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Asymptotically autonomous dynamics for fractional subcritical nonclassical diffusion equations driven by nonlinear colored noise

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Abstract

The asymptotically autonomous dynamics in \(H^s(\mathbb R^n)\) for any \(s\in (0,1)\) and \(n\in \mathbb {N}\) are discussed for a class of highly nonlinear nonclassical diffusion equations perturbed by colored noise. The main feature of this model is that the time-dependent drift and diffusion terms have subcritical and superlinear polynomial growth of orders p and q, respectively, satisfying

$$\begin{aligned} 2\le 2q<p<\infty \text { if }n=1\text { and }s\in [\frac{1}{2},1);\text { otherwise, }\, 2\le 2q<p<\frac{2n}{n-2s}. \end{aligned}$$

The number \(\frac{2n}{n-2s}\) is called the fractional critical Sobolev embedding exponent. Under this setting, we prove the existence, time-dependent uniform compactness and asymptotically autonomous convergence of pathwise random attractors of the equations in \(H^s(\mathbb R^n)\) when the time-dependent nonlinearities satisfy some new conditions. The time-dependent uniform pullback asymptotical compactness of the solution operators in \(H^s(\mathbb R^n)\) is proved by virtue of a cut-off technique [38], a spectral decomposition approach and uniform estimates in a time-uniformly tempered attracting universe in order to overcome several difficulties caused by the lack of compact Sobolev embeddings on unbounded domains, the weak dissipativeness of the systems and the unknown measurability of attractors.

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Notes

  1. \(\delta \) is called the correlation time of the colored noise-

References

  1. Ball, J.M.: Continuity properties and global attractors of generalized semiflows and the Navier-Stokes equations. J. Nonlinear Sci. 7, 475–502 (1997)

    MATH  Google Scholar 

  2. Ball, J.M.: Global attractors for damped semilinear wave equations. Discrete Contin. Dyn. Syst. 10(1), 31–52 (2004)

    MATH  Google Scholar 

  3. Cui, H.Y., Langa, J.A.: Uniform attractors for non-autonomous random dynamical systems. J. Differ. Equ. 263(2), 1225–1268 (2017)

    MATH  Google Scholar 

  4. Chen, P.Y., Wang, B.X., Zhang, X.P.: Dynamics of fractional nonclassical diffusion equations with delay driven by additive noise on \(\mathbb{R} ^n\). Discrete Contin. Dyn. Syst. Ser. B 27(9), 5129–5159 (2022)

    MATH  Google Scholar 

  5. Chen, P.Y., Wang, R.H., Zhang, X.P.: Long-time dynamics of fractional nonclassical diffusion equations with nonlinear colored noise and delay on unbounded domains. Bull. Sci. Math. 173, 103071 (2021)

    MATH  Google Scholar 

  6. Chen, P.Y., Wang, B.X., Wang, R.H., Zhang, X.P.: Multivalued random dynamics of Benjamin-Bona-Mahony equations driven by nonlinear colored noise on unbounded domains, 2022, Mathematische Annalen. https://doi.org/10.1007/s00208-022-02400-0.

  7. Chen, Z., Wang, B.X.: Invariant measures of fractional stochastic delay reaction-diffusion equations on unbounded domains. Nonlinearity 34(6), 3969–4016 (2021)

    MATH  Google Scholar 

  8. Camassa, R., Holm, D.D.: An integrable shallow water equation with peaked solitons. Phys. Rev. Lett. 71, 1661–1665 (1993)

    MATH  Google Scholar 

  9. Caraballo, T., Langa, J.A., Robinson, J.C.: Upper semicontinuity of attractors for small random perturbations of dynamical systems. Commun. Part. Differ. Equ. 23(9–10), 1557–1581 (1998)

    MATH  Google Scholar 

  10. Caraballo, T., Guo, B.L., Tuan, N.H., Wang, R.H.: Asymptotically autonomous robustness of random attractors for a class of weakly dissipative stochastic wave equations on unbounded domains, Proc. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci. 151(6), 1700-1730 (2021)

  11. Caraballo, T., Márquez-Durán, A.M., Rivero, F.: Well-posedness and asymptotic behavior of a nonclassical nonautonomous diffusion equation with delay. Int. J. Bifur. Chaos 25(14), 1540021 (2015)

    MATH  Google Scholar 

  12. Caraballo, T., Márquez-Durán, A.M.: Existence, uniqueness and asymptotic behavior of solutions for a nonclassical diffusion equation with delay. Dyn. Partial Differ. Equ. 10(3), 267–281 (2013)

    MATH  Google Scholar 

  13. Caraballo, T., Márquez-Durán, A.M., Rivero, F.: Asymptotic behaviour of a non-classical and non-autonomous diffusion equation containing some hereditary characteristic. Discrete Contin. Dyn. Syst. Ser. B 22(5), 1817–1833 (2017)

    MATH  Google Scholar 

  14. Chueshov, I., Lasiecka, I.: Long-time behavior of second order evolution equations with nonlinear damping. Mem. Am. Math. Soc. 195(912), 1–183 (2008)

