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March  2019, 24(3): 1049-1077. doi: 10.3934/dcdsb.2019006

Robustness of dynamically gradient multivalued dynamical systems

Rubén Caballero 1, , Alexandre N. Carvalho 2, , Pedro Marín-Rubio 3, and  José Valero 4,

1. 

Centro de Investigación Operativa, Universidad Miguel Hernández de Elche, 03202-Elche, Alicante, Spain
2. 

Instituto de Ciências Matemáticas e de Computaçao, Universidade de São Paulo, Campus de São Carlos, Caixa Postal 668, 13560-970 São Carlos, SP, Brazil
3. 

Departamento de Ecuaciones Diferenciales y Análisis Numérico, Universidad de Sevilla, c/Tarfia s/n, 41012-Sevilla, Spain
4. 

Centro de Investigación Operativa, Universidad Miguel Hernández de Elche, 03202-Elche, Alicante, Spain

To Professor Valery Melnik, in Memoriam

Received  January 2018 Revised  May 2018 Published  March 2019 Early access  January 2019

Fund Project: The authors of this work have been partially funded by the following projects: R. Caballero is a fellow of Programa de FPU del Ministerio de Educación, Cultura y Deporte, reference FPU15/03080; P. Marín-Rubio was supported by projects PHB2010-0002-PC (Ministerio de Educación-DGPU), MTM2015-63723-P (MINECO/FEDER, EU) and P12-FQM-1492 (Junta de Andalucía); A. N. Carvalho was supported by grant 2018/10997-6 from FAPESP-Brazil and grant 303929/2015-4 from CNPq-Brazil; and J.Valero by projects MTM2015-63723-P, MTM2016-74921-P (MINECO/FEDER, EU) and P12-FQM-1492 (Junta de Andalucía).

In this paper we study the robustness of dynamically gradient multivalued semiflows. As an application, we describe the dynamical properties of a family of Chafee-Infante problems approximating a differential inclusion studied in [3], proving that the weak solutions of these problems generate a dynamically gradient multivalued semiflow with respect to suitable Morse sets.

Keywords: Attractors, reaction-diffusion equations, stability, dynamically gradient multivalued semiflows, Morse decomposition, set-valued dynamical systems.
Mathematics Subject Classification: 37B25, 35B40, 35B41, 35K55, 37L30, 58C06.
Citation: Rubén Caballero, Alexandre N. Carvalho, Pedro Marín-Rubio, José Valero. Robustness of dynamically gradient multivalued dynamical systems. Discrete and Continuous Dynamical Systems - B, 2019, 24 (3) : 1049-1077. doi: 10.3934/dcdsb.2019006
References:
[1]

E. R. Aragão-Costa, T. Caraballo, A. N. Carvalho and J. A. Langa, Stability of gradient semigroups under perturbations, Nonlinearity, 24 (2011), 2099-2117.  doi: 10.1088/0951-7715/24/7/010.

[2]

E. R. Aragão-Costa, T. Caraballo, A. N. Carvalho and J. A. Langa, Non-autonomous Morse-decomposition and Lyapunov functions for gradient-like processes, Transactions of the American Mathematical Society, 365 (2013), 5277-5312.  doi: 10.1090/S0002-9947-2013-05810-2.

[3]

J. M. Arrieta, A. Rodríguez-Bernal and J. Valero, Dynamics of a reaction-diffusion equation with a discontinuous nonlinearity, International Journal of Bifurcation and Chaos, 16 (2006), 2965-2984.  doi: 10.1142/S0218127406016586.

[4]

J. M. Ball, Continuity properties and global attractors of generalized semiflows and the Navier-Stokes equations, Journal of Nonlinear Science, 7 (1997), 475-502.  doi: 10.1007/s003329900037.

[5]

V. Barbu, Nonlinear Semigroups and Differential Equations in Banach Spaces, Editura Academiei, Bucuresti, 1976.

[6]

T. Caraballo, P. Marín-Rubio and J. Robinson, A comparison between two theories for multi-valued semiflows and their asymptotic behaviour, Set-Valued Analysis, 11 (2003), 297-322.  doi: 10.1023/A:1024422619616.

[7]

A. N. Carvalho and J. A. Langa, An extension of the concept of gradient semigroups which is stable under perturbation, J. Differential Equations, 246 (2009), 2646-2668.  doi: 10.1016/j.jde.2009.01.007.

