# American Institute of Mathematical Sciences

## Non-autonomous stochastic evolution equations of parabolic type with nonlocal initial conditions

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

* Corresponding author: Pengyu Chen

Received  May 2020 Published  October 2020

Fund Project: Research supported by National Natural Science Foundation of China (No. 12061063), Project of NWNU-LKQN2019-3, Project of NWNU-LKQN2019-13 and China Scholarship Council (No. 201908625016)

In this paper, we study the non-autonomous stochastic evolution equations of parabolic type with nonlocal initial conditions in Hilbert spaces, where the operators in linear part (possibly unbounded) depend on time $t$ and generate an evolution family. New existence result of mild solutions is established under more weaker conditions by introducing a new Green's function. The discussions are based on Schauder's fixed-point theorem as well as the theory of evolution family. At last, an example is also given to illustrate the feasibility of our theoretical results. 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, Xuping Zhang. Non-autonomous stochastic evolution equations of parabolic type with nonlocal initial conditions. Discrete & Continuous Dynamical Systems - B, doi: 10.3934/dcdsb.2020308
##### References:
 [1] P. Acquistapace, Evolution operators and strong solution of abstract parabolic equations, Differential Integral Equations, 1 (1988), 433-457.   Google Scholar [2] P. Acquistapace and B. Terreni, A unified approach to abstract linear parabolic equations, Rend. Semin. Mat. Univ. Padova, 78 (1987), 47-107.   Google Scholar [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.  Google Scholar [4] J. Bao, Z. Hou and C. Yuan, Stability in distribution of mild solutions to stochastic partial differential equations, Proc. Amer. Math. Soci., 138 (2010), 2169-2180.  doi: 10.1090/S0002-9939-10-10230-5.  Google Scholar [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.  Google Scholar [6] P. Chen and Y. 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Math., 66 (2015), 63-76.  doi: 10.1007/s13348-014-0106-y.  Google Scholar [11] 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.  Google Scholar [12] 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.  Google Scholar [13] P. Chen, X. Zhang, Y. Li, Study on fractional non-autonomous evolution equations with delay, Comput. Math. Appl., 73 (2017), 794-803. doi: 10.1016/j.camwa.2017.01.009.  Google Scholar [14] P. Chen, X. Zhang and Y. Li, A blowup alternative result for fractional nonautonomous evolution equation of Volterra type, Commun. Pure Appl. Anal., 17 (2018), 1975-1992.  doi: 10.3934/cpaa.2018094.  Google Scholar [15] 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.  Google Scholar [16] 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.  Google Scholar [17] 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.  Google Scholar [18] 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.  Google Scholar [19] P. Chen, Y. Li and X. Zhang, Cauchy problem for stochastic non-autonomous evolution equations governed by noncompact evolution families, Discrete Contin. Dyn. Syst. Ser. B. doi: 10.3934/dcdsb.2020171.  Google Scholar [20] 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.  Google Scholar [21] 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.  Google Scholar [22] G. Da Prato and J. Zabczyk, Stochastic Equations in Infinite Dimensions, Cambridge University Press, Cambridge, 1992.  doi: 10.1017/CBO9780511666223.  Google Scholar [23] K. Deng, Exponential decay of solutions of semilinear parabolic equations with nonlocal initial conditions, J. Math. Anal. Appl., 179 (1993), 630-637.  doi: 10.1006/jmaa.1993.1373.  Google Scholar [24] M. M. EI-Borai, O. L. Mostafa and H. M. Ahmed, Asymptotic stability of some stochastic evolution equations, Appl. Math. Comput., 144 (2003), 273-286.  doi: 10.1016/S0096-3003(02)00406-X.  Google Scholar [25] 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.  Google Scholar [26] 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.  Google Scholar [27] S. Farahi and T. Guendouzi, Approximate controllability of fractional neutral stochastic evolution equations with nonlocal conditions, Results. Math., 65 (2014), 501-521.  