# American Institute of Mathematical Sciences

May  2018, 17(3): 729-749. doi: 10.3934/cpaa.2018038

## Random attractors for stochastic parabolic equations with additive noise in weighted spaces

 1 School of Science, Hohai University, Nanjing, Jiangsu 210098, China 2 School of Mathematics and Information Science, Shandong Technology and Business University, Yantai, Shandong 264005, China 3 Department of Mathematics, Brigham Young University, Provo, Utah 84602, USA

* Corresponding author: Xiaojun Li

Received  March 2017 Revised  August 2017 Published  January 2018

In this paper, we establish the existence of random attractors for stochastic parabolic equations driven by additive noise as well as deterministic non-autonomous forcing terms in weighted Lebesgue spaces $L_{\delta}^r(\mathcal{O})$, where $1<r<\infty ,\ \delta$ is the distance from $x$ to the boundary. The nonlinearity $f(x,u)$ of equation depending on the spatial variable does not have the bound on the derivative in $u$, and then causes critical exponent. In both subcritical and critical cases, we get the well-posedness and dissipativeness of the problem under consideration and, by smoothing property of heat semigroup in weighted space, the asymptotical compactness of random dynamical system corresponding to the original system.

Citation: Xiaojun Li, Xiliang Li, Kening Lu. Random attractors for stochastic parabolic equations with additive noise in weighted spaces. Communications on Pure & Applied Analysis, 2018, 17 (3) : 729-749. doi: 10.3934/cpaa.2018038
##### References:
 [1] L. Arnold, Random Dynamical Systems, Springer-Verlag, 1998. [2] J. M. Arrieta and A. N. Carvalho, Abstract parabolic problems with critial nonlinearities and applications to Navier-Stokes and heat equatins, Trans. Amer. Math. Soc., 352 (1999), 285-310. [3] A. V. Babin and M. I. Vishik, Attractors of Evolutionary Equations, North-Holland, Amsterdam, 1992. [4] P. W. Bates, K. Lu and B. Wang, Random attractors for stochastic reaction-diffusion equations on unbounded domains, J. Differential Equations, 246 (2009), 845-869. [5] P. W. Bates, K. Lu and B. Wang, Tempered random attractors for parabolic equations in weighted spaces, J. Math. Phys., 54 (2013), 081505. [6] H. Brezis and T. Cazenave, A nonlinear heat equation with singular initial data, J. Anal. Math., 68 (1996), 277-304. [7] D. Cao, C. Sun and M. Yang, Dynamics for a stochastic reaction-diffusion equation with additive noise, J. Differential Equations, 259 (2015), 838-872. [8] T. Caraballo, G. Lukaszewicz and J. Real, Pullback attractors for asymptotically compact nonautonomous dynamical systems, Nonlinear Anal., 64 (2006), 484-498. [9] A. N. Carvalho and J. W. Cholewa, Continuation and asymptotics of solutions to semilinear parabolic equations with critical nonlinearities, J. Math. Anal. Appl., 310 (2005), 557-578. [10] D. Cheban, P. E. Kloeden and B. Schmalfuß, The relationship between pullback, forwards and global attractors of nonautonomous dynamical systems, Nonlinear Dyn. Syst. Theory, 2 (2002), 9-28. [11] M. D. Chekrouna, E. Parkb and R. Temam, The Stampacchia maximum principle for stochastic partial differential equations and applications, J. Differential Equations, 260 (2016), 2926-2972. [12] V. V. Chepyzhov and M. I. Vishik, Attractors for Equations of