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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 |
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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