Random dynamics of non-autonomous semi-linear degenerate parabolic equations on <inline-formula><tex-math id="M1">\begin{document} $\mathbb{R}^N$ \end{document}</tex-math></inline-formula> driven by an unbounded additive noise

Wenqiang Zhao

doi:10.3934/dcdsb.2018065

Previous Article
Mechanism for the color transition of the Belousov-Zhabotinsky reaction catalyzed by cerium ions and ferroin
DCDS-B Home
This Issue
Next Article
A new flexible discrete-time model for stable populations

August 2018, 23(6): 2499-2526. doi: 10.3934/dcdsb.2018065

Random dynamics of non-autonomous semi-linear degenerate parabolic equations on $\mathbb{R}^N$ driven by an unbounded additive noise

Wenqiang Zhao ^,

School of Mathematics and Statistics, Chongqing Technology and Business University, Chongqing 400067, China

* Corresponding author: Wenqiang Zhao

Received May 2017 Revised September 2017 Published February 2018

Fund Project: This work was supported by CTBU Grant 1751041, Chongqing NSF Grant of China cstc2016jcyjA0262 and China NSF Grant 11601046

In this paper, we study the dynamics of a non-autonomous semi-linear degenerate parabolic equation on $\mathbb{R}^N$ driven by an unbounded additive noise. The nonlinearity has $(p,q)$-exponent growth and the degeneracy means that the diffusion coefficient $σ$ is unbounded and allowed to vanish at some points. Firstly we prove the existence of pullback attractor in $L^2(\mathbb{R}^N)$ by using a compact embedding of the weighted Sobolev space. Secondly we establish the higher-attraction of the pullback attractor in $L^δ(\mathbb{R}^N)$, which implies that the cocycle is absorbing in $L^δ(\mathbb{R}^N)$ after a translation by the complete orbit, for arbitrary $δ∈[2,∞)$. Thirdly we verify that the derived $L^2$-pullback attractor is in fact a compact attractor in $L^p(\mathbb{R}^N)\cap L^q(\mathbb{R}^N)\cap D_0^{1,2}(\mathbb{R}^N,σ)$, mainly by means of the estimate of difference of solutions instead of the usual truncation method.

Keywords: Non-autonomous degenerate semilinear parabolic equations, unbounded additive noise, pullback attractor, higher-order attraction, asymptotically compact.

Mathematics Subject Classification: Primary: 60H15, 35B40, 35B41; Secondary: 37H10.

Citation: Wenqiang Zhao. Random dynamics of non-autonomous semi-linear degenerate parabolic equations on $\mathbb{R}^N$ driven by an unbounded additive noise. Discrete & Continuous Dynamical Systems - B, 2018, 23 (6) : 2499-2526. doi: 10.3934/dcdsb.2018065

References:

