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May  2018, 23(3): 1011-1036. doi: 10.3934/dcdsb.2018140

Robustness of time-dependent attractors in H1-norm for nonlocal problems

Tomás Caraballo , Marta Herrera-Cobos and  Pedro Marín-Rubio

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

To Professor Igor Chueshov, in Memoriam

Received  March 2017 Revised  May 2017 Published  May 2018 Early access  February 2018

Fund Project: Partially funded by the projects MTM2015-63723-P (MINECO/FEDER, EU) and P12-FQM-1492 (Junta de Andalucía).

In this paper, the existence of regular pullback attractors as well as their upper semicontinuous behaviour in H1-norm are analysed for a parameterized family of non-autonomous nonlocal reaction-diffusion equations without uniqueness, improving previous results [Nonlinear Dyn. 84 (2016), 35-50].

Keywords: Non-autonomous nonlocal reaction-diffusion equations, pullback attractors, regularity of attractors, upper semicontinuous behaviour.
Mathematics Subject Classification: Primary: 35B41, 35B65, 35K57, 35Q92, 37L30.
Citation: Tomás Caraballo, Marta Herrera-Cobos, Pedro Marín-Rubio. Robustness of time-dependent attractors in H1-norm for nonlocal problems. Discrete and Continuous Dynamical Systems - B, 2018, 23 (3) : 1011-1036. doi: 10.3934/dcdsb.2018140
References:
[1]

A. Andami Ovono, Asymptotic behaviour for a diffusion equation governed by nonlocal interactions, Electron. J. Differential Equations, 134 (2010), 1-16. 

[2]

M. Anguiano, Attractors for Nonlinear and Non-Autonomous Parabolic PDEs in Unbounded Domains, PhD-thesis, Universidad de Sevilla, 2011.

[3]

M. Anguiano, P. E. Kloeden and T. Lorenz, Asymptotic behaviour of nonlocal reaction-diffusion equations, Nonlinear Anal., 73 (2010), 3044-3057.  doi: 10.1016/j.na.2010.06.073.

[4]

J. Billingham, Dynamics of a strongly nonlocal reaction-diffusion population model, Nonlinearity, 17 (2004), 313-346.  doi: 10.1088/0951-7715/17/1/018.

[5]

T. Caraballo, M. Herrera-Cobos and P. Marín-Rubio, Long-time behaviour of a non-autonomous parabolic equation with nonlocal diffusion and sublinear terms, Nonlinear Anal., 121 (2015), 3-18.  doi: 10.1016/j.na.2014.07.011.

[6]

T. Caraballo, M. Herrera-Cobos and P. Marín-Rubio, Robustness of nonautonomous attractors for a family of nonlocal reaction-diffusion equations without uniqueness, Nonlinear Dyn., 84 (2016), 35-50.  doi: 10.1007/s11071-015-2200-4.

[7]

T. Caraballo, M. Herrera-Cobos and P. Marín-Rubio, Time-dependent attractors for nonautonomous nonlocal reaction-diffusion equations, Proc. Roy. Soc. Edinburgh Sect. A, To appear.

[8]

T. Caraballo and P. E. Kloeden, Non-autonomous attractors for integro-differential evolution equations, Discrete Contin. Dyn. Syst. Ser. S, 2 (2009), 17-36.  doi: 10.3934/dcdss.2009.2.17.

[9]

T. Caraballo, G. Lukaszewicz and J. Real, Pullback attractors for asymptotically compact non-autonomous dynamical systems, Nonlinear Anal., 64 (2006), 484-498.  doi: 10.1016/j.na.2005.03.111.

[10]

T. Caraballo, G. Lukaszewicz and J. Real, Pullback attractors for non-autonomous 2D-Navier-Stokes equations in some unbounded domains, C. R. Math. Acad. Sci. Paris, 342 (2006), 263-268.  doi: 10.1016/j.crma.2005.12.015.

[11]

N. H. Chang and M. Chipot, Nonlinear nonlocal evolution problems, Rev. R. Acad. Cienc. Exactas Fís. Nat. Ser. A Mat. RACSAM, 97 (2003), 423-445. 

[12]

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

[13]

M. Chipot and B. Lovat, On the asymptotic behaviour of some nonlocal problems, Positivity, 3 (1999), 65-81.  doi: 10.1023/A:1009706118910.

[14]

M. Chipot and L. Molinet, Asymptotic behaviour of some nonlocal diffusion problems, Appl. Anal., 80 (2001), 273-315.  doi: 10.1080/00036810108840994.

[15]

M. Chipot and T. Savistka, Nonlocal p-Laplace equations depending on the $L^p$ norm of the gradient, Adv. Differential Equations, 19 (2014), 997-1020. 

