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Long term dynamics of second order-in-time stochastic evolution equations with state-dependent delay
Robustness of time-dependent attractors in H1-norm for nonlocal problems
Departamento de Ecuaciones Diferenciales y Análisis Numérico, Universidad de Sevilla, c/Tarfia s/n, 41012 Sevilla, Spain |
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].
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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