[1]	C. T. Anh and T. Q. Bao, Pullback attractors for a non-autonomous semi-linear degenerate parabolic equation, Glasg. Math. J., 52 (2010), 537-554. doi: 10.1017/S0017089510000418. Google Scholar
[2]	C. T. Anh and L. T. Thuy, Global attractors for a class of semilinear degenerate parabolic equations on $\mathbb{R}^N$, Bull. Pol. Acad. Sci. Math., 61 (2013), 47-65. doi: 10.4064/ba61-1-6. Google Scholar
[3]	L. Arnold, Random Dynamical System, Springer-Verlag, Berlin, 1998. Google Scholar
[4]	T. Bartsch and Z. Liu, On a supperlinear elliptic $p$-Laplacian equation, J. Differential Equations, 198 (2004), 149-179. doi: 10.1016/j.jde.2003.08.001. Google Scholar
[5]	P. W. Bates, H. Lisei and K. Lu, Attractors for stochastic lattice dynamical systems, Stoch. Dyn., 6 (2006), 1-21. doi: 10.1142/S0219493706001621. Google Scholar
[6]	P. Caldiroli and R. Musina, On a variational degenerate elliptic problem, Nonlinear Differ. Equ. Appl., 7 (2000), 187-199. doi: 10.1007/s000300050004. Google Scholar
[7]	A. N. Carvalho, J. A. Langa and J. C. Robinson, Attractors for Infinite-Dimensional Non-autonomous Dynamical Systems, Appl. Math. Sciences, vol. 184, Springer, 2013. Google Scholar
[8]	D. Cao, C. Sun and M. Yang, Dynamics for a stochastic reaction-diffusion equation with additive noise, J. Differential Equations, 259 (2015), 838-872. doi: 10.1016/j.jde.2015.02.020. Google Scholar
[9]	I. Chueshov, Monotone Random Systems Theory and Applications, Springer-Verlag, Berlin, 2002. Google Scholar
[10]	I. Chueshov and B. Schmalfuß, Parabolic stochastic partial differential equations with dynamical boundary conditions, Differential Integral Equtions, 17 (2004), 751-780. Google Scholar
[11]	H. Crauel and F. Flandoli, Attractors for random dynamical systems, Probab. Theory Related Fields, 100 (1994), 365-393. doi: 10.1007/BF01193705. Google Scholar
[12]	H. Crauel, A. Debussche and F. Flandoli, Random attractors, J. Dynam. Differential Equations, 9 (1997), 307-341. doi: 10.1007/BF02219225. Google Scholar
[13]	H. Crauel, G. Dimitroff and M. Scheutzow, Criteria for strong and weak random attractors, J. Dyn. Differ. Equ., 21 (2009), 233-247. doi: 10.1007/s10884-009-9135-8. Google Scholar
[14]	H. Crauel and P. E. Kloeden, Nonautonomous and random attractors, Jahresber. Dtsch. Math.-Ver., 117 (2015), 173-206. doi: 10.1365/s13291-015-0115-0. Google Scholar
[15]	H. Cui and Y. Li, Existence and upper semicontinuity of random attractors for stochastic degenerate parabolic equations with multiplicative noises, Appl. Math. Comput., 271 (2015), 777-789. doi: 10.1016/j.amc.2015.09.031. Google Scholar
[16]	G. Da Prato and Z. Jerzy, Stochastic Equations in Infinite Dimensions, Cambridge University Press, Cambridge, 1992. Google Scholar
[17]	R. Dautray and J. L. Lions, Mathematical Analysis and Numerical Methods for Science and Technology, Vol. Ⅰ: Physical origins and classical methods, Springer-Verlag, Berlin, 1990. Google Scholar
