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September  2020, 28(3): 1357-1374. doi: 10.3934/era.2020072

Strong $ (L^2,L^\gamma\cap H_0^1) $-continuity in initial data of nonlinear reaction-diffusion equation in any space dimension

Hongyong Cui 1, , Peter E. Kloeden 2, and  Wenqiang Zhao 3,

1. 

School of Mathematics and Statistics, Huazhong University of Science and Technology, Wuhan 430074, China
2. 

Mathematisches Institut, Universität Tübingen, D-72076 Tübingen, Germany
3. 

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

Received  March 2020 Published  July 2020

In this paper we study the continuity in initial data of a classical reaction-diffusion equation with arbitrary $ p>2 $ order nonlinearity and in any space dimension $ N \geqslant 1 $. It is proved that the weak solutions can be $ (L^2, L^\gamma\cap H_0^1) $-continuous in initial data for arbitrarily large $ \gamma \geqslant 2 $ (independent of the physical parameters of the system), i.e., can converge in the norm of any $ L^\gamma\cap H_0^1 $ as the corresponding initial values converge in $ L^2 $. In fact, the system is shown to be $ (L^2, L^\gamma\cap H_0^1) $-smoothing in a H$ \ddot{\rm o} $lder way. Applying this to the global attractor we find that, with external forcing only in $ L^2 $, the attractor $ \mathscr{A} $ attracts bounded subsets of $ L^2 $ in the norm of any $ L^\gamma\cap H_0^1 $, and that every translation set $ \mathscr{A}-z_0 $ of $ \mathscr{A} $ for any $ z_0\in \mathscr{A} $ is a finite dimensional compact subset of $ L^\gamma\cap H_0^1 $. The main technique we employ is a combination of a Moser iteration and a decomposition of the nonlinearity, by which the interpolation inequalities are avoided and the new continuity result is obtained without any restrictions on the order $ p>2 $ of the nonlinearity and the space dimension $ N \geqslant 1 $.

Keywords: Smoothing property, global attractor, regularity, fractal dimension.
Mathematics Subject Classification: 35B40, 35B41, 37L30.
Citation: Hongyong Cui, Peter E. Kloeden, Wenqiang Zhao. Strong $ (L^2,L^\gamma\cap H_0^1) $-continuity in initial data of nonlinear reaction-diffusion equation in any space dimension. Electronic Research Archive, 2020, 28 (3) : 1357-1374. doi: 10.3934/era.2020072
References:
[1]

A. V. Babin and M. I. Vishik, Attractors of Evolution Equations, Studies in Mathematics and its Applications, 25, North-Holland Publishing Co., Amsterdam, 1992.  Google Scholar

[2]

T. Bartsch and Z. Liu, On a superlinear elliptic $p$-Laplacian equation, J. Differential Equations, 198 (2004), 149-175.  doi: 10.1016/j.jde.2003.08.001.  Google Scholar

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T. Caraballo and S. Sonner, Random pullback exponential attractors: General existence results for random dynamical systems in Banach spaces, Discrete Contin. Dyn. Syst., 37 (2017), 6383-6403.  doi: 10.3934/dcds.2017277.  Google Scholar

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H. Cui, J. A. Langa and Y. Li, Measurability of random attractors for quasi strong-to-weak continuous random dynamical systems, J. Dynam. Differential Equations, 30 (2018), 1873-1898.  doi: 10.1007/s10884-017-9617-z.  Google Scholar

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H. Cui, Y. Li and J. Yin, Existence and upper semicontinuity of bi-spatial pullback attractors for smoothing cocycles, Nonlinear Anal., 128 (2015), 303-324.  doi: 10.1016/j.na.2015.08.009.  Google Scholar

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L. C. Evans, Partial Differential Equations, Graduate Studies in Mathematics, 19, American Mathematical Society, Providence, RI, 2010. doi: 10.1090/gsm/019.  Google Scholar

