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

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

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 $(H_0^1 (\Omega))^N$.
Citation: Oleksiy V. Kapustyan, Pavlo O. Kasyanov, José Valero. Regular solutions and global attractors for reaction-diffusion systems without uniqueness. Communications on Pure and 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, Collegue de France Seminar, Vol.VII, Research Notes in Math $n^o$ 122, Pitman (1985), 11-34.

[2]

A. V. Babin and M. I. Vishik, Attractors of Evolution Equations, Nauka, Moscow, 1989.

[3]

V. V. Chepyzhov and M. I. Vishik, Attractors for Equations of Mathematical Physics, American Mathematical Society, Providence, Rhode Island, 2002.

[4]

M. I. Vishik, S. V. Zelik and V. V. Chepyzhov, Strong trajectory attractor of dissipative reaction-diffusion system, Doklady RAN, 435 (2010), 155-159. doi: 10.1134/S1064562410060086.

[5]

J. M. Ball, Global attractors for damped semilinear wave equations, Discrete Contin. Dyn. Syst., 10 (2004), 31-52. 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, Set-valued Analysis, 11 (2003), 297-322. doi: 10.1023/A:1024422619616.

[7]

P. Brunovsky and B. Fiedler, Connecting orbits in scalar reaction diffusion equations, Dynamics Reported, 1 (1988), 57-89.

[8]

N. V. Gorban, O. V. Kapustyan and P. O. Kasyanov, Uniform trajectory attractor for non-autonomous reaction-diffusion equations with Caratheodory's nonlinearity, Nonlinear Analysis, 98 (2014), 13-26 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, Solid Mechanics and Its Applications, 211 (2013), 205-220.

[10]

N. V. Gorban, P. O. Kasyanov, O. V. Kapustyan and L. S. Paliichuk, On global attractors for autonomous wave equation with discontinuous nonlinearity, Solid Mechanics and Its Applications, 211 (2014), 221-237.

[11]

J. K. Hale, Asymptotic Behavior of Dissipative Systems, American Mathematical Society, Providence, 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, Kyiv, 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, Discrete Contin. Dyn. Syst., 34 (2014), 4155-4182. 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, Set-Valued Var. Anal., 20 (2012), 445-465. 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, J. Math. Anal. Appl., 357 (2009), 254-272. 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, Internat. J. Bifur. Chaos, 20 (2010), 2723-2734. 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, Solid Mechanics and Its Applications, 211 (2014), 163-180.

[18]

P. O. Kasyanov, Multivalued dynamics of solutions of an autonomous differential-operator inclusion with pseudomonotone nonlinearity, Cybernetics and Systems Analysis, 47 (2011), 800-811.

[19]

P. O. Kasyanov, Multivalued dynamics of solutions of autonomous operator differential equations with pseudomonotone nonlinearity, Mathematical Notes, 92 (2012), 205-218.

[20]

P. O. Kasyanov et al., Regularity of weak solutions and their attractors for a parabolic feedback control problem, Set-Valued and Variational Analysis, 21 (2013), 271-282. doi: 10.1007/s11228-013-0233-8.

[21]

J. L. Lions, Quelques méthodes de résolution des problèmes aux limites non linéaires, Gauthier-Villar, Paris, 1969.

[22]

V. S. Melnik and J. Valero, On attractors of multi-valued semi-flows and differential inclusions, Set-Valued Anal., 6 (1998), 83-111. doi: 10.1023/A:1008608431399.

[23]

C. Rocha, Properties of the attractor of a scalar parabolic PDE, J. Dynamics Differential Equations, 3 (1991), 575-591. doi: 10.1007/BF01049100.

[24]

C. Rocha and B. Fiedler, Heteroclinic orbits of semilinear parabolic equations, J. Differential. Equations, 125 (1996), 239-281. 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, New York, 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, J. Math. Anal. Appl., 323 (2006), 614-633. 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, Rend. Accad. Naz. Sci. XL Mem. Mat. Appl., 24 (2000), 1-25.

[29]

M. Z. Zgurovsky et al., Long-time behavior of solutions for quasilinear hyperbolic hemivariational inequalities with application to piezoelectricity problem, Applied Mathematics Letters, 25 (2012), 1569-1574. 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, Berlin, 2012, 330 pp.