[16]

M. Chipot and M. Siegwart, On the asymptotic behaviour of some nonlocal mixed boundary value problems, in Nonlinear Analysis and applications: To V. Lakshmikantam on his 80th birthday, Kluwer Acad. Publ., Dordrecht, 1/2 (2003), 431-449.

[17]

M. Chipot, V. Valente and G. V. Caffarelli, Remarks on a nonlocal problem involving the Dirichlet energy, Rend. Sem. Mat. Univ. Padova, 110 (2003), 199-220. 

[18]

M. Chipot and S. Zheng, Asymptotic behavior of solutions to nonlinear parabolic equations with nonlocal terms, Asymptot. Anal., 45 (2005), 301-312. 

[19]

I. Chueshov, Monotone Random Systems Theory and Applications, Springer-Verlag, Berlin, 2002.

[20]

I. Chueshov and L. S. Pankratov, Upper semicontinuity of attractors of semilinear parabolic equations with asymptotically degenerating coefficients, Mat. Fiz. Anal. Geom., 6 (1999), 158-181. 

[21]

H. Crauel, A. Debussche and F. Flandoli, Random attractors, J. Dynam. Differential Equations, 9 (1997), 307-341.  doi: 10.1007/BF02219225.

[22]

R. Dautray and J. L. Lions, Analyse Mathématique et Calcul Numérique pour les Sciences et les Techniques, Masson, Paris, 1988.

[23]

P. Freitas, Nonlocal reaction-diffusion equations, Differential equations with applications to biology (Halifax, NS, 1997), Fields Inst. Commun., Amer. Math. Soc., Providence, RI, 21 (1999), 187-204.

[24]

J. García-Luengo, P. Marín-Rubio and J. Real, Pullback attractors in V for non-autonomous 2D-Navier-Stokes equations and their tempered behaviour, J. Differential Equations, 252 (2012), 4333-4356.  doi: 10.1016/j.jde.2012.01.010.

[25]

J. García-Melián and J. D. Rossi, A logistic equation with refuge and nonlocal diffusion, Commun. Pure Appl. Anal., 8 (2009), 2037-2053.  doi: 10.3934/cpaa.2009.8.2037.

[26]

D. Gilbarg and N. S. Trudinger, Elliptic Partial Differential Equations of Second Order, Reprint of the 1998 edition, Springer-Verlag, Berlin, 2001.

[27]

A. V. Kapustyan, V. S. Melnik and J. Valero, Attractors of multivalued dynamical processes generated by phase-field equations, Internat. J. Bifur. Chaos Appl. Sci. Engrg., 13 (2003), 1969-1983.  doi: 10.1142/S0218127403007801.

[28]

P. E. Kloeden, Pullback attractors of nonautonomous semidynamical systems, Stoch. Dyn., 3 (2003), 101-112.  doi: 10.1142/S0219493703000632.

[29]

P. E. Kloeden and M. Rasmussen, Nonautonomous Dynamical Systems, American Mathematical Society, Providence, RI, 2011.

[30]

P. E. Kloeden and B. Schmalfuß, Nonautonomous systems, cocycle attractors and variable time-step discretization, Numer. Algorithms, 14 (1997), 141-152.  doi: 10.1023/A:1019156812251.

[31]

P. E. Kloeden and B. Schmalfuß, Asymptotic behaviour of nonautonomous difference inclusions, Systems Control Lett., 33 (1998), 275-280.  doi: 10.1016/S0167-6911(97)00107-2.

[32] O. A. Ladyzhenskaya and N. N. Ural'tseva, Linear and Quasilinear Elliptic Equations, Academic Press, New York-London, 1968. 
[33]

J. L. Lions, Quelques Méthodes de Résolution des Problèmes aux Limites Non Lineaires, Dunod, Paris, 1969.

[34]

P. Marín-Rubio, Attractors for parametric delay differential equations without uniqueness and their upper semicontinuous behaviour, Nonlinear Anal., 68 (2008), 3166-3174.  doi: 10.1016/j.na.2007.03.011.

[35]

P. Marín-Rubio, G. Planas and J. Real, Asymptotic behaviour of a phase-field model with three coupled equations without uniqueness, J. Differential Equations, 246 (2009), 4632-4652.  doi: 10.1016/j.jde.2009.01.021.

[36]

P. Marín-Rubio and J. Real, Pullback attractors for 2D-Navier-Stokes equations with delays in continuous and sub-linear operators, Discrete Contin. Dyn. Syst., 26 (2010), 989-1006. 

[37]

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.

[38]

G. R. Sell, Nonautonomous differential equations and dynamical systems, Trans. Amer. Math. Soc., 127 (1967), 263-283.  doi: 10.1090/S0002-9947-1967-0212314-4.

[39]

G. Stampacchia, Le problème de Dirichlet pour les équations elliptiques du second ordre á coefficients discontinus, Ann. Inst. Fourier, 15 (1965), 189-258.  doi: 10.5802/aif.204.

