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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. [2] V. V. Chepyzhov and M. I. Vishik, Attractors for equations of mathematical physics, Amer. Math. Soc., Providence, Rhode Island, 2002. [3] H. Crauel, P. 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. [4] H. Crauel, P. 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. [5] J. K. Hale, Asymptotic Behavior of Dissipative Systems, American Mathematical Society, Providence, 1988. [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. [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. [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. [9] P. E. Kloeden, P. 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. [10] P. E. Kloeden, C. Pötzsche and M. Rasmussen, Limitations of pullback attractors of processes, J. Difference Eqns. Applns., 18 (2012), 693-701. doi: 10.1080/10236198.2011.578070. [11] P. E. Kloeden and M. Rasmussen, Nonautonomous Dynamical Systems, Amer. Math. Soc., Providence, 2011. doi: 10.1090/surv/176. [12] P. E. Kloeden, J. 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. [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. [14] J. P. Lasalle, The Stability of Dynamical Systems, SIAM-CBMS, Philadelphia, 1976. [15] M. I. Vishik, Asymptotic Behaviour of Solutions of Evolutionary Equations, Cambridge University Press, Cambridge, 1992.

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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. [2] V. V. Chepyzhov and M. I. Vishik, Attractors for equations of mathematical physics, Amer. Math. Soc., Providence, Rhode Island, 2002. [3] H. Crauel, P. 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. [4] H. Crauel, P. 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. [5] J. K. Hale, Asymptotic Behavior of Dissipative Systems, American Mathematical Society, Providence, 1988. [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. [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. [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. [9] P. E. Kloeden, P. 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. [10] P. E. Kloeden, C. Pötzsche and M. Rasmussen, Limitations of pullback attractors of processes, J. Difference Eqns. Applns., 18 (2012), 693-701. doi: 10.1080/10236198.2011.578070. [11] P. E. Kloeden and M. Rasmussen, Nonautonomous Dynamical Systems, Amer. Math. Soc., Providence, 2011. doi: 10.1090/surv/176. [12] P. E. Kloeden, J. 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. [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. [14] J. P. Lasalle, The Stability of Dynamical Systems, SIAM-CBMS, Philadelphia, 1976. [15] M. I. Vishik, Asymptotic Behaviour of Solutions of Evolutionary Equations, Cambridge University Press, Cambridge, 1992.
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