doi: 10.3934/dcdsb.2021212
Online First

Online First articles are published articles within a journal that have not yet been assigned to a formal issue. This means they do not yet have a volume number, issue number, or page numbers assigned to them, however, they can still be found and cited using their DOI (Digital Object Identifier). Online First publication benefits the research community by making new scientific discoveries known as quickly as possible.

Readers can access Online First articles via the “Online First” tab for the selected journal.

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 & Continuous Dynamical Systems - B, doi: 10.3934/dcdsb.2021212
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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[4]

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

[5]

A. Haraux, Systuèmes Dynamiques Dissipatifs et Applications, Paris, Masson, 1991.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[9]

S. Lu, Attractors for nonautonomous reaction-diffusion systems with symbols without strong translation compactness, Asymptot. Anal., 54 (2007), 197-210.   Google Scholar

[10]

S. LuH. 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.  Google Scholar

[11]

S. MaX. 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.  Google Scholar

[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.  Google Scholar

[13]

S. MaC. 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.  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]

R. Temam, Infinite Dimensional Dynamical Systems in Mechanics and Physics, 2nd edition, Berlin, Springer, 1997. doi: 10.1007/978-1-4612-0645-3.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[18]

J. XuZ. 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.  Google Scholar

[19]

X.-G. YangM. 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.  Google Scholar

[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.  Google Scholar

show all references

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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[4]

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

[5]

A. Haraux, Systuèmes Dynamiques Dissipatifs et Applications, Paris, Masson, 1991.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[9]

S. Lu, Attractors for nonautonomous reaction-diffusion systems with symbols without strong translation compactness, Asymptot. Anal., 54 (2007), 197-210.   Google Scholar

[10]

S. LuH. 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.  Google Scholar

[11]

S. MaX. 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.  Google Scholar

[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.  Google Scholar

[13]

S. MaC. 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.  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]

R. Temam, Infinite Dimensional Dynamical Systems in Mechanics and Physics, 2nd edition, Berlin, Springer, 1997. doi: 10.1007/978-1-4612-0645-3.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[18]

J. XuZ. 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.  Google Scholar

[19]

X.-G. YangM. 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.  Google Scholar

[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.  Google Scholar

[1]

Yejuan Wang, Peter E. Kloeden. The uniform attractor of a multi-valued process generated by reaction-diffusion delay equations on an unbounded domain. Discrete & Continuous Dynamical Systems, 2014, 34 (10) : 4343-4370. doi: 10.3934/dcds.2014.34.4343

[2]

Elena Beretta, Cecilia Cavaterra. Identifying a space dependent coefficient in a reaction-diffusion equation. Inverse Problems & Imaging, 2011, 5 (2) : 285-296. doi: 10.3934/ipi.2011.5.285

[3]

M. Grasselli, V. Pata. A reaction-diffusion equation with memory. Discrete & Continuous Dynamical Systems, 2006, 15 (4) : 1079-1088. doi: 10.3934/dcds.2006.15.1079

[4]

Oleksiy V. Kapustyan, Pavlo O. Kasyanov, José Valero. Structure and regularity of the global attractor of a reaction-diffusion equation with non-smooth nonlinear term. Discrete & Continuous Dynamical Systems, 2014, 34 (10) : 4155-4182. doi: 10.3934/dcds.2014.34.4155

[5]

Linfang Liu, Xianlong Fu, Yuncheng You. Pullback attractor in $H^{1}$ for nonautonomous stochastic reaction-diffusion equations on $\mathbb{R}^n$. Discrete & Continuous Dynamical Systems - B, 2017, 22 (10) : 3629-3651. doi: 10.3934/dcdsb.2017143

[6]

Alexey Cheskidov, Songsong Lu. The existence and the structure of uniform global attractors for nonautonomous Reaction-Diffusion systems without uniqueness. Discrete & Continuous Dynamical Systems - S, 2009, 2 (1) : 55-66. doi: 10.3934/dcdss.2009.2.55

[7]