Mathematical Physics, Amer. Math. Soc. Colloq. Publ., Vol. 49, Amer. Math. Soc., Providence, RI, 2002. [13] I. Chueshov, Monotone Random Systems Theory and Applications, Lecture Notes in Mathematics, vol. 1779,2002. [14] H. Crauel, A. Debussche and F. Flandoli, Random attractors, J. Dynam. Differential Equations, 9 (1997), 307-341. [15] H. Crauel and F. Flandoli, Attractor for random dynamical systems, Probab. Theory Related Fields, 100 (1994), 365-393. [16] H. Cui, Langa and A. José, Uniform attractors for non-autonomous random dynamical systems, J. Differential Equations, 263 (2017), 1225-1268. [17] G. Da Prato and J. Zabczyk, Stochastic Equations in Infinite Dimensions, Cambridge Univ. Press, 1992. [18] M. Fila, P. Souplet and F. B. Weissler, Linear and nonlinear heat equations in $L^q_{\delta }$ spaces and universal bounds for global solutions, Math. Ann., 320 (2001), 87-113. [19] F. Flandoli and B. Schmalfuß, Random attractors for the 3D stochastic Navier-Stokes equation with multiplicative noise, Stoch. Stoch. Rep., 59 (1996), 21-45. [20] J. K. Hale, Asymptotic Behavior of Dissipative Systems, Amer. Math. Soc., Providence. RI, 1988. [21] M. Loayza, The heat equation with singular nonlinearity and singular initial data, J. Differential Equations, 229 (2006), 509-528. [22] X. Li, Non-autonomous parabolic problems with singular initial data in weighted spaces, Rocky Mountain J. Math., 42 (2012), 1215-1245. [23] X. Li and L. Ren, Dynamics of non-autonomous parabolic problems with subcritical and critical nonlinearities, Bull. Sci. math., 138 (2014), 540-564. [24] X. Li and S. Ruan, Attractors for non-autonomous parabolic problems with singular initial data, J. Differential Equations, 251 (2011), 728-757. [25] B. K. Øksendal, Stochastic Differential Equations: An Introduction with Applications, Springer-Verlay, New York, 1995, 4th ed. [26] P. Quittner and P. Souplet, A priori estimates and existence for elliptic systems via bootstrap in weighted Lebesgue spaces, Arch. Rational Mech. Anal., 174 (2004), 49-81. [27] G. R. Sell and Y. C. You, Dynamics of Evolutionary Equations, Springer-Verlay, New York, 2002. [28] R. Temam, Infinite-dimensional Dynamical Systems in Mechanics and Physics, Springer-Verlag, New York, 1997. [29] B. Wang, Sufficient and necessary criteria for existence of pullback attractors for non-compact random dynamical systems, J. Differential Equations, 253 (2012), 1544-1583. [30] X. Wang, K. Lu and B. Wang, Random attractors for delay parabolic equations with additive noise and deterministic nonautonomous forcing, SIAM J. Appl. Dyn. Syst., 14 (2015), 1018-1047. [31] Y. Wang and J. Wang, Pullback attractors for multi-valued non-compact random dynamical systems generated by reaction-diffusion equations on an unbounded domain, J. Differential Equations, 259 (2015), 728-776. [32] M. Yang and P. E. Kloeden, Random attractors for stochastic semi-linear degenerate parabolic equations., Nonlinear Anal. Real World Appl., 12 (2011), 2811-2821. [33] W. Zhao, ${\rm{H}}^1$-random attractorsfor stochastic reaction diffusion equations with additive noise, Nonlinear Anal., 84 (2013), 61-72. [34] W. Zhao and Y. Li, Random attractors for stochastic semi-linear degenerate parabolic equations with additive noises, Dyn. Partial Differ. Equ., 11 (2014), 269-298.