[18]	F. Flandoli and B. Schmalfuß, Random attractors for the stochastic Navier-Stokes equation with multiplicative noise, Stoch. Stoch. Rep., 59 (1996), 21-45. doi: 10.1080/17442509608834083. Google Scholar
[19]	F. Flandoli, B. Gess and M. Scheutzow, Synchronization by noise, Probab. Theory Related Fields, 168 (2017), 511-556. doi: 10.1007/s00440-016-0716-2. Google Scholar
[20]	N. I. Karachalios and N. B. Zographopoulos, On the dynamics of a degenerate parabolic equation: Global bifurcation of stationary states and convergence, Calc. Var. Partial Differential Equations, 25 (2006), 361-393. doi: 10.1007/s00526-005-0347-4. Google Scholar
[21]	A. Krause and B. Wang, Pullback attractors of non-autonomous stochastic degenerate parabolic equations on unbounded domains, J. Math. Anal. Appl., 417 (2014), 1018-1038. doi: 10.1016/j.jmaa.2014.03.037. Google Scholar
[22]	A. Krause, M. Lewis and B. Wang, Dynamics of the non-autonomous stochastic $p$-Laplace equation driven by multiplicative noise, Appl. Math. Comput., 246 (2014), 365-376. doi: 10.1016/j.amc.2014.08.033. Google Scholar
[23]	X. Li, C. Sun and N. Zhang, Dynamics for a non-autonomous degenerate parabolic equation in $D_0^{1}(Ω,σ)$, Discrete Contin. Dyn. Syst., 36 (2016), 7063-7079. doi: 10.3934/dcds.2016108. Google Scholar
[24]	Y. Li and J. Yin, A modiffied proof of pullback attractors in a Sobolev space for stochastic FitzHugh-Nagumo equations, Discrete Contin. Dyn. Syst., 21 (2016), 1203-1223. doi: 10.3934/dcdsb.2016.21.1203. Google Scholar
[25]	Y. Li and B. Guo, Random attractors for quasi-continuous random dynamical systems and applications to stochastic reaction-diffusion equations, J. Differential Equations, 245 (2008), 1775-1800. doi: 10.1016/j.jde.2008.06.031. Google Scholar
[26]	Y. Li, A. Gu and J. Li, Existence and continuity of bi-spatial random attractors and application to stochasitic semilinear Laplacian equations, J. Differential Equations, 258 (2015), 504-534. doi: 10.1016/j.jde.2014.09.021. Google Scholar
[27]	W. Niu, Global attractors for degenerate semilinear parabolic equations, Nonlinear Anal., 77 (2013), 158-170. doi: 10.1016/j.na.2012.09.010. Google Scholar
[28]	J. C. Robinson, Infinite-Dimensional Dyanmical Systems: An Introduction to Dissipative Parabolic PDEs and the Theory of Global Attractors, Cambridge University Press, Cambridge, 2001. Google Scholar
[29]	M. Scheutzow, Comparsion of various concepts of a random attractor: A case study, Arch. Math., 78 (2002), 233-240. doi: 10.1007/s00013-002-8241-1. Google Scholar
[30]	B. Schmalfuß, Backward cocycle and attractors of stochastic differential equations, in: V. Reitmann, T. Riedrich, N. Koksch (Eds.), International Seminar on Applied MathematicsNonlinear Dynamics: Attractor Approximation and Global Behavior, Technische Universität, Dresden, (1992), 185-192. Google Scholar
[31]	B. Schmalfuß, Attractors for the nonautonomous dynamical systems, in:International Conference on Differential Equations, vol.1, 2, World Sci. Publishing, River Edge, NJ, (2000), 684-689. Google Scholar
