# American Institute of Mathematical Sciences

doi: 10.3934/dcdsb.2021212
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## Uniform attractors for nonautonomous reaction-diffusion equations with the nonlinearity in a larger symbol space

 Department of Mathematics, Nanjing University, Nanjing 210093, China

* Corresponding author: Chengkui Zhong

Received  March 2021 Revised  June 2021 Early access August 2021

Fund Project: The work is supported by the NSFC(11731005)

Existence and structure of the uniform attractors for reaction-diffusion equations with the nonlinearity in a weaker topology space are considered. Firstly, a weaker symbol space is defined and an example is given as well, showing that the compactness can be easier obtained in this space. Then the existence of solutions with new symbols is presented. Finally, the existence and structure of the uniform attractor are obtained by proving the $(L^{2}\times \Sigma, L^{2})$-continuity of the processes generated by solutions.

Citation: Xiangming Zhu, Chengkui Zhong. Uniform attractors for nonautonomous reaction-diffusion equations with the nonlinearity in a larger symbol space. Discrete and Continuous Dynamical Systems - B, doi: 10.3934/dcdsb.2021212
##### References:
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##### References:
 [1] C. T. Anh and N. V. Quang, Uniform attractors for nonautonomous parabolic equations involving weighted p-Laplacian operators, Ann. Polon. Math., 98 (2010), 251-271.  doi: 10.4064/ap98-3-5. [2] A. N. Carvalho, J. A. Langa and J. C. Robinson, Attractors for Infinite-Dimensional Non-Autonomous Dynamical Systems, Applied Mathematical Sciences, 182, Springer, New York, 2013. doi: 10.1007/978-1-4614-4581-4. [3] V. V. Chepyzhov and M. I. Vishik, Attractors for Equations of Mathematical Physics, American Mathematical Society Colloquium Publications, 49, American Mathematical Society, Providence, RI, 2002. doi: 10.1051/cocv:2002056. [4] H. Crauel and F. Flandoli, Attractors for random dynamical systems, Probab. Theory Related Fields, 100 (1994), 365-393.  doi: 10.1007/BF01193705. [5] A. Haraux, Systuèmes Dynamiques Dissipatifs et Applications, Paris, Masson, 1991. [6] P. E. Kloeden and M. Rasmussen, Nonautonomous Dynamical Systems, Mathematical Surveys and Monographs, vol. 176, American Mathematical Soc., 2011. doi: 10.1090/surv/176. [7] X. Li, Uniform random attractors for 2D non-autonomous stochastic Navier-Stokes equations, J. Differential Equations, 276 (2021), 1-42.  doi: 10.1016/j.jde.2020.12.014. [8] S. Lu, Attractors for nonautonomous 2D Navier-Stokes equations with less regular normal forces, J. Differential Equations, 230 (2006), 196-212.  doi: 10.1016/j.jde.2006.07.009. [9] S. Lu, Attractors for nonautonomous reaction-diffusion systems with symbols without strong translation compactness, Asymptot. Anal., 54 (2007), 197-210. [10] S. Lu, H. Wu and C. Zhong, Attractors for nonautonomous 2D Navier-Stokes equations with normal external forces, Discrete Contin. Dyn. Syst., 13 (2005), 701-719.  doi: 10.3934/dcds.2005.13.701. [11] S. Ma, X. Cheng and H. Li, Attractors for non-autonomous wave equations with a new class of external forces, J. Math. Anal. Appl., 337 (2008), 808-820.  doi: 10.1016/j.jmaa.2007.03.108. [12] S. Ma and C. Zhong, The attractors for weakly damped non-autonomous hyperbolic equations with a new class of external forces, Discrete Contin. Dyn. Syst., 18 (2007), 53-70.  doi: 10.3934/dcds.2007.18.53. [13] S. Ma, C. Zhong and H. Song, Attractors for nonautonomous 2D Navier-Stokes equations with less regular symbols, Nonlinear Anal., 71 (2009), 4215-4222.  doi: 10.1016/j.na.2009.02.107. [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. [15] R. Temam, Infinite Dimensional Dynamical Systems in Mechanics and Physics, 2nd edition, Berlin, Springer, 1997. doi: 10.1007/978-1-4612-0645-3. [16] J. Valero, Characterization of the attractor for nonautonomous reaction-diffusion equations with discontinuous nonlinearity, J. Differential Equations, 275 (2021), 270-308.  doi: 10.1016/j.jde.2020.11.036. [17] Y. Xie, K. Zhu and C. Sun, The existence of uniform attractors for non-autonomous reaction diffusion equations on the whole space, J. Math. Phys., 53(2012), 082703, 11 pp. doi: 10.1063/1.4746693. [18] J. Xu, Z. Zhang and T. Caraballo, Non-autonomous nonlocal partial differential equations with delay and memory, J. Differential Equations, 270 (2021), 505-546.  doi: 10.1016/j.jde.2020.07.037. [19] X.-G. Yang, M. J. D. Nascimento and M. L. Pelicer, Uniform attractors for non-autonomous plate equations with p-Laplacian perturbation and critical nonlinearities, Discrete Contin. Dyn. Syst., 40 (2020), 1937-1961.  doi: 10.3934/dcds.2020100. [20] S. Zelik, Strong uniform attractors for non-autonomous dissipative PDEs with non translationcompact external force, Discrete Contin. Dyn. Syst. Ser. B, 20 (2015), 781-810.  doi: 10.3934/dcdsb.2015.20.781.
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