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Random Attractor, Invariant Measures, and Ergodicity of Lattice p-Laplacian Equations Driven by Superlinear Noise

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Abstract

A highly nonlinear lattice p-Laplacian equation driven by superlinear noise is considered. By using an appropriate stopping time technique and the dissipativeness of the nonlinear drift terms, we establish the global existence and uniqueness of the solutions in \(C\bigg ([\tau ,\infty ), L^2(\Omega , \ell ^2)\bigg )\cap L^p\bigg (\Omega , L_{\text {loc}}^p((\tau ,\infty ), \ell ^p )\bigg ) \) for any \(p>2\) when the coefficient of the noise has a superlinear growth order \(q\in [2,p)\). By the theory of mean random dynamical systems recently developed in Kloeden and Lorenz (J Differ Equ 253:1422–1438, 2012) and Wang (J Dyn Differ Equ 31:2177–2204, 2019), we prove that the nonautonomous system has a unique mean random attractor in the Bochner space \(L^2(\Omega , \ell ^2)\). When the drift and diffusion terms satisfy certain conditions, we show that the autonomous system has a unique, ergodic, mixing, and stable invariant probability measure in \(\ell ^2\). The idea of uniform tail-estimates is employed to establish the tightness of a family of distribution laws of the solutions in order to overcome the lack of compactness in infinite lattice as well as the infinite-dimensionalness of \(\ell ^2\). This work deepens and extends the results in Wang and Wang (Stoch Process Appl 130:7431–7462, 2020) where the coefficient of the noise grows linearly rather than superlinearly.

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References

  1. Arnold, L.: Stochastic Differential Equations: Theory and Applications. Wiley, New York (1974)

    MATH  Google Scholar 

  2. Arnold, L.: Random Dynamical Systems. Springer, Berlin (1998)

    Book  MATH  Google Scholar 

  3. Bates, P.W., Lisei, H., Lu, K.: Attractors for stochastic lattice dynamical systems. Stoch. Dyn. 6, 1–21 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  4. Bates, P.W., Lu, K., Wang, B.: Attractors of non-autonomous stochastic lattice systems in weighted spaces. Physica D 289, 32–50 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  5. Bell, J., Cosner, C.: Threshold behaviour and propagation for nonlinear differential-difference systems motivated by modeling myelinated axons. Quart. Appl. Math. 42, 1–14 (1984)

    Article  MathSciNet  MATH  Google Scholar 

  6. Brzeźniak, Z., Ondreját, M., Seidler, J.: Invariant measures for stochastic nonlinear beam and wave equations. J. Differ. Equ. 260, 4157–4179 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  7. Brzeźniak, Z., Motyl, E., Ondrejat, M.: Invariant measure for the stochastic Navier–Stokes equations in unbounded 2D domains. Ann. Probab. 45, 3145–3201 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  8. Caraballo, T., Morillas, F., Valerom, J.: Attractors of stochastic lattice dynamical systems with a multiplicative noise and non-Lipschitz nonlinearities. J. Differ. Equ. 253, 667–693 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  9. Caraballo, T., Han, X., Schmalfuss, B., Valero, J.: Random attractors for stochastic lattice dynamical systems with infinite multiplicative white noise. Nonlinear Anal. 130, 255–278 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  10. Caraballo, T., Guo, B., Tuan, N.H., Wang, R.: 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, 1700–1730 (2021)

    MathSciNet  MATH  Google Scholar 

  11. Chen, L., Dong, Z., Jiang, J., Zhai, J.: On limiting behavior of stationary measures for stochastic evolution systems with small noisy intensity. Sci. China Math. 63, 1463–1504 (2020)

    Article  MathSciNet  MATH  Google Scholar 

  12. Cheng, C., Feng, Z., Su, Y.: Global stability of traveling wave fronts for a reaction–diffusion system with a quiescent stage on a one-dimensional spatial lattice. Appl. Anal. 97, 2920–2940 (2018)

    Article  MathSciNet  MATH  Google Scholar 

  13. Chow, S.N., Mallet-Paret, J.: Pattern formation and spatial chaos in lattice dynamical systems, I, II. IEEE Trans. Circuits Syst. 42, 746–751 (1995)

