Robustness of time-dependent attractors in <i>H</i><sup>1</sup>-norm for nonlocal problems

Tomás Caraballo; Marta Herrera-Cobos; Pedro Marín-Rubio

doi:10.3934/dcdsb.2018140

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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 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 & 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. Google Scholar
[2]	M. Anguiano, Attractors for Nonlinear and Non-Autonomous Parabolic PDEs in Unbounded Domains, PhD-thesis, Universidad de Sevilla, 2011. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[12]	V. V. Chepyzhov and M. I. Vishik, Attractors for Equations of Mathematical Physics, American Mathematical Society, Providence, RI, 2002. Google Scholar
[13]	M. Chipot and B. Lovat, On the asymptotic behaviour of some nonlocal problems, Positivity, 3 (1999), 65-81. doi: 10.1023/A:1009706118910. Google Scholar
[14]	M. Chipot and L. Molinet, Asymptotic behaviour of some nonlocal diffusion problems, Appl. Anal., 80 (2001), 273-315. doi: 10.1080/00036810108840994. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[18]	M. Chipot and S. Zheng, Asymptotic behavior of solutions to nonlinear parabolic equations with nonlocal terms, Asymptot. Anal., 45 (2005), 301-312. Google Scholar
[19]	I. Chueshov, Monotone Random Systems Theory and Applications, Springer-Verlag, Berlin, 2002. Google Scholar
[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. Google Scholar
[21]	H. Crauel, A. Debussche and F. Flandoli, Random attractors, J. Dynam. Differential Equations, 9 (1997), 307-341. doi: 10.1007/BF02219225. Google Scholar
[22]	R. Dautray and J. L. Lions, Analyse Mathématique et Calcul Numérique pour les Sciences et les Techniques, Masson, Paris, 1988. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[26]	D. Gilbarg and N. S. Trudinger, Elliptic Partial Differential Equations of Second Order, Reprint of the 1998 edition, Springer-Verlag, Berlin, 2001. Google Scholar
[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. Google Scholar
[28]	P. E. Kloeden, Pullback attractors of nonautonomous semidynamical systems, Stoch. Dyn., 3 (2003), 101-112. doi: 10.1142/S0219493703000632. Google Scholar
[29]	P. E. Kloeden and M. Rasmussen, Nonautonomous Dynamical Systems, American Mathematical Society, Providence, RI, 2011. Google Scholar
[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. Google Scholar
[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. Google Scholar
[32]	O. A. Ladyzhenskaya and N. N. Ural'tseva, Linear and Quasilinear Elliptic Equations, Academic Press, New York-London, 1968. Google Scholar
[33]	J. L. Lions, Quelques Méthodes de Résolution des Problèmes aux Limites Non Lineaires, Dunod, Paris, 1969. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[41]	R. Temam, Infinite-Dimensional Dynamical Systems in Mechanics and Physics, 2nd.ed., Springer, New York, 1997. Google Scholar
[42]	D. Werner, Funktionalanalysis, Springer-Verlag, Berlin, 2005. Google Scholar

show all references

References:

[1]	A. Andami Ovono, Asymptotic behaviour for a diffusion equation governed by nonlocal interactions, Electron. J. Differential Equations, 134 (2010), 1-16. Google Scholar
[2]	M. Anguiano, Attractors for Nonlinear and Non-Autonomous Parabolic PDEs in Unbounded Domains, PhD-thesis, Universidad de Sevilla, 2011. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[12]	V. V. Chepyzhov and M. I. Vishik, Attractors for Equations of Mathematical Physics, American Mathematical Society, Providence, RI, 2002. Google Scholar
[13]	M. Chipot and B. Lovat, On the asymptotic behaviour of some nonlocal problems, Positivity, 3 (1999), 65-81. doi: 10.1023/A:1009706118910. Google Scholar
[14]	M. Chipot and L. Molinet, Asymptotic behaviour of some nonlocal diffusion problems, Appl. Anal., 80 (2001), 273-315. doi: 10.1080/00036810108840994. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[18]	M. Chipot and S. Zheng, Asymptotic behavior of solutions to nonlinear parabolic equations with nonlocal terms, Asymptot. Anal., 45 (2005), 301-312. Google Scholar
[19]	I. Chueshov, Monotone Random Systems Theory and Applications, Springer-Verlag, Berlin, 2002. Google Scholar
[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. Google Scholar
[21]	H. Crauel, A. Debussche and F. Flandoli, Random attractors, J. Dynam. Differential Equations, 9 (1997), 307-341. doi: 10.1007/BF02219225. Google Scholar
[22]	R. Dautray and J. L. Lions, Analyse Mathématique et Calcul Numérique pour les Sciences et les Techniques, Masson, Paris, 1988. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[26]	D. Gilbarg and N. S. Trudinger, Elliptic Partial Differential Equations of Second Order, Reprint of the 1998 edition, Springer-Verlag, Berlin, 2001. Google Scholar
[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. Google Scholar
[28]	P. E. Kloeden, Pullback attractors of nonautonomous semidynamical systems, Stoch. Dyn., 3 (2003), 101-112. doi: 10.1142/S0219493703000632. Google Scholar
[29]	P. E. Kloeden and M. Rasmussen, Nonautonomous Dynamical Systems, American Mathematical Society, Providence, RI, 2011. Google Scholar
[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. Google Scholar
[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. Google Scholar
[32]	O. A. Ladyzhenskaya and N. N. Ural'tseva, Linear and Quasilinear Elliptic Equations, Academic Press, New York-London, 1968. Google Scholar
[33]	J. L. Lions, Quelques Méthodes de Résolution des Problèmes aux Limites Non Lineaires, Dunod, Paris, 1969. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[41]	R. Temam, Infinite-Dimensional Dynamical Systems in Mechanics and Physics, 2nd.ed., Springer, New York, 1997. Google Scholar
[42]	D. Werner, Funktionalanalysis, Springer-Verlag, Berlin, 2005. Google Scholar

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

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