Regular solutions and global attractors for reaction-diffusion systems without uniqueness

Previous Article
Non-isothermal viscous Cahn-Hilliard equation with inertial term and dynamic boundary conditions
CPAA Home
This Issue
Next Article
Reaction-diffusion equations with a switched--off reaction zone

September 2014, 13(5): 1891-1906. doi: 10.3934/cpaa.2014.13.1891

Regular solutions and global attractors for reaction-diffusion systems without uniqueness

Oleksiy V. Kapustyan ^1, , Pavlo O. Kasyanov ^2, and José Valero ^3,

1.	Taras Shevchenko National University of Kyiv, 60, Volodymyrska Street, 01601, Kyiv, Ukraine
2.	Institute for Applied System Analysis, National Technical University of Ukraine "KPI", Kyiv
3.	Universidad Miguel Hernández, Centro de Investigación Operativa, Avda. Universidad s/n, Elche (Alicante), 03202

Received September 2013 Revised September 2013 Published June 2014

Abstract
Related Papers

In this paper we study the structural properties of global attractors of multi-valued semiflows generated by regular solutions of reaction-diffusion system without uniqueness of the Cauchy problem. Under additional gradient-like condition on the nonlinear term we prove that the global attractor coincides with the unstable manifold of the set of stationary points, and with the stable one when we consider only bounded complete trajectories. As an example we consider a generalized Fitz-Hugh-Nagumo system. We also suggest additional conditions, which provide that the global attractor is a bounded set in $(L^\infty(\Omega))^N$ and compact in $(H_0^1 (\Omega))^N$.

Keywords: global attractor, multi-valued dynamical system, Reaction-diffusion system, unstable manifold, Fitz-Hugh-Nagumo system..

Mathematics Subject Classification: Primary: 35B40, 35B41, 35K55; Secondary: 37B25, 58C0.

Citation: Oleksiy V. Kapustyan, Pavlo O. Kasyanov, José Valero. Regular solutions and global attractors for reaction-diffusion systems without uniqueness. Communications on Pure & Applied Analysis, 2014, 13 (5) : 1891-1906. doi: 10.3934/cpaa.2014.13.1891

References:

[1]	A. V. Babin, M. I. Vishik, Attracteurs maximaux dans les équations aux dérivées partielles, Nonlinear partial differential equations and their applications,, \emph{Collegue de France Seminar}, (1985), 11.
[2]	A. V. Babin and M. I. Vishik, Attractors of Evolution Equations,, Nauka, (1989).
[3]	V. V. Chepyzhov and M. I. Vishik, Attractors for Equations of Mathematical Physics,, American Mathematical Society, (2002).
[4]	M. I. Vishik, S. V. Zelik and V. V. Chepyzhov, Strong trajectory attractor of dissipative reaction-diffusion system,, \emph{Doklady RAN}, 435 (2010), 155. doi: 10.1134/S1064562410060086.
[5]	J. M. Ball, Global attractors for damped semilinear wave equations,, \emph{Discrete Contin. Dyn. Syst., 10 (2004), 31. doi: 10.3934/dcds.2004.10.31.
[6]	T. Caraballo, P. Marin-Rubio and J. Robinson, A comparison between two theories for multivalued semiflows and their asymptotic behavior,, \emph{Set-valued Analysis}, 11 (2003), 297. doi: 10.1023/A:1024422619616.
[7]	P. Brunovsky and B. Fiedler, Connecting orbits in scalar reaction diffusion equations,, \emph{Dynamics Reported, 1 (1988), 57.
[8]	N. V. Gorban, O. V. Kapustyan and P. O. Kasyanov, Uniform trajectory attractor for non-autonomous reaction-diffusion equations with Caratheodory's nonlinearity,, \emph{Nonlinear Analysis}, 98 (2014), 13. doi: 10.1016/j.na.2013.12.004.
[9]	N. V. Gorban and P. O. Kasyanov, On regularity of all weak solutions and their attractors for reaction-diffusion inclusion in unbounded domain,, \emph{Solid Mechanics and Its Applications}, 211 (2013), 205.
[10]	N. V. Gorban, P. O. Kasyanov, O. V. Kapustyan and L. S. Paliichuk, On global attractors for autonomous wave equation with discontinuous nonlinearity,, \emph{Solid Mechanics and Its Applications}, 211 (2014), 221.
[11]	J. K. Hale, Asymptotic Behavior of Dissipative Systems,, American Mathematical Society, (1988).
[12]	O. V. Kapustyan, V. S. Melnik, J. Valero and V. V. Yasinsky, Global Attractors of Multivalued Dynamical Systems and Evolution Equations Without Uniqueness,, Naukova Dumka, (2008).
[13]	O. V. Kapustyan, P. O. Kasyanov and J. Valero, Structure and regularity of the global attractor of a reaction-diffusion equation with non-smooth nonlinear term,, \emph{Discrete Contin. Dyn. Syst.}, 34 (2014), 4155. doi: 10.3934/dcds.2014.34.4155.
[14]	O. V. Kapustyan, A. V. Pankov and J. Valero, On global attractors of multivalued semiflows generated by the 3D Bénard system,, \emph{Set-Valued Var. Anal.}, 20 (2012), 445. doi: 10.1007/s11228-011-0197-5.
[15]	O. V. Kapustyan and J. Valero, On the Kneser property for the complex Ginzburg-Landau equation and the Lotka-Volterra system with diffusion,, \emph{J. Math. Anal. Appl.}, 357 (2009), 254. doi: 10.1016/j.jmaa.2009.04.010.
[16]	O. V. Kapustyan and J. Valero, Comparison between trajectory and global attractors for evolution systems without uniqueness of solutions,, \emph{Internat. J. Bifur. Chaos}, 20 (2010), 2723. doi: 10.1142/S0218127410027313.
[17]	O. V. Kapustyan, P. O. Kasyanov, J. Valero and M. Z. Zgurovsky, Structure of uniform global attractor for general non-autonomous reaction-diffusion system,, \emph{Solid Mechanics and Its Applications}, 211 (2014), 163.
[18]	P. O. Kasyanov, Multivalued dynamics of solutions of an autonomous differential-operator inclusion with pseudomonotone nonlinearity,, \emph{Cybernetics and Systems Analysis}, 47 (2011), 800.
[19]	P. O. Kasyanov, Multivalued dynamics of solutions of autonomous operator differential equations with pseudomonotone nonlinearity,, \emph{Mathematical Notes}, 92 (2012), 205.
[20]	P. O. Kasyanov et al., Regularity of weak solutions and their attractors for a parabolic feedback control problem,, \emph{Set-Valued and Variational Analysis}, 21 (2013), 271. doi: 10.1007/s11228-013-0233-8.
[21]	J. L. Lions, Quelques méthodes de résolution des problèmes aux limites non linéaires,, Gauthier-Villar, (1969).
[22]	V. S. Melnik and J. Valero, On attractors of multi-valued semi-flows and differential inclusions,, \emph{Set-Valued Anal.}, 6 (1998), 83. doi: 10.1023/A:1008608431399.
[23]	C. Rocha, Properties of the attractor of a scalar parabolic PDE,, \emph{J. Dynamics Differential Equations, 3 (1991), 575. doi: 10.1007/BF01049100.
[24]	C. Rocha and B. Fiedler, Heteroclinic orbits of semilinear parabolic equations,, \emph{J. Differential. Equations, 125 (1996), 239. doi: 10.1006/jdeq.1996.0031.
[25]	G. R. Sell and Y. You, Dynamics of Evolutionary Equations,, Springer, (2002). doi: 10.1007/978-1-4757-5037-9.
[26]	R. Temam, Infinite-dimensional Dynamical Systems in Mechanics and Physics,, Springer-Verlag, (1997). doi: 10.1007/978-1-4612-0645-3.
[27]	J. Valero and O. V. Kapustyan, On the connectedness and asymptotic behaviour of solutions of reaction-diffusion systems,, \emph{J. Math. Anal. Appl.}, 323 (2006), 614. doi: 10.1016/j.jmaa.2005.10.042.
[28]	S. Zelik, The attractor for a nonlinear reaction-diffusion system with a supercritical nonlinearity and it's dimension,, \emph{Rend. Accad. Naz. Sci. XL Mem. Mat. Appl., 24 (2000), 1.
[29]	M. Z. Zgurovsky et al., Long-time behavior of solutions for quasilinear hyperbolic hemivariational inequalities with application to piezoelectricity problem,, \emph{Applied Mathematics Letters}, 25 (2012), 1569. doi: 10.1016/j.aml.2012.01.016.
[30]	M. Z. Zgurovsky and P. O. Kasyanov, O. V. Kapustyan, J. Valero and N. V. Zadoianchuk, Evolution inclusions and variation inequalities for earth data processing III. Long-time behavior of evolution inclusions solutions in Earth data analysis,, Springer, (2012).