[32]	C. Sun, L. Yuan and J. Shi, Higher-order integrability for a semilinear reaction-diffusion equation with distribution derivatives, Appl. Math. Lett., 26 (2013), 949-956. doi: 10.1016/j.aml.2013.04.010. Google Scholar
[33]	C. Sun and W. Tan, Non-autonomous reaction-diffusion model with dynamic boundary conditions, J. Math. Anal. Appl., 443 (2016), 1007-1032. doi: 10.1016/j.jmaa.2016.05.054. Google Scholar
[34]	R. Temam, Infinite-Dimensional Dynamical Systems in Mechanics and Physics, Springer-Verlag, New York, 1997. Google Scholar
[35]	B. Wang, Random attractors for non-autonomous stochastic wave euqations with multiplicative noises, Discrete Contin. Dyn. Syst., 34 (2014), 269-300. doi: 10.3934/dcds.2014.34.269. Google Scholar
[36]	B. Wang, Suffcient and necessary criteria for existence of pullback attractors for non-compact random dynamical systems, J. Differential Equations, 253 (2012), 1544-1583. doi: 10.1016/j.jde.2012.05.015. Google Scholar
[37]	M. Yang and P. E. Kloeden, Random attractors for stochastic semi-linear degenerate parabolic equations, Nonlinear Anal. Real World Appl., 12 (2011), 2811-2821. doi: 10.1016/j.nonrwa.2011.04.007. Google Scholar
[38]	J. Yin, Y. Li and H. Zhao, Random attractors for stochastic semi-linear degenerate parabolic equations with additive noise in $L^q$, Appl. Math. Comput., 225 (2013), 526-540. doi: 10.1016/j.amc.2013.09.051. Google Scholar
[39]	J. Yin, Y. Li and H. Cui, Box-counting dimensions and upper semicontinuities of bi-spatial attractors for stochastic degenerate parabolic equations onanunbounded domain, J. Math. Anal. Appl., 450 (2017), 1180-1207. doi: 10.1016/j.jmaa.2017.01.064. Google Scholar
[40]	W. Zhao, Regularity of random attractors for a degenerate parabolic equations driven by additive noises, Appl. Math. Comput., 239 (2014), 358-374. doi: 10.1016/j.amc.2014.04.106. Google Scholar
[41]	W. Zhao and Y. Zhang, Compactness and attracting of random attractors for non-autonomous stochastic lattice dynamical systems in weighted space $\ell_ρ^p$, Appl. Math. Comput., 291 (2016), 226-243. doi: 10.1016/j.amc.2016.06.045. Google Scholar
[42]	W. Zhao, Regularity of random attractors for a stochastic degenerate parabolic equation driven by multiplicative noise, Acta Math. Sci. Ser. B Engl. Ed., 36 (2016), 409-427. doi: 10.1016/S0252-9602(16)30009-1. Google Scholar
[43]	W. Zhao and Y. Li, $(L^2, L^p)$-random attractors for stochastic reaction-diffusion equation on unbounded domains, Nonlinear Anal., 75 (2012), 485-502. doi: 10.1016/j.na.2011.08.050. Google Scholar
[44]	W. Zhao, Long-time random dynamics of stochastic parabolic $p$-Laplacian equations on $\mathbb{R}^N$, Nonlinear Anal., 152 (2017), 196-219. doi: 10.1016/j.na.2017.01.004. Google Scholar
[45]	K. Zhu and F. Zhou, Continuity and pullback attractors for a non-autonomous reaction-diffusion equation in $\mathbb{R}^N$, Comput. Math. Appl., 71 (2016), 2089-2105. doi: 10.1016/j.camwa.2016.04.004. Google Scholar

