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doi: 10.3934/dcdsb.2020252

Upper and weak-lower semicontinuity of pullback attractors to impulsive evolution processes

1. 

Departamento de Matemática, Universidade Federal de Santa Catarina, Campus Florianópolis - Brasil, Florianópolis, 88040-900, SC, Brazil

2. 

Departamento de Estatística, Análise Matemática e Optimización, & Instituto de Matemáticas, Universidade de Santiago de Compostela, Santiago de Compostela, Spain

* Corresponding author: Matheus C. Bortolan

Received  May 2020 Published  August 2020

Fund Project: The first author was partially supported by CNPq, project # 407635/2016-5. The second author was partially supported by the predoctoral contact BES-2017-082334

In this paper, following the work done in [11], we deal with the upper and weak-lower semicontinuity of pullback attractors for impulsive evolution processes. We first deal with the upper semicontinuity, presenting the abstract theory and applying it to uniform perturbations of a nonautonomous integrate-and-fire neuron model. We also present the abstract theory of weak-lower semicontinuity, and finish with an improvement of [11,Subsection 4.2], proving an invariance property for impulsive pullback $ \omega $-limits with weaker assumptions.

Citation: Matheus C. Bortolan, José Manuel Uzal. Upper and weak-lower semicontinuity of pullback attractors to impulsive evolution processes. Discrete & Continuous Dynamical Systems - B, doi: 10.3934/dcdsb.2020252
References:
[1]

S. B. Angenent, The Morse-Smale property for a semilinear parabolic equation, J. Differential Equations, 62 (1986), 427-442.  doi: 10.1016/0022-0396(86)90093-8.  Google Scholar

[2]

E. R. Aragão-CostaT. CaraballoA. 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.  Google Scholar

[3]

E. R. Aragão-CostaA. N. CarvalhoP. Marín-Rubio and G. Planas, Gradient-like nonlinear semigroups with infinitely many equilibria and applications to cascade systems, Topol. Methods Nonlinear Anal., 42 (2013), 345-376.   Google Scholar

[4]

J. M. Arrieta, A. N. Carvalho and A. Rodríguez-Bernal, Upper semicontinuity for attractors of parabolic problems with localized large diffusion and nonlinear boundary conditions, J. Differential Equations, 168 (2000), 33–59, Special issue in celebration of Jack K. Hale's 70th birthday, Part 1 (Atlanta, GA/Lisbon, 1998). doi: 10.1006/jdeq.2000.3876.  Google Scholar

[5]

P. W. Bates, K. Lu and C. Zeng, Existence and persistence of invariant manifolds for semiflows in Banach space, Mem. Amer. Math. Soc., 135 (1998), viii+129 pp. doi: 10.1090/memo/0645.  Google Scholar

[6]

E. M. Bonotto and M. Federson, Limit sets and the Poincaré-Bendixson theorem in impulsive semidynamical systems, J. Differential Equations, 244 (2008), 2334-2349.  doi: 10.1016/j.jde.2008.02.007.  Google Scholar

[7]

E. M. BonottoM. C. BortolanT. Caraballo and R. Collegari, Upper and lower semicontinuity of impulsive cocycle attractors for impulsive nonautonomous systems, Journal of Dynamics and Differential Equations, 1 (2019), 1-25.   Google Scholar

[8]

E. M. BonottoM. C. BortolanR. Collegari and R. Czaja, Semicontinuity of attractors for impulsive dynamical systems, J. Differential Equations, 261 (2016), 4338-4367.  doi: 10.1016/j.jde.2016.06.024.  Google Scholar

[9]

E. M. Bonotto and P. Kalita, On attractors of generalized semiflows with impulses, J. Geom. Anal., 30 (2020), 1412-1449.  doi: 10.1007/s12220-019-00143-0.  Google Scholar

[10]

M. C. Bortolan, A. N. Carvalho and J. A. Langa, Attractors Under Autonomous and Non-Autonomous Perturbations, Mathematical Surveys and Monographs 246, American Mathematical Society, 2020. https://bookstore.ams.org/surv-246/ Google Scholar

