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March  2019, 24(3): 1259-1271. doi: 10.3934/dcdsb.2019015

Forward attracting sets of reaction-diffusion equations on variable domains

School of Mathematics and Statistics, and Hubei Key Laboratory of Engineering Modeling and Scientific Computing, Huazhong University of Science and Technology, Wuhan 430074, China

1Corresponding author

Dedicated to the memory of V. S. Mel’nik.

Received  October 2017 Revised  February 2018 Published  January 2019

Reaction-diffusion equations on time-variable domains are instrinsically nonautonomous even if the coefficients in the equation do not depend explicitly on time. Thus the appropriate asymptotic concepts, such as attractors, are nonautonomous. Forward attracting sets based on omega-limit sets are considered in this paper. These are related to the Vishik uniform attractor but are not as restrictive since they depend only on the dynamics in the distant future. They are usually not invariant. Here it is shown that they are asymptotically positively invariant, in general, and, if the future dynamics is appropriately uniform, also asymptotically negatively invariant as well as upper semi continuous dependence in a parameter will be established. These results also apply to reaction-diffusion equations on a fixed domain.

Citation: Peter E. Kloeden, Meihua Yang. Forward attracting sets of reaction-diffusion equations on variable domains. Discrete & Continuous Dynamical Systems - B, 2019, 24 (3) : 1259-1271. doi: 10.3934/dcdsb.2019015
References:
[1]

A. N. Carvalho, J. A. Langa and J. C. Robinson, Attractors of Infinite Dimensional Nonautonomous Dynamical Systems, Applied Mathematical Sciences, 182. Springer, New York, 2013. doi: 10.1007/978-1-4614-4581-4.  Google Scholar

[2]

V. V. Chepyzhov and M. I. Vishik, Attractors for equations of mathematical physics, Amer. Math. Soc., Providence, Rhode Island, 2002.  Google Scholar

[3]

H. CrauelP. E. Kloeden and J. Real, Stochastic partial differential equations on time-varying domains, Boletín de la Sociedad Española de Matemática Aplicada., 51 (2010), 41-48.  doi: 10.1007/bf03322552.  Google Scholar

[4]

H. CrauelP. E. Kloeden and M. Yang, Random attractors of stochastic reaction-diffusion equations on variable domains, Stochastics & Dynamics, 11 (2011), 301-314.  doi: 10.1142/S0219493711003292.  Google Scholar

[5]

J. K. Hale, Asymptotic Behavior of Dissipative Systems, American Mathematical Society, Providence, 1988.  Google Scholar

[6]

P. E. Kloeden, Asymptotic invariance and the approximation of nonautonomous forward attracting sets, J. Comput. Dynamics, 3 (2016), 179-189.  doi: 10.3934/jcd.2016009.  Google Scholar

[7]

P. E. Kloeden and T. Lorenz, Construction of nonautonomous forward attractors, Proc. Amer. Mat. Soc., 144 (2016), 259-268.  doi: 10.1090/proc/12735.  Google Scholar

[8]

P. E. Kloeden, T. Lorenz and M. Yang, Forward attractors in discrete time nonautonomous dynamical systems, in Differential and Difference Equations with Applications, Springer Proceedings in Mathematics & Statistics, 164, Editors: O. Dosly, P.E, Kloeden, S. Pinelas; Springer, Heidelberg, (2016), 313–322. doi: 10.1007/978-3-319-32857-7_29.  Google Scholar

[9]

P. E. KloedenP. Marín-Rubio and J. Real, Pullback attractors for a semilinear heat equation in a non-cylindrical domain, J. Differential Eqns., 244 (2008), 2062-2090.  doi: 10.1016/j.jde.2007.10.031.  Google Scholar

[10]

P. E. KloedenC. Pötzsche and M. Rasmussen, Limitations of pullback attractors of processes, J. Difference Eqns. Applns., 18 (2012), 693-701.  doi: 10.1080/10236198.2011.578070.  Google Scholar

[11]

P. E. Kloeden and M. Rasmussen, Nonautonomous Dynamical Systems, Amer. Math. Soc., Providence, 2011. doi: 10.1090/surv/176.  Google Scholar

[12]

