American Institute of Mathematical Sciences

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

Structure and regularity of the global attractor of a reaction-diffusion equation with non-smooth nonlinear term

 1 Kyiv National Taras Shevchenko University, 01033-Kyiv, Ukraine 2 Institute for Applied System Analysis, National Technical University of Ukraine "KPI", Kyiv, Ukraine 3 Centro de Investigación Operativa, Universidad Miguel Hernández de Elche, Avda. de la Universidad, s/n, 03202 Elche

Received  September 2012 Revised  January 2013 Published  April 2014

In this paper we study the structure of the global attractor for a reaction-diffusion equation in which uniqueness of the Cauchy problem is not guarantied. We prove that the global attractor can be characterized using either the unstable manifold of the set of stationary points or the stable one but considering in this last case only solutions in the set of bounded complete trajectories.
Citation: 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 and Continuous Dynamical Systems, 2014, 34 (10) : 4155-4182. doi: 10.3934/dcds.2014.34.4155
References:
 [1] M. Anguiano, T. Caraballo, J. Real and J. Valero, Pullback attractors for reaction-diffusion equations in some unbounded domains with an $H^{-1}$-valued non-autonomous forcing term and without uniqueness of solutions, Discrete Contin. Dyn. Syst., Series B, 14 (2010), 307-326. doi: 10.3934/dcdsb.2010.14.307. [2] J. M. Arrieta, A. Rodríguez-Bernal and J. Valero, Dynamics of a reaction-diffusion equation with a discontinuous nonlinearity, Internat. J. Bifur. Chaos, 16 (2006), 2695-2984. doi: 10.1142/S0218127406016586. [3] A. V. Babin and M. I. Vishik, Attracteurs maximaux dans les équations aux dérivées partielles, in Nonlinear Partial Differential Equations and their Applications, Collegue de France Seminar, Vol. VII (Paris, 1983-1984), Research Notes in Math., 122, Pitman, Boston, MA, 1985, 11-34. [4] A. V. Babin and M. I. Vishik, Attractors of Evolution Equations, Nauka, Moscow, 1989. [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] P. Brunovsky and B. Fiedler, Connecting orbits in scalar reaction diffusion equations, Dynamics Reported, 1 (1988), 57-89. [7] T. Caraballo, P. Marín-Rubio and J. C.Robinson, A comparison between two theories for multivalued semiflows and their asymptotic behaviour, Set-Valued Anal., 11 (2003), 297-322. doi: 10.1023/A:1024422619616. [8] V. V. Chepyzhov and M. I.Vishik, Attractors for Equations of Mathematical Physics, American Mathematical Society, Providence, Rhode Island, 2002. [9] N. V. Gorban, O. V. Kapustyan and P. O. Kasyanov, Uniform trajectory attractor for non-autonomous reaction-diffusion equations with Carathodorys nonlinearity, Nonlinear Analysis, 98 (2014), 13-26. doi: 10.1016/j.na.2013.12.004. [10] A. V. Kapustyan, Global attractors for nonautonomous reaction-diffusion equation, Differential Equations, 38 (2002), 1467-1471. doi: 10.1023/A:1022378831393. [11] 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. [12] A. 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. [13] O. V. Kapustyan, P. O. Kasyanov and J. Valero, Structure of uniform global attractor for general non-autonomous reaction-diffusion system, in Continuous and Distributed Systems: Theory and Applications (eds. M. Z. Zgurovsky and V. A. Sadovninchniy), Solid Mechanics and Its Applications, 211, Springer, 2014, 163-180. doi: 10.1007/978-3-319-03146-0_12. [14] A. V. Kapustyan and J. Valero, 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. [15] A. 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] P. O. Kasyanov, Multivalued dynamics of solutions of autonomous operator differential equations with pseudomonotone nonlinearity, Math. Notes, 92 (2012), 205-218. doi: 10.1134/S0001434612070231. [18] P. O. Kasyanov, L. Toscano and N. V. Zadoianchuk, Regularity of weak solutions and their attractors for a parabolic feedback control problem, Set-Valued Var. Anal., 21 (2013), 271-282. doi: 10.1007/s11228-013-0233-8. [19] J. K. Hale, Asymptotic Behavior of Dissipative Systems, American Mathematical Society, Providence, 1988. [20] D. Henry, Some infinite-dimensional Morse-Smale systems defined by parabolic partial differential equations, J. Differential Equations, 59 (1985), 165-205. doi: 10.1016/0022-0396(85)90153-6. [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, Examples of attractors in scalar reaction-diffusion equations, J. Differential Equations, 73 (1988), 178-195. doi: 10.1016/0022-0396(88)90124-6. [24] C. Rocha, Properties of the attractor of a scalar parabolic PDE, J. Dynamics Differential Equations, 3 (1991), 575-591. doi: 10.1007/BF01049100. [25] C. Rocha and B. Fiedler, Heteroclinic orbits of semilinear parabolic equations, J. Differential. Equations, 125 (1996), 239-281. doi: 10.1006/jdeq.1996.0031. [26] G. R. Sell and Y. You, Dynamics of Evolutionary Equations, Springer, 2002. [27] R. Temam, Infinite-Dimensional Dynamical Systems in Mechanics and Physics, Springer-Verlag, New York, 1997. [28] E. Zeidler, Nonlinear Functional Analysis and Its Applications II, Springer, New York, 1990. doi: 10.1007/978-1-4612-4838-5. [29] M. Z. Zgurovsky, P. O. Kasyanov, O. V. Kapustyan, J. Valero and J. V. Zadoianchuk, Evolution Inclusions and Variation Inequalities for Earth Data Processing III. Long-Time Behavior of Evolution Inclusions Solutions in Earth Data Analysis, Series: Advances in Mechanics and Mathematics, 27, Springer, Berlin, 2012.