[11]

M. C. Bortolan and J. M. Uzal, Pullback attractors to impulsive evolution processes: Applications to differential equations and tube conditions, Discrete Contin. Dyn. Syst., 40 (2020), 2791-2826.  doi: 10.3934/dcds.2020150.  Google Scholar

[12]

R. Brette and W. Gerstner, Adaptive exponential integrate-and-fire model as an effective description of neuronal activity, Journal of Neurophysiology, 94 (2005), 3637-3642.  doi: 10.1152/jn.00686.2005.  Google Scholar

[13]

T. Caraballo and J. A. Langa, On the upper semicontinuity of cocycle attractors for non-autonomous and random dynamical systems, Dyn. Contin. Discrete Impuls. Syst. Ser. A Math. Anal., 10 (2003), 491-513.   Google Scholar

[14]

T. CaraballoJ. A. Langa and J. C. Robinson, Upper semicontinuty of attractors for small random perturbations of dynamical systems, Comm. Partial Differential Equations, 23 (1998), 1557-1581.  doi: 10.1080/03605309808821394.  Google Scholar

[15]

A. N. de Carvalho and G. Hines, Lower semicontinuity of attractors for gradient systems, Dynam. Systems Appl., 9 (2000), 37-50.   Google Scholar

[16]

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.  Google Scholar

[17]

A. N. CarvalhoJ. A. Langa and J. C. Robinson, Lower semicontinuity of attractors for non-autonomous dynamical systems, Ergodic Theory and Dynamical Systems, 29 (2009), 1765-1780.  doi: 10.1017/S0143385708000850.  Google Scholar

[18]

A. N. Carvalho, J. A. Langa and J. C. Robinson, Attractors for Infinite-Dimensional Non-Autonomous Dynamical Systems, Applied Mathematical Sciences, 182 Springer, New York, 2013. doi: 10.1007/978-1-4614-4581-4.  Google Scholar

[19]

C. Ding, Lyapunov quasi-stable trajectories, Fund. Math., 220 (2013), 139-154.  doi: 10.4064/fm220-2-4.  Google Scholar

[20]

C. Ding, Limit sets in impulsive semidynamical systems, Topol. Methods Nonlinear Anal., 43 (2014), 97-115.  doi: 10.12775/TMNA.2014.007.  Google Scholar

[21]

B. Ding, S. Pan and C. Ding, The index of impulsive periodic orbits, Nonlinear Anal., 192 (2020), 111659, 9 pp. doi: 10.1016/j.na.2019.111659.  Google Scholar

[22]

J. K. HaleX.-B. Lin and G. Raugel, Upper semicontinuity of attractors for approximations of semigroups and partial differential equations, Math. Comp., 50 (1988), 89-123.  doi: 10.1090/S0025-5718-1988-0917820-X.  Google Scholar

[23]

J. K. Hale and G. Raugel, Lower semicontinuity of attractors of gradient systems and applications, Ann. Mat. Pura Appl. (4), 154 (1989), 281-326.  doi: 10.1007/BF01790353.  Google Scholar

[24]

J. K. Hale and G. Raugel, Lower semicontinuity of the attractor for a singularly perturbed hyperbolic equation, J. Dynam. Differential Equations, 2 (1990), 19-67.  doi: 10.1007/BF01047769.  Google Scholar

[25]

D. Henry, Invariant Manifolds Near a Fixed Point, Handwritten notes, IME-USP. Google Scholar

[26]

D. Henry, Geometric Theory of Semilinear Parabolic Equations, Lecture Notes in Mathematics, 840. Springer-Verlag, Berlin-New York, 1981.  Google Scholar

[27]

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.  Google Scholar

[28]

J. P. Keener, F. C. Hoppensteadt and J. Rinzel, Integrate-and-fire models of nerve membrane response to oscillatory input, SIAM J. Appl. Math., 41 (1981), 503–517. https://www.jstor.org/stable/2101456?seq=1#metadata_info_tab_contents doi: 10.1137/0141042.  Google Scholar