[40]

Z. Szymańska, C. Morales-Rodrigo, M. Lachowicz and M. A. J. Chaplain, Mathematical modelling of cancer invasion of tissue: The role and effect of nonlocal interaction, Math. Models Methods Appl. Sci., 19 (2009), 257-281.  doi: 10.1142/S0218202509003425.

[41]

R. Temam, Infinite-Dimensional Dynamical Systems in Mechanics and Physics, 2nd.ed., Springer, New York, 1997.

[42]

D. Werner, Funktionalanalysis, Springer-Verlag, Berlin, 2005.

show all references

To Professor Igor Chueshov, in Memoriam

References:
[1]

A. Andami Ovono, Asymptotic behaviour for a diffusion equation governed by nonlocal interactions, Electron. J. Differential Equations, 134 (2010), 1-16. 

[2]

M. Anguiano, Attractors for Nonlinear and Non-Autonomous Parabolic PDEs in Unbounded Domains, PhD-thesis, Universidad de Sevilla, 2011.

[3]

M. Anguiano, P. E. Kloeden and T. Lorenz, Asymptotic behaviour of nonlocal reaction-diffusion equations, Nonlinear Anal., 73 (2010), 3044-3057.  doi: 10.1016/j.na.2010.06.073.

[4]

J. Billingham, Dynamics of a strongly nonlocal reaction-diffusion population model, Nonlinearity, 17 (2004), 313-346.  doi: 10.1088/0951-7715/17/1/018.

[5]

T. Caraballo, M. Herrera-Cobos and P. Marín-Rubio, Long-time behaviour of a non-autonomous parabolic equation with nonlocal diffusion and sublinear terms, Nonlinear Anal., 121 (2015), 3-18.  doi: 10.1016/j.na.2014.07.011.

[6]

T. Caraballo, M. Herrera-Cobos and P. Marín-Rubio, Robustness of nonautonomous attractors for a family of nonlocal reaction-diffusion equations without uniqueness, Nonlinear Dyn., 84 (2016), 35-50.  doi: 10.1007/s11071-015-2200-4.

[7]

T. Caraballo, M. Herrera-Cobos and P. Marín-Rubio, Time-dependent attractors for nonautonomous nonlocal reaction-diffusion equations, Proc. Roy. Soc. Edinburgh Sect. A, To appear.

[8]

T. Caraballo and P. E. Kloeden, Non-autonomous attractors for integro-differential evolution equations, Discrete Contin. Dyn. Syst. Ser. S, 2 (2009), 17-36.  doi: 10.3934/dcdss.2009.2.17.

[9]

T. Caraballo, G. Lukaszewicz and J. Real, Pullback attractors for asymptotically compact non-autonomous dynamical systems, Nonlinear Anal., 64 (2006), 484-498.  doi: 10.1016/j.na.2005.03.111.

[10]

T. Caraballo, G. Lukaszewicz and J. Real, Pullback attractors for non-autonomous 2D-Navier-Stokes equations in some unbounded domains, C. R. Math. Acad. Sci. Paris, 342 (2006), 263-268.  doi: 10.1016/j.crma.2005.12.015.

[11]

N. H. Chang and M. Chipot, Nonlinear nonlocal evolution problems, Rev. R. Acad. Cienc. Exactas Fís. Nat. Ser. A Mat. RACSAM, 97 (2003), 423-445. 

[12]

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

[13]

M. Chipot and B. Lovat, On the asymptotic behaviour of some nonlocal problems, Positivity, 3 (1999), 65-81.  doi: 10.1023/A:1009706118910.

[14]

M. Chipot and L. Molinet, Asymptotic behaviour of some nonlocal diffusion problems, Appl. Anal., 80 (2001), 273-315.  doi: 10.1080/00036810108840994.

[15]

M. Chipot and T. Savistka, Nonlocal p-Laplace equations depending on the $L^p$ norm of the gradient, Adv. Differential Equations, 19 (2014), 997-1020. 

[16]

M. Chipot and M. Siegwart, On the asymptotic behaviour of some nonlocal mixed boundary value problems, in Nonlinear Analysis and applications: To V. Lakshmikantam on his 80th birthday, Kluwer Acad. Publ., Dordrecht, 1/2 (2003), 431-449.

[17]

M. Chipot, V. Valente and G. V. Caffarelli, Remarks on a nonlocal problem involving the Dirichlet energy, Rend. Sem. Mat. Univ. Padova, 110 (2003), 199-220. 

[18]

M. Chipot and S. Zheng, Asymptotic behavior of solutions to nonlinear parabolic equations with nonlocal terms, Asymptot. Anal., 45 (2005), 301-312. 