show all references

References:

[1]	C. T. Anh and T. Q. Bao, Pullback attractors for a non-autonomous semi-linear degenerate parabolic equation, Glasg. Math. J., 52 (2010), 537-554. doi: 10.1017/S0017089510000418. Google Scholar
[2]	C. T. Anh and L. T. Thuy, Global attractors for a class of semilinear degenerate parabolic equations on $\mathbb{R}^N$, Bull. Pol. Acad. Sci. Math., 61 (2013), 47-65. doi: 10.4064/ba61-1-6. Google Scholar
[3]	L. Arnold, Random Dynamical System, Springer-Verlag, Berlin, 1998. Google Scholar
[4]	T. Bartsch and Z. Liu, On a supperlinear elliptic $p$-Laplacian equation, J. Differential Equations, 198 (2004), 149-179. doi: 10.1016/j.jde.2003.08.001. Google Scholar
[5]	P. W. Bates, H. Lisei and K. Lu, Attractors for stochastic lattice dynamical systems, Stoch. Dyn., 6 (2006), 1-21. doi: 10.1142/S0219493706001621. Google Scholar
[6]	P. Caldiroli and R. Musina, On a variational degenerate elliptic problem, Nonlinear Differ. Equ. Appl., 7 (2000), 187-199. doi: 10.1007/s000300050004. Google Scholar
[7]	A. N. Carvalho, J. A. Langa and J. C. Robinson, Attractors for Infinite-Dimensional Non-autonomous Dynamical Systems, Appl. Math. Sciences, vol. 184, Springer, 2013. Google Scholar
[8]	D. Cao, C. Sun and M. Yang, Dynamics for a stochastic reaction-diffusion equation with additive noise, J. Differential Equations, 259 (2015), 838-872. doi: 10.1016/j.jde.2015.02.020. Google Scholar
[9]	I. Chueshov, Monotone Random Systems Theory and Applications, Springer-Verlag, Berlin, 2002. Google Scholar
[10]	I. Chueshov and B. Schmalfuß, Parabolic stochastic partial differential equations with dynamical boundary conditions, Differential Integral Equtions, 17 (2004), 751-780. Google Scholar
[11]	H. Crauel and F. Flandoli, Attractors for random dynamical systems, Probab. Theory Related Fields, 100 (1994), 365-393. doi: 10.1007/BF01193705. Google Scholar
[12]	H. Crauel, A. Debussche and F. Flandoli, Random attractors, J. Dynam. Differential Equations, 9 (1997), 307-341. doi: 10.1007/BF02219225. Google Scholar
[13]	H. Crauel, G. Dimitroff and M. Scheutzow, Criteria for strong and weak random attractors, J. Dyn. Differ. Equ., 21 (2009), 233-247. doi: 10.1007/s10884-009-9135-8. Google Scholar
[14]	H. Crauel and P. E. Kloeden, Nonautonomous and random attractors, Jahresber. Dtsch. Math.-Ver., 117 (2015), 173-206. doi: 10.1365/s13291-015-0115-0. Google Scholar
[15]	H. Cui and Y. Li, Existence and upper semicontinuity of random attractors for stochastic degenerate parabolic equations with multiplicative noises, Appl. Math. Comput., 271 (2015), 777-789. doi: 10.1016/j.amc.2015.09.031. Google Scholar
[16]	G. Da Prato and Z. Jerzy, Stochastic Equations in Infinite Dimensions, Cambridge University Press, Cambridge, 1992. Google Scholar
[17]	R. Dautray and J. L. Lions, Mathematical Analysis and Numerical Methods for Science and Technology, Vol. Ⅰ: Physical origins and classical methods, Springer-Verlag, Berlin, 1990. Google Scholar
