# 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 & Continuous Dynamical Systems - A, 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., 14 (2010), 307.  doi: 10.3934/dcdsb.2010.14.307.  Google Scholar [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.  doi: 10.1142/S0218127406016586.  Google Scholar [3] A. V. Babin and M. I. Vishik, Attracteurs maximaux dans les équations aux dérivées partielles,, in Nonlinear Partial Differential Equations and their Applications, (1985), 1983.   Google Scholar [4] A. V. Babin and M. I. Vishik, Attractors of Evolution Equations,, Nauka, (1989).   Google Scholar [5] J. M. Ball, Global attractors for damped semilinear wave equations,, Discrete Contin. Dyn. Syst., 10 (2004), 31.  doi: 10.3934/dcds.2004.10.31.  Google Scholar [6] P. Brunovsky and B. Fiedler, Connecting orbits in scalar reaction diffusion equations,, Dynamics Reported, 1 (1988), 57.   Google Scholar [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.  doi: 10.1023/A:1024422619616.  Google Scholar [8] V. V. Chepyzhov and M. I.Vishik, Attractors for Equations of Mathematical Physics,, American Mathematical Society, (2002).   Google Scholar [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.  doi: 10.1016/j.na.2013.12.004.  Google Scholar [10] A. V. Kapustyan, Global attractors for nonautonomous reaction-diffusion equation,, Differential Equations, 38 (2002), 1467.  doi: 10.1023/A:1022378831393.  Google Scholar [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, (2008).   Google Scholar [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.  doi: 10.1007/s11228-011-0197-5.  Google Scholar [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), (2014), 163.  doi: 10.1007/978-3-319-03146-0_12.  Google Scholar [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.  doi: 10.1016/j.jmaa.2005.10.042.  Google Scholar [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.  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,, Internat. J. Bifur. Chaos, 20 (2010), 2723.  doi: 10.1142/S0218127410027313.  Google Scholar [17] P. O. Kasyanov, Multivalued dynamics of solutions of autonomous operator differential equations with pseudomonotone nonlinearity,, Math. Notes, 92 (2012), 205.  doi: 10.1134/S0001434612070231.  Google Scholar [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.  doi: 10.1007/s11228-013-0233-8.  Google Scholar [19] J. K. Hale, Asymptotic Behavior of Dissipative Systems,, American Mathematical Society, (1988).   Google Scholar [20] D. Henry, Some infinite-dimensional Morse-Smale systems defined by parabolic partial differential equations,, J. Differential Equations, 59 (1985), 165.  doi: 10.1016/0022-0396(85)90153-6.  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,, Set-Valued Anal., 6 (1998), 83.  doi: 10.1023/A:1008608431399.  Google Scholar [23] C. Rocha, Examples of attractors in scalar reaction-diffusion equations,, J. Differential Equations, 73 (1988), 178.  doi: 10.1016/0022-0396(88)90124-6.  Google Scholar [24] C. Rocha, Properties of the attractor of a scalar parabolic PDE,, J. Dynamics Differential Equations, 3 (1991), 575.  doi: 10.1007/BF01049100.  Google Scholar [25] C. Rocha and B. Fiedler, Heteroclinic orbits of semilinear parabolic equations,, J. Differential. Equations, 125 (1996), 239.  doi: 10.1006/jdeq.1996.0031.  Google Scholar [26] G. R. Sell and Y. You, Dynamics of Evolutionary Equations,, Springer, (2002).   Google Scholar [27] R. Temam, Infinite-Dimensional Dynamical Systems in Mechanics and Physics,, Springer-Verlag, (1997).   Google Scholar [28] E. Zeidler, Nonlinear Functional Analysis and Its Applications II,, Springer, (1990).  doi: 10.1007/978-1-4612-4838-5.  Google Scholar [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, (2012).   Google Scholar

show all references

##### 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., 14 (2010), 307.  doi: 10.3934/dcdsb.2010.14.307.  Google Scholar [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.  doi: 10.1142/S0218127406016586.  Google Scholar [3] A. V. Babin and M. I. Vishik, Attracteurs maximaux dans les équations aux dérivées partielles,, in Nonlinear Partial Differential Equations and their Applications, (1985), 1983.   Google Scholar [4] A. V. Babin and M. I. Vishik, Attractors of Evolution Equations,, Nauka, (1989).   Google Scholar [5] J. M. Ball, Global attractors for damped semilinear wave equations,, Discrete Contin. Dyn. Syst., 10 (2004), 31.  doi: 10.3934/dcds.2004.10.31.  Google Scholar [6] P. Brunovsky and B. Fiedler, Connecting orbits in scalar reaction diffusion equations,, Dynamics Reported, 1 (1988), 57.   Google Scholar [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.  doi: 10.1023/A:1024422619616.  Google Scholar [8] V. V. Chepyzhov and M. I.Vishik, Attractors for Equations of Mathematical Physics,, American Mathematical Society, (2002).   Google Scholar [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.  doi: 10.1016/j.na.2013.12.004.  Google Scholar [10] A. V. Kapustyan, Global attractors for nonautonomous reaction-diffusion equation,, Differential Equations, 38 (2002), 1467.  doi: 10.1023/A:1022378831393.  Google Scholar [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, (2008).   Google Scholar [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.  doi: 10.1007/s11228-011-0197-5.  Google Scholar [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), (2014), 163.  doi: 10.1007/978-3-319-03146-0_12.  Google Scholar [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.  doi: 10.1016/j.jmaa.2005.10.042.  Google Scholar [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.  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,, Internat. J. Bifur. Chaos, 20 (2010), 2723.  doi: 10.1142/S0218127410027313.  Google Scholar [17] P. O. Kasyanov, Multivalued dynamics of solutions of autonomous operator differential equations with pseudomonotone nonlinearity,, Math. Notes, 92 (2012), 205.  doi: 10.1134/S0001434612070231.  Google Scholar [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.  doi: 10.1007/s11228-013-0233-8.  Google Scholar [19] J. K. Hale, Asymptotic Behavior of Dissipative Systems,, American Mathematical Society, (1988).   Google Scholar [20] D. Henry, Some infinite-dimensional Morse-Smale systems defined by parabolic partial differential equations,, J. Differential Equations, 59 (1985), 165.  doi: 10.1016/0022-0396(85)90153-6.  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,, Set-Valued Anal., 6 (1998), 83.  doi: 10.1023/A:1008608431399.  Google Scholar [23] C. Rocha, Examples of attractors in scalar reaction-diffusion equations,, J. Differential Equations, 73 (1988), 178.  doi: 10.1016/0022-0396(88)90124-6.  Google Scholar [24] C. Rocha, Properties of the attractor of a scalar parabolic PDE,, J. Dynamics Differential Equations, 3 (1991), 575.  doi: 10.1007/BF01049100.  Google Scholar [25] C. Rocha and B. Fiedler, Heteroclinic orbits of semilinear parabolic equations,, J. Differential. Equations, 125 (1996), 239.  doi: 10.1006/jdeq.1996.0031.  Google Scholar [26] G. R. Sell and Y. You, Dynamics of Evolutionary Equations,, Springer, (2002).   Google Scholar [27] R. Temam, Infinite-Dimensional Dynamical Systems in Mechanics and Physics,, Springer-Verlag, (1997).   Google Scholar [28] E. Zeidler, Nonlinear Functional Analysis and Its Applications II,, Springer, (1990).  doi: 10.1007/978-1-4612-4838-5.  Google Scholar [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, (2012).   Google Scholar
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