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

Mengyu Cheng; Zhenxin Liu

doi:10.3934/dcdsb.2021026

Article Contents

2021, Volume 26, Issue 12: 6425-6462. Doi: 10.3934/dcdsb.2021026 open access

This issue Previous Article The multi-patch logistic equation Next Article Dynamics of a vector-host model under switching environments

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

Mengyu Cheng and
Zhenxin Liu^,

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

^* Corresponding author: Zhenxin Liu
^* Corresponding author: Zhenxin Liu

Received: November 2020

Early access: January 2021

Published: December 2021

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

Abstract Full Text(HTML) Related Papers Cited by

Abstract

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.

Keywords:

Mathematics Subject Classification: Primary: 37B20, 37L15, 60H15; Secondary: 35B10, 35B15, 35B35.

Citation:

FullText(HTML)

Related Papers

Cited by

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.
[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.
[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.
[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.
[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.
[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.
[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.
[8]	S. Bochner, Curvature and Betti numbers in real and complex vector bundles, Rend. Semin. Mat. Univ. Politec. Torino, 15 (1955/1956), 225-253.
[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.
[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).
[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.
[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.
[13]	F. Chen, Y. Han, Y. 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.
[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.
[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.
[16]	I. Ciotir, A Trotter-type theorem for nonlinear stochastic equations in variational formulation and homogenization, Differential Integral Equations, 24 (2011), 371-388.
[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.
[18]	G. Da Prato, M. Röckner, B. 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.
[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.
[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.
[21]	A. Es-Sarhir, M. 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.
[22]	J. Favard, Sur les équations différentielles linéaires à coefficients presque-périodiques, Acta Math., 51 (1928), 31–81 (in French). doi: 10.1007/BF02545660.
[23]	J. Favard, Lecons sur les Fonctions Presque-Périodiques., Gauthier-Villars, Paris, 1933.
[24]	A. M. Fink, Almost Periodic Differential Equations, Lecture Notes in Math., vol. 377, Springer-Verlag, Berlin-New York, 1974. viii+336 pp.
[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.
[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.
[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.
[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.
[29]	B. Gess, W. 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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[37]	M. Kamenskii, O. 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.
[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)
[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.
[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.
[41]	Y. Li, Z. 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.
[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).
[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.
[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.
[45]	X. Liu and Z. Liu, Poisson stable solutions for stochastic differential equations with Lévy noise, Acta Math. Sin. (Engl. Ser.), to appear.
[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.
[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.
[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).
[49]	K. R. Parthasarathy, Probability Measures on Metric Spaces, Probability and Mathematical Statistics, No. 3 Academic Press, Inc., New York-London, 1967.
[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.
[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.
[52]	M. Röckner, B. Schmuland and X. Zhang, Yamada-Watanabe Theorem for stochastic evolution equations in infinite dimensions, Cond. Matt. Phys., 11 (2008), 247-259.
[53]	R. J. Sacker and G. R. Sell, Lifting properties in skew-product flows with applications to differential equations, Mem. Amer. Math. Soc., 11 (1977), iv+67 pp. doi: 10.1090/memo/0190.
[54]	W. Shen and Y. Yi, Almost automorphic and almost periodic dynamics in skew-product semiflows, Mem. Amer. Math. Soc., 136 (1998), x+93 pp. doi: 10.1090/memo/0647.
[55]	E. R. van Kampen, Almost periodic functions and compact groups, Ann. of Math., 37 (1936), 78-91. doi: 10.2307/1968688.
[56]	W. A. Veech, Almost automorphic functions on groups, Amer. J. Math., 87 (1965), 719-751. doi: 10.2307/2373071.
[57]	J. von Neumann, Almost periodic functions in a group. I, Trans. Amer. Math. Soc., 36 (1934), 445-492. doi: 10.2307/1989792.
[58]	Y. Wang and Z. Liu, Almost periodic solutions for stochastic differential equations with Lévy noise, Nonlinearity, 25 (2012), 2803-2821. doi: 10.1088/0951-7715/25/10/2803.
[59]	T. Yoshizawa, Stability Theory and the Existence of Periodic Solutions and Almost Periodic Solutions, Applied Mathematical Sciences, Vol. 14. Springer-Verlag, New York-Heidelberg, 1975. vii+233 pp.
[60]	X. Zhang, On stochastic evolution equations with non-Lipschitz coefficients, Stoch. Dyn., 9 (2009), 549-595. doi: 10.1142/S0219493709002774.