December  2021, 26(12): 6425-6462. doi: 10.3934/dcdsb.2021026

Periodic, almost periodic and almost automorphic solutions for SPDEs with monotone coefficients

School of Mathematical Sciences, Dalian University of Technology, Dalian 116024, China

* Corresponding author: Zhenxin Liu

Received  November 2020 Published  December 2021 Early access  January 2021

Fund Project: The authors sincerely thank the referee for his/her careful reading of the paper. This work is partially supported by NSFC Grants 11522104, 11871132, 11925102, and Xinghai Jieqing and DUT19TD14 funds from Dalian University of Technology

In this paper, we use the variational approach to investigate recurrent properties of solutions for stochastic partial differential equations, which is in contrast to the previous semigroup framework. Consider stochastic differential equations with monotone coefficients. Firstly, we establish the continuous dependence on initial values and coefficients for solutions, which is interesting in its own right. Secondly, we prove the existence of recurrent solutions, which include periodic, almost periodic and almost automorphic solutions. Then we show that these recurrent solutions are globally asymptotically stable in square-mean sense. Finally, for illustration of our results we give two applications, i.e. stochastic reaction diffusion equations and stochastic porous media equations.

Citation: Mengyu Cheng, Zhenxin Liu. Periodic, almost periodic and almost automorphic solutions for SPDEs with monotone coefficients. Discrete & Continuous Dynamical Systems - B, 2021, 26 (12) : 6425-6462. doi: 10.3934/dcdsb.2021026
References:
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S. Agmon, Lectures on Elliptic Boundary Value Problems, Prepared for publication by B. Frank Jones, Jr. with the assistance of George W. Batten, Jr. Van Nostrand Mathematical Studies, No. 2 D. Van Nostrand Co., Inc., Princeton, N.J.-Toronto-London, 1965. v+291 pp.  Google Scholar

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B. GessW. Liu and M. Röckner, Random attractors for a class of stochastic partial differential equations driven by general additive noise, J. Differential Equations, 251 (2011), 1225-1253.  doi: 10.1016/j.jde.2011.02.013.  Google Scholar

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B. Gess and M. Röckner, Stochastic variational inequalities and regularity for degenerate stochastic partial differential equations, Trans. Amer. Math. Soc., 369 (2017), 3017-3045.  doi: 10.1090/tran/6981.  Google Scholar

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B. Gess and J. M. Tölle, Stability of solutions to stochastic partial differential equations, J. Differential Equations, 260 (2016), 4973-5025.  doi: 10.1016/j.jde.2015.11.039.  Google Scholar

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M. KamenskiiO. Mellah and P. Raynaud de Fitte, Weak averaging of semilinear stochastic differential equations with almost periodic coefficients, J. Math. Anal. Appl., 427 (2015), 336-364.  doi: 10.1016/j.jmaa.2015.02.036.  Google Scholar

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show all references

References:
[1]

S. Agmon, Lectures on Elliptic Boundary Value Problems, Prepared for publication by B. Frank Jones, Jr. with the assistance of George W. Batten, Jr. Van Nostrand Mathematical Studies, No. 2 D. Van Nostrand Co., Inc., Princeton, N.J.-Toronto-London, 1965. v+291 pp.  Google Scholar

[2]

L. Amerio and G. Prouse, Almost-Periodic Functions and Functional Equations, Van Nostrand Reinhold Co., New York-Toronto, Ont.-Melbourne, 1971. viii+184 pp.  Google Scholar

[3]

L. Arnold and C. Tudor, Stationary and almost periodic solutions of almost periodic affine stochastic differential equations, Stochastics Stochastics Rep., 64 (1998), 177-193.  doi: 10.1080/17442509808834163.  Google Scholar

[4]

V. Barbu and G. Da Prato, Ergodicity for nonlinear stochastic equations in variational formulation, Appl. Math. Optim., 53 (2006), 121-139.  doi: 10.1007/s00245-005-0838-x.  Google Scholar

[5]

