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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 |
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.
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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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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.
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T. Bartsch and Z. Liu,
On a supperlinear elliptic $p$-Laplacian equation, J. Differential Equations, 198 (2004), 149-179.
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P. W. Bates, H. Lisei and K. Lu,
Attractors for stochastic lattice dynamical systems, Stoch. Dyn., 6 (2006), 1-21.
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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. |
[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. |
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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. |
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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.
|
[11] |
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. 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.
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G. Da Prato and Z. Jerzy, Stochastic Equations in Infinite Dimensions, Cambridge University Press, Cambridge, 1992. |
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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.
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[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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