[18]	F. Flandoli and B. Schmalfuß, Random attractors for the stochastic Navier-Stokes equation with multiplicative noise, Stoch. Stoch. Rep., 59 (1996), 21-45. doi: 10.1080/17442509608834083. Google Scholar
[19]	F. Flandoli, B. Gess and M. Scheutzow, Synchronization by noise, Probab. Theory Related Fields, 168 (2017), 511-556. doi: 10.1007/s00440-016-0716-2. Google Scholar
[20]	N. I. Karachalios and N. B. Zographopoulos, On the dynamics of a degenerate parabolic equation: Global bifurcation of stationary states and convergence, Calc. Var. Partial Differential Equations, 25 (2006), 361-393. doi: 10.1007/s00526-005-0347-4. Google Scholar
[21]	A. Krause and B. Wang, Pullback attractors of non-autonomous stochastic degenerate parabolic equations on unbounded domains, J. Math. Anal. Appl., 417 (2014), 1018-1038. doi: 10.1016/j.jmaa.2014.03.037. Google Scholar
[22]	A. Krause, M. Lewis and B. Wang, Dynamics of the non-autonomous stochastic $p$-Laplace equation driven by multiplicative noise, Appl. Math. Comput., 246 (2014), 365-376. doi: 10.1016/j.amc.2014.08.033. Google Scholar
[23]	X. Li, C. Sun and N. Zhang, Dynamics for a non-autonomous degenerate parabolic equation in $D_0^{1}(Ω,σ)$, Discrete Contin. Dyn. Syst., 36 (2016), 7063-7079. doi: 10.3934/dcds.2016108. Google Scholar
[24]	Y. Li and J. Yin, A modiffied proof of pullback attractors in a Sobolev space for stochastic FitzHugh-Nagumo equations, Discrete Contin. Dyn. Syst., 21 (2016), 1203-1223. doi: 10.3934/dcdsb.2016.21.1203. Google Scholar
[25]	Y. Li and B. Guo, Random attractors for quasi-continuous random dynamical systems and applications to stochastic reaction-diffusion equations, J. Differential Equations, 245 (2008), 1775-1800. doi: 10.1016/j.jde.2008.06.031. Google Scholar
[26]	Y. Li, A. Gu and J. Li, Existence and continuity of bi-spatial random attractors and application to stochasitic semilinear Laplacian equations, J. Differential Equations, 258 (2015), 504-534. doi: 10.1016/j.jde.2014.09.021. Google Scholar
[27]	W. Niu, Global attractors for degenerate semilinear parabolic equations, Nonlinear Anal., 77 (2013), 158-170. doi: 10.1016/j.na.2012.09.010. Google Scholar
[28]	J. C. Robinson, Infinite-Dimensional Dyanmical Systems: An Introduction to Dissipative Parabolic PDEs and the Theory of Global Attractors, Cambridge University Press, Cambridge, 2001. Google Scholar
[29]	M. Scheutzow, Comparsion of various concepts of a random attractor: A case study, Arch. Math., 78 (2002), 233-240. doi: 10.1007/s00013-002-8241-1. Google Scholar
[30]	B. Schmalfuß, Backward cocycle and attractors of stochastic differential equations, in: V. Reitmann, T. Riedrich, N. Koksch (Eds.), International Seminar on Applied MathematicsNonlinear Dynamics: Attractor Approximation and Global Behavior, Technische Universität, Dresden, (1992), 185-192. Google Scholar
[31]	B. Schmalfuß, Attractors for the nonautonomous dynamical systems, in:International Conference on Differential Equations, vol.1, 2, World Sci. Publishing, River Edge, NJ, (2000), 684-689. Google Scholar