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References:
 [1] M. Anguiano, T. Caraballo, J. Real and J. Valero, Pullback attractors for reaction-diffusion equations in some unbounded domains with an $H^{-1}$-valued non-autonomous forcing term and without uniqueness of solutions, Discrete Contin. Dyn. Syst., Series B, 14 (2010), 307-326. doi: 10.3934/dcdsb.2010.14.307. [2] J. M. Arrieta, A. Rodríguez-Bernal and J. Valero, Dynamics of a reaction-diffusion equation with a discontinuous nonlinearity, Internat. J. Bifur. Chaos, 16 (2006), 2695-2984. doi: 10.1142/S0218127406016586. [3] A. V. Babin and M. I. Vishik, Attracteurs maximaux dans les équations aux dérivées partielles, in Nonlinear Partial Differential Equations and their Applications, Collegue de France Seminar, Vol. VII (Paris, 1983-1984), Research Notes in Math., 122, Pitman, Boston, MA, 1985, 11-34. [4] A. V. Babin and M. I. Vishik, Attractors of Evolution Equations, Nauka, Moscow, 1989. [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] P. Brunovsky and B. Fiedler, Connecting orbits in scalar reaction diffusion equations, Dynamics Reported, 1 (1988), 57-89. [7] T. Caraballo, P. Marín-Rubio and J. C.Robinson, A comparison between two theories for multivalued semiflows and their asymptotic behaviour, Set-Valued Anal., 11 (2003), 297-322. doi: 10.1023/A:1024422619616. [8] V. V. Chepyzhov and M. I.Vishik, Attractors for Equations of Mathematical Physics, American Mathematical Society, Providence, Rhode Island, 2002. [9] N. V. Gorban, O. V. Kapustyan and P. O. Kasyanov, Uniform trajectory attractor for non-autonomous reaction-diffusion equations with Carathodorys nonlinearity, Nonlinear Analysis, 98 (2014), 13-26. doi: 10.1016/j.na.2013.12.004. [10] A. V. Kapustyan, Global attractors for nonautonomous reaction-diffusion equation, Differential Equations, 38 (2002), 1467-1471. doi: 10.1023/A:1022378831393. [11] 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. [12] A. 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. [13] O. V. Kapustyan, P. O. Kasyanov and J. Valero, Structure of uniform global attractor for general non-autonomous reaction-diffusion system, in Continuous and Distributed Systems: Theory and Applications (eds. M. Z. Zgurovsky and V. A. Sadovninchniy), Solid Mechanics and Its Applications, 211, Springer, 2014, 163-180. doi: 10.1007/978-3-319-03146-0_12. [14] A. V. Kapustyan and J. Valero, 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. [15] A. 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] P. O. Kasyanov, Multivalued dynamics of solutions of autonomous operator differential equations with pseudomonotone nonlinearity, Math. Notes, 92 (2012), 205-218. doi: 10.1134/S0001434612070231. [18] P. O. Kasyanov, L. Toscano and N. V. Zadoianchuk, Regularity of weak solutions and their attractors for a parabolic feedback control problem, Set-Valued Var. Anal., 21 (2013), 271-282. doi: 10.1007/s11228-013-0233-8. [19] J. K. Hale, Asymptotic Behavior of Dissipative Systems, American Mathematical Society, Providence, 1988. [20] D. Henry, Some infinite-dimensional Morse-Smale systems defined by parabolic partial differential equations, J. Differential Equations, 59 (1985), 165-205. doi: 10.1016/0022-0396(85)90153-6. [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, Examples of attractors in scalar reaction-diffusion equations, J. Differential Equations, 73 (1988), 178-195. doi: 10.1016/0022-0396(88)90124-6. [24] C. Rocha, Properties of the attractor of a scalar parabolic PDE, J. Dynamics Differential Equations, 3 (1991), 575-591. doi: 10.1007/BF01049100. [25] C. Rocha and B. Fiedler, Heteroclinic orbits of semilinear parabolic equations, J. Differential. Equations, 125 (1996), 239-281. doi: 10.1006/jdeq.1996.0031. [26] G. R. Sell and Y. You, Dynamics of Evolutionary Equations, Springer, 2002. [27] R. Temam, Infinite-Dimensional Dynamical Systems in Mechanics and Physics, Springer-Verlag, New York, 1997. [28] E. Zeidler, Nonlinear Functional Analysis and Its Applications II, Springer, New York, 1990. doi: 10.1007/978-1-4612-4838-5. [29] M. Z. Zgurovsky, P. O. Kasyanov, O. V. Kapustyan, J. Valero and J. V. Zadoianchuk, Evolution Inclusions and Variation Inequalities for Earth Data Processing III. Long-Time Behavior of Evolution Inclusions Solutions in Earth Data Analysis, Series: Advances in Mechanics and Mathematics, 27, Springer, Berlin, 2012.
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