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March  2021, 26(3): 1531-1547. doi: 10.3934/dcdsb.2020171

Cauchy problem for stochastic non-autonomous evolution equations governed by noncompact evolution families

Pengyu Chen , , Yongxiang Li and  Xuping Zhang

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

* Corresponding author: Pengyu Chen

Received  September 2019 Revised  December 2019 Published  March 2021 Early access  May 2020

Fund Project: Research supported by National Natural Science Foundations of China (No. 11501455, No. 11661071), Science Research Project for Colleges and Universities of Gansu Province (No. 2019B-047), Project of NWNU-LKQN2019-3, Project of NWNU-LKQN2019-13 and China Scholarship Council (No. 201908625016)

This paper investigates the Cauchy problem to a class of stochastic non-autonomous evolution equations of parabolic type governed by noncompact evolution families in Hilbert spaces. Combining the theory of evolution families, the fixed point theorem with respect to convex-power condensing operator and a new estimation technique of the measure of noncompactness, we established some new existence results of mild solutions under the situation that the nonlinear function satisfy some appropriate local growth condition and a noncompactness measure condition. Our results generalize and improve some previous results on this topic, since the strong restriction on the constants in the condition of noncompactness measure is completely deleted, and also the condition of uniformly continuity of the nonlinearity is not required. At last, as samples of applications, we consider the Cauchy problem to a class of stochastic non-autonomous partial differential equation of parabolic type.

Keywords: Stochastic non-autonomous evolution equations, Cauchy problem, measure of noncompactness, Wiener process, noncompact evolution family.
Mathematics Subject Classification: Primary: 34F05; Secondary: 47J35, 60H15.
Citation: Pengyu Chen, Yongxiang Li, Xuping Zhang. Cauchy problem for stochastic non-autonomous evolution equations governed by noncompact evolution families. Discrete and Continuous Dynamical Systems - B, 2021, 26 (3) : 1531-1547. doi: 10.3934/dcdsb.2020171
References:
[1]

P. Acquistapace, Evolution operators and strong solutions of abstract linear parabolic equations, Differential Integral Equations, 1 (1988), 433-457. 

[2]

P. Acquistapace and B. Terreni, A unified approach to abstract linear nonautonomous parabolic equations, Rend. Sem. 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. and K. Goebel, Measures of Noncompactness in Banach Spaces, Lecture Notes in Pure and Applied Mathematics, 60, Marcel Dekker, Inc., New York, 1980.

[5]

J. Bao, Z. Hou and C. Yuan, Stability in distribution of mild solutions to stochastic partial differential equations, Proc. Amer. Math. Soc., 138 (2010), 2169-2180.  doi: 10.1090/S0002-9939-10-10230-5.

[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.

[7]

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.

[8]

P. Chen, X. Zhang and 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.

[9]

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.

[10]

P. Chen, X. Zhang and Y. Li, Non-autonomous parabolic evolution equations with non-instantaneous impulses governed by noncompact evolution families, J. Fixed Point Theory Appl., 21 (2019), 17pp. doi: 10.1007/s11784-019-0719-6.

[11]

P. Chen, X. Zhang and Y. Li, Non-autonomous evolution equations of parabolic type with non-instantaneous impulses, Mediterr. J. Math., 16 (2019), 14pp. doi: 10.1007/s00009-019-1384-0.

[12]

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.

[13]

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.

[14]

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.

[15] G. Da Prato and J. Zabczyk, Stochastic Equations in Infinite Dimensions, Encyclopedia of Mathematics and its Applications, 44, Cambridge University Press, Cambridge, 1992.  doi: 10.1017/CBO9780511666223.
[16]

K. Deimling, Nonlinear Functional Analysis, Springer-Verlag, Berlin, 1985. doi: 10.1007/978-3-662-00547-7.

[17]

X. Fu, Existence of solutions for non-autonomous functional evolution equations with nonlocal conditions, Electron. J. Differential Equations, 2012 (2012), 15pp.

[18]

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.

[19]

W. Grecksch and C. Tudor, Stochastic Evolution Equations. A Hilbert Space Approach, Mathematical Research, 85, Akademie-Verlag, Berlin, 1995.

[20]

D. J. Guo, Solutions of nonlinear integrodifferential equations of mixed type in Banach spaces, J. Appl. Math. Simulation, 2 (1989), 1-11.  doi: 10.1155/S1048953389000018.

[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] V. Lakshmikantham and S. Leela, Nonlinear Differential Equations in Abstract Spaces, International Series in Nonlinear Mathematics: Theory, Methods and Applications, 2, Pergamon Press, Oxford-New York, 1981. 
[23]

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.

[24]

K. Liu, Stability of Infinite Dimensional Stochastic Differential Equations with Applications, Chapman & Hall/CRC Monographs and Surveys in Pure and Applied Mathematics, 135, Chapman & Hall/CRC, Boca Raton, FL, 2006. doi: 10.1201/9781420034820.

[25]

L. Liu, F. Guo, C. Wu and Y. Wu, Existence theorems of global solutions for nonlinear Volterra type integral equations in Banach spaces, J. Math. Anal. Appl., 309 (2005), 638-649.  doi: 10.1016/j.jmaa.2004.10.069.

