# American Institute of Mathematical Sciences

May  2022, 27(5): 2483-2499. doi: 10.3934/dcdsb.2021142

## Center manifolds for ill-posed stochastic evolution equations

 School of Mathematics, South China University of Technology, Guangzhou 510640, China

* Corresponding author: Caibin Zeng

Received  December 2020 Revised  March 2021 Published  May 2022 Early access  May 2021

The aim of this paper is to develop a center manifold theory for a class of stochastic partial differential equations with a non-dense domain through the Lyapunov-Perron method. We construct a novel variation of constants formula by the resolvent operator to formulate the integrated solutions. Moreover, we impose an additional condition involving a non-decreasing map to deduce the required estimate since Young's convolution inequality is not applicable. As an application, we present a stochastic parabolic equation to illustrate the obtained results.

Citation: Zonghao Li, Caibin Zeng. Center manifolds for ill-posed stochastic evolution equations. Discrete and Continuous Dynamical Systems - B, 2022, 27 (5) : 2483-2499. doi: 10.3934/dcdsb.2021142
##### References:
 [1] W. Arendt, Resolvent positive operators, Proc. London Math. Soc., 54 (1987), 321-349.  doi: 10.1112/plms/s3-54.2.321. [2] W. Arendt, Vector valued Laplace transforms and Cauchy problems, Israel J. Math., 59 (1987), 327-352.  doi: 10.1007/BF02774144. [3] L. Arnold, Random Dynamical Systems, Springer, 1998. doi: 10.1007/978-3-662-12878-7. [4] P. Boxler, How to construct stochastic center manifolds on the level of vector fields, in Lyapunov Exponents (eds. L. Arnold, H. Crauel and J.-P. Eckmann), Springer, 1486 (1991), 141–158. doi: 10.1007/BFb0086664. [5] T. Caraballo, J. Duan, K. Lu and B. Schmalfuß, Invariant manifolds for random and stochastic partial differential equations, Adv. Nonlinear Stud., 10 (2010), 23-52.  doi: 10.1515/ans-2010-0102. [6] T. Caraballo, J. A. Langa and J. A. Robinson, A stochastic pitchfork bifurcation in a reaction-diffusion equation, R. Soc. Lond. Proc. Ser. A, 457 (2001), 2041-2061.  doi: 10.1098/rspa.2001.0819. [7] C. Castaing and M. Valadier, Convex Analysis and Measurable Multifunctions, Springer, 1977. [8] X. Chen, A. J. Roberts and J. Duan, Centre manifolds for stochastic evolution equations, J. Difference Equ. Appl., 21 (2015), 606-632.  doi: 10.1080/10236198.2015.1045889. [9] G. Da Prato and E. Sinestrari, Differential operators with non-dense domain, Ann. Scuola Norm-Sci., 14 (1987), 285-344. [10] J. Duan, K. Lu and B. Schmalfuß, Invariant manifolds for stochastic partial differential equations, Ann. Probab., 31 (2003), 2109-2135.  doi: 10.1214/aop/1068646380. [11] J. Duan, K. Lu and B. Schmalfuß, Smooth stable and unstable manifolds for stochastic evolutionary equations, J. Dynam. Differential Equations, 16 (2004), 949-972.  doi: 10.1007/s10884-004-7830-z. [12] T. Gallay, A center-stable manifold theorem for differential equations in Banach spaces, Comm. Math. Phys., 152 (1993), 249-268.  doi: 10.1007/BF02098299. [13] K. Lu and B. Schmalfuß, Invariant manifolds for stochastic wave equation, J. Differential Equations, 236 (2007), 460-492.  doi: 10.1016/j.jde.2006.09.024. [14] P. Magal and S. Ruan, On integrated semigroups and age-structured models in ${\mathcal{L}}^p$ space, Differential Integral Equations, 20 (2007), 197-239. [15] P. Magal and S. Ruan, On semilinear Cauchy problems with non-dense domain, Adv. Difference Equations, 14 (2009), 1041-1084. [16] P. Magal and S. Ruan, Center manifolds for semilinear equations with non-dense domain and applications to Hopf bifurcation in age structured models, Mem. Amer. Math. Soc., 202 (2009), vi+71 pp. doi: 10.1090/S0065-9266-09-00568-7. [17] P. Magal and O. Seydi, Variation of constants formula and exponential dichotomy for non-autonomous non densely defined Cauchy problems, arXiv: 1608.07079 [18] S.-E. A. Mohammed and M. K. R. Scheutzow, The stable manifold theorem for stochastic differential equations, Ann. Probab., 27 (1999), 615-652.  doi: 10.1214/aop/1022677380. [19] A. Neamtu, Random invariant manifolds for ill-posed stochastic evolution equations, Stoch. Dyn., 20 (2020), 2050013, 31pp. doi: 10.1142/S0219493720500136. [20] A. Pazy, Semigroups of Linear Operator and Applications to Partial Differential Equations, Springer, 1983. doi: 10.1007/978-1-4612-5561-1. [21] J. Shen and C. Zeng, Invariant foliations for stochastic partial differential equations with non-dense domain, submitted. [22] L. Shi, Smooth convergence of random center manifolds for SPDEs in varying phase spaces, J. Differential Equations, 269 (2020), 1963-2011.  doi: 10.1016/j.jde.2020.01.028. [23] H. R. Thieme, Semiflows generated by Lipschitz perturbations of non-densely defined operators, Differential Integral Equations, 3 (1990), 1035-1066. [24] H. R. Thieme, "Integrated semigroups" and integrated solutions to abstract Cauchy problems, J. Math. Anal. Appl., 152 (1990), 416-447.  doi: 10.1016/0022-247X(90)90074-P.

