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Robustness of dynamically gradient multivalued dynamical systems
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 |
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 [
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] | |
[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] | |
[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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