    MATH  Google Scholar 

  15. Chueshov, I., Lasiecka, I.: Von Karman Evolution Equations: Well-posedness and Long Time Dynamics, Springer (2010)

  16. Chueshov, I.: Dynamics of Quasi-Stable Dissipative Systems, Springer (2015)

  17. Di Nezza, E., Palatucci, G., Valdinoci, E.: Hitchhiker’s guide to the fractional Sobolev spaces. Bull. Sci. Math. 136, 521–573 (2012)

    MATH  Google Scholar 

  18. Gerstner, W., Kistler, W., Naud, R., Paninski, L.: Neuronal Dynamics: From Single Neurons to Networks and Models of Cognition. Cambridge University Press, Cambridge (2014)

    Google Scholar 

  19. Gu, A.H., Guo, B.L., Wang, B.X.: Long term behavior of random Navier-Stokes equations driven by colored noise. Discrete Contin. Dyn. Syst. Ser. B 25(7), 2495–2532 (2020)

    MATH  Google Scholar 

  20. Gu, A.H., Li, D.S., Wang, B.X., Yang, H.: Regularity of random attractors for fractional stochastic reaction-diffusion equations on \(\mathbb{R} ^n\). J. Differ. Equ. 264(12), 7094–7137 (2018)

    MATH  Google Scholar 

  21. Gu, A.H., Wang, B.X.: Asymptotic behavior of random FitzHugh-Nagumo systems driven by colored noise. Discrete Contin. Dyn. Syst. Ser. B 23(4), 1689–1720 (2018)

    MATH  Google Scholar 

  22. Hale, J.K., Raugel, G.: A damped hyperbolic equation on thin domains. Trans. Amer. Math. Soc. 329, 185–219 (1992)

    MATH  Google Scholar 

  23. Häunggi, P., Jung, P.: Colored Noise in Dynamical Systems, Advances in Chemical Physics, vol. 89. John Wiley & Sons, Inc., Hoboken, NJ (1994)

    Google Scholar 

  24. van Kampen, N.: Stochastic Processes in Physics and Chemistry. North-Holland, Amsterdam (1992)

    MATH  Google Scholar 

  25. Kuttler, K., Aifantis, E.C.: Quasilinear evolution equations in nonclassical diffusion. SIAM J. Math. Anal. 19, 110–120 (1998)

    MATH  Google Scholar 

  26. Kloeden, P.E., Langa, J.A., Flattening, squeezing and the existence of random attractors, Proc. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci. 463, 163-181 (2007)

  27. Kloeden, P.E., Simsen, J.: Attractors of asymptotically autonomous quasi-linear parabolic equation with spatially variable exponents. J. Math. Anal. Appl. 425(2), 911–918 (2015)

    MATH  Google Scholar 

  28. Kloeden, P.E., Simsen, J., Simsen, M.S.: Asymptotically autonomous multivalued cauchy problems with spatially variable exponents. J. Math. Anal. Appl. 445(1), 513–531 (2017)

    MATH  Google Scholar 

  29. Kloeden, P.E., Rasmussen, M: Nonautonomous Dynamical Systems, American Mathematical Society (2011)

  30. Klosek-Dygas, M., Matkowsky, B., Schuss, Z.: Colored noise in dynamical systems. SIAM J. Appl. Math. 48, 425–441 (1988)

    MATH  Google Scholar 

  31. Li, Y.R., Gu, A.H.: Li, J: Existence and continuity of bi-spatial random attractors and application to stochastic semilinear Laplacian equations. J. Differ. Equ. 258(2), 504–534 (2015)

    MATH  Google Scholar 

  32. Li, Y.R., She, L.B., Wang, R.H.: Asymptotically autonomous dynamics for parabolic equations. J. Math. Anal. Appl. 459(2), 1106–1123 (2018)

    MATH  Google Scholar 

  33. Li, Y.R., Wang, R.H., She, L.B.: Backward controllability of pullback trajectory attractors with applications to multi-valued Jeffreys-Oldroyd equations. Evolution Equ. Control Theory 7(4), 617–637 (2018)

    MATH  Google Scholar 

  34. Morosi, C., Pizzocchero, L.: On the constants for some fractional Gagliardo-Nirenberg and Sobolev inequalities. Expo. Math. 36, 32–77 (2018)

    MATH  Google Scholar 

  35. Robinson, J.C.: Infinite-dimensional Dynamical Systems: An Introduction to Dissipative Parabolic PDEs and the Theory of Global Attractors. Cambridge University Press, Cambridge (2001)

  36. Robinson, J.C.: Stability of random attractors under perturbation and approximation. J. Differ. Equ. 186(2), 652–669 (2002)

