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

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.

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
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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.

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L. Arnold, Random Dynamical System, Springer-Verlag, Berlin, 1998.

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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.

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P. W. BatesH. Lisei and K. Lu, Attractors for stochastic lattice dynamical systems, Stoch. Dyn., 6 (2006), 1-21. doi: 10.1142/S0219493706001621.

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P. Caldiroli and R. Musina, On a variational degenerate elliptic problem, Nonlinear Differ. Equ. Appl., 7 (2000), 187-199. doi: 10.1007/s000300050004.

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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.

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D. CaoC. 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.

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I. Chueshov, Monotone Random Systems Theory and Applications, Springer-Verlag, Berlin, 2002.

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I. Chueshov and B. Schmalfuß, Parabolic stochastic partial differential equations with dynamical boundary conditions, Differential Integral Equtions, 17 (2004), 751-780.

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H. Crauel and F. Flandoli, Attractors for random dynamical systems, Probab. Theory Related Fields, 100 (1994), 365-393. doi: 10.1007/BF01193705.

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H. CrauelA. Debussche and F. Flandoli, Random attractors, J. Dynam. Differential Equations, 9 (1997), 307-341. doi: 10.1007/BF02219225.

[13]

H. CrauelG. 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. FlandoliB. 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. KrauseM. 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. LiC. 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. LiA. 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. SunL. 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. YinY. 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. YinY. 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. BatesH. 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. CaoC. 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. CrauelA. Debussche and F. Flandoli, Random attractors, J. Dynam. Differential Equations, 9 (1997), 307-341. doi: 10.1007/BF02219225.

[13]

H. CrauelG. 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. FlandoliB. 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. KrauseM. 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. LiC. 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. LiA. 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. SunL. 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. YinY. 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. YinY. 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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