# American Institute of Mathematical Sciences

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

 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 $N \geqslant 1$.

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
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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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##### 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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