    Article  MathSciNet  Google Scholar 

  14. Chua, L.O., Roska, T.: The CNN paradigm. IEEE Trans. Circuits Syst. 40, 147–156 (1993)

    Article  MATH  Google Scholar 

  15. Da Prato, G., Zabczyk, J.: Stochastic Equations in Infinite Dimensions. Cambridge University Press, Cambridge (1992)

    Book  MATH  Google Scholar 

  16. Da Prato, G., Zabczyk, J.: Ergodicity for Infinite Dimensional Systems, London Mathematical Society Lecture Notes, 229. Cambridge University Press, Cambridge (1996)

    Book  Google Scholar 

  17. Damascelli, L.: Comparison theorems for some quasilinear degenerate elliptic operators and applications to symmetry and monotonicity results. Ann. Inst. H. Poincaré Anal. Non Linéaire 15, 493–516 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  18. Debussche, A.: Ergodicity Results for the Stochastic Navier–Stokes Equations: An Introduction, Topics in Mathematical Fluid Mechanics, Volume 2073 of the Series Lecture Notes in Mathematics, pp. 23–108. Springer, Berlin (2013)

    Google Scholar 

  19. Erneux, T., Nicolis, G.: Propagating waves in discrete bistable reaction systems. Physica D 67, 237–244 (1993)

    Article  MathSciNet  MATH  Google Scholar 

  20. Gu, A., Li, Y.: Dynamic behavior of stochastic \(p\)-Laplacian-type lattice equations. Stoch. Dyn. 17, 1750040 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  21. Gu, A., Li, Y., Li, J.: Random attractor for stochastic lattice Fitzhugh–Nagumo system driven by \(\alpha \)-stable Lévy noises. Int. J. Bifurcat. Chaos 24, 1450123 (2014)

    Article  MATH  Google Scholar 

  22. Han, X.: Asymptotic behaviors for second order stochastic lattice dynamical systems on \({\mathbb{Z} }^k\) in weighted spaces. J. Math. Anal. Appl. 397, 242–254 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  23. Han, X., Kloeden, P.E.: Asymptotic behavior of a neural field lattice model with a Heaviside operator. Physica D 389, 1–12 (2019)

    Article  MathSciNet  MATH  Google Scholar 

  24. Han, X., Shen, W., Zhou, S.: Random attractors for stochastic lattice dynamical systems in weighted spaces. J. Differ. Equ. 250, 1235–1266 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  25. Kapval, R.: Discrete models for chemically reacting systems. J. Math. Chem. 6, 113–163 (1991)

    Article  MathSciNet  Google Scholar 

  26. Katok, A., Hasselblatt, B.: Introduction to the Modern Theory of Dynamical Systems. Cambridge University Press, Berlin (1996)

    MATH  Google Scholar 

  27. Keener, J.P.: Propagation and its failure in coupled systems of discrete excitable cells. SIAM J. Appl. Math. 47, 556–572 (1987)

    Article  MathSciNet  MATH  Google Scholar 

  28. Kim, J.: Periodic and invariant measures for stochastic wave equations. Electron. J. Differ. Equ. 2004(05), 1–30 (2004)

    MathSciNet  Google Scholar 

  29. Kim, J.: On the stochastic Burgers equation with polynomial nonlinearity in the real line. Discret. Contin. Dyn. Syst. Ser. B 6, 835–866 (2006)

    MathSciNet  MATH  Google Scholar 

  30. Kim, J.: On the stochastic Benjamin–Ono equation. J. Differ. Equ. 228, 737–768 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  31. Kim, J.: Invariant measures for a stochastic nonlinear Schrodinger equation. Indiana Univ. Math. J. 55, 687–717 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  32. Kloeden, P.E., Lorenz, T.: Mean-square random dynamical systems. J. Differ. Equ. 253, 1422–1438 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  33. Li, Y., Yi, Y.: Systematic measures of biological networks I: invariant measures and entropy. Commun. Pure Appl. Math. 69, 1777–1811 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  34. Li, D., Wang, B., Wang, X.: Limiting behavior of invariant measures of stochastic delay lattice systems. J. Dyn. Differ. Equ. 34, 1453–1487 (2022)

    Article  MathSciNet  MATH  Google Scholar 

  35. Mihailescu, E.: Ergodic properties for some non-expanding non-reversible systems. Nonlinear Anal. 73, 3779–3787 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  36. Misiats, O., Stanzhytskyi, O., Yip, N.: Existence and uniqueness of invariant measures for stochastic reaction–diffusion equations in unbounded domains. J. Theor. Probab. 29, 996–1026 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  37. Mohan, M.T.: Well posedness, large deviations and ergodicity of the stochastic 2D Oldroyd model of order one. Stoch. Process. Appl. 130, 4513–4562 (2020)

