• Previous Article
    Mathematical analysis of an in-host model of viral dynamics with spatial heterogeneity
  • DCDS-B Home
  • This Issue
  • Next Article
    Mathematical and numerical analysis of a mathematical model of mixed immunotherapy and chemotherapy of cancer
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 & 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. 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. doi: 10.1142/S0218202504003234.

[3]

M. Efendiev, Finite and Infinite Dimensional Attractors for Evolution Equations of Mathematical Physics,, Gakkōtosho Co., (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. 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. 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. doi: 10.1098/rspa.2005.1554.

[7]

M. Grasselli and V. Pata, Uniform attractors of nonautonomous dynamical systems with memory,, in Evolution equations, 50 (2002), 155.

[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. 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 $\mathbbR^n$,, Acta Math. Sin. (Engl. Ser.), 27 (2011), 1121. 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, (2008), 103. doi: 10.1016/S1874-5717(08)00003-0.

[11]

V. Pata, Stability and exponential stability in linear viscoelasticity,, Milan J. Math., 77 (2009), 333. 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. 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. doi: 10.1016/j.jmaa.2006.03.049.

[14]

G. R. Sell and Y. You, Dynamics of Evolutionary Equations,, Springer-Verlag, (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. 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. 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. 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. 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. 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. 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. doi: 10.1142/S0218202504003234.

[3]

M. Efendiev, Finite and Infinite Dimensional Attractors for Evolution Equations of Mathematical Physics,, Gakkōtosho Co., (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. 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. 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. doi: 10.1098/rspa.2005.1554.

[7]

M. Grasselli and V. Pata, Uniform attractors of nonautonomous dynamical systems with memory,, in Evolution equations, 50 (2002), 155.

[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. 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 $\mathbbR^n$,, Acta Math. Sin. (Engl. Ser.), 27 (2011), 1121. 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, (2008), 103. doi: 10.1016/S1874-5717(08)00003-0.

[11]

V. Pata, Stability and exponential stability in linear viscoelasticity,, Milan J. Math., 77 (2009), 333. 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. 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. doi: 10.1016/j.jmaa.2006.03.049.

[14]

G. R. Sell and Y. You, Dynamics of Evolutionary Equations,, Springer-Verlag, (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. 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. 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. 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. 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. doi: 10.1002/cpa.10068.

[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 - A, 2014, 34 (10) : 4343-4370. doi: 10.3934/dcds.2014.34.4343

[2]

S.V. Zelik. The attractor for a nonlinear hyperbolic equation in the unbounded domain. Discrete & Continuous Dynamical Systems - A, 2001, 7 (3) : 593-641. doi: 10.3934/dcds.2001.7.593

[3]

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 - A, 2014, 34 (10) : 4155-4182. doi: 10.3934/dcds.2014.34.4155

[4]

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

[5]

Martino Prizzi. A remark on reaction-diffusion equations in unbounded domains. Discrete & Continuous Dynamical Systems - A, 2003, 9 (2) : 281-286. doi: 10.3934/dcds.2003.9.281

[6]

Brahim Alouini, Olivier Goubet. Regularity of the attractor for a Bose-Einstein equation in a two dimensional unbounded domain. Discrete & Continuous Dynamical Systems - B, 2014, 19 (3) : 651-677. doi: 10.3934/dcdsb.2014.19.651

[7]

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

[8]

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

[9]

Guo Lin, Haiyan Wang. Traveling wave solutions of a reaction-diffusion equation with state-dependent delay. Communications on Pure & Applied Analysis, 2016, 15 (2) : 319-334. doi: 10.3934/cpaa.2016.15.319

[10]

Maria do Carmo Pacheco de Toledo, Sergio Muniz Oliva. A discretization scheme for an one-dimensional reaction-diffusion equation with delay and its dynamics. Discrete & Continuous Dynamical Systems - A, 2009, 23 (3) : 1041-1060. doi: 10.3934/dcds.2009.23.1041

[11]

Boris Andreianov, Halima Labani. Preconditioning operators and $L^\infty$ attractor for a class of reaction-diffusion systems. Communications on Pure & Applied Analysis, 2012, 11 (6) : 2179-2199. doi: 10.3934/cpaa.2012.11.2179

[12]

Wei Wang, Anthony Roberts. Macroscopic discrete modelling of stochastic reaction-diffusion equations on a periodic domain. Discrete & Continuous Dynamical Systems - A, 2011, 31 (1) : 253-273. doi: 10.3934/dcds.2011.31.253

[13]

Peter Poláčik, Eiji Yanagida. Stable subharmonic solutions of reaction-diffusion equations on an arbitrary domain. Discrete & Continuous Dynamical Systems - A, 2002, 8 (1) : 209-218. doi: 10.3934/dcds.2002.8.209

[14]

Narcisa Apreutesei, Vitaly Volpert. Reaction-diffusion waves with nonlinear boundary conditions. Networks & Heterogeneous Media, 2013, 8 (1) : 23-35. doi: 10.3934/nhm.2013.8.23

[15]

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

[16]

Keng Deng, Yixiang Wu. Asymptotic behavior for a reaction-diffusion population model with delay. Discrete & Continuous Dynamical Systems - B, 2015, 20 (2) : 385-395. doi: 10.3934/dcdsb.2015.20.385

[17]

Peter E. Kloeden, Thomas Lorenz. Pullback attractors of reaction-diffusion inclusions with space-dependent delay. Discrete & Continuous Dynamical Systems - B, 2017, 22 (5) : 1909-1964. doi: 10.3934/dcdsb.2017114

[18]

Dalibor Pražák. Exponential attractor for the delayed logistic equation with a nonlinear diffusion. Conference Publications, 2003, 2003 (Special) : 717-726. doi: 10.3934/proc.2003.2003.717

[19]

Brahim Alouini. Finite dimensional global attractor for a Bose-Einstein equation in a two dimensional unbounded domain. Communications on Pure & Applied Analysis, 2015, 14 (5) : 1781-1801. doi: 10.3934/cpaa.2015.14.1781

[20]

José A. Langa, James C. Robinson, Aníbal Rodríguez-Bernal, A. Suárez, A. Vidal-López. Existence and nonexistence of unbounded forwards attractor for a class of non-autonomous reaction diffusion equations. Discrete & Continuous Dynamical Systems - A, 2007, 18 (2/3) : 483-497. doi: 10.3934/dcds.2007.18.483

2016 Impact Factor: 0.994

Metrics

  • PDF downloads (2)
  • HTML views (0)
  • Cited by (0)

Other articles
by authors

[Back to Top]