[29]

W. M. Oliva, Morse-smale semiflows, openness and $A$-stability, Fields Institute Communications, Amer. Math. Soc., Providence, RI, 31 (2002), 285–307.  Google Scholar

[30]

G. Raugel, Stabilité d'une équation parabolique de Morse-Smale perturbée de manière singulière en une équation hyperbolique, C. R. Acad. Sci. Paris Sér. I Math. 310 (1990), 85–88.  Google Scholar

[31]

S. Smale, Morse inequalities for a dynamical system, Bull. Amer. Math. Soc., 66 (1960), 43-49.  doi: 10.1090/S0002-9904-1960-10386-2.  Google Scholar

show all references

References:
[1]

S. B. Angenent, The Morse-Smale property for a semilinear parabolic equation, J. Differential Equations, 62 (1986), 427-442.  doi: 10.1016/0022-0396(86)90093-8.  Google Scholar

[2]

E. R. Aragão-CostaT. CaraballoA. 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.  Google Scholar

[3]

E. R. Aragão-CostaA. N. CarvalhoP. Marín-Rubio and G. Planas, Gradient-like nonlinear semigroups with infinitely many equilibria and applications to cascade systems, Topol. Methods Nonlinear Anal., 42 (2013), 345-376.   Google Scholar

[4]

J. M. Arrieta, A. N. Carvalho and A. Rodríguez-Bernal, Upper semicontinuity for attractors of parabolic problems with localized large diffusion and nonlinear boundary conditions, J. Differential Equations, 168 (2000), 33–59, Special issue in celebration of Jack K. Hale's 70th birthday, Part 1 (Atlanta, GA/Lisbon, 1998). doi: 10.1006/jdeq.2000.3876.  Google Scholar

[5]

P. W. Bates, K. Lu and C. Zeng, Existence and persistence of invariant manifolds for semiflows in Banach space, Mem. Amer. Math. Soc., 135 (1998), viii+129 pp. doi: 10.1090/memo/0645.  Google Scholar

[6]

E. M. Bonotto and M. Federson, Limit sets and the Poincaré-Bendixson theorem in impulsive semidynamical systems, J. Differential Equations, 244 (2008), 2334-2349.  doi: 10.1016/j.jde.2008.02.007.  Google Scholar

[7]

E. M. BonottoM. C. BortolanT. Caraballo and R. Collegari, Upper and lower semicontinuity of impulsive cocycle attractors for impulsive nonautonomous systems, Journal of Dynamics and Differential Equations, 1 (2019), 1-25.   Google Scholar

[8]

E. M. BonottoM. C. BortolanR. Collegari and R. Czaja, Semicontinuity of attractors for impulsive dynamical systems, J. Differential Equations, 261 (2016), 4338-4367.  doi: 10.1016/j.jde.2016.06.024.  Google Scholar

[9]

E. M. Bonotto and P. Kalita, On attractors of generalized semiflows with impulses, J. Geom. Anal., 30 (2020), 1412-1449.  doi: 10.1007/s12220-019-00143-0.  Google Scholar

[10]

M. C. Bortolan, A. N. Carvalho and J. A. Langa, Attractors Under Autonomous and Non-Autonomous Perturbations, Mathematical Surveys and Monographs 246, American Mathematical Society, 2020. https://bookstore.ams.org/surv-246/ Google Scholar

[11]

M. C. Bortolan and J. M. Uzal, Pullback attractors to impulsive evolution processes: Applications to differential equations and tube conditions, Discrete Contin. Dyn. Syst., 40 (2020), 2791-2826.  doi: 10.3934/dcds.2020150.  Google Scholar

[12]

R. Brette and W. Gerstner, Adaptive exponential integrate-and-fire model as an effective description of neuronal activity, Journal of Neurophysiology, 94 (2005), 3637-3642.  doi: 10.1152/jn.00686.2005.  Google Scholar

[13]