[26]

L. Liu, C. Wu and F. Guo, Existence theorems of global solutions of initial value problems for nonlinear integrodifferential equations of mixed type in Banach spaces and applications, Comput. Math. Appl., 47 (2004), 13-22.  doi: 10.1016/S0898-1221(04)90002-8.

[27]

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.

[28]

X. Mao, Stochastic Differential Equations and Their Applications, Horwood Publishing Limited, Chichester, 1997.

[29]

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

[30]

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.

[31]

K. Sobczyk, Stochastic Differential Equations. With Applications to Physics and Engineering, Mathematics and its Applications, 40, Kluwer Academic Publishers Group, Dordrecht, 1991. doi: 10.1007/978-94-011-3712-6.

[32]

J. X. Sun and X. Y. Zhang, A fixed point theorem for convex-power condensing operators and its applications to abstract semilinear evolution equations, Acta Math. Sinica (Chin. Ser.), 48 (2005), 439-446. 

[33]

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.

[34]

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.

[35]

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.

[36]

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.

[37]

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.

[38]

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.

show all references

References:
[1]

P. Acquistapace, Evolution operators and strong solutions of abstract linear parabolic equations, Differential Integral Equations, 1 (1988), 433-457. 

[2]

P. Acquistapace and B. Terreni, A unified approach to abstract linear nonautonomous parabolic equations, Rend. Sem. 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. and K. Goebel, Measures of Noncompactness in Banach Spaces, Lecture Notes in Pure and Applied Mathematics, 60, Marcel Dekker, Inc., New York, 1980.

[5]

J. Bao, Z. Hou and C. Yuan, Stability in distribution of mild solutions to stochastic partial differential equations, Proc. Amer. Math. Soc., 138 (2010), 2169-2180.  doi: 10.1090/S0002-9939-10-10230-5.

[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.

[7]

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.

[8]

P. Chen, X. Zhang and 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.

[9]

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.

[10]

P. Chen, X. Zhang and Y. Li, Non-autonomous parabolic evolution equations with non-instantaneous impulses governed by noncompact evolution families, J. Fixed Point Theory Appl., 21 (2019), 17pp. doi: 10.1007/s11784-019-0719-6.

[11]

P. Chen, X. Zhang and Y. Li, Non-autonomous evolution equations of parabolic type with non-instantaneous impulses, Mediterr. J. Math., 16 (2019), 14pp. doi: 10.1007/s00009-019-1384-0.

[12]

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.

[13]

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.

[14]

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.

[15] G. Da Prato and J. Zabczyk, Stochastic Equations in Infinite Dimensions, Encyclopedia of Mathematics and its Applications, 44, Cambridge University Press, Cambridge, 1992.  doi: 10.1017/CBO9780511666223.
[16]

K. Deimling, Nonlinear Functional Analysis, Springer-Verlag, Berlin, 1985. doi: 10.1007/978-3-662-00547-7.

[17]

X. Fu, Existence of solutions for non-autonomous functional evolution equations with nonlocal conditions, Electron. J. Differential Equations, 2012 (2012), 15pp.

[18]

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.

[19]

W. Grecksch and C. Tudor, Stochastic Evolution Equations. A Hilbert Space Approach, Mathematical Research, 85, Akademie-Verlag, Berlin, 1995.

[20]

D. J. Guo, Solutions of nonlinear integrodifferential equations of mixed type in Banach spaces, J. Appl. Math. Simulation, 2 (1989), 1-11.  doi: 10.1155/S1048953389000018.

[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] V. Lakshmikantham and S. Leela, Nonlinear Differential Equations in Abstract Spaces, International Series in Nonlinear Mathematics: Theory, Methods and Applications, 2, Pergamon Press, Oxford-New York, 1981. 
[23]

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.

[24]

K. Liu, Stability of Infinite Dimensional Stochastic Differential Equations with Applications, Chapman & Hall/CRC Monographs and Surveys in Pure and Applied Mathematics, 135, Chapman & Hall/CRC, Boca Raton, FL, 2006. doi: 10.1201/9781420034820.

[25]

L. Liu, F. Guo, C. Wu and Y. Wu, Existence theorems of global solutions for nonlinear Volterra type integral equations in Banach spaces, J. Math. Anal. Appl., 309 (2005), 638-649.  doi: 10.1016/j.jmaa.2004.10.069.

[26]

L. Liu, C. Wu and F. Guo, Existence theorems of global solutions of initial value problems for nonlinear integrodifferential equations of mixed type in Banach spaces and applications, Comput. Math. Appl., 47 (2004), 13-22.  doi: 10.1016/S0898-1221(04)90002-8.

[27]

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.

[28]

X. Mao, Stochastic Differential Equations and Their Applications, Horwood Publishing Limited, Chichester, 1997.

[29]

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

[30]

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.

[31]

K. Sobczyk, Stochastic Differential Equations. With Applications to Physics and Engineering, Mathematics and its Applications, 40, Kluwer Academic Publishers Group, Dordrecht, 1991. doi: 10.1007/978-94-011-3712-6.

[32]

J. X. Sun and X. Y. Zhang, A fixed point theorem for convex-power condensing operators and its applications to abstract semilinear evolution equations, Acta Math. Sinica (Chin. Ser.), 48 (2005), 439-446. 

[33]

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.

[34]

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.

[35]

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.

[36]

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.

[37]

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.

[38]

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.

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