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July  2017, 22(5): 1801-1816. doi: 10.3934/dcdsb.2017107

Global attractor for a nonlocal p-Laplacian equation without uniqueness of solution

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

Departamento de Ecuaciones Diferenciales y Análisis Numérico, Universidad de Sevilla, Apdo. de Correos 1160,41080-Sevilla, Spain

To the memory of Pablo Cobos, Marta's grandfather, with sorrow and love

Received  February 2016 Revised  June 2016 Published  March 2017

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 solution for a $p$-Laplacian parabolic equation with nonlocal diffusion is established. To do this, we make use of a change of variable which transforms the original problem into a nonlocal one but with local diffusion. Since the uniqueness of solution is unknown, the asymptotic behaviour of the solutions is analysed in a multi-valued framework. Namely, the existence of the compact global attractor in $L^2(Ω)$ is ensured.

Keywords: Nonlocal p-Laplacian equation, compact global attractor, asymptotic compactness, multi-valued dynamical systems.
Mathematics Subject Classification: Primary: 35B41, 35K55, 35Q92, 37L3.
Citation: Tomás Caraballo, Marta Herrera-Cobos, Pedro Marín-Rubio. Global attractor for a nonlocal p-Laplacian equation without uniqueness of solution. Discrete & Continuous Dynamical Systems - B, 2017, 22 (5) : 1801-1816. doi: 10.3934/dcdsb.2017107
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]

A. Andami Ovono and A. Rougirel, Elliptic equations with diffusion parameterized by the range of nonlocal interactions, Ann. Mat. Pura Appl.(4), 189 (2010), 163-183.  doi: 10.1007/s10231-009-0104-y.  Google Scholar

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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

[4]

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

[5]

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

[6]

Y. Chen, S. Levine and M. Rao, Variable exponent, linear growth functionals in image restoration, SIAM J. Appl. Math., 66 (2006), 1383-1406.  doi: 10.1137/050624522.  Google Scholar

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M. Chipot and F. J. S. A. Corrêa, Boundary layer solutions to functional elliptic equations, Bull. Braz. Math. Soc. (N.S.), 40 (2009), 381-393.  doi: 10.1007/s00574-009-0017-9.  Google Scholar

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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

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M. Chipot and L. Molinet, Asymptotic behaviour of some nonlocal diffusion problems, Appl. Anal., 80 (2001), 279-315.  doi: 10.1080/00036810108840994.  Google Scholar

[10]

M. Chipot and J. F. Rodrigues, On a class of nonlocal nonlinear elliptic problems, RAIRO Modél. Math. Anal. Numér., 26 (1992), 447-467.  doi: 10.1051/m2an/1992260304471.  Google Scholar

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M. Chipot and P. Roy, Existence results for some functional elliptic equations, Differential Integral Equations, 27 (2014), 289-300.   Google Scholar

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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

[13]

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, pp. 431-449, Kluwer Acad. Publ. , Dordrecht, 2003. Google Scholar

[14]

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

[15]

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

[16]

E. A. Coddington and N. Levinson, Theory of Ordinary Differential Equations, McGraw-Hill Book Company, Inc. , New York-Toronto-London, 1955. Google Scholar

[17]

F. J. S. A. Corrêa, On positive solutions of nonlocal and nonvariational elliptic problems, Nonlinear Anal., 59 (2004), 1147-1155.  doi: 10.1016/S0362-546X(04)00322-0.  Google Scholar

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F. J. S. A. Corrêa, S. B. de Menezes and J. Ferreira, On a class of problems involving a nonlocal operator, Appl. Math. Comput., 147 (2004), 475-489.  doi: 10.1016/S0096-3003(02)00740-3.  Google Scholar

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R. Dautray and J. L. Lions, Analyse Mathématique et Calcul Numérique pour les Sciences et les Techniques Masson, Paris, 1987. Google Scholar

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J. Furter and M. Grinfeld, Local vs. nonlocal interactions in population dynamics, J. Math. Biol., 27 (1989), 65-80.  doi: 10.1007/BF00276081.  Google Scholar

[21]

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

[22]

D. Hilhorst and J. F. Rodrigues, On a nonlocal diffusion equation with discontinuous reaction, Adv. Differential Equations, 5 (2000), 657-680.   Google Scholar

[23]

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

[24]

A. V. Kapustyan and J. Valero, On the connectedness and asymptotic behaviour of solutions of reaction-diffusion systems, J. Math. Anal. Appl., 323 (2006), 614-633.  doi: 10.1016/j.jmaa.2005.10.042.  Google Scholar

[25]

A. V. Kapustyan and J. Valero, Weak and strong attractors fo the 3D Navier-Stokes system, J. Differential Equations, 240 (2007), 249-278.  doi: 10.1016/j.jde.2007.06.008.  Google Scholar

[26]

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

[27]

B. Lovat, Études de Quelques Problémes Paraboliques Non Locaux Thése, Université de Metz, 1995. Google Scholar

[28]

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

[