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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

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.   Google Scholar

[2]

A. V. Babin and M. I. Vishik, Attractors of Evolution Equations,, Nauka, (1989).   Google Scholar

[3]

V. V. Chepyzhov and M. I. Vishik, Attractors for Equations of Mathematical Physics,, American Mathematical Society, (2002).   Google Scholar

[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.  Google Scholar

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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.  Google Scholar

[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.  Google Scholar

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P. Brunovsky and B. Fiedler, Connecting orbits in scalar reaction diffusion equations,, \emph{Dynamics Reported, 1 (1988), 57.   Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[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.   Google Scholar

[11]

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

[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).   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[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.   Google Scholar

[19]

P. O. Kasyanov, Multivalued dynamics of solutions of autonomous operator differential equations with pseudomonotone nonlinearity,, \emph{Mathematical Notes}, 92 (2012), 205.   Google Scholar

[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.  Google Scholar

[21]

J. L. Lions, Quelques méthodes de résolution des problèmes aux limites non linéaires,, Gauthier-Villar, (1969).   Google Scholar

[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.  Google Scholar

[23]

C. Rocha, Properties of the attractor of a scalar parabolic PDE,, \emph{J. Dynamics Differential Equations, 3 (1991), 575.  doi: 10.1007/BF01049100.  Google Scholar

[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.  Google Scholar

[25]

G. R. Sell and Y. You, Dynamics of Evolutionary Equations,, Springer, (2002).  doi: 10.1007/978-1-4757-5037-9.  Google Scholar

[26]

R. Temam, Infinite-dimensional Dynamical Systems in Mechanics and Physics,, Springer-Verlag, (1997).  doi: 10.1007/978-1-4612-0645-3.  Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[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).   Google Scholar

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.   Google Scholar

[2]

A. V. Babin and M. I. Vishik, Attractors of Evolution Equations,, Nauka, (1989).   Google Scholar

[3]

V. V. Chepyzhov and M. I. Vishik, Attractors for Equations of Mathematical Physics,, American Mathematical Society, (2002).   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[7]

P. Brunovsky and B. Fiedler, Connecting orbits in scalar reaction diffusion equations,, \emph{Dynamics Reported, 1 (1988), 57.   Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[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.   Google Scholar

[11]

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

[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).   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[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.   Google Scholar

[19]

P. O. Kasyanov, Multivalued dynamics of solutions of autonomous operator differential equations with pseudomonotone nonlinearity,, \emph{Mathematical Notes}, 92 (2012), 205.   Google Scholar

[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.  Google Scholar

[21]

J. L. Lions, Quelques méthodes de résolution des problèmes aux limites non linéaires,, Gauthier-Villar, (1969).   Google Scholar

[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.  Google Scholar

[23]

C. Rocha, Properties of the attractor of a scalar parabolic PDE,, \emph{J. Dynamics Differential Equations, 3 (1991), 575.  doi: 10.1007/BF01049100.  Google Scholar

[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.  Google Scholar

[25]

G. R. Sell and Y. You, Dynamics of Evolutionary Equations,, Springer, (2002).  doi: 10.1007/978-1-4757-5037-9.  Google Scholar

[26]

R. Temam, Infinite-dimensional Dynamical Systems in Mechanics and Physics,, Springer-Verlag, (1997).  doi: 10.1007/978-1-4612-0645-3.  Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[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).   Google Scholar

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