Contin. Dyn. Syst., 39 (2019), 683-706.
doi: 10.3934/dcds.2019028. |
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X. Mao, Stochastic Differential Equations and Applications, Second Edition, Horwood Publishing Limited, Chichester, 2008.
doi: 10.1533/9780857099402. |
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S. E. A. Mohammed, Stochastic Functional Differential Equations, Research Notes in Mathematics, 99, Pitman, Boston, 1984. |
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C. Morosi and L. Pizzocchero,
On the constants for some fractional Gagliardo-Nirenberg and Sobolev inequalities, Expo. Math., 36 (2018), 32-77.
doi: 10.1016/j.exmath.2017.08.007. |
[39] |
X. Ros-Oton and J. Serra,
The Dirichlet problem for the fractional Laplacian: Regularity up to the boundary, J. Math. Pures Appl., 101 (2014), 275-302.
doi: 10.1016/j.matpur.2013.06.003. |
[40] |
R. Servadei and E. Valdinoci,
On the spectrum of two different fractional operators, Proc. Roy. Soc. Edinburgh Sect. A, 144 (2014), 831-855.
doi: 10.1017/S0308210512001783. |
[41] |
M. Sui and Y. Wang,
Upper semicontinuity of pullback attractors for lattice nonclassical diffusion delay equations under singular perturbations, Appl. Math. Comput., 242 (2014), 315-327.
doi: 10.1016/j.amc.2014.05.045. |
[42] |
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. |
[43] |
B. Wang,
Asymptotic behavior of stochastic wave equations with critical exponents on $\mathbb{R}^{3}$, Tran. Amer. Math. Soc., 363 (2011), 3639-3663.
doi: 10.1090/S0002-9947-2011-05247-5. |
[44] |
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. |
[45] |
B. Wang, Existence and upper semicontinuity of attractors for stochastic equations with deterministic non-autonomous terms, Stoch. Dyn., 14 (2014), 1450009, 31 pp.
doi: 10.1142/S0219493714500099. |
[46] |
B. Wang,
Asymptotic behavior of non-autonomous fractional stochastic reaction-diffusion equations, Nonlinear Anal., 158 (2017), 60-82.
doi: 10.1016/j.na.2017.04.006. |
[47] |
B. Wang,
Weak pullback attractors for mean random dynamical systems in Bochner spaces, J. Dynam. Differential Equations, 31 (2019), 2177-2204.
doi: 10.1007/s10884-018-9696-5. |
[48] |
B. Wang,
Dynamics of fractional stochastic reaction-diffusion equations on unbounded domains driven by nonlinear noise, J. Differential Equations, 268 (2019), 1-59.
doi: 10.1016/j.jde.2019.08.007. |
[49] |
R. Wang, Y. Li and B. Wang,
Random dynamics of fractional nonclassical diffusion equations driven by colored noise, Discrete Contin. Dyn. Syst., 39 (2019), 4091-4126.
doi: 10.3934/dcds.2019165. |
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X. Wang, K. Lu and B. Wang,
Random attractors for delay parabolic equations with additive noise and deterministic nonautonomous forcing, SIAM J. Appl. Dyn. Syst., 14 (2015), 1018-1047.
|
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X. Wang, K. Lu and B. Wang,
Exponential stability of non-autonomous stochastic delay lattice systems with multiplicative noise, J. Dynam. Differential Equations, 28 (2016), 1309-1335.
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R. Wang, L. 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. |
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L. Wang and D. Xu,
Asymptotic behavior of a class of reaction-diffusion equations with delays, J. Math. Anal. Appl., 281 (2003), 439-453.
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J. Wu, Theory and Applications of Partial Functional Differential Equations, Springer, New York, 1996.
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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.
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show all references
References:
[1] |
A. Adili and B. Wang,
Random attractors for stochastic FitzHugh-Nagumo systems driven by deterministic non-autonomous forcing, Discrete Contin. Dyn. Syst. Ser. B, 18 (2013), 643-666.
doi: 10.3934/dcdsb.2013.18.643. |
[2] |
L. Arnold, Random Dynamical Systems, Springer-Verlag, New York, 1998.
doi: 10.1007/978-3-662-12878-7. |
[3] |
P. W. Bates, K. 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. |
[4] |
P. W. Bates, K. Lu and B. Wang, Tempered random attractors for parabolic equations in weighted spaces, J. Math. Phys., 54 (2013), 081505, 26 pp.
doi: 10.1063/1.4817597. |
[5] |
P. W. Bates, K. Lu and B. Wang,
Attractors of non-autonomous stochastic lattice systems in weighted spaces, Phys. D, 289 (2014), 32-50.
doi: 10.1016/j.physd.2014.08.004. |
[6] |
L. A. Caffarelli, J.-M. Roquejoffre and Y. Sire,
Variational problems for free boundaries for the fractional Laplacian, J. Eur. Math. Soc., 12 (2010), 1151-1179.
doi: 10.4171/JEMS/226. |
[7] |
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. |
[8] |
T. Caraballo, M. J. Garrido-Atienza, B. 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. Caraballo, M. J. Garrido-Atienza and T. Taniguchi,
The existence and exponential behavior of solutions to stochastic delay evolution equations with a fractional Brownian motion, Nonlinear Anal., 74 (2011), 3671-3684.