[19]

I. Chueshov, Monotone Random Systems Theory and Applications, Springer-Verlag, Berlin, 2002.

[20]

I. Chueshov and L. S. Pankratov, Upper semicontinuity of attractors of semilinear parabolic equations with asymptotically degenerating coefficients, Mat. Fiz. Anal. Geom., 6 (1999), 158-181. 

[21]

H. Crauel, A. Debussche and F. Flandoli, Random attractors, J. Dynam. Differential Equations, 9 (1997), 307-341.  doi: 10.1007/BF02219225.

[22]

R. Dautray and J. L. Lions, Analyse Mathématique et Calcul Numérique pour les Sciences et les Techniques, Masson, Paris, 1988.

[23]

P. Freitas, Nonlocal reaction-diffusion equations, Differential equations with applications to biology (Halifax, NS, 1997), Fields Inst. Commun., Amer. Math. Soc., Providence, RI, 21 (1999), 187-204.

[24]

J. García-Luengo, P. Marín-Rubio and J. Real, Pullback attractors in V for non-autonomous 2D-Navier-Stokes equations and their tempered behaviour, J. Differential Equations, 252 (2012), 4333-4356.  doi: 10.1016/j.jde.2012.01.010.

[25]

J. García-Melián and J. D. Rossi, A logistic equation with refuge and nonlocal diffusion, Commun. Pure Appl. Anal., 8 (2009), 2037-2053.  doi: 10.3934/cpaa.2009.8.2037.

[26]

D. Gilbarg and N. S. Trudinger, Elliptic Partial Differential Equations of Second Order, Reprint of the 1998 edition, Springer-Verlag, Berlin, 2001.

[27]

A. V. Kapustyan, V. S. Melnik and J. Valero, Attractors of multivalued dynamical processes generated by phase-field equations, Internat. J. Bifur. Chaos Appl. Sci. Engrg., 13 (2003), 1969-1983.  doi: 10.1142/S0218127403007801.

[28]

P. E. Kloeden, Pullback attractors of nonautonomous semidynamical systems, Stoch. Dyn., 3 (2003), 101-112.  doi: 10.1142/S0219493703000632.

[29]

P. E. Kloeden and M. Rasmussen, Nonautonomous Dynamical Systems, American Mathematical Society, Providence, RI, 2011.

[30]

P. E. Kloeden and B. Schmalfuß, Nonautonomous systems, cocycle attractors and variable time-step discretization, Numer. Algorithms, 14 (1997), 141-152.  doi: 10.1023/A:1019156812251.

[31]

P. E. Kloeden and B. Schmalfuß, Asymptotic behaviour of nonautonomous difference inclusions, Systems Control Lett., 33 (1998), 275-280.  doi: 10.1016/S0167-6911(97)00107-2.

[32] O. A. Ladyzhenskaya and N. N. Ural'tseva, Linear and Quasilinear Elliptic Equations, Academic Press, New York-London, 1968. 
[33]

J. L. Lions, Quelques Méthodes de Résolution des Problèmes aux Limites Non Lineaires, Dunod, Paris, 1969.

[34]

P. Marín-Rubio, Attractors for parametric delay differential equations without uniqueness and their upper semicontinuous behaviour, Nonlinear Anal., 68 (2008), 3166-3174.  doi: 10.1016/j.na.2007.03.011.

[35]

P. Marín-Rubio, G. Planas and J. Real, Asymptotic behaviour of a phase-field model with three coupled equations without uniqueness, J. Differential Equations, 246 (2009), 4632-4652.  doi: 10.1016/j.jde.2009.01.021.

[36]

P. Marín-Rubio and J. Real, Pullback attractors for 2D-Navier-Stokes equations with delays in continuous and sub-linear operators, Discrete Contin. Dyn. Syst., 26 (2010), 989-1006. 

[37]

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.

[38]

G. R. Sell, Nonautonomous differential equations and dynamical systems, Trans. Amer. Math. Soc., 127 (1967), 263-283.  doi: 10.1090/S0002-9947-1967-0212314-4.

[39]

G. Stampacchia, Le problème de Dirichlet pour les équations elliptiques du second ordre á coefficients discontinus, Ann. Inst. Fourier, 15 (1965), 189-258.  doi: 10.5802/aif.204.

[40]

Z. Szymańska, C. Morales-Rodrigo, M. Lachowicz and M. A. J. Chaplain, Mathematical modelling of cancer invasion of tissue: The role and effect of nonlocal interaction, Math. Models Methods Appl. Sci., 19 (2009), 257-281.  doi: 10.1142/S0218202509003425.

[41]

R. Temam, Infinite-Dimensional Dynamical Systems in Mechanics and Physics, 2nd.ed., Springer, New York, 1997.

[42]

D. Werner, Funktionalanalysis, Springer-Verlag, Berlin, 2005.

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