doi: 10.1007/s00025-013-0362-2.  Google Scholar [28] W. E. Fitzgibbon, Semilinear functional equations in Banach space, J. Differential Equations, 29 (1978), 1-14.  doi: 10.1016/0022-0396(78)90037-2.  Google Scholar [29] X. Fu, Existence of solutions for non-autonomous functional evolution equations with nonlocal conditions, Electron. J. Differential Equations, 2012 (2012), No. 110, 15 pp.  Google Scholar [30] 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.  Google Scholar [31] W. Grecksch and C. Tudor, Stochastic Evolution Equations: A Hilbert Space Approach, Akademic Verlag, Berlin, 1995.  Google Scholar [32] D. Henry, Geometric Theory of Semilinear Parabolic Equations, Lecture Notes in Math., vol. 840, Springer-Verlag, New York, 1981.  Google Scholar [33] J. Liang, J. Liu and T.-J. Xiao, Nonlocal Cauchy problems governed by compact operator families, Nonlinear Anal., 57 (2004), 183-189.  doi: 10.1016/j.na.2004.02.007.  Google Scholar [34] 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.  Google Scholar [35] K. Liu, Stability of Infinite Dimensional Stochastic Differential Equations with Applications, Chapman and Hall/CRC, Boca Raton, FL, 2006.  Google Scholar [36] J. Luo, Fixed points and exponential stability of mild solutions of stochastic partial differential equations with delays, J. Math. Anal. Appl., 342 (2008), 753-760.  doi: 10.1016/j.jmaa.2007.11.019.  Google Scholar [37] X. Mao, Stochastic Differential Equations and Their Applications, Horwood Publishing Ltd., Chichester, 1997.  Google Scholar [38] M. McKibben, Discoving Evolution Equations with Applications, Vol. I Deterministic Models, Chapman and Hall/CRC Appl. Math. Nonlinear Sci. Ser., 2011. Google Scholar [39] A. Pazy, Semigroups of Linear Operators and Applications to Partial Differential Equations, Springer-Verlag, New York, 1983. doi: 10.1007/978-1-4612-5561-1.  Google Scholar [40] Y. Ren, Q. Zhou and L. Chen, Existence, uniqueness and stability of mild solutions for time-dependent stochastic evolution equations with poisson jumps and infinite delay, J. Optim. Theory Appl., 149 (2011), 315-331.  doi: 10.1007/s10957-010-9792-0.  Google Scholar [41] R. Sakthivel, Y. Ren, A. Debbouche and N. I. Mahmudov, Approximate controllability of fractional stochastic differential inclusions with nonlocal conditions, Appl. Anal., 95 (2016), 2361-2382.  doi: 10.1080/00036811.2015.1090562.  Google Scholar [42] K. Sobczyk, Stochastic Differential Equations with Applications to Physics and Engineering, Kluwer Academic Publishers, Dordrecht, 1991. doi: 10.1007/978-94-011-3712-6.  Google Scholar [43] H. Tanabe, Functional Analytic Methods for Partial Differential Equations, Marcel Dekker, New York, USA, 1997.  Google Scholar [44] T. Taniguchi, K. Liu and A. Truman, Existence, uniqueness and asymptotic behavior of mild solutions to stochastic functional differential equations in Hilbert spaces, J. Differential Equations, 181 (2002), 72-91.  doi: 10.1006/jdeq.2001.4073.  Google Scholar [45] I. I. Vrabie, Delay evolution equations with mixed nonlocal plus local initial conditions, Commun. Contemp. Math., 17 (2015), 1350035. doi: 10.1142/S0219199713500351.  Google Scholar [46] R.-N. Wang, K. Ezzinbi and P.-X. Zhu, Non-autonomous impulsive Cauchy problems of parabolic type involving nonlocal initial conditions, J. Integral Equations Appl., 26 (2014), 275-299.  doi: 10.1216/JIE-2014-26-2-275.  Google Scholar [47] 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.  Google Scholar [48] X. Zhang, P. Chen, A. Abdelmonem and Y. Li, Fractional stochastic evolution equations with nonlocal initial conditions and noncompact semigroups, Stochastics, 90 (2018), 1005-1022.  doi: 10.1080/17442508.2018.1466885.  Google Scholar [49] X. Zhang, P. Chen, A. Abdelmonem and Y. Li, Mild solution of stochastic partial differential equation with nonlocal conditions and noncompact semigroups, Math. Slovaca, 69 (2019), 111-124.  doi: 10.1515/ms-2017-0207.  Google Scholar [50] B. Zhu, L. Liu and Y. Wu, Local and global existence of mild solutions for a class of nonlinear fractional reaction-diffusion equations with delay, Appl. Math. Lett., 61 (2016), 73-79.  doi: 10.1016/j.aml.2016.05.010.  Google Scholar