show all references

References:

[1]	A. V. Babin, M. I. Vishik, Attracteurs maximaux dans les équations aux dérivées partielles, Nonlinear partial differential equations and their applications,, \emph{Collegue de France Seminar}, (1985), 11.
[2]	A. V. Babin and M. I. Vishik, Attractors of Evolution Equations,, Nauka, (1989).
[3]	V. V. Chepyzhov and M. I. Vishik, Attractors for Equations of Mathematical Physics,, American Mathematical Society, (2002).
[4]	M. I. Vishik, S. V. Zelik and V. V. Chepyzhov, Strong trajectory attractor of dissipative reaction-diffusion system,, \emph{Doklady RAN}, 435 (2010), 155. doi: 10.1134/S1064562410060086.
[5]	J. M. Ball, Global attractors for damped semilinear wave equations,, \emph{Discrete Contin. Dyn. Syst., 10 (2004), 31. doi: 10.3934/dcds.2004.10.31.
[6]	T. Caraballo, P. Marin-Rubio and J. Robinson, A comparison between two theories for multivalued semiflows and their asymptotic behavior,, \emph{Set-valued Analysis}, 11 (2003), 297. doi: 10.1023/A:1024422619616.
[7]	P. Brunovsky and B. Fiedler, Connecting orbits in scalar reaction diffusion equations,, \emph{Dynamics Reported, 1 (1988), 57.
[8]	N. V. Gorban, O. V. Kapustyan and P. O. Kasyanov, Uniform trajectory attractor for non-autonomous reaction-diffusion equations with Caratheodory's nonlinearity,, \emph{Nonlinear Analysis}, 98 (2014), 13. doi: 10.1016/j.na.2013.12.004.
[9]	N. V. Gorban and P. O. Kasyanov, On regularity of all weak solutions and their attractors for reaction-diffusion inclusion in unbounded domain,, \emph{Solid Mechanics and Its Applications}, 211 (2013), 205.
[10]	N. V. Gorban, P. O. Kasyanov, O. V. Kapustyan and L. S. Paliichuk, On global attractors for autonomous wave equation with discontinuous nonlinearity,, \emph{Solid Mechanics and Its Applications}, 211 (2014), 221.
[11]	J. K. Hale, Asymptotic Behavior of Dissipative Systems,, American Mathematical Society, (1988).
[12]	O. V. Kapustyan, V. S. Melnik, J. Valero and V. V. Yasinsky, Global Attractors of Multivalued Dynamical Systems and Evolution Equations Without Uniqueness,, Naukova Dumka, (2008).
[13]	O. V. Kapustyan, P. O. Kasyanov and J. Valero, Structure and regularity of the global attractor of a reaction-diffusion equation with non-smooth nonlinear term,, \emph{Discrete Contin. Dyn. Syst.}, 34 (2014), 4155. doi: 10.3934/dcds.2014.34.4155.
[14]	O. V. Kapustyan, A. V. Pankov and J. Valero, On global attractors of multivalued semiflows generated by the 3D Bénard system,, \emph{Set-Valued Var. Anal.}, 20 (2012), 445. doi: 10.1007/s11228-011-0197-5.
[15]	O. V. Kapustyan and J. Valero, On the Kneser property for the complex Ginzburg-Landau equation and the Lotka-Volterra system with diffusion,, \emph{J. Math. Anal. Appl.}, 357 (2009), 254. doi: 10.1016/j.jmaa.2009.04.010.
