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

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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 August 2018 Early access 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 and 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.
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[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.
[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.
[6]	P. Caldiroli and R. Musina, On a variational degenerate elliptic problem, Nonlinear Differ. Equ. Appl., 7 (2000), 187-199. doi: 10.1007/s000300050004.
[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.
[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.
[9]	I. Chueshov, Monotone Random Systems Theory and Applications, Springer-Verlag, Berlin, 2002.
[10]	I. Chueshov and B. Schmalfuß, Parabolic stochastic partial differential equations with dynamical boundary conditions, Differential Integral Equtions, 17 (2004), 751-780.
[11]	H. Crauel and F. Flandoli, Attractors for random dynamical systems, Probab. Theory Related Fields, 100 (1994), 365-393. doi: 10.1007/BF01193705.
[12]	H. Crauel, A. Debussche and F. Flandoli, Random attractors, J. Dynam. Differential Equations, 9 (1997), 307-341. doi: 10.1007/BF02219225.
[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.
[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.
[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.
[16]	G. Da Prato and Z. Jerzy, Stochastic Equations in Infinite Dimensions, Cambridge University Press, Cambridge, 1992.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[27]	W. Niu, Global attractors for degenerate semilinear parabolic equations, Nonlinear Anal., 77 (2013), 158-170. doi: 10.1016/j.na.2012.09.010.
[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.
[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.
[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.
[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.
[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.
[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.
[34]	R. Temam, Infinite-Dimensional Dynamical Systems in Mechanics and Physics, Springer-Verlag, New York, 1997.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.

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.
[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.
[3]	L. Arnold, Random Dynamical System, Springer-Verlag, Berlin, 1998.
[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.
[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.
[6]	P. Caldiroli and R. Musina, On a variational degenerate elliptic problem, Nonlinear Differ. Equ. Appl., 7 (2000), 187-199. doi: 10.1007/s000300050004.
[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.
[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.
[9]	I. Chueshov, Monotone Random Systems Theory and Applications, Springer-Verlag, Berlin, 2002.
[10]	I. Chueshov and B. Schmalfuß, Parabolic stochastic partial differential equations with dynamical boundary conditions, Differential Integral Equtions, 17 (2004), 751-780.
[11]	H. Crauel and F. Flandoli, Attractors for random dynamical systems, Probab. Theory Related Fields, 100 (1994), 365-393. doi: 10.1007/BF01193705.
[12]	H. Crauel, A. Debussche and F. Flandoli, Random attractors, J. Dynam. Differential Equations, 9 (1997), 307-341. doi: 10.1007/BF02219225.
[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.
[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.
[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.
[16]	G. Da Prato and Z. Jerzy, Stochastic Equations in Infinite Dimensions, Cambridge University Press, Cambridge, 1992.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[27]	W. Niu, Global attractors for degenerate semilinear parabolic equations, Nonlinear Anal., 77 (2013), 158-170. doi: 10.1016/j.na.2012.09.010.
[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.
[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.
[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.
[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.
[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.
[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.
[34]	R. Temam, Infinite-Dimensional Dynamical Systems in Mechanics and Physics, Springer-Verlag, New York, 1997.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.

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