# American Institute of Mathematical Sciences

June  2021, 41(6): 2725-2737. doi: 10.3934/dcds.2020383

## Non-autonomous stochastic evolution equations with nonlinear noise and nonlocal conditions governed by noncompact evolution families

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

Received  April 2020 Revised  July 2020 Published  June 2021 Early access  November 2020

Fund Project: Research supported by the National Natural Science Foundations of China (No. 12061063, No. 11661071), Project of NWNU-LKQN2019-3 and China Scholarship Council (No. 201908625016)

In this paper, we investigate the non-autonomous stochastic evolution equations of parabolic type with nonlinear noise and nonlocal initial conditions in Hilbert spaces, where the operators in linear part depend on time $t$ and generate an noncompact evolution family. New existence result of mild solutions is established under some weaker growth and measure of noncompactness conditions on nonlinear functions and nonlocal term. The discussions are based on Sadovskii's fixed-point theorem as well as the theory of evolution family. At last, as a sample of application, the obtained abstract result is applied to a class of non-autonomous stochastic partial differential equations of parabolic type with nonlocal initial conditions. The result obtained in this paper is a supplement to the existing literature and essentially extends some existing results in this area.

Citation: Pengyu Chen. Non-autonomous stochastic evolution equations with nonlinear noise and nonlocal conditions governed by noncompact evolution families. Discrete and Continuous Dynamical Systems, 2021, 41 (6) : 2725-2737. doi: 10.3934/dcds.2020383
##### References:
 [1] P. Acquistapace, Evolution operators and strong solution of abstract parabolic equations, Differential Integral Equations, 1 (1988), 433-457. [2] P. Acquistapace and B. Terreni, A unified approach to abstract linear parabolic equations, Rend. Semin. Mat. Univ. Padova, 78 (1987), 47-107. [3] H. Amann, Parabolic evolution equations and nonlinear boundary conditions, J. Differential Equations, 72 (1988), 201-269.  doi: 10.1016/0022-0396(88)90156-8. [4] J. Banaś and K. Goebel, Measures of Noncompactness in Banach Spaces, In Lecture Notes in Pure and Applied Mathematics, Volume 60, Marcel Dekker, New York, 1980. [5] L. Byszewski, Application of preperties of the right hand sides of evolution equations to an investigation of nonlocal evolution problems, Nonlinear Anal., 33 (1998), 413-426.  doi: 10.1016/S0362-546X(97)00594-4. [6] P. Chen, A. Abdelmonem and Y. Li, Global existence and asymptotic stability of mild solutions for stochastic evolution equations with nonlocal initial conditions, J. Integral Equations Appl., 29 (2017), 325-348.  doi: 10.1216/JIE-2017-29-2-325. [7] P. Chen and Y. Li, Monotone iterative technique for a class of semilinear evolution equations with nonlocal conditions, Results Math., 63 (2013), 731-744.  doi: 10.1007/s00025-012-0230-5. [8] P. Chen and Y. Li, Nonlocal Cauchy problem for fractional stochastic evolution equations in Hilbert spaces, Collect. Math., 66 (2015), 63-76.  doi: 10.1007/s13348-014-0106-y. [9] P. Chen, X. Zhang and Y. Li, Nonlocal problem for fractional stochastic evolution equations with solution operators, Fract. Calcu. Appl. Anal., 19 (2016), 1507-1526.  doi: 10.1515/fca-2016-0078. [10] P. Chen, X. Zhang and Y. Li, Approximate controllability of non-autonomous evolution system with nonlocal conditions, J. Dyn. Control. Syst., 26 (2020), 1-16.  