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##### References:
 [1] L. Arnold, Random Dynamical Systems, Springer-Verlag, 1998. [2] J. M. Arrieta and A. N. Carvalho, Abstract parabolic problems with critial nonlinearities and applications to Navier-Stokes and heat equatins, Trans. Amer. Math. Soc., 352 (1999), 285-310. [3] A. V. Babin and M. I. Vishik, Attractors of Evolutionary Equations, North-Holland, Amsterdam, 1992. [4] P. W. Bates, K. Lu and B. Wang, Random attractors for stochastic reaction-diffusion equations on unbounded domains, J. Differential Equations, 246 (2009), 845-869. [5] P. W. Bates, K. Lu and B. Wang, Tempered random attractors for parabolic equations in weighted spaces, J. Math. Phys., 54 (2013), 081505. [6] H. Brezis and T. Cazenave, A nonlinear heat equation with singular initial data, J. Anal. Math., 68 (1996), 277-304. [7] D. Cao, C. Sun and M. Yang, Dynamics for a stochastic reaction-diffusion equation with additive noise, J. Differential Equations, 259 (2015), 838-872. [8] T. Caraballo, G. Lukaszewicz and J. Real, Pullback attractors for asymptotically compact nonautonomous dynamical systems, Nonlinear Anal., 64 (2006), 484-498. [9] A. N. Carvalho and J. W. Cholewa, Continuation and asymptotics of solutions to semilinear parabolic equations with critical nonlinearities, J. Math. Anal. Appl., 310 (2005), 557-578. [10] D. Cheban, P. E. Kloeden and B. Schmalfuß, The relationship between pullback, forwards and global attractors of nonautonomous dynamical systems, Nonlinear Dyn. Syst. Theory, 2 (2002), 9-28. [11] M. D. Chekrouna, E. Parkb and R. Temam, The Stampacchia maximum principle for stochastic partial differential equations and applications, J. Differential Equations, 260 (2016), 2926-2972. [12] V. V. Chepyzhov and M. I. Vishik, Attractors for Equations of Mathematical Physics, Amer. Math. Soc. Colloq. Publ., Vol. 49, Amer. Math. Soc., Providence, RI, 2002. [13] I. Chueshov, Monotone Random Systems Theory and Applications, Lecture Notes in Mathematics, vol. 1779,2002. [14] H. Crauel, A. Debussche and F. Flandoli, Random attractors, J. Dynam. Differential Equations, 9 (1997), 307-341. [15] H. Crauel and F. Flandoli, Attractor for random dynamical systems, Probab. Theory Related Fields, 100 (1994), 365-393. [16] H. Cui, Langa and A. José, Uniform attractors for non-autonomous random dynamical systems, J. Differential Equations, 263 (2017), 1225-1268. [17] G. Da Prato and J. Zabczyk, Stochastic Equations in Infinite Dimensions, Cambridge Univ. Press, 1992. [18] M. Fila, P. Souplet and F. B. Weissler, Linear and nonlinear heat equations in $L^q_{\delta }$ spaces and universal bounds for global solutions, Math. Ann., 320 (2001), 87-113. [19] F. Flandoli and B. Schmalfuß, Random attractors for the 3D stochastic Navier-Stokes equation with multiplicative noise, Stoch. Stoch. Rep., 59 (1996), 21-45. [20] J. K. Hale, Asymptotic Behavior of Dissipative Systems, Amer. Math. Soc., Providence. RI, 1988. [21] M. Loayza, The heat equation with singular nonlinearity and singular initial data, J. Differential Equations, 229 (2006), 509-528. [22] X. Li, Non-autonomous parabolic problems with singular initial data in weighted spaces, Rocky Mountain J. Math., 42 (2012), 1215-1245. [23] X. Li and L. Ren, Dynamics of non-autonomous parabolic problems with subcritical and critical nonlinearities, Bull. Sci. math., 138 (2014), 540-564. [24] X. Li and S. Ruan, Attractors for non-autonomous parabolic problems with singular initial data, J. Differential Equations, 251 (2011), 728-757. [25] B. K. Øksendal, Stochastic Differential Equations: An Introduction with Applications, Springer-Verlay, New York, 1995, 4th ed. [26] P. Quittner and P. Souplet, A priori estimates and existence for elliptic systems via bootstrap in weighted Lebesgue spaces, Arch. Rational Mech. Anal., 174 (2004), 49-81. [27] G. R. Sell and Y. C. You, Dynamics of Evolutionary Equations, Springer-Verlay, New York, 2002. [28] R. Temam, Infinite-dimensional Dynamical Systems in Mechanics and Physics, Springer-Verlag, New York, 1997. [29] B. Wang, Sufficient and necessary criteria for existence of pullback attractors for non-compact random dynamical systems, J. Differential Equations, 253 (2012), 1544-1583. [30] X. Wang, K. Lu and B. Wang, Random attractors for delay parabolic equations with additive noise and deterministic nonautonomous forcing, SIAM J. Appl. Dyn. Syst., 14 (2015), 1018-1047. [31] Y. Wang and J. Wang, Pullback attractors for multi-valued non-compact random dynamical systems generated by reaction-diffusion equations on an unbounded domain, J. Differential Equations, 259 (2015), 728-776. [32] M. Yang and P. E. Kloeden, Random attractors for stochastic semi-linear degenerate parabolic equations., Nonlinear Anal. Real World Appl., 12 (2011), 2811-2821. [33] W. Zhao, ${\rm{H}}^1$-random attractorsfor stochastic reaction diffusion equations with additive noise, Nonlinear Anal., 84 (2013), 61-72. [34] W. Zhao and Y. Li, Random attractors for stochastic semi-linear degenerate parabolic equations with additive noises, Dyn. Partial Differ. Equ., 11 (2014), 269-298.
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