V. Barbu and M. Röckner, Stochastic variational inequalities and applications to the total variation flow perturbed by linear multiplicative noise, Arch. Ration. Mech. Anal., 209 (2013), 797-834.  doi: 10.1007/s00205-013-0632-x.  Google Scholar

[6]

P. H. Bezandry and T. Diagana, Existence of almost periodic solutions to some stochastic differential equations, Appl. Anal., 86 (2007), 819-827.  doi: 10.1080/00036810701397788.  Google Scholar

[7]

S. Bochner, Beiträge zur theorie der fastperiodischen funktionen, I. Funktionen einer Variablen, Math. Ann., 96 (1927), 119–147 (in German). doi: 10.1007/BF01209156.  Google Scholar

[8]

S. Bochner, Curvature and Betti numbers in real and complex vector bundles, Rend. Semin. Mat. Univ. Politec. Torino, 15 (1955/1956), 225-253.   Google Scholar

[9]

S. Bochner, A new approach to almost periodicity, Proc. Nat. Acad. Sci. U.S.A., 48 (1962), 2039-2043.  doi: 10.1073/pnas.48.12.2039.  Google Scholar

[10]

V. I. Bogachev, G. Da Prato and M. Röckner, Invariant measures of generalized stochastic equations of porous media, Dokl. Akad. Nauk, 396 (2004), 7–11 (in Russian).  Google Scholar

[11]

H. Bohr, Zur theorie der fastperiodischen funktionen. I, Acta Math., 45 (1924), 29–127; II, Acta Math., 46 (1925), 101–214; III, Acta Math., 47 (1926), 237–281. (All in German) doi: 10.1007/BF02543859.  Google Scholar

[12]

D. Cheban and Z. Liu, Periodic, qusi-periodic, almost periodic, almost automorphic, Birkhoff recurrent and Poisson stable solutions for stochastic differential equations, J. Differential Equations, 269 (2020), 3652-3685.  doi: 10.1016/j.jde.2020.03.014.  Google Scholar

[13]

F. ChenY. HanY. Li and X. Yang, Periodic solutions of Fokker-Planck equations, J. Differential Equations, 263 (2017), 285-298.  doi: 10.1016/j.jde.2017.02.032.  Google Scholar

[14]

Z. Chen and W. Lin, Square-mean weighted pseudo almost automorphic solutions for non-autonomous stochastic evolution equations, J. Math. Pures Appl., 100 (2013), 476-504.  doi: 10.1016/j.matpur.2013.01.010.  Google Scholar

[15]

I. Ciotir, A Trotter type result for the stochastic porous media equations, Nonlinear Anal., 71 (2009), 5606-5615.  doi: 10.1016/j.na.2009.04.054.  Google Scholar

[16]

I. Ciotir, A Trotter-type theorem for nonlinear stochastic equations in variational formulation and homogenization, Differential Integral Equations, 24 (2011), 371-388.   Google Scholar

[17]

I. Ciotir and J. M. Tölle, Convergence of invariant measures for singular stochastic diffusion equations, Stochastic Process. Appl., 123 (2013), 1178-1181.  doi: 10.1016/j.spa.2012.10.009.  Google Scholar

[18]

G. Da PratoM. RöcknerB. L. Rozovskii and F. Wang, Strong solutions of stochastic generalized porous media equations: Existence, uniqueness, and ergodicity, Comm. Partial Differential Equations, 31 (2006), 277-291.  doi: 10.1080/03605300500357998.  Google Scholar

[19]

G. Da Prato and C. Tudor, Periodic and almost periodic solutions for semilinear stochastic equations, Stochastic Anal. Appl., 13 (1995), 13-33.  doi: 10.1080/07362999508809380.  Google Scholar

[20] R. M. Dudley, Real Analysis and Probability, Revised reprint of the 1989 original. Cambridge Studies in Advanced Mathematics, 74. Cambridge University Press, Cambridge, 2002.  doi: 10.1017/CBO9780511755347.  Google Scholar
[21]