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, Collegue de France Seminar, Vol.VII, Research Notes in Math $n^o$ 122, Pitman (1985), 11-34.

[2]

A. V. Babin and M. I. Vishik, Attractors of Evolution Equations, Nauka, Moscow, 1989.

[3]

V. V. Chepyzhov and M. I. Vishik, Attractors for Equations of Mathematical Physics, American Mathematical Society, Providence, Rhode Island, 2002.

[4]

M. I. Vishik, S. V. Zelik and V. V. Chepyzhov, Strong trajectory attractor of dissipative reaction-diffusion system, Doklady RAN, 435 (2010), 155-159. doi: 10.1134/S1064562410060086.

[5]

J. M. Ball, Global attractors for damped semilinear wave equations, Discrete Contin. Dyn. Syst., 10 (2004), 31-52. 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, Set-valued Analysis, 11 (2003), 297-322. doi: 10.1023/A:1024422619616.

[7]

P. Brunovsky and B. Fiedler, Connecting orbits in scalar reaction diffusion equations, Dynamics Reported, 1 (1988), 57-89.

[8]

N. V. Gorban, O. V. Kapustyan and P. O. Kasyanov, Uniform trajectory attractor for non-autonomous reaction-diffusion equations with Caratheodory's nonlinearity, Nonlinear Analysis, 98 (2014), 13-26 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, Solid Mechanics and Its Applications, 211 (2013), 205-220.

[10]

N. V. Gorban, P. O. Kasyanov, O. V. Kapustyan and L. S. Paliichuk, On global attractors for autonomous wave equation with discontinuous nonlinearity, Solid Mechanics and Its Applications, 211 (2014), 221-237.

[11]

J. K. Hale, Asymptotic Behavior of Dissipative Systems, American Mathematical Society, Providence, 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, Kyiv, 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, Discrete Contin. Dyn. Syst., 34 (2014), 4155-4182. 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, Set-Valued Var. Anal., 20 (2012), 445-465. 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, J. Math. Anal. Appl., 357 (2009), 254-272. 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, Internat. J. Bifur. Chaos, 20 (2010), 2723-2734. 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, Solid Mechanics and Its Applications, 211 (2014), 163-180.

[18]

P. O. Kasyanov, Multivalued dynamics of solutions of an autonomous differential-operator inclusion with pseudomonotone nonlinearity, Cybernetics and Systems Analysis, 47 (2011), 800-811.

[19]

P. O. Kasyanov, Multivalued dynamics of solutions of autonomous operator differential equations with pseudomonotone nonlinearity, Mathematical Notes, 92 (2012), 205-218.

[20]

P. O. Kasyanov et al., Regularity of weak solutions and their attractors for a parabolic feedback control problem, Set-Valued and Variational Analysis, 21 (2013), 271-282. doi: 10.1007/s11228-013-0233-8.

[21]

J. L. Lions, Quelques méthodes de résolution des problèmes aux limites non linéaires, Gauthier-Villar, Paris, 1969.

[22]

V. S. Melnik and J. Valero, On attractors of multi-valued semi-flows and differential inclusions, Set-Valued Anal., 6 (1998), 83-111. doi: 10.1023/A:1008608431399.

[23]

C. Rocha, Properties of the attractor of a scalar parabolic PDE, J. Dynamics Differential Equations, 3 (1991), 575-591. doi: 10.1007/BF01049100.

[24]

C. Rocha and B. Fiedler, Heteroclinic orbits of semilinear parabolic equations, J. Differential. Equations, 125 (1996), 239-281. 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, New York, 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, J. Math. Anal. Appl., 323 (2006), 614-633. 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, Rend. Accad. Naz. Sci. XL Mem. Mat. Appl., 24 (2000), 1-25.

[29]

M. Z. Zgurovsky et al., Long-time behavior of solutions for quasilinear hyperbolic hemivariational inequalities with application to piezoelectricity problem, Applied Mathematics Letters, 25 (2012), 1569-1574. 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, Berlin, 2012, 330 pp.

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