P. E. KloedenJ. Real and C. Y. Sun, Pullback attractors for a semilinear heat equation on time-varying domains, J. Differential Eqns., 246 (2009), 4702-4730.  doi: 10.1016/j.jde.2008.11.017.  Google Scholar

[13]

P. E. Kloeden and M. Yang, Forward attraction in nonautonomous difference equations, J. Difference Eqns. Applns., 22 (2016), 513-525.  doi: 10.1080/10236198.2015.1107550.  Google Scholar

[14]

J. P. Lasalle, The Stability of Dynamical Systems, SIAM-CBMS, Philadelphia, 1976.  Google Scholar

[15]

M. I. Vishik, Asymptotic Behaviour of Solutions of Evolutionary Equations, Cambridge University Press, Cambridge, 1992.  Google Scholar

show all references

References:
[1]

A. N. Carvalho, J. A. Langa and J. C. Robinson, Attractors of Infinite Dimensional Nonautonomous Dynamical Systems, Applied Mathematical Sciences, 182. Springer, New York, 2013. doi: 10.1007/978-1-4614-4581-4.  Google Scholar

[2]

V. V. Chepyzhov and M. I. Vishik, Attractors for equations of mathematical physics, Amer. Math. Soc., Providence, Rhode Island, 2002.  Google Scholar

[3]

H. CrauelP. E. Kloeden and J. Real, Stochastic partial differential equations on time-varying domains, Boletín de la Sociedad Española de Matemática Aplicada., 51 (2010), 41-48.  doi: 10.1007/bf03322552.  Google Scholar

[4]

H. CrauelP. E. Kloeden and M. Yang, Random attractors of stochastic reaction-diffusion equations on variable domains, Stochastics & Dynamics, 11 (2011), 301-314.  doi: 10.1142/S0219493711003292.  Google Scholar

[5]

J. K. Hale, Asymptotic Behavior of Dissipative Systems, American Mathematical Society, Providence, 1988.  Google Scholar

[6]

P. E. Kloeden, Asymptotic invariance and the approximation of nonautonomous forward attracting sets, J. Comput. Dynamics, 3 (2016), 179-189.  doi: 10.3934/jcd.2016009.  Google Scholar

[7]

P. E. Kloeden and T. Lorenz, Construction of nonautonomous forward attractors, Proc. Amer. Mat. Soc., 144 (2016), 259-268.  doi: 10.1090/proc/12735.  Google Scholar

[8]

P. E. Kloeden, T. Lorenz and M. Yang, Forward attractors in discrete time nonautonomous dynamical systems, in Differential and Difference Equations with Applications, Springer Proceedings in Mathematics & Statistics, 164, Editors: O. Dosly, P.E, Kloeden, S. Pinelas; Springer, Heidelberg, (2016), 313–322. doi: 10.1007/978-3-319-32857-7_29.  Google Scholar

[9]

P. E. KloedenP. Marín-Rubio and J. Real, Pullback attractors for a semilinear heat equation in a non-cylindrical domain, J. Differential Eqns., 244 (2008), 2062-2090.  doi: 10.1016/j.jde.2007.10.031.  Google Scholar

[10]

P. E. KloedenC. Pötzsche and M. Rasmussen, Limitations of pullback attractors of processes, J. Difference Eqns. Applns., 18 (2012), 693-701.  doi: 10.1080/10236198.2011.578070.  Google Scholar

[11]

P. E. Kloeden and M. Rasmussen, Nonautonomous Dynamical Systems, Amer. Math. Soc., Providence, 2011. doi: 10.1090/surv/176.  Google Scholar

[12]

P. E. KloedenJ. Real and C. Y. Sun, Pullback attractors for a semilinear heat equation on time-varying domains, J. Differential Eqns., 246 (2009), 4702-4730.  doi: 10.1016/j.jde.2008.11.017.  Google Scholar

[13]

P. E. Kloeden and M. Yang, Forward attraction in nonautonomous difference equations, J. Difference Eqns. Applns., 22 (2016), 513-525.  doi: 10.1080/10236198.2015.1107550.  Google Scholar

[14]

J. P. Lasalle, The Stability of Dynamical Systems, SIAM-CBMS, Philadelphia, 1976.  Google Scholar

[15]

M. I. Vishik, Asymptotic Behaviour of Solutions of Evolutionary Equations, Cambridge University Press, Cambridge, 1992.  Google Scholar

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