doi: 10.1016/j.na.2011.02.047. |
[10] |
T. Caraballo and A. M. Márquez-Durán,
Existence, uniqueness and asymptotic behavior of solutions for a nonclassical diffusion equation with delay, Dyn. Partial Differ. Equ., 10 (2013), 267-281.
doi: 10.4310/DPDE.2013.v10.n3.a3. |
[11] |
T. Caraballo, M. J. Garrido-Atienza, B. Schmalfuss and J. Valero,
Attractors for a random evolution equation with infinite memory: Theoretical results, Discrete Contin. Dyn. Syst. Ser. B, 22 (2017), 1779-1800.
doi: 10.3934/dcdsb.2017106. |
[12] |
P. Chen, Y. Li and X. Zhang, Cauchy problem for stochastic non-autonomous evolution equations governed by noncompact evolution families, Discrete Contin. Dyn. Syst. Ser. B, published online, 2020.
doi: 10.3934/dcdsb.2020171. |
[13] |
P. Chen, X. Zhang and Y. Li,
Existence and approximate controllability of fractional evolution equations with nonlocal conditions via resolvent operators, Fract. Calcu. Appl. Anal., 23 (2020), 268-291.
doi: 10.1515/fca-2020-0011. |
[14] |
Z. Chen and B. Wang, Invariant measures of fractional stochastic delay reaction-diffusion equations on unbounded domains, submitted. Google Scholar |
[15] |
H. Crauel, A. Debussche and F. Flandoli,
Random attractors, J. Dynam. Differential Equations, 9 (1997), 307-341.
doi: 10.1007/BF02219225. |
[16] |
E. Di Nezza, G. Palatucci and E. Valdinoci,
Hitchhiker's guide to the fractional Sobolev spaces, Bull. Sci. Math., 136 (2012), 521-573.
doi: 10.1016/j.bulsci.2011.12.004. |
[17] |
J. Duan and B. Schmalfuss,
The 3D quasigeostrophic fluid dynamics under random forcing on boundary, Commun. Math. Sci., 1 (2003), 133-151.
|
[18] |
M. J. Garrido-Atienza, A. Ogrowsky and B. Schmalfuss,
Random differential equations with random delays, Stoch. Dyn., 11 (2011), 369-388.
doi: 10.1142/S0219493711003358. |
[19] |
M. J. Garrido-Atienza and B. Schmalfuss,
Ergodicity of the infinite dimensional fractional Brownian motion, J. Dynam. Differential Equations, 23 (2011), 671-681.
doi: 10.1007/s10884-011-9222-5. |
[20] |
B. Gess,
Random attractors for singular stochastic evolution equations, J. Differential Equations, 255 (2013), 524-559.
doi: 10.1016/j.jde.2013.04.023. |
[21] |
A. Gu, D. Li, B. Wang and H. Yang,
Regularity of random attractors for fractional stochastic reaction-diffusion equations on $\mathbb{R}^n$, J. Differential Equations, 264 (2018), 7094-7137.
doi: 10.1016/j.jde.2018.02.011. |
[22] |
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. |
[23] |
X. Han, P. E. Kloden and B. Usman,
Upper semi-continuous convergence of attractors for a Hopfield-type lattice model, Nonlinearity, 33 (2020), 1881-1906.
doi: 10.1088/1361-6544/ab6813. |
[24] |
J. Huang, T. Shen and Y. Li,
Dynamics of stochastic fractional Boussinesq equations, Discrete Contin. Dyn. Syst. Ser. B, 20 (2015), 2051-2067.
doi: 10.3934/dcdsb.2015.20.2051. |
[25] |
P. E. Kloeden,
Upper semicontinuity of attractors of delay differential equations in the delay, Bull. Austral. Math. Soc., 73 (2006), 299-306.
doi: 10.1017/S0004972700038880. |
[26] |
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. |
[27] |
P. E. Kloeden and T. Lorenz,
Pullback attractors of reaction-diffusion inclusions with space-dependent delay, Discrete Contin. Dyn. Syst. Ser. B, 22 (2017), 1909-1964.
doi: 10.3934/dcdsb.2017114. |
[28] |
D. Li, K. Lu, B. Wang and X. Wang,
Limiting dynamics for non-autonomous stochastic retarded reaction-diffusion equations on thin domains, Discrete Contin. Dyn. Syst., 39 (2019), 3717-3747.
doi: 10.3934/dcds.2019151. |
[29] |
D. Li, L. Shi and X. Wang,
Long term behavior of stochastic discrete complex Ginzburg-Landau equations with time delays in weighted spaces, Discrete Contin. Dyn. Syst. Ser. B, 24 (2019), 5121-5148.
doi: 10.3934/dcdsb.2019046. |
[30] |
Y. Li and Y. Wang,
The existence and asymptotic behavior of solutions to fractional stochastic evolution equations with infinite delay, J. Differential Equations, 266 (2019), 3514-3558.