[16]	O. V. Kapustyan and J. Valero, Comparison between trajectory and global attractors for evolution systems without uniqueness of solutions,, \emph{Internat. J. Bifur. Chaos}, 20 (2010), 2723. doi: 10.1142/S0218127410027313.
[17]	O. V. Kapustyan, P. O. Kasyanov, J. Valero and M. Z. Zgurovsky, Structure of uniform global attractor for general non-autonomous reaction-diffusion system,, \emph{Solid Mechanics and Its Applications}, 211 (2014), 163.
[18]	P. O. Kasyanov, Multivalued dynamics of solutions of an autonomous differential-operator inclusion with pseudomonotone nonlinearity,, \emph{Cybernetics and Systems Analysis}, 47 (2011), 800.
[19]	P. O. Kasyanov, Multivalued dynamics of solutions of autonomous operator differential equations with pseudomonotone nonlinearity,, \emph{Mathematical Notes}, 92 (2012), 205.
[20]	P. O. Kasyanov et al., Regularity of weak solutions and their attractors for a parabolic feedback control problem,, \emph{Set-Valued and Variational Analysis}, 21 (2013), 271. doi: 10.1007/s11228-013-0233-8.
[21]	J. L. Lions, Quelques méthodes de résolution des problèmes aux limites non linéaires,, Gauthier-Villar, (1969).
[22]	V. S. Melnik and J. Valero, On attractors of multi-valued semi-flows and differential inclusions,, \emph{Set-Valued Anal.}, 6 (1998), 83. doi: 10.1023/A:1008608431399.
[23]	C. Rocha, Properties of the attractor of a scalar parabolic PDE,, \emph{J. Dynamics Differential Equations, 3 (1991), 575. doi: 10.1007/BF01049100.
[24]	C. Rocha and B. Fiedler, Heteroclinic orbits of semilinear parabolic equations,, \emph{J. Differential. Equations, 125 (1996), 239. doi: 10.1006/jdeq.1996.0031.
[25]	G. R. Sell and Y. You, Dynamics of Evolutionary Equations,, Springer, (2002). doi: 10.1007/978-1-4757-5037-9.
[26]	R. Temam, Infinite-dimensional Dynamical Systems in Mechanics and Physics,, Springer-Verlag, (1997). doi: 10.1007/978-1-4612-0645-3.
[27]	J. Valero and O. V. Kapustyan, On the connectedness and asymptotic behaviour of solutions of reaction-diffusion systems,, \emph{J. Math. Anal. Appl.}, 323 (2006), 614. doi: 10.1016/j.jmaa.2005.10.042.
[28]	S. Zelik, The attractor for a nonlinear reaction-diffusion system with a supercritical nonlinearity and it's dimension,, \emph{Rend. Accad. Naz. Sci. XL Mem. Mat. Appl., 24 (2000), 1.
[29]	M. Z. Zgurovsky et al., Long-time behavior of solutions for quasilinear hyperbolic hemivariational inequalities with application to piezoelectricity problem,, \emph{Applied Mathematics Letters}, 25 (2012), 1569. doi: 10.1016/j.aml.2012.01.016.
[30]	M. Z. Zgurovsky and P. O. Kasyanov, O. V. Kapustyan, J. Valero and N. V. Zadoianchuk, Evolution inclusions and variation inequalities for earth data processing III. Long-time behavior of evolution inclusions solutions in Earth data analysis,, Springer, (2012).