    MATH  Google Scholar 

  37. Uhlenbeck, G., Ornstein, L.: On the theory of Brownian motion. Phys. Rev. 36, 823841 (1930)

    Google Scholar 

  38. Wang, B.X.: Attractors for reaction-diffusion equations in unbounded domains. Phys. D 128(1), 41–52 (1999)

    MATH  Google Scholar 

  39. Wang, B.X.: Sufficient and necessary criteria for existence of pullback attractors for non-compact random dynamical systems. J. Differ. Equ. 253(5), 1544–1583 (2012)

    MATH  Google Scholar 

  40. Wang, B.X.: Dynamics of fractional stochastic reaction-diffusion equations on unbounded domains driven by nonlinear noise. J. Differ. Equ. 268(1), 1–59 (2019)

    MATH  Google Scholar 

  41. Wang, B.X.: Asymptotic behavior of stochastic wave equations with critical exponents on \(\mathbb{R} ^{3}\). Tran. Amer. Math. Soc. 363, 3639–3663 (2011)

    MATH  Google Scholar 

  42. Wang, B.X.: Random attractors for non-autonomous stochastic wave equations with multiplicative noise. Discrete Contin. Dyn. Syst. 34(1), 269–300 (2014)

    MATH  Google Scholar 

  43. Wang, R.H., Li, Y.R.: Asymptotic autonomy of kernel sections for Newton-Boussinesq equations on unbounded zonary domains. Dyn. Partial Differ. Equ. 16, 295–316 (2019)

    MATH  Google Scholar 

  44. Wang, R.H., Li, Y.R., Wang, B.X.: Random dynamics of fractional nonclassical diffusion equations driven by colored noise. Discrete Contin. Dyn. Syst. 39(7), 4091–4126 (2019)

    MATH  Google Scholar 

  45. Wang, R.H., Shi, L., Wang, B.X.: Asymptotic behavior of fractional nonclassical diffusion equations driven by nonlinear colored noise on \(\mathbb{R} ^N\). Nonlinearity 32(11), 4524–4556 (2019)

    MATH  Google Scholar 

  46. Wang, R.H., Li, Y.R., Wang, B.X.: Bi-spatial pullback attractors of fractional nonclassical diffusion equations on unbounded domains with \((p, q)\)-growth nonlinearities. Appl. Math. Optim. 84, 425–461 (2021)

    MATH  Google Scholar 

  47. Wang, R.H., Guo, B.L., Wang, B.X.: Well-posedness and dynamics of fractional FitzHugh-Nagumo systems on \(\mathbb{R} ^N\) driven by nonlinear noise. Sci. China Math. 64(11), 2395–2436 (2021)

    MATH  Google Scholar 

  48. Wang, S.L., Li, Y.R.: Longtime robustness of pullback random attractors for stochastic magneto-hydrodynamics equations. Physica D 382–383(1), 46–57 (2018)

    MATH  Google Scholar 

  49. Wang, Y.H., Zhu, Z.L., Li, P.R.: Regularity of pullback attractors for nonautonomous nonclassical diffusion equations. J. Math. Anal. Appl. 459(1), 16–31 (2018)

    MATH  Google Scholar 

  50. Wang, L.Z., Wang, Y.H., Qin, Y.M.: Upper semi-continuity of attractors for nonclassical diffusion equations in \(H^1(\mathbb{R} ^3)\). Appl. Math. Comput. 240(1), 51–61 (2014)

    Google Scholar 

  51. Wang, X.C., Xu, R.Z.: Global existence and finite time blowup for a nonlocal semilinear pseudo-parabolic equation. Adv. Nonlinear Anal. 10(1), 261–288 (2021)

    MATH  Google Scholar 

  52. Xu, R.Z., Su, J.: Global existence and finite time blow-up for a class of semilinear pseudo-parabolic equations. J. Funct. Anal. 264(12), 2732–2763 (2013)

    MATH  Google Scholar 

  53. Xu, J.H., Caraballo, T.: Long time behavior of stochastic nonlocal partial differential equations and Wong-Zakai approximations. SIAM J. Math. Anal. 54, 2792–2844 (2022)

    MATH  Google Scholar 

  54. Zhao, W.Q., Song, S.Z.: Dynamics of stochastic nonclassical diffusion equations on unbounded domains. Electronic J. Differ. Equ. 282, 1–22 (2015)

    MATH  Google Scholar 

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Acknowledgements

Fuzhi Li was supported by the NSFC (12201415) and the Natural Science Foundation of Jiangxi Province (20202BABL211006). M. M. Freitas thank the CNPq for financial support through the project Attractors and asymptotic behavior of nonlinear evolution equations by Grant 313081/2021-2.

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  1. School of Mathematics & Computer Science, Shangrao Normal University, Shangrao, 334001, China

    Fuzhi Li

  2. Faculty of Mathematics, Federal University of Pará, Raimundo Santana Street, Salinópolis, PA, 68721-000, Brazil

    Mirelson M. Freitas

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  1. Fuzhi Li
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Li, F., Freitas, M.M. Asymptotically autonomous dynamics for fractional subcritical nonclassical diffusion equations driven by nonlinear colored noise. Fract Calc Appl Anal 26, 414–438 (2023). https://doi.org/10.1007/s13540-022-00112-5

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