Zhaosheng Feng. Traveling waves to a reaction-diffusion equation. Conference Publications, 2007, 2007 (Special) : 382-390. doi: 10.3934/proc.2007.2007.382

[8]

Nick Bessonov, Gennady Bocharov, Tarik Mohammed Touaoula, Sergei Trofimchuk, Vitaly Volpert. Delay reaction-diffusion equation for infection dynamics. Discrete & Continuous Dynamical Systems - B, 2019, 24 (5) : 2073-2091. doi: 10.3934/dcdsb.2019085

[9]

Razvan Gabriel Iagar, Ariel Sánchez. Eternal solutions for a reaction-diffusion equation with weighted reaction. Discrete & Continuous Dynamical Systems, 2021  doi: 10.3934/dcds.2021160

[10]

Perla El Kettani, Danielle Hilhorst, Kai Lee. A stochastic mass conserved reaction-diffusion equation with nonlinear diffusion. Discrete & Continuous Dynamical Systems, 2018, 38 (11) : 5615-5648. doi: 10.3934/dcds.2018246

[11]

Henri Berestycki, Nancy Rodríguez. A non-local bistable reaction-diffusion equation with a gap. Discrete & Continuous Dynamical Systems, 2017, 37 (2) : 685-723. doi: 10.3934/dcds.2017029

[12]

Maho Endo, Yuki Kaneko, Yoshio Yamada. Free boundary problem for a reaction-diffusion equation with positive bistable nonlinearity. Discrete & Continuous Dynamical Systems, 2020, 40 (6) : 3375-3394. doi: 10.3934/dcds.2020033

[13]

Elena Trofimchuk, Sergei Trofimchuk. Admissible wavefront speeds for a single species reaction-diffusion equation with delay. Discrete & Continuous Dynamical Systems, 2008, 20 (2) : 407-423. doi: 10.3934/dcds.2008.20.407

[14]

Tomás Caraballo, José A. Langa, James C. Robinson. Stability and random attractors for a reaction-diffusion equation with multiplicative noise. Discrete & Continuous Dynamical Systems, 2000, 6 (4) : 875-892. doi: 10.3934/dcds.2000.6.875

[15]

Michio Urano, Kimie Nakashima, Yoshio Yamada. Transition layers and spikes for a reaction-diffusion equation with bistable nonlinearity. Conference Publications, 2005, 2005 (Special) : 868-877. doi: 10.3934/proc.2005.2005.868

[16]

Tarik Mohammed Touaoula, Mohammed Nor Frioui, Nikolay Bessonov, Vitaly Volpert. Dynamics of solutions of a reaction-diffusion equation with delayed inhibition. Discrete & Continuous Dynamical Systems - S, 2020, 13 (9) : 2425-2442. doi: 10.3934/dcdss.2020193

[17]

Takanori Ide, Kazuhiro Kurata, Kazunaga Tanaka. Multiple stable patterns for some reaction-diffusion equation in disrupted environments. Discrete & Continuous Dynamical Systems, 2006, 14 (1) : 93-116. doi: 10.3934/dcds.2006.14.93

[18]

Samira Boussaïd, Danielle Hilhorst, Thanh Nam Nguyen. Convergence to steady state for the solutions of a nonlocal reaction-diffusion equation. Evolution Equations & Control Theory, 2015, 4 (1) : 39-59. doi: 10.3934/eect.2015.4.39

[19]

Mohamed Ouzahra. Approximate controllability of the semilinear reaction-diffusion equation governed by a multiplicative control. Discrete & Continuous Dynamical Systems - B, 2021  doi: 10.3934/dcdsb.2021081

[20]

Vladimir V. Chepyzhov, Mark I. Vishik. Trajectory attractor for reaction-diffusion system with diffusion coefficient vanishing in time. Discrete & Continuous Dynamical Systems, 2010, 27 (4) : 1493-1509. doi: 10.3934/dcds.2010.27.1493

2020 Impact Factor: 1.327

Metrics

  • PDF downloads (117)
  • HTML views (152)
  • Cited by (0)

Other articles
by authors

[Back to Top]