    Article  MathSciNet  MATH  Google Scholar 

  38. Mohan, M.T., Sakthivel, K., Sritharan, S.S.: Ergodicity for the 3D stochastic Navier–Stokes equations perturbed by Lévy noise. Math. Nachr. 292, 1056–1088 (2019)

    Article  MathSciNet  MATH  Google Scholar 

  39. Wang, B.: Attractors for reaction–diffusion equations in unbounded domains. Physica D 128, 41–52 (1999)

    Article  MathSciNet  MATH  Google Scholar 

  40. Wang, B.: Dynamics of systems on infinite lattices. J. Differ. Equ. 221, 224–245 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  41. Wang, B.: Weak pullback attractors for mean random dynamical systems in bochner spaces. J. Dyn. Differ. Equ. 31, 2177–2204 (2019)

    Article  MathSciNet  MATH  Google Scholar 

  42. Wang, B.: Dynamics of stochastic reaction-diffusion lattice system driven by nonlinear noise. J. Math. Anal. Appl. 477, 104–132 (2019)

    Article  MathSciNet  MATH  Google Scholar 

  43. Wang, R.: Long-time dynamics of stochastic lattice plate equations with nonlinear noise and damping. J. Dyn. Differ. Equ. 33, 767–803 (2021)

    Article  MathSciNet  MATH  Google Scholar 

  44. Wang, R., Wang, B.: Random dynamics of \(p\)-Laplacian lattice systems driven by infinite-dimensional nonlinear noise. Stoch. Process. Appl. 130, 7431–7462 (2020)

    Article  MathSciNet  MATH  Google Scholar 

  45. Wang, R., Wang, B.: Global well-posedness and long-term behavior of discrete reaction–diffusion equations driven by superlinear noise. Stoch. Anal. Appl. 39, 667–696 (2021)

    Article  MathSciNet  MATH  Google Scholar 

  46. Wang, X., Lu, K., Wang, B.: Exponential stability of non-autonomous stochastic delay lattice systems with multiplicative noise. J. Dyn. Differ. Equ. 28, 1309–1335 (2016)

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  50. Zhang, C., Zhao, L.: The attractors for \(2\)nd-order stochastic delay lattice systems. Discret. Contin. Dyn. Syst. 37, 575–590 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  51. Zhao, W., Zhang, Y.: Compactness and attracting of random attractors for non-autonomous stochastic lattice dynamical systems in weighted space \(\ell ^p_\rho \). Appl. Math. Comput. 291, 226–243 (2016)

    MathSciNet  Google Scholar 

  52. Zhao, L., Zhang, C., Li, D.: Global attractor for a class of retarded lattice dynamical systems. J. Math. Anal. Appl. 425, 178–193 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  53. Zhao, M., Zhou, S., Sheng, F.: Random attractor for nonautonomous stochastic Boussinesq lattice equations with additive white noises. Acta Math. Sci. Ser. A 38, 924–940 (2018)

    MathSciNet  MATH  Google Scholar 

  54. Zhou, S.: Random exponential attractor for cocycle and application to non-autonomous stochastic lattice systems with multiplicative white noise. J. Differ. Equ. 263, 2247–2279 (2017)

    Article  MathSciNet  MATH  Google Scholar 

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Acknowledgements

The first author would like to express his deep thanks to Professor Renhai Wang for helpful discussions. The authors would like to thank the referees for their valuable comments and suggestions, which improved the quality of this paper.

Funding

Pengyu Chen was supported by the National Natural Science Foundation of China (No. 12061063), the Outstanding Youth Science Fund of Gansu Province (No. 21JR7RA159) and Project of NWNU-LKQN2019-3. M.M. Freitas thanks the CNPq for financial support through the project Attractors and asymptotic behavior of non-linear evolution equations by Grant 313081/2021-2. Xuping Zhang was supported by Project of NWNU-LKQN2019-13.

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Authors and Affiliations

  1. Department of Mathematics, Northwest Normal University, Lanzhou, 730070, China

    Pengyu Chen & Xuping Zhang

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

    Mirelson M. Freitas

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  1. Pengyu Chen
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Chen, P., Freitas, M.M. & Zhang, X. Random Attractor, Invariant Measures, and Ergodicity of Lattice p-Laplacian Equations Driven by Superlinear Noise. J Geom Anal 33, 98 (2023). https://doi.org/10.1007/s12220-022-01175-9

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