T. Caraballo and J. A. Langa, On the upper semicontinuity of cocycle attractors for non-autonomous and random dynamical systems, Dyn. Contin. Discrete Impuls. Syst. Ser. A Math. Anal., 10 (2003), 491-513.   Google Scholar

[14]

T. CaraballoJ. A. Langa and J. C. Robinson, Upper semicontinuty of attractors for small random perturbations of dynamical systems, Comm. Partial Differential Equations, 23 (1998), 1557-1581.  doi: 10.1080/03605309808821394.  Google Scholar

[15]

A. N. de Carvalho and G. Hines, Lower semicontinuity of attractors for gradient systems, Dynam. Systems Appl., 9 (2000), 37-50.   Google Scholar

[16]

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.  Google Scholar

[17]

A. N. CarvalhoJ. A. Langa and J. C. Robinson, Lower semicontinuity of attractors for non-autonomous dynamical systems, Ergodic Theory and Dynamical Systems, 29 (2009), 1765-1780.  doi: 10.1017/S0143385708000850.  Google Scholar

[18]

A. N. Carvalho, J. A. Langa and J. C. Robinson, Attractors for Infinite-Dimensional Non-Autonomous Dynamical Systems, Applied Mathematical Sciences, 182 Springer, New York, 2013. doi: 10.1007/978-1-4614-4581-4.  Google Scholar

[19]

C. Ding, Lyapunov quasi-stable trajectories, Fund. Math., 220 (2013), 139-154.  doi: 10.4064/fm220-2-4.  Google Scholar

[20]

C. Ding, Limit sets in impulsive semidynamical systems, Topol. Methods Nonlinear Anal., 43 (2014), 97-115.  doi: 10.12775/TMNA.2014.007.  Google Scholar

[21]

B. Ding, S. Pan and C. Ding, The index of impulsive periodic orbits, Nonlinear Anal., 192 (2020), 111659, 9 pp. doi: 10.1016/j.na.2019.111659.  Google Scholar

[22]

J. K. HaleX.-B. Lin and G. Raugel, Upper semicontinuity of attractors for approximations of semigroups and partial differential equations, Math. Comp., 50 (1988), 89-123.  doi: 10.1090/S0025-5718-1988-0917820-X.  Google Scholar

[23]

J. K. Hale and G. Raugel, Lower semicontinuity of attractors of gradient systems and applications, Ann. Mat. Pura Appl. (4), 154 (1989), 281-326.  doi: 10.1007/BF01790353.  Google Scholar

[24]

J. K. Hale and G. Raugel, Lower semicontinuity of the attractor for a singularly perturbed hyperbolic equation, J. Dynam. Differential Equations, 2 (1990), 19-67.  doi: 10.1007/BF01047769.  Google Scholar

[25]

D. Henry, Invariant Manifolds Near a Fixed Point, Handwritten notes, IME-USP. Google Scholar

[26]

D. Henry, Geometric Theory of Semilinear Parabolic Equations, Lecture Notes in Mathematics, 840. Springer-Verlag, Berlin-New York, 1981.  Google Scholar

[27]

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.  Google Scholar

[28]

J. P. Keener, F. C. Hoppensteadt and J. Rinzel, Integrate-and-fire models of nerve membrane response to oscillatory input, SIAM J. Appl. Math., 41 (1981), 503–517. https://www.jstor.org/stable/2101456?seq=1#metadata_info_tab_contents doi: 10.1137/0141042.  Google Scholar

[29]

W. M. Oliva, Morse-smale semiflows, openness and $A$-stability, Fields Institute Communications, Amer. Math. Soc., Providence, RI, 31 (2002), 285–307.  Google Scholar

[30]

G. Raugel, Stabilité d'une équation parabolique de Morse-Smale perturbée de manière singulière en une équation hyperbolique, C. R. Acad. Sci. Paris Sér. I Math. 310 (1990), 85–88.  Google Scholar

[31]

S. Smale, Morse inequalities for a dynamical system, Bull. Amer. Math. Soc., 66 (1960), 43-49.  doi: 10.1090/S0002-9904-1960-10386-2.  Google Scholar

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