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##### References:
 [1] W. Arendt, Resolvent positive operators, Proc. London Math. Soc., 54 (1987), 321-349.  doi: 10.1112/plms/s3-54.2.321. [2] W. Arendt, Vector valued Laplace transforms and Cauchy problems, Israel J. Math., 59 (1987), 327-352.  doi: 10.1007/BF02774144. [3] L. Arnold, Random Dynamical Systems, Springer, 1998. doi: 10.1007/978-3-662-12878-7. [4] P. Boxler, How to construct stochastic center manifolds on the level of vector fields, in Lyapunov Exponents (eds. L. Arnold, H. Crauel and J.-P. Eckmann), Springer, 1486 (1991), 141–158. doi: 10.1007/BFb0086664. [5] T. Caraballo, J. Duan, K. Lu and B. Schmalfuß, Invariant manifolds for random and stochastic partial differential equations, Adv. Nonlinear Stud., 10 (2010), 23-52.  doi: 10.1515/ans-2010-0102. [6] T. Caraballo, J. A. Langa and J. A. Robinson, A stochastic pitchfork bifurcation in a reaction-diffusion equation, R. Soc. Lond. Proc. Ser. A, 457 (2001), 2041-2061.  doi: 10.1098/rspa.2001.0819. [7] C. Castaing and M. Valadier, Convex Analysis and Measurable Multifunctions, Springer, 1977. [8] X. Chen, A. J. Roberts and J. Duan, Centre manifolds for stochastic evolution equations, J. Difference Equ. Appl., 21 (2015), 606-632.  doi: 10.1080/10236198.2015.1045889. [9] G. Da Prato and E. Sinestrari, Differential operators with non-dense domain, Ann. Scuola Norm-Sci., 14 (1987), 285-344. [10] J. Duan, K. Lu and B. Schmalfuß, Invariant manifolds for stochastic partial differential equations, Ann. Probab., 31 (2003), 2109-2135.  doi: 10.1214/aop/1068646380. [11] J. Duan, K. Lu and B. Schmalfuß, Smooth stable and unstable manifolds for stochastic evolutionary equations, J. Dynam. Differential Equations, 16 (2004), 949-972.  doi: 10.1007/s10884-004-7830-z. [12] T. Gallay, A center-stable manifold theorem for differential equations in Banach spaces, Comm. Math. Phys., 152 (1993), 249-268.  doi: 10.1007/BF02098299. [13] K. Lu and B. Schmalfuß, Invariant manifolds for stochastic wave equation, J. Differential Equations, 236 (2007), 460-492.  doi: 10.1016/j.jde.2006.09.024. [14] P. Magal and S. Ruan, On integrated semigroups and age-structured models in ${\mathcal{L}}^p$ space, Differential Integral Equations, 20 (2007), 197-239. [15] P. Magal and S. Ruan, On semilinear Cauchy problems with non-dense domain, Adv. Difference Equations, 14 (2009), 1041-1084. [16] P. Magal and S. Ruan, Center manifolds for semilinear equations with non-dense domain and applications to Hopf bifurcation in age structured models, Mem. Amer. Math. Soc., 202 (2009), vi+71 pp. doi: 10.1090/S0065-9266-09-00568-7. [17] P. Magal and O. Seydi, Variation of constants formula and exponential dichotomy for non-autonomous non densely defined Cauchy problems, arXiv: 1608.07079 [18] S.-E. A. Mohammed and M. K. R. Scheutzow, The stable manifold theorem for stochastic differential equations, Ann. Probab., 27 (1999), 615-652.  doi: 10.1214/aop/1022677380. [19] A. Neamtu, Random invariant manifolds for ill-posed stochastic evolution equations, Stoch. Dyn., 20 (2020), 2050013, 31pp. doi: 10.1142/S0219493720500136. [20] A. Pazy, Semigroups of Linear Operator and Applications to Partial Differential Equations, Springer, 1983. doi: 10.1007/978-1-4612-5561-1. [21] J. Shen and C. Zeng, Invariant foliations for stochastic partial differential equations with non-dense domain, submitted. [22] L. Shi, Smooth convergence of random center manifolds for SPDEs in varying phase spaces, J. Differential Equations, 269 (2020), 1963-2011.  doi: 10.1016/j.jde.2020.01.028. [23] H. R. Thieme, Semiflows generated by Lipschitz perturbations of non-densely defined operators, Differential Integral Equations, 3 (1990), 1035-1066. [24] H. R. Thieme, "Integrated semigroups" and integrated solutions to abstract Cauchy problems, J. Math. Anal. Appl., 152 (1990), 416-447.  doi: 10.1016/0022-247X(90)90074-P.
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