[32]	C. Sun, L. Yuan and J. Shi, Higher-order integrability for a semilinear reaction-diffusion equation with distribution derivatives, Appl. Math. Lett., 26 (2013), 949-956. doi: 10.1016/j.aml.2013.04.010. Google Scholar
[33]	C. Sun and W. Tan, Non-autonomous reaction-diffusion model with dynamic boundary conditions, J. Math. Anal. Appl., 443 (2016), 1007-1032. doi: 10.1016/j.jmaa.2016.05.054. Google Scholar
[34]	R. Temam, Infinite-Dimensional Dynamical Systems in Mechanics and Physics, Springer-Verlag, New York, 1997. Google Scholar
[35]	B. Wang, Random attractors for non-autonomous stochastic wave euqations with multiplicative noises, Discrete Contin. Dyn. Syst., 34 (2014), 269-300. doi: 10.3934/dcds.2014.34.269. Google Scholar
[36]	B. Wang, Suffcient and necessary criteria for existence of pullback attractors for non-compact random dynamical systems, J. Differential Equations, 253 (2012), 1544-1583. doi: 10.1016/j.jde.2012.05.015. Google Scholar
[37]	M. Yang and P. E. Kloeden, Random attractors for stochastic semi-linear degenerate parabolic equations, Nonlinear Anal. Real World Appl., 12 (2011), 2811-2821. doi: 10.1016/j.nonrwa.2011.04.007. Google Scholar
[38]	J. Yin, Y. Li and H. Zhao, Random attractors for stochastic semi-linear degenerate parabolic equations with additive noise in $L^q$, Appl. Math. Comput., 225 (2013), 526-540. doi: 10.1016/j.amc.2013.09.051. Google Scholar
[39]	J. Yin, Y. Li and H. Cui, Box-counting dimensions and upper semicontinuities of bi-spatial attractors for stochastic degenerate parabolic equations onanunbounded domain, J. Math. Anal. Appl., 450 (2017), 1180-1207. doi: 10.1016/j.jmaa.2017.01.064. Google Scholar
[40]	W. Zhao, Regularity of random attractors for a degenerate parabolic equations driven by additive noises, Appl. Math. Comput., 239 (2014), 358-374. doi: 10.1016/j.amc.2014.04.106. Google Scholar
[41]	W. Zhao and Y. Zhang, Compactness and attracting of random attractors for non-autonomous stochastic lattice dynamical systems in weighted space $\ell_ρ^p$, Appl. Math. Comput., 291 (2016), 226-243. doi: 10.1016/j.amc.2016.06.045. Google Scholar
[42]	W. Zhao, Regularity of random attractors for a stochastic degenerate parabolic equation driven by multiplicative noise, Acta Math. Sci. Ser. B Engl. Ed., 36 (2016), 409-427. doi: 10.1016/S0252-9602(16)30009-1. Google Scholar
[43]	W. Zhao and Y. Li, $(L^2, L^p)$-random attractors for stochastic reaction-diffusion equation on unbounded domains, Nonlinear Anal., 75 (2012), 485-502. doi: 10.1016/j.na.2011.08.050. Google Scholar
[44]	W. Zhao, Long-time random dynamics of stochastic parabolic $p$-Laplacian equations on $\mathbb{R}^N$, Nonlinear Anal., 152 (2017), 196-219. doi: 10.1016/j.na.2017.01.004. Google Scholar
[45]	K. Zhu and F. Zhou, Continuity and pullback attractors for a non-autonomous reaction-diffusion equation in $\mathbb{R}^N$, Comput. Math. Appl., 71 (2016), 2089-2105. doi: 10.1016/j.camwa.2016.04.004. Google Scholar