[8]

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

[9]

H. B. da Costa and J. Valero, Morse decompositions and Lyapunov functions for dynamically gradient multivalued semiflows, Nonlinear Dyn., 84 (2016), 19-34.  doi: 10.1007/s11071-015-2193-z.

[10]

P. Gruber, Convex and Discrete Geometry, Springer, 2007.

[11]

D. Henry, Geometric Theory of Semilinear Parabolic Equations, Springer-Verlag, Berlin-New York, 1981.

[12]

D. Henry, Some infinite-dimensional Morse-Smale systems defined by parabolic partial differential equations, J. Diff. Eqs., 59 (1985), 165-205.  doi: 10.1016/0022-0396(85)90153-6.

[13]

O. V. Kapustyan, P. O. Kasyanov and J. Valero, Structure and regularity of the global attractor of a reacction-diffusion equation with non-smooth nonlinear term, Discrete Continuous Dynamical Systems, 34 (2014), 4155-4182.  doi: 10.3934/dcds.2014.34.4155.

[14]

O. V. Kapustyan, P. O. Kasyanov and J. Valero, Structure of the global attractor for weak solutions of a reaction-diffusion equation, Information Sciences, 9 (2015), 2257-2264. 

[15]

O. V. Kapustyan, V. Pankov and J. Valero, On global attractors of multivalued semiflows generated by the 3D Bénard system, Set-Valued and Variational Analysis, 20 (2012), 445-465.  doi: 10.1007/s11228-011-0197-5.

[16]

D. Li, Morse decompositions for general dynamical systems and differential inclusions with applications to control systems, SIAM Journal on Control and Optimization, 46 (2007), 35-60.  doi: 10.1137/060662101.

[17]

J. L. Lions, Quelques Méthodes de Résolution des Problèmes aux Limites non Linéaires, Gauthier-Villar, Paris, 1969.

[18]

S. Mazzini, Atratores Para o Problema de Chafee-Infante, PhD-thesis, Universidade de São Paulo, 1997.

[19]

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

[20] J. C. Robinson, Infinite-Dimensional Dynamical Systems: An Introduction to Dissipative Parabilic PDEs and the Theory of Global Attractors, Cambridge University Press, Cambridge, UK, 2001.  doi: 10.1007/978-94-010-0732-0.
[21]

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

[22]

R. Temam, Navier-Stokes Equations, North-Holland, Amsterdam-New York, 1979.

[23]

A. Tolstonogov, On solutions of evolution inclusions I, Siberian Math. J., 33 (1992), 500-511.  doi: 10.1007/BF00970899.

[24]

J. Valero, On locally compact attractors of dynamical systems, J. Math. Anal. Appl., 237 (1999), 43-54.  doi: 10.1006/jmaa.1999.6446.

[25]

J. Valero, Attractors of parabolic equations without uniqueness, J. Dynamics Differential Equations, 13 (2001), 711-744.  doi: 10.1023/A:1016642525800.

[26]

J. Valero, On the Kneser property for some parabolic problems, Topology Appl., 153 (2005), 975-989.  doi: 10.1016/j.topol.2005.01.025.

[27]

J. Valero and A. V. Kapustyan, On the connectedness and symptotic behaviour of solutions of reaction-diffusion systems, J. Math. Anal. Appl., 323 (2006), 614-633.  doi: 10.1016/j.jmaa.2005.10.042.

[28] A. Wayne and D. Varberg, Convex Functions, Academic Press, Elsevier, 1973. 
[29]

K. Yosida, Functinoal Analysis, Springer-Verlag, Berlin, 1965.

show all references

To Professor Valery Melnik, in Memoriam

References:
[1]

E. R. Aragão-Costa, T. Caraballo, A. N. Carvalho and J. A. Langa, Stability of gradient semigroups under perturbations, Nonlinearity, 24 (2011), 2099-2117.  doi: 10.1088/0951-7715/24/7/010.

[2]

E. R. Aragão-Costa, T. Caraballo, A. N. Carvalho and J. A. Langa, Non-autonomous Morse-decomposition and Lyapunov functions for gradient-like processes, Transactions of the American Mathematical Society, 365 (2013), 5277-5312.  doi: 10.1090/S0002-9947-2013-05810-2.

[3]

J. M. Arrieta, A. Rodríguez-Bernal and J. Valero, Dynamics of a reaction-diffusion equation with a discontinuous nonlinearity, International Journal of Bifurcation and Chaos, 16 (2006), 2965-2984.  doi: 10.1142/S0218127406016586.