[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]	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
[3]	Takashi Kajiwara. The sub-supersolution method for the FitzHugh-Nagumo type reaction-diffusion system with heterogeneity. Discrete & Continuous Dynamical Systems - A, 2018, 38 (5) : 2441-2465. doi: 10.3934/dcds.2018101
[4]	Takashi Kajiwara. A Heteroclinic Solution to a Variational Problem Corresponding to FitzHugh-Nagumo type Reaction-Diffusion System with Heterogeneity. Communications on Pure & Applied Analysis, 2017, 16 (6) : 2133-2156. doi: 10.3934/cpaa.2017106
[5]	B. Ambrosio, M. A. Aziz-Alaoui, V. L. E. Phan. Global attractor of complex networks of reaction-diffusion systems of Fitzhugh-Nagumo type. Discrete & Continuous Dynamical Systems - B, 2018, 23 (9) : 3787-3797. doi: 10.3934/dcdsb.2018077
[6]	Thomas I. Seidman. Interface conditions for a singular reaction-diffusion system. Discrete & Continuous Dynamical Systems - S, 2009, 2 (3) : 631-643. doi: 10.3934/dcdss.2009.2.631
[7]	Nejib Mahmoudi. Single-point blow-up for a multi-component reaction-diffusion system. Discrete & Continuous Dynamical Systems - A, 2018, 38 (1) : 209-230. doi: 10.3934/dcds.2018010
[8]	Jifa Jiang, Junping Shi. Dynamics of a reaction-diffusion system of autocatalytic chemical reaction. Discrete & Continuous Dynamical Systems - A, 2008, 21 (1) : 245-258. doi: 10.3934/dcds.2008.21.245
[9]	Lili Du, Chunlai Mu, Zhaoyin Xiang. Global existence and blow-up to a reaction-diffusion system with nonlinear memory. Communications on Pure & Applied Analysis, 2005, 4 (4) : 721-733. doi: 10.3934/cpaa.2005.4.721
[10]	Shu-Xiang Huang, Fu-Cai Li, Chun-Hong Xie. Global existence and blow-up of solutions to a nonlocal reaction-diffusion system. Discrete & Continuous Dynamical Systems - A, 2003, 9 (6) : 1519-1532. doi: 10.3934/dcds.2003.9.1519
[11]	Sebastian Aniţa, William Edward Fitzgibbon, Michel Langlais. Global existence and internal stabilization for a reaction-diffusion system posed on non coincident spatial domains. Discrete & Continuous Dynamical Systems - B, 2009, 11 (4) : 805-822. doi: 10.3934/dcdsb.2009.11.805
[12]	Sze-Bi Hsu, Junping Shi, Feng-Bin Wang. Further studies of a reaction-diffusion system for an unstirred chemostat with internal storage. Discrete & Continuous Dynamical Systems - B, 2014, 19 (10) : 3169-3189. doi: 10.3934/dcdsb.2014.19.3169
[13]	Nicolas Bacaër, Cheikh Sokhna. A reaction-diffusion system modeling the spread of resistance to an antimalarial drug. Mathematical Biosciences & Engineering, 2005, 2 (2) : 227-238. doi: 10.3934/mbe.2005.2.227
[14]	W. E. Fitzgibbon, M. Langlais, J.J. Morgan. A reaction-diffusion system modeling direct and indirect transmission of diseases. Discrete & Continuous Dynamical Systems - B, 2004, 4 (4) : 893-910. doi: 10.3934/dcdsb.2004.4.893
[15]	José-Francisco Rodrigues, Lisa Santos. On a constrained reaction-diffusion system related to multiphase problems. Discrete & Continuous Dynamical Systems - A, 2009, 25 (1) : 299-319. doi: 10.3934/dcds.2009.25.299
[16]	Haomin Huang, Mingxin Wang. The reaction-diffusion system for an SIR epidemic model with a free boundary. Discrete & Continuous Dynamical Systems - B, 2015, 20 (7) : 2039-2050. doi: 10.3934/dcdsb.2015.20.2039
[17]	Sebastian Aniţa, Vincenzo Capasso. Stabilization of a reaction-diffusion system modelling malaria transmission. Discrete & Continuous Dynamical Systems - B, 2012, 17 (6) : 1673-1684. doi: 10.3934/dcdsb.2012.17.1673
[18]	Michaël Bages, Patrick Martinez. Existence of pulsating waves in a monostable reaction-diffusion system in solid combustion. Discrete & Continuous Dynamical Systems - B, 2010, 14 (3) : 817-869. doi: 10.3934/dcdsb.2010.14.817
[19]	José-Francisco Rodrigues, João Lita da Silva. On a unilateral reaction-diffusion system and a nonlocal evolution obstacle problem. Communications on Pure & Applied Analysis, 2004, 3 (1) : 85-95. doi: 10.3934/cpaa.2004.3.85
[20]	Qiang Liu, Zhichang Guo, Chunpeng Wang. Renormalized solutions to a reaction-diffusion system applied to image denoising. Discrete & Continuous Dynamical Systems - B, 2016, 21 (6) : 1839-1858. doi: 10.3934/dcdsb.2016025

2017 Impact Factor: 0.884

American Institute of Mathematical Sciences