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##### References:
 [1] P. Acquistapace, Evolution operators and strong solution of abstract parabolic equations, Differential Integral Equations, 1 (1988), 433-457.   Google Scholar [2] P. Acquistapace and B. Terreni, A unified approach to abstract linear parabolic equations, Rend. Semin. Mat. Univ. Padova, 78 (1987), 47-107.   Google Scholar [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.  Google Scholar [4] J. Bao, Z. Hou and C. Yuan, Stability in distribution of mild solutions to stochastic partial differential equations, Proc. Amer. Math. Soci., 138 (2010), 2169-2180.  doi: 10.1090/S0002-9939-10-10230-5.  Google Scholar [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.  Google Scholar [6] 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.  Google Scholar [7] P. Chen and Y. Li, Existence of mild solutions for fractional evolution equations with mixed monotone nonlocal conditions, Z. Angew. Math. Phys., 65 (2014), 711-728.  doi: 10.1007/s00033-013-0351-z.  Google Scholar [8] P. Chen, X. Zhang and Y. Li, Approximation technique for fractional evolution equations with nonlocal integral conditions, Mediterr. J. Math., 14 (2017), Art. 226. doi: 10.1007/s00009-017-1029-0.  Google Scholar [9] P. Chen, Y. Li and X. Zhang, On the initial value problem of fractional stochastic evolution equations in Hilbert spaces, Commun. Pure Appl. Anal., 14 (2015), 1817-1840.  doi: 10.3934/cpaa.2015.14.1817.  Google Scholar [10] 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.  Google Scholar [11] 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.  Google Scholar [12] 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.  Google Scholar [13] P. Chen, X. Zhang, Y. Li, Study on fractional non-autonomous evolution equations with delay, Comput. Math. Appl., 73 (2017), 794-803. doi: 10.1016/j.camwa.2017.01.009.  Google Scholar [14] P. Chen, X. Zhang and Y. Li, A blowup alternative result for fractional nonautonomous evolution equation of Volterra type, Commun. Pure Appl. Anal., 17 (2018), 1975-1992.  doi: 10.3934/cpaa.2018094.  Google Scholar [15] 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.  Google Scholar [16] 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.  Google Scholar [17] 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.  Google Scholar [18] 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.  Google Scholar [19] P. Chen, Y. Li and X. Zhang, Cauchy problem for stochastic non-autonomous evolution equations governed by noncompact evolution families, Discrete Contin. Dyn. Syst. Ser. B. doi: 10.3934/dcdsb.2020171.  Google Scholar [20] 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.  Google Scholar [21] 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.  Google Scholar [22] G. Da Prato and J. Zabczyk, Stochastic Equations in Infinite Dimensions, Cambridge University Press, Cambridge, 1992.  doi: 10.1017/CBO9780511666223.  Google Scholar [23] K. Deng, Exponential decay of solutions of semilinear parabolic equations with nonlocal initial conditions, J. Math. Anal. Appl., 179 (1993), 630-637.  doi: 10.1006/jmaa.1993.1373.  Google Scholar [24] M. M. EI-Borai, O. L. Mostafa and H. M. Ahmed, Asymptotic stability of some stochastic evolution equations, Appl. Math. Comput., 144 (2003), 273-286.  doi: 10.1016/S0096-3003(02)00406-X.  Google Scholar [25] 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.  Google Scholar [26] 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.  Google Scholar [27] S. Farahi and T. Guendouzi, Approximate controllability of fractional neutral stochastic evolution equations with nonlocal conditions, Results. Math., 65 (2014), 501-521.  doi: 10.1007/s00025-013-0362-2.  Google Scholar [28] W. E. Fitzgibbon, Semilinear functional equations in Banach space, J. Differential Equations, 29 (1978), 1-14.  doi: 10.1016/0022-0396(78)90037-2.  