[4]

J. M. Ball, Continuity properties and global attractors of generalized semiflows and the Navier-Stokes equations, Journal of Nonlinear Science, 7 (1997), 475-502.  doi: 10.1007/s003329900037.

[5]

V. Barbu, Nonlinear Semigroups and Differential Equations in Banach Spaces, Editura Academiei, Bucuresti, 1976.

[6]

T. Caraballo, P. Marín-Rubio and J. Robinson, A comparison between two theories for multi-valued semiflows and their asymptotic behaviour, Set-Valued Analysis, 11 (2003), 297-322.  doi: 10.1023/A:1024422619616.

[7]

A. N. Carvalho and J. A. Langa, An extension of the concept of gradient semigroups which is stable under perturbation, J. Differential Equations, 246 (2009), 2646-2668.  doi: 10.1016/j.jde.2009.01.007.

[8]

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

[9]

H. B. da Costa and J. Valero, Morse decompositions and Lyapunov functions for dynamically gradient multivalued semiflows, Nonlinear Dyn., 84 (2016), 19-34.  doi: 10.1007/s11071-015-2193-z.

[10]

P. Gruber, Convex and Discrete Geometry, Springer, 2007.

[11]

D. Henry, Geometric Theory of Semilinear Parabolic Equations, Springer-Verlag, Berlin-New York, 1981.

[12]

D. Henry, Some infinite-dimensional Morse-Smale systems defined by parabolic partial differential equations, J. Diff. Eqs., 59 (1985), 165-205.  doi: 10.1016/0022-0396(85)90153-6.

[13]

O. V. Kapustyan, P. O. Kasyanov and J. Valero, Structure and regularity of the global attractor of a reacction-diffusion equation with non-smooth nonlinear term, Discrete Continuous Dynamical Systems, 34 (2014), 4155-4182.  doi: 10.3934/dcds.2014.34.4155.

[14]

O. V. Kapustyan, P. O. Kasyanov and J. Valero, Structure of the global attractor for weak solutions of a reaction-diffusion equation, Information Sciences, 9 (2015), 2257-2264. 

[15]

O. V. Kapustyan, V. Pankov and J. Valero, On global attractors of multivalued semiflows generated by the 3D Bénard system, Set-Valued and Variational Analysis, 20 (2012), 445-465.  doi: 10.1007/s11228-011-0197-5.

[16]

D. Li, Morse decompositions for general dynamical systems and differential inclusions with applications to control systems, SIAM Journal on Control and Optimization, 46 (2007), 35-60.  doi: 10.1137/060662101.

[17]

J. L. Lions, Quelques Méthodes de Résolution des Problèmes aux Limites non Linéaires, Gauthier-Villar, Paris, 1969.

[18]

S. Mazzini, Atratores Para o Problema de Chafee-Infante, PhD-thesis, Universidade de São Paulo, 1997.

[19]

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

[20] J. C. Robinson, Infinite-Dimensional Dynamical Systems: An Introduction to Dissipative Parabilic PDEs and the Theory of Global Attractors, Cambridge University Press, Cambridge, UK, 2001.  doi: 10.1007/978-94-010-0732-0.
[21]

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

[22]

R. Temam, Navier-Stokes Equations, North-Holland, Amsterdam-New York, 1979.

[23]

A. Tolstonogov, On solutions of evolution inclusions I, Siberian Math. J., 33 (1992), 500-511.  doi: 10.1007/BF00970899.

[24]

J. Valero, On locally compact attractors of dynamical systems, J. Math. Anal. Appl., 237 (1999), 43-54.  doi: 10.1006/jmaa.1999.6446.

[25]

J. Valero, Attractors of parabolic equations without uniqueness, J. Dynamics Differential Equations, 13 (2001), 711-744.  doi: 10.1023/A:1016642525800.

[26]

J. Valero, On the Kneser property for some parabolic problems, Topology Appl., 153 (2005), 975-989.  doi: 10.1016/j.topol.2005.01.025.

[27]

J. Valero and A. V. Kapustyan, On the connectedness and symptotic behaviour of solutions of reaction-diffusion systems, J. Math. Anal. Appl., 323 (2006), 614-633.  doi: 10.1016/j.jmaa.2005.10.042.

[28] A. Wayne and D. Varberg, Convex Functions, Academic Press, Elsevier, 1973. 
[29]

K. Yosida, Functinoal Analysis, Springer-Verlag, Berlin, 1965.

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