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June  2016, 21(4): 1259-1277. doi: 10.3934/dcdsb.2016.21.1259

## Attractors and entropy bounds for a nonlinear RDEs with distributed delay in unbounded domains

 1 Department of Mathematical Analysis, Charles University, Prague, Sokolovská 83, 186 75 Prague 8 2 Department of Mathematical Analysis, Charles University, Prague, Sokolovská 83, 186 75 Praha 8, Czech Republic

Received  May 2015 Revised  December 2015 Published  March 2016

A nonlinear reaction-diffusion problem with a general, both spatially and delay distributed reaction term is considered in an unbounded domain $\mathbb{R}^N$. The existence of a unique weak solution is proved. A locally compact attractor together with entropy bound is also established.
Citation: Dalibor Pražák, Jakub Slavík. Attractors and entropy bounds for a nonlinear RDEs with distributed delay in unbounded domains. Discrete and Continuous Dynamical Systems - B, 2016, 21 (4) : 1259-1277. doi: 10.3934/dcdsb.2016.21.1259
##### References:
 [1] J. M. Arrieta, J. W. Cholewa, T. Dlotko and A. Rodríguez-Bernal, Dissipative parabolic equations in locally uniform spaces, Math. Nachr., 280 (2007), 1643-1663. doi: 10.1002/mana.200510569. [2] J. M. Arrieta, A. Rodríguez-Bernal, J. W. Cholewa and T. Dlotko, Linear parabolic equations in locally uniform spaces, Math. Models Methods Appl. Sci., 14 (2004), 253-293. doi: 10.1142/S0218202504003234. [3] M. Efendiev, Finite and Infinite Dimensional Attractors for Evolution Equations of Mathematical Physics, Gakkōtosho Co., Ltd., Tokyo, 2010. [4] M. A. Efendiev and S. V. Zelik, The attractor for a nonlinear reaction-diffusion system in an unbounded domain, Comm. Pure Appl. Math., 54 (2001), 625-688. doi: 10.1002/cpa.1011. [5] M. Fabrizio, C. Giorgi and V. Pata, A new approach to equations with memory, Arch. Ration. Mech. Anal., 198 (2010), 189-232. doi: 10.1007/s00205-010-0300-3. [6] T. Faria, W. Huang and J. Wu, Travelling waves for delayed reaction-diffusion equations with global response, Proc. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci., 462 (2006), 229-261. doi: 10.1098/rspa.2005.1554. [7] M. Grasselli and V. Pata, Uniform attractors of nonautonomous dynamical systems with memory, in Evolution equations, semigroups and functional analysis (eds. A. Lorenzi and B. Ruf), Birkhäuser, Basel, 50 (2002), 155-178. [8] M. Grasselli, D. Pražák and G. Schimperna, Attractors for nonlinear reaction-diffusion systems in unbounded domains via the method of short trajectories, J. Differential Equations, 249 (2010), 2287-2315. doi: 10.1016/j.jde.2010.06.001. [9] X. Li and Z. X. Li, The Global Attractor of a Non-Local PDE Model with Delay for Population Dynamics in $\mathbb{R}^{N}$, Acta Math. Sin. (Engl. Ser.), 27 (2011), 1121-1136. doi: 10.1007/s10114-011-8539-7. [10] 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 (eds. C. M. Dafermos and M. Pokorný), Elsevier/North-Holland, Amsterdam, 2008, 103-200. doi: 10.1016/S1874-5717(08)00003-0. [11] V. Pata, Stability and exponential stability in linear viscoelasticity, Milan J. Math., 77 (2009), 333-360. doi: 10.1007/s00032-009-0098-3. [12] D. Pražák, Exponential attractors for abstract parabolic systems with bounded delay, Bull. Austral. Math. Soc., 76 (2007), 285-295. doi: 10.1017/S0004972700039666. [13] A. V. Rezounenko, Partial diffferential equations with discrete and distributed state-dependent delays, J. Math. Anal. Appl., 326 (2007), 1031-1045. doi: 10.1016/j.jmaa.2006.03.049. [14] G. R. Sell and Y. You, Dynamics of Evolutionary Equations, Springer-Verlag, New York, 2002. doi: 10.1007/978-1-4757-5037-9. [15] Y. Wang and P. E. Kloeden, The uniform attractor of a multi-valued process generated by reaction-diffusion delay equations on an unbounded domain, Discrete Continuous Dynam. Systems - A, 34 (2014), 4343-4370. doi: 10.3934/dcds.2014.34.4343. [16] Y. Wang, L. Wang and W. Zhao, Pullback attractors for nonautonomous reaction-diffusion equations in unbounded domains, J. Math. Anal. Appl., 336 (2007), 330-347. doi: 10.1016/j.jmaa.2007.02.081. [17] Z. Wang, W. Li and S. Ruan, Travelling wave fronts in reaction-diffusion systems with spatio-temporal delays, J. Differential Equations, 222 (2006), 185-232. doi: 10.1016/j.jde.2005.08.010. [18] T. Yi, Y. Chen and J. Wu, Global dynamics of delayed reaction-diffusion equations in unbounded domains, Z. Angew. Math. Phys., 63 (2012), 793-812. doi: 10.1007/s00033-012-0224-x. [19] S. V. Zelik, Attractors of reaction-diffusion systems in unbounded domains and their spatial complexity, Comm. Pure Appl. Math., 56 (2003), 584-637. doi: 10.1002/cpa.10068.