[1]	Rodrigo Samprogna, Tomás Caraballo. Pullback attractor for a dynamic boundary non-autonomous problem with Infinite Delay. Discrete & Continuous Dynamical Systems - B, 2018, 23 (2) : 509-523. doi: 10.3934/dcdsb.2017195
[2]	T. Caraballo, J. A. Langa, J. Valero. Structure of the pullback attractor for a non-autonomous scalar differential inclusion. Discrete & Continuous Dynamical Systems - S, 2016, 9 (4) : 979-994. doi: 10.3934/dcdss.2016037
[3]	Yangrong Li, Renhai Wang, Jinyan Yin. Backward compact attractors for non-autonomous Benjamin-Bona-Mahony equations on unbounded channels. Discrete & Continuous Dynamical Systems - B, 2017, 22 (7) : 2569-2586. doi: 10.3934/dcdsb.2017092
[4]	Yangrong Li, Shuang Yang. Backward compact and periodic random attractors for non-autonomous sine-Gordon equations with multiplicative noise. Communications on Pure & Applied Analysis, 2019, 18 (3) : 1155-1175. doi: 10.3934/cpaa.2019056
[5]	José A. Langa, James C. Robinson, Aníbal Rodríguez-Bernal, A. Suárez, A. Vidal-López. Existence and nonexistence of unbounded forwards attractor for a class of non-autonomous reaction diffusion equations. Discrete & Continuous Dynamical Systems - A, 2007, 18 (2&3) : 483-497. doi: 10.3934/dcds.2007.18.483
[6]	Lu Yang, Meihua Yang, Peter E. Kloeden. Pullback attractors for non-autonomous quasi-linear parabolic equations with dynamical boundary conditions. Discrete & Continuous Dynamical Systems - B, 2012, 17 (7) : 2635-2651. doi: 10.3934/dcdsb.2012.17.2635
[7]	Angelo Favini, Georgy A. Sviridyuk, Alyona A. Zamyshlyaeva. One Class of Sobolev Type Equations of Higher Order with Additive "White Noise". Communications on Pure & Applied Analysis, 2016, 15 (1) : 185-196. doi: 10.3934/cpaa.2016.15.185
[8]	María Anguiano, Tomás Caraballo, José Real, José Valero. Pullback attractors for reaction-diffusion equations in some unbounded domains with an $H^{-1}$-valued non-autonomous forcing term and without uniqueness of solutions. Discrete & Continuous Dynamical Systems - B, 2010, 14 (2) : 307-326. doi: 10.3934/dcdsb.2010.14.307
[9]	Hong Lu, Jiangang Qi, Bixiang Wang, Mingji Zhang. Random attractors for non-autonomous fractional stochastic parabolic equations on unbounded domains. Discrete & Continuous Dynamical Systems - A, 2019, 39 (2) : 683-706. doi: 10.3934/dcds.2019028
[10]	Ting Li. Pullback attractors for asymptotically upper semicompact non-autonomous multi-valued semiflows. Communications on Pure & Applied Analysis, 2007, 6 (1) : 279-285. doi: 10.3934/cpaa.2007.6.279
[11]	Peter E. Kloeden, Jacson Simsen. Pullback attractors for non-autonomous evolution equations with spatially variable exponents. Communications on Pure & Applied Analysis, 2014, 13 (6) : 2543-2557. doi: 10.3934/cpaa.2014.13.2543
[12]	Zhijian Yang, Yanan Li. Upper semicontinuity of pullback attractors for non-autonomous Kirchhoff wave equations. Discrete & Continuous Dynamical Systems - B, 2019, 24 (9) : 4899-4912. doi: 10.3934/dcdsb.2019036
[13]	Cung The Anh, Le Van Hieu, Nguyen Thieu Huy. Inertial manifolds for a class of non-autonomous semilinear parabolic equations with finite delay. Discrete & Continuous Dynamical Systems - A, 2013, 33 (2) : 483-503. doi: 10.3934/dcds.2013.33.483
[14]	Alexandre Nolasco de Carvalho, Marcelo J. D. Nascimento. Singularly non-autonomous semilinear parabolic problems with critical exponents. Discrete & Continuous Dynamical Systems - S, 2009, 2 (3) : 449-471. doi: 10.3934/dcdss.2009.2.449
[15]	Wen Tan. The regularity of pullback attractor for a non-autonomous p-Laplacian equation with dynamical boundary condition. Discrete & Continuous Dynamical Systems - B, 2019, 24 (2) : 529-546. doi: 10.3934/dcdsb.2018194
[16]	Bixiang Wang. Random attractors for non-autonomous stochastic wave equations with multiplicative noise. Discrete & Continuous Dynamical Systems - A, 2014, 34 (1) : 269-300. doi: 10.3934/dcds.2014.34.269
[17]	Shengfan Zhou, Min Zhao. Fractal dimension of random attractor for stochastic non-autonomous damped wave equation with linear multiplicative white noise. Discrete & Continuous Dynamical Systems - A, 2016, 36 (5) : 2887-2914. doi: 10.3934/dcds.2016.36.2887
[18]	Zhaojuan Wang, Shengfan Zhou. Random attractor and random exponential attractor for stochastic non-autonomous damped cubic wave equation with linear multiplicative white noise. Discrete & Continuous Dynamical Systems - A, 2018, 38 (9) : 4767-4817. doi: 10.3934/dcds.2018210
[19]	Feliz Minhós, Hugo Carrasco. Solvability of higher-order BVPs in the half-line with unbounded nonlinearities. Conference Publications, 2015, 2015 (special) : 841-850. doi: 10.3934/proc.2015.0841
[20]	Julia García-Luengo, Pedro Marín-Rubio, José Real, James C. Robinson. Pullback attractors for the non-autonomous 2D Navier--Stokes equations for minimally regular forcing. Discrete & Continuous Dynamical Systems - A, 2014, 34 (1) : 203-227. doi: 10.3934/dcds.2014.34.203

American Institute of Mathematical Sciences

Random dynamics of non-autonomous semi-linear degenerate parabolic equations on $\mathbb{R}^N$ driven by an unbounded additive noise

References:

References:

Tools

Metrics

Other articles
by authors

Tools

Tools

Metrics

Other articles
by authors

American Institute of Mathematical Sciences

Random dynamics of non-autonomous semi-linear degenerate parabolic equations on $\mathbb{R}^N$ driven by an unbounded additive noise

References:

References:

Tools

Metrics

Other articlesby authors

Tools

Tools

Metrics

Other articlesby authors

Other articles
by authors

Other articles
by authors