Google Scholar [29] X. Fu, Existence of solutions for non-autonomous functional evolution equations with nonlocal conditions, Electron. J. Differential Equations, 2012 (2012), No. 110, 15 pp.  Google Scholar [30] 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.  Google Scholar [31] W. Grecksch and C. Tudor, Stochastic Evolution Equations: A Hilbert Space Approach, Akademic Verlag, Berlin, 1995.  Google Scholar [32] D. Henry, Geometric Theory of Semilinear Parabolic Equations, Lecture Notes in Math., vol. 840, Springer-Verlag, New York, 1981.  Google Scholar [33] J. Liang, J. Liu and T.-J. Xiao, Nonlocal Cauchy problems governed by compact operator families, Nonlinear Anal., 57 (2004), 183-189.  doi: 10.1016/j.na.2004.02.007.  Google Scholar [34] 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.  Google Scholar [35] K. Liu, Stability of Infinite Dimensional Stochastic Differential Equations with Applications, Chapman and Hall/CRC, Boca Raton, FL, 2006.  Google Scholar [36] J. Luo, Fixed points and exponential stability of mild solutions of stochastic partial differential equations with delays, J. Math. Anal. Appl., 342 (2008), 753-760.  doi: 10.1016/j.jmaa.2007.11.019.  Google Scholar [37] X. Mao, Stochastic Differential Equations and Their Applications, Horwood Publishing Ltd., Chichester, 1997.  Google Scholar [38] M. McKibben, Discoving Evolution Equations with Applications, Vol. I Deterministic Models, Chapman and Hall/CRC Appl. Math. Nonlinear Sci. Ser., 2011. Google Scholar [39] A. Pazy, Semigroups of Linear Operators and Applications to Partial Differential Equations, Springer-Verlag, New York, 1983. doi: 10.1007/978-1-4612-5561-1.  Google Scholar [40] Y. Ren, Q. Zhou and L. Chen, Existence, uniqueness and stability of mild solutions for time-dependent stochastic evolution equations with poisson jumps and infinite delay, J. Optim. Theory Appl., 149 (2011), 315-331.  doi: 10.1007/s10957-010-9792-0.  Google Scholar [41] R. Sakthivel, Y. Ren, A. Debbouche and N. I. Mahmudov, Approximate controllability of fractional stochastic differential inclusions with nonlocal conditions, Appl. Anal., 95 (2016), 2361-2382.  doi: 10.1080/00036811.2015.1090562.  Google Scholar [42] K. Sobczyk, Stochastic Differential Equations with Applications to Physics and Engineering, Kluwer Academic Publishers, Dordrecht, 1991. doi: 10.1007/978-94-011-3712-6.  Google Scholar [43] H. Tanabe, Functional Analytic Methods for Partial Differential Equations, Marcel Dekker, New York, USA, 1997.  Google Scholar [44] T. Taniguchi, K. Liu and A. Truman, Existence, uniqueness and asymptotic behavior of mild solutions to stochastic functional differential equations in Hilbert spaces, J. Differential Equations, 181 (2002), 72-91.  doi: 10.1006/jdeq.2001.4073.  Google Scholar [45] I. I. Vrabie, Delay evolution equations with mixed nonlocal plus local initial conditions, Commun. Contemp. Math., 17 (2015), 1350035. doi: 10.1142/S0219199713500351.  Google Scholar [46] R.-N. Wang, K. Ezzinbi and P.-X. Zhu, Non-autonomous impulsive Cauchy problems of parabolic type involving nonlocal initial conditions, J. Integral Equations Appl., 26 (2014), 275-299.  doi: 10.1216/JIE-2014-26-2-275.  Google Scholar [47] 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.  Google Scholar [48] X. Zhang, P. Chen, A. Abdelmonem and Y. Li, Fractional stochastic evolution equations with nonlocal initial conditions and noncompact semigroups, Stochastics, 90 (2018), 1005-1022.  doi: 10.1080/17442508.2018.1466885.  Google Scholar [49] X. Zhang, P. Chen, A. Abdelmonem and Y. Li, Mild solution of stochastic partial differential equation with nonlocal conditions and noncompact semigroups, Math. Slovaca, 69 (2019), 111-124.  doi: 10.1515/ms-2017-0207.  Google Scholar [50] B. Zhu, L. Liu and Y. Wu, Local and global existence of mild solutions for a class of nonlinear fractional reaction-diffusion equations with delay, Appl. Math. Lett., 61 (2016), 73-79.  doi: 10.1016/j.aml.2016.05.010.  Google Scholar
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