show all references

##### References:
 [1] J. M. Arrieta, J. W. Cholewa, T. Dlotko and A. Rodríguez-Bernal, Dissipative parabolic equations in locally uniform spaces, Math. Nachr., 280 (2007), 1643-1663. doi: 10.1002/mana.200510569. [2] J. M. Arrieta, A. Rodríguez-Bernal, J. W. Cholewa and T. Dlotko, Linear parabolic equations in locally uniform spaces, Math. Models Methods Appl. Sci., 14 (2004), 253-293. doi: 10.1142/S0218202504003234. [3] M. Efendiev, Finite and Infinite Dimensional Attractors for Evolution Equations of Mathematical Physics, Gakkōtosho Co., Ltd., Tokyo, 2010. [4] M. A. Efendiev and S. V. Zelik, The attractor for a nonlinear reaction-diffusion system in an unbounded domain, Comm. Pure Appl. Math., 54 (2001), 625-688. doi: 10.1002/cpa.1011. [5] M. Fabrizio, C. Giorgi and V. Pata, A new approach to equations with memory, Arch. Ration. Mech. Anal., 198 (2010), 189-232. doi: 10.1007/s00205-010-0300-3. [6] T. Faria, W. Huang and J. Wu, Travelling waves for delayed reaction-diffusion equations with global response, Proc. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci., 462 (2006), 229-261. doi: 10.1098/rspa.2005.1554. [7] M. Grasselli and V. Pata, Uniform attractors of nonautonomous dynamical systems with memory, in Evolution equations, semigroups and functional analysis (eds. A. Lorenzi and B. Ruf), Birkhäuser, Basel, 50 (2002), 155-178. [8] M. Grasselli, D. Pražák and G. Schimperna, Attractors for nonlinear reaction-diffusion systems in unbounded domains via the method of short trajectories, J. Differential Equations, 249 (2010), 2287-2315. doi: 10.1016/j.jde.2010.06.001. [9] X. Li and Z. X. Li, The Global Attractor of a Non-Local PDE Model with Delay for Population Dynamics in $\mathbb{R}^{N}$, Acta Math. Sin. (Engl. Ser.), 27 (2011), 1121-1136. doi: 10.1007/s10114-011-8539-7. [10] 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 (eds. C. M. Dafermos and M. Pokorný), Elsevier/North-Holland, Amsterdam, 2008, 103-200. doi: 10.1016/S1874-5717(08)00003-0. [11] V. Pata, Stability and exponential stability in linear viscoelasticity, Milan J. Math., 77 (2009), 333-360. doi: 10.1007/s00032-009-0098-3. [12] D. Pražák, Exponential attractors for abstract parabolic systems with bounded delay, Bull. Austral. Math. Soc., 76 (2007), 285-295. doi: 10.1017/S0004972700039666. [13] A. V. Rezounenko, Partial diffferential equations with discrete and distributed state-dependent delays, J. Math. Anal. Appl., 326 (2007), 1031-1045. doi: 10.1016/j.jmaa.2006.03.049. [14] G. R. Sell and Y. You, Dynamics of Evolutionary Equations, Springer-Verlag, New York, 2002. doi: 10.1007/978-1-4757-5037-9. [15] Y. Wang and P. E. Kloeden, The uniform attractor of a multi-valued process generated by reaction-diffusion delay equations on an unbounded domain, Discrete Continuous Dynam. Systems - A, 34 (2014), 4343-4370. doi: 10.3934/dcds.2014.34.4343. [16] Y. Wang, L. Wang and W. Zhao, Pullback attractors for nonautonomous reaction-diffusion equations in unbounded domains, J. Math. Anal. Appl., 336 (2007), 330-347. doi: 10.1016/j.jmaa.2007.02.081. [17] Z. Wang, W. Li and S. Ruan, Travelling wave fronts in reaction-diffusion systems with spatio-temporal delays, J. Differential Equations, 222 (2006), 185-232. doi: 10.1016/j.jde.2005.08.010. [18] T. Yi, Y. Chen and J. Wu, Global dynamics of delayed reaction-diffusion equations in unbounded domains, Z. Angew. Math. Phys., 63 (2012), 793-812. doi: 10.1007/s00033-012-0224-x. [19] S. V. Zelik, Attractors of reaction-diffusion systems in unbounded domains and their spatial complexity, Comm. Pure Appl. Math., 56 (2003), 584-637. doi: 10.1002/cpa.10068.
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