doi: 10.1016/j.jde.2018.09.009. |
[31] |
D. Li, B. Wang and X. Wang, Random dynamics of fractional stochastic reaction-diffusion equations on $\mathbb{R}^{n}$ without uniqueness, J. Math. Phys., 60 (2019), 072704, 21 pp.
doi: 10.1063/1.5063840. |
[32] |
H. Lu, P. W. Bates, S. Lü and M. Zhang,
Dynamics of 3-D fractional complex Ginzburg-Landau equation, J. Differential Equations, 259 (2015), 5276-5301.
doi: 10.1016/j.jde.2015.06.028. |
[33] |
H. Lu, P. W. Bates, S. Lü and M. Zhang,
Dynamics of the 3D fractional Ginzburg-Landau equation with multiplicative noise on an unbounded domain, Commun. Math. Sci., 14 (2016), 273-295.
|
[34] |
H. Lu, P. W. Bates, J. Xin and M. Zhang,
Asymptotic behavior of stochastic fractional power dissipative equations on $\mathbb{R}^{n}$, Nonlinear Anal., 128 (2015), 176-198.
doi: 10.1016/j.na.2015.06.033. |
[35] |
H. Lu, J. Qi, B. Wang and M. Zhang,
Random attractors for non-autonomous fractional stochastic parabolic equations on unbounded domains, Discrete Contin. Dyn. Syst., 39 (2019), 683-706.
doi: 10.3934/dcds.2019028. |
[36] |
X. Mao, Stochastic Differential Equations and Applications, Second Edition, Horwood Publishing Limited, Chichester, 2008.
doi: 10.1533/9780857099402. |
[37] |
S. E. A. Mohammed, Stochastic Functional Differential Equations, Research Notes in Mathematics, 99, Pitman, Boston, 1984. |
[38] |
C. Morosi and L. Pizzocchero,
On the constants for some fractional Gagliardo-Nirenberg and Sobolev inequalities, Expo. Math., 36 (2018), 32-77.
doi: 10.1016/j.exmath.2017.08.007. |
[39] |
X. Ros-Oton and J. Serra,
The Dirichlet problem for the fractional Laplacian: Regularity up to the boundary, J. Math. Pures Appl., 101 (2014), 275-302.
doi: 10.1016/j.matpur.2013.06.003. |
[40] |
R. Servadei and E. Valdinoci,
On the spectrum of two different fractional operators, Proc. Roy. Soc. Edinburgh Sect. A, 144 (2014), 831-855.
doi: 10.1017/S0308210512001783. |
[41] |
M. Sui and Y. Wang,
Upper semicontinuity of pullback attractors for lattice nonclassical diffusion delay equations under singular perturbations, Appl. Math. Comput., 242 (2014), 315-327.
doi: 10.1016/j.amc.2014.05.045. |
[42] |
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. |
[43] |
B. Wang,
Asymptotic behavior of stochastic wave equations with critical exponents on $\mathbb{R}^{3}$, Tran. Amer. Math. Soc., 363 (2011), 3639-3663.
doi: 10.1090/S0002-9947-2011-05247-5. |
[44] |
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. |
[45] |
B. Wang, Existence and upper semicontinuity of attractors for stochastic equations with deterministic non-autonomous terms, Stoch. Dyn., 14 (2014), 1450009, 31 pp.
doi: 10.1142/S0219493714500099. |
[46] |
B. Wang,
Asymptotic behavior of non-autonomous fractional stochastic reaction-diffusion equations, Nonlinear Anal., 158 (2017), 60-82.
doi: 10.1016/j.na.2017.04.006. |
[47] |
B. Wang,
Weak pullback attractors for mean random dynamical systems in Bochner spaces, J. Dynam. Differential Equations, 31 (2019), 2177-2204.
doi: 10.1007/s10884-018-9696-5. |
[48] |
B. Wang,
Dynamics of fractional stochastic reaction-diffusion equations on unbounded domains driven by nonlinear noise, J. Differential Equations, 268 (2019), 1-59.
doi: 10.1016/j.jde.2019.08.007. |
[49] |
R. Wang, Y. Li and B. Wang,
Random dynamics of fractional nonclassical diffusion equations driven by colored noise, Discrete Contin. Dyn. Syst., 39 (2019), 4091-4126.
doi: 10.3934/dcds.2019165. |
[50] |
X. Wang, K. Lu and B. Wang,
Random attractors for delay parabolic equations with additive noise and deterministic nonautonomous forcing, SIAM J. Appl. Dyn. Syst., 14 (2015), 1018-1047.
|
[51] |
X. Wang, K. Lu and B. Wang,
Exponential stability of non-autonomous stochastic delay lattice systems with multiplicative noise, J. Dynam. Differential Equations, 28 (2016), 1309-1335.
doi: 10.1007/s10884-015-9448-8. |
[52] |
R. Wang, L. 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. |
[53] |
L. Wang and D. Xu,
Asymptotic behavior of a class of reaction-diffusion equations with delays, J. Math. Anal. Appl., 281 (2003), 439-453.
doi: 10.1016/S0022-247X(03)00112-4. |
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