doi: 10.1007/s10883-018-9423-x. [11] P. Chen, X. Zhang and Y. Li, Fractional non-autonomous evolution equation with nonlocal conditions, J. Pseudo-Differ. Oper. Appl., 10 (2019), 955-973.  doi: 10.1007/s11868-018-0257-9. [12] P. Chen, X. Zhang and Y. Li, Cauchy problem for fractional non-autonomous evolution equations, Banach J. Math. Anal., 14 (2020), 559-584.  doi: 10.1007/s43037-019-00008-2. [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] J. Cui, L. Yan and X. Wu, Nonlocal Cauchy problem for some stochastic integro-differential equations in Hilbert spaces, J. Korean Stat. Soci., 41 (2012), 279-290.  doi: 10.1016/j.jkss.2011.10.001. [15] R. F. Curtain and P. L. Falb, Stochastic differential equations in Hilbert space, J. Differential Equations, 10 (1971), 412-430.  doi: 10.1016/0022-0396(71)90004-0. [16] G. Da Prato and J. Zabczyk, Stochastic Equations in Infinite Dimensions, Cambridge University Press, Cambridge, 1992.  doi: 10.1017/CBO9780511666223. [17] K. Deimling, Nonlinear Functional Analysis, Springer-Verlag, New York, 1985. doi: 10.1007/978-3-662-00547-7. [18] K. Ezzinbi, X. Fu and K. Hilal, Existence and regularity in the $\alpha$-norm for some neutral partial differential equations with nonlocal conditions, Nonlinear Anal., 67 (2007), 1613-1622.  doi: 10.1016/j.na.2006.08.003. [19] Z. Fan and G. Li, Existence results for semilinear differential equations with nonlocal and impulsive conditions, J. Funct. Anal., 258 (2010), 1709-1727.  doi: 10.1016/j.jfa.2009.10.023. [20] X. Fu, Approximate controllability of semilinear non-autonomous evolution systems with state-dependent delay, Evol. Equ. Control Theory, 6 (2017), 517-534.  doi: 10.3934/eect.2017026. [21] H.-P. Heinz, On the behaviour of measure of noncompactness with respect to differentiation and integration of vector-valued functions, Nonlinear Anal., 7 (1983), 1351-1371.  doi: 10.1016/0362-546X(83)90006-8. [22] D. Henry, Geometric Theory of Semilinear Parabolic Equations, Lecture Notes in Math., vol. 840, Springer-Verlag, New York, 1981. [23] J. Liang, J. H. Liu and T.-J. Xiao, Nonlocal impulsive problems for nonlinear differential equations in Banach spaces, Math. Comput. Modelling, 49 (2009), 798-804.  doi: 10.1016/j.mcm.2008.05.046. [24] J. Liang, J. H. Liu and T.-J. Xiao, Nonlocal Cauchy problems for nonautonomous evolution equations, Commun. Pure Appl. Anal., 5 (2006), 529-535.  doi: 10.3934/cpaa.2006.5.529. [25] K. Liu, Stability of Infinite Dimensional Stochastic Differential Equations with Applications, Chapman and Hall, London, 2006. [26] X. Mao, Stochastic Differential Equations and their Applications, Horwood Publishing Ltd., Chichester, 1997. [27] A. Pazy, Semigroups of Linear Operators and Applications to Partial Differential Equations, Springer-Verlag, Berlin, 1983. doi: 10.1007/978-1-4612-5561-1. [28] I. I. Vrabie, Delay evolution equations with mixed nonlocal plus local initial conditions, Commun. Contemp. Math., 17 (2015), 1350035. doi: 10.1142/S0219199713500351. [29] J. Wang, Approximate mild solutions of fractional stochastic evolution equations in Hilbert spaces, Appl. Math. Comput., 256 (2015), 315-323.  doi: 10.1016/j.amc.2014.12.155. [30] R. N. Wang and P. X. Zhu, Non-autonomous evolution inclusions with nonlocal history conditions: Global integral solutions, Nonlinear Anal., 85 (2013), 180-191.  doi: 10.1016/j.na.2013.02.026.

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##### References:
 [1] P. Acquistapace, Evolution operators and strong solution of abstract parabolic equations, Differential Integral Equations, 1 (1988), 433-457. [2] P. Acquistapace and B. Terreni, A unified approach to abstract linear parabolic equations, Rend. Semin. Mat. Univ. Padova, 78 (1987), 47-107. [3] H. Amann, Parabolic evolution equations and nonlinear boundary conditions, J. Differential Equations, 72 (1988), 201-269.  doi: 10.1016/0022-0396(88)90156-8. [4] J. Banaś and K. Goebel, Measures of Noncompactness in Banach Spaces, In Lecture Notes in Pure and Applied Mathematics, Volume 60, Marcel Dekker, New York, 1980. [5] L. Byszewski, Application of preperties of the right hand sides of evolution equations to an investigation of nonlocal evolution problems, Nonlinear Anal., 33 (1998), 413-426.  doi: 10.1016/S0362-546X(97)00594-4. [6] P. Chen, A. Abdelmonem and Y. Li, Global existence and asymptotic stability of mild solutions for stochastic evolution equations with nonlocal initial conditions, J. Integral Equations Appl., 29 (2017), 325-348.  doi: 10.1216/JIE-2017-29-2-325. [7] P. Chen and Y. Li, Monotone iterative technique for a class of semilinear evolution equations with nonlocal conditions, Results Math., 63 (2013), 731-744.  doi: 10.1007/s00025-012-0230-5. [8] P. Chen and Y. Li, Nonlocal Cauchy problem for fractional stochastic evolution equations in Hilbert spaces, Collect. Math., 66 (2015), 63-76.  doi: 10.1007/s13348-014-0106-y. [9] P. Chen, X. Zhang and Y. Li, Nonlocal problem for fractional stochastic evolution equations with solution operators, Fract. Calcu. Appl. Anal., 19 (2016), 1507-1526.  doi: 10.1515/fca-2016-0078. [10] P. Chen, X. Zhang and Y. Li, Approximate controllability of non-autonomous evolution system with nonlocal conditions, J. Dyn. Control. Syst., 26 (2020), 1-16.  doi: 10.1007/s10883-018-9423-x. [11] P. Chen, X. Zhang and Y. Li, Fractional non-autonomous evolution equation with nonlocal conditions, J. Pseudo-Differ. Oper. Appl., 10 (2019), 955-973.  doi: 10.1007/s11868-018-0257-9. [12] P. Chen, X. Zhang and Y. Li, Cauchy problem for fractional non-autonomous evolution equations, Banach J. Math. Anal., 14 (2020), 559-584.  doi: 10.1007/s43037-019-00008-2. [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] J. Cui, L. Yan and X. Wu, Nonlocal Cauchy problem for some stochastic integro-differential equations in Hilbert spaces, J. Korean Stat. Soci., 41 (2012), 279-290.  doi: 10.1016/j.jkss.2011.10.001. [15] R. F. Curtain and P. L. Falb, Stochastic differential equations in Hilbert space, J. Differential Equations, 10 (1971), 412-430.  doi: 10.1016/0022-0396(71)90004-0. [16] G. Da Prato and J. Zabczyk, Stochastic Equations in Infinite Dimensions, Cambridge University Press, Cambridge, 1992.  doi: 10.1017/CBO9780511666223. [17] K. Deimling, Nonlinear Functional Analysis, Springer-Verlag, New York, 1985. doi: 10.1007/978-3-662-00547-7. [18] K. Ezzinbi, X. Fu and K. Hilal, Existence and regularity in the $\alpha$-norm for some neutral partial differential equations with nonlocal conditions, Nonlinear Anal., 67 (2007), 1613-1622.  doi: 10.1016/j.na.2006.08.003. [19] Z. Fan and G. Li, Existence results for semilinear differential equations with nonlocal and impulsive conditions, J. Funct. Anal., 258 (2010), 1709-1727.  doi: 10.1016/j.jfa.2009.10.023. [20] X. Fu, Approximate controllability of semilinear non-autonomous evolution systems with state-dependent delay, Evol. Equ. Control Theory, 6 (2017), 517-534.  doi: 10.3934/eect.2017026. [21] H.-P. Heinz, On the behaviour of measure of noncompactness with respect to differentiation and integration of vector-valued functions, Nonlinear Anal., 7 (1983), 1351-1371.  doi: 10.1016/0362-546X(83)90006-8. [22] D. Henry, Geometric Theory of Semilinear Parabolic Equations, Lecture Notes in Math., vol. 840, Springer-Verlag, New York, 1981. [23] J. Liang, J. H. Liu and T.-J. Xiao, Nonlocal impulsive problems for nonlinear differential equations in Banach spaces, Math. Comput. Modelling, 49 (2009), 798-804.  doi: 10.1016/j.mcm.2008.05.046. [24] J. Liang, J. H. Liu and T.-J. Xiao, Nonlocal Cauchy problems for nonautonomous evolution equations, Commun. Pure Appl. Anal., 5 (2006), 529-535.  doi: 10.3934/cpaa.2006.5.529. [25] K. Liu, Stability of Infinite Dimensional Stochastic Differential Equations with Applications, Chapman and Hall, London, 2006. [26] X. Mao, Stochastic Differential Equations and their Applications, Horwood Publishing Ltd., Chichester, 1997. [27] A. Pazy, Semigroups of Linear Operators and Applications to Partial Differential Equations, Springer-Verlag, Berlin, 1983. doi: 10.1007/978-1-4612-5561-1. [28] I. I. Vrabie, Delay evolution equations with mixed nonlocal plus local initial conditions, Commun. Contemp. Math., 17 (2015), 1350035. doi: 10.1142/S0219199713500351. [29] J. Wang, Approximate mild solutions of fractional stochastic evolution equations in Hilbert spaces, Appl. Math. Comput., 256 (2015), 315-323.  doi: 10.1016/j.amc.2014.12.155. [30] R. N. Wang and P. X. Zhu, Non-autonomous evolution inclusions with nonlocal history conditions: Global integral solutions, Nonlinear Anal., 85 (2013), 180-191.  doi: 10.1016/j.na.2013.02.026.
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