A. Es-SarhirM. K. von Renesse and W. Stannat, Estimates for the ergodic measure and polynomial stability of plane stochastic curve shortening flow, NoDEA Nonlinear Differential Equations Appl., 19 (2012), 663-675.  doi: 10.1007/s00030-011-0146-x.  Google Scholar

[22]

J. Favard, Sur les équations différentielles linéaires à coefficients presque-périodiques, Acta Math., 51 (1928), 31–81 (in French). doi: 10.1007/BF02545660.  Google Scholar

[23]

J. Favard, Lecons sur les Fonctions Presque-Périodiques., Gauthier-Villars, Paris, 1933. Google Scholar

[24]

A. M. Fink, Almost Periodic Differential Equations, Lecture Notes in Math., vol. 377, Springer-Verlag, Berlin-New York, 1974. viii+336 pp.  Google Scholar

[25]

M. Fu and Z. Liu, Square-mean almost automorphic solutions for some stochastic differential equations, Proc. Amer. Math. Soc., 138 (2010), 3689-3701.  doi: 10.1090/S0002-9939-10-10377-3.  Google Scholar

[26]

P. Gao, Some periodic type solutions for stochastic reaction-diffusion equation with cubic nonlinearities, Comput. Math. Appl., 74 (2017), 2281-2297.  doi: 10.1016/j.camwa.2017.07.005.  Google Scholar

[27]

B. Gess, Random attractors for singular stochastic evolution equations, J. Differential Equations, 255 (2013), 524-559.  doi: 10.1016/j.jde.2013.04.023.  Google Scholar

[28]

B. Gess, Random attractors for degenerate stochastic partial differential equations, J. Dynam. Differential Equations, 25 (2013), 121-157.  doi: 10.1007/s10884-013-9294-5.  Google Scholar

[29]

B. GessW. Liu and M. Röckner, Random attractors for a class of stochastic partial differential equations driven by general additive noise, J. Differential Equations, 251 (2011), 1225-1253.  doi: 10.1016/j.jde.2011.02.013.  Google Scholar

[30]

B. Gess and M. Röckner, Singular-degenerate multivalued stochastic fast diffusion equations, SIAM J. Math. Anal., 47 (2015), 4058-4090.  doi: 10.1137/151003726.  Google Scholar

[31]

B. Gess and M. Röckner, Stochastic variational inequalities and regularity for degenerate stochastic partial differential equations, Trans. Amer. Math. Soc., 369 (2017), 3017-3045.  doi: 10.1090/tran/6981.  Google Scholar

[32]

B. Gess and J. M. Tölle, Multi-valued, singular stochastic evolution inclusions, J. Math. Pures Appl. (9), 101 (2014), 789–827. doi: 10.1016/j.matpur.2013.10.004.  Google Scholar

[33]

B. Gess and J. M. Tölle, Stability of solutions to stochastic partial differential equations, J. Differential Equations, 260 (2016), 4973-5025.  doi: 10.1016/j.jde.2015.11.039.  Google Scholar

[34]

A. Halanay, Periodic and almost periodic solutions to affine stochastic systems., Proceedings of the Eleventh International Conference on Nonlinear Oscillations (Budapest, 1987), 94–101, János Bolyai Math. Soc., Budapest, 1987.  Google Scholar

[35]

M. Ji, W. Qi, Z. Shen and Y. Yi, Existence of periodic probability solutions to Fokker-Planck equations with applications, J. Funct. Anal., 277 (2019), Art. 108281, 41 pp. doi: 10.1016/j.jfa.2019.108281.  Google Scholar

[36]

R. A. Johnson, A linear, almost periodic equation with an almost automorphic solution, Proc. Amer. Math. Soc., 82 (1981), 199-205.  doi: 10.1090/S0002-9939-1981-0609651-0.  Google Scholar

[37]

M. KamenskiiO. Mellah and P. Raynaud de Fitte, Weak averaging of semilinear stochastic differential equations with almost periodic coefficients, J. Math. Anal. Appl., 427 (2015), 336-364.  doi: 10.1016/j.jmaa.2015.02.036.  Google Scholar

[38]

R. Has'minskiǐ, Stochastic Stability of Differential Equations, Translated from the Russian by D. Louvish. Sijthoff & Noordhoff, Alphen aan den Rijn–Germantown, Md., 1980. xvi+344 pp. (see also 2nd ed., Springer, New York, 2012. xviii+339)  Google Scholar

[39]

N. V. Krylov and B. L. Rozovskiǐ, Stochastic evolution equations, Current Problems in Mathematics, Vol. 14 (Russian), Akad. Nauk SSSR, Vsesoyuz. Inst. Nauchn. i Tekhn. Informatsii, Moscow, 256 (1979), 71–147.  Google Scholar

[40] B. M. Levitan and V. V. Zhikov, Almost Periodic Functions and Differential Equations, Translated from the Russian by L. W. Longdon. Cambridge University Press, Cambridge-New York, 1982.   Google Scholar
[41]

Y. LiZ. Liu and W. Wang, Almost periodic solutions and stable solutions for stochastic differential equations, Discrete Contin. Dyn. Syst. Ser. B, 24 (2019), 5927-5944.  doi: 10.3934/dcdsb.2019113.  Google Scholar

[42]

J. L. Lions, Équations Différentielles Opérationelles et Problémes aux Limites, Die Grundlehren der mathematischen Wissenschaften, Bd. 111 Springer-Verlag, Berlin-Göttingen-Heidelberg, 1961. ix+292 pp (in French).  Google Scholar

[43]

W. Liu, Invariance of subspaces under the solution flow of SPDE, Infin. Dimens. Anal. Quantum Probab. Relat. Top., 13 (2010), 87-98.  doi: 10.1142/S021902571000395X.  Google Scholar

[44]

W. Liu and J. M. Tölle, Existence and uniqueness of invariant measures for stochastic evolution equations with weakly dissipative drifts, Electron. Commun. Probab., 16 (2011), 447-457.  doi: 10.1214/ECP.v16-1643.  Google Scholar

[45]

X. Liu and Z. Liu, Poisson stable solutions for stochastic differential equations with Lévy noise, Acta Math. Sin. (Engl. Ser.), to appear. Google Scholar

[46]

Z. Liu and K. Sun, Almost automorphic solutions for stochastic differential equations driven by Lévy noise, J. Funct. Anal., 266 (2014), 1115-1149.  doi: 10.1016/j.jfa.2013.11.011.  Google Scholar

[47]

Z. Liu and W. Wang, Favard separation method for almost periodic stochastic differential equations, J. Differential Equations, 260 (2016), 8109-8136.  doi: 10.1016/j.jde.2016.02.019.  Google Scholar

[48]

E. Pardoux, Équations aux dérivées partielles stochastiques de type monotone, Séminaire sur les Équations aux Dérivées Partielles (1974–1975), III, Exp. No. 2, 10 pp. Collége de France, Paris, 1975 (in French).  Google Scholar

[49] K. R. Parthasarathy, Probability Measures on Metric Spaces, Probability and Mathematical Statistics, No. 3 Academic Press, Inc., New York-London, 1967.   Google Scholar
[50]

A. Pazy, Semigroups of Linear Operators and Applications to Partial Differential Equations., Applied Mathematical Sciences, 44. Springer-Verlag, New York, 1983. viii+279 pp. doi: 10.1007/978-1-4612-5561-1.  Google Scholar

[51]

C. Prévôt and M. Röckner, A Concise Course on Stochastic Partial Differential Equations, Lecture Notes in Mathematics, 1905. Springer, Berlin, 2007. vi+144 pp.  Google Scholar

[52]

M. RöcknerB. Schmuland and X. Zhang, Yamada-Watanabe Theorem for stochastic evolution equations in infinite dimensions, Cond. Matt. Phys., 11 (2008), 247-259.   Google Scholar

[53]

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