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Y. Li, A. Gu and J. Li, Existence and continuity of bi-spatial random attractors and application to stochastic semilinear Laplacian equations, J. Differential Equations, 258 (2015), 504-534.  doi: 10.1016/j.jde.2014.09.021.  Google Scholar

[11]

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

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A. Miranville and S. Zelik, Attractors for dissipative partial differential equations in bounded and unbounded domains, in Handbook of Differential Equations: Evolutionary Equations. Vol. IV, Handb. Differ. Equ., Elsevier/North-Holland, Amsterdam, 103–200. doi: 10.1016/S1874-5717(08)00003-0.  Google Scholar

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J. C. Robinson, Infinite-Dimensional Dynamical Systems. An Introduction to Dissipative Parabolic PDEs and the Theory of Global Attractors, Cambridge Texts in Applied Mathematics, Cambridge University Press, Cambridge, 2001. doi: 10.1007/978-94-010-0732-0.  Google Scholar

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J. C. Robinson, Dimensions, Embeddings, and Attractors, Cambridge Tracts in Mathematics, 186, Cambridge University Press, Cambridge, 2011. doi: 10.1017/CBO9780511933912.  Google Scholar

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A. Shirikyan and S. Zelik, Exponential attractors for random dynamical systems and applications, Stoch. Partial Differ. Equ. Anal. Comput., 1 (2013), 241-281.  doi: 10.1007/s40072-013-0007-1.  Google Scholar

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C. Sun, Asymptotic regularity for some dissipative equations, J. Differential Equations, 248 (2010), 342-362.  doi: 10.1016/j.jde.2009.08.007.  Google Scholar

[18]

T. Trujillo and B. Wang, Continuity of strong solutions of the reaction-diffusion equation in initial data, Nonlinear Anal., 69 (2008), 2525-2532.  doi: 10.1016/j.na.2007.08.032.  Google Scholar

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S. Zelik, The attractor for a nonlinear reaction-diffusion system with a supercritical nonlinearity and its dimension, Rend. Accad. Naz. Sci. XL Mem. Mat. Appl. (5), 24 (2000), 1–25.  Google Scholar

[20]

W. Zhao, Random dynamics of stochastic $p$-Laplacian equations on $\mathbb{R}^N$ with an unbounded additive noise, J. Math. Anal. Appl., 455 (2017), 1178-1203.  doi: 10.1016/j.jmaa.2017.06.025.  Google Scholar

[21]

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

[22]

C.-K. Zhong, M.-H. Yang and C.-Y. Sun, The existence of global attractors for the norm-to-weak continuous semigroup and application to the nonlinear reaction-diffusion equations, J. Differential Equations, 223 (2006), 367-399.  doi: 10.1016/j.jde.2005.06.008.  Google Scholar

[23]

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]

A. V. Babin and M. I. Vishik, Attractors of Evolution Equations, Studies in Mathematics and its Applications, 25, North-Holland Publishing Co., Amsterdam, 1992.  Google Scholar

[2]

T. Bartsch and Z. Liu, On a superlinear elliptic $p$-Laplacian equation, J. Differential Equations, 198 (2004), 149-175.  doi: 10.1016/j.jde.2003.08.001.  Google Scholar

[3]

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

[4]

T. Caraballo and S. Sonner, Random pullback exponential attractors: General existence results for random dynamical systems in Banach spaces, Discrete Contin. Dyn. Syst., 37 (2017), 6383-6403.  doi: 10.3934/dcds.2017277.  Google Scholar

[5]

M. Coti Zelati and P. Kalita, Minimality properties of set-valued processes and their pullback attractors, SIAM J. Math. Anal., 47 (2015), 1530-1561.  doi: 10.1137/140978995.  Google Scholar

[6]

H. Cui, J. A. Langa and Y. Li, Measurability of random attractors for quasi strong-to-weak continuous random dynamical systems, J. Dynam. Differential Equations, 30 (2018), 1873-1898.  doi: 10.1007/s10884-017-9617-z.  Google Scholar

[7]

H. Cui, Y. Li and J. Yin, Existence and upper semicontinuity of bi-spatial pullback attractors for smoothing cocycles, Nonlinear Anal., 128 (2015), 303-324.  doi: 10.1016/j.na.2015.08.009.  Google Scholar

[8]

A. Eden, C. Foias, B. Nicolaenko and R. Temam, Exponential Attractors for Dissipative Evolution Equations, RAM: Research in Applied Mathematics, 37, Masson, Paris; John Wiley & Sons, Ltd., Chichester, 1994.  Google Scholar

[9]

L. C. Evans, Partial Differential Equations, Graduate Studies in Mathematics, 19, American Mathematical Society, Providence, RI, 2010. doi: 10.1090/gsm/019.  Google Scholar

[10]

Y. Li, A. Gu and J. Li, Existence and continuity of bi-spatial random attractors and application to stochastic semilinear Laplacian equations, J. Differential Equations, 258 (2015), 504-534.  doi: 10.1016/j.jde.2014.09.021.  Google Scholar

[11]

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

[12]

A. Miranville and S. Zelik, Attractors for dissipative partial differential equations in bounded and unbounded domains, in Handbook of Differential Equations: Evolutionary Equations. Vol. IV, Handb. Differ. Equ., Elsevier/North-Holland, Amsterdam, 103–200. doi: 10.1016/S1874-5717(08)00003-0.  Google Scholar

[13]

V. Pata and S. Zelik, A result on the existence of global attractors for semigroups of closed operators, Commun. Pure Appl. Anal., 6 (2007), 481-486.  doi: 10.3934/cpaa.2007.6.481.  Google Scholar

[14]

J. C. Robinson, Infinite-Dimensional Dynamical Systems. An Introduction to Dissipative Parabolic PDEs and the Theory of Global Attractors, Cambridge Texts in Applied Mathematics, Cambridge University Press, Cambridge, 2001. doi: 10.1007/978-94-010-0732-0.  Google Scholar

[15]

J. C. Robinson, Dimensions, Embeddings, and Attractors, Cambridge Tracts in Mathematics, 186, Cambridge University Press, Cambridge, 2011. doi: 10.1017/CBO9780511933912.  Google Scholar

[16]

A. Shirikyan and S. Zelik, Exponential attractors for random dynamical systems and applications, Stoch. Partial Differ. Equ. Anal. Comput., 1 (2013), 241-281.  doi: 10.1007/s40072-013-0007-1.  Google Scholar

[17]

C. Sun, Asymptotic regularity for some dissipative equations, J. Differential Equations, 248 (2010), 342-362.  doi: 10.1016/j.jde.2009.08.007.  Google Scholar

[18]

T. Trujillo and B. Wang, Continuity of strong solutions of the reaction-diffusion equation in initial data, Nonlinear Anal., 69 (2008), 2525-2532.  doi: 10.1016/j.na.2007.08.032.  Google Scholar

[19]

S. Zelik, The attractor for a nonlinear reaction-diffusion system with a supercritical nonlinearity and its dimension, Rend. Accad. Naz. Sci. XL Mem. Mat. Appl. (5), 24 (2000), 1–25.  Google Scholar

[20]

W. Zhao, Random dynamics of stochastic $p$-Laplacian equations on $\mathbb{R}^N$ with an unbounded additive noise, J. Math. Anal. Appl., 455 (2017), 1178-1203.  doi: 10.1016/j.jmaa.2017.06.025.  Google Scholar

[21]

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

[22]

C.-K. Zhong, M.-H. Yang and C.-Y. Sun, The existence of global attractors for the norm-to-weak continuous semigroup and application to the nonlinear reaction-diffusion equations, J. Differential Equations, 223 (2006), 367-399.  doi: 10.1016/j.jde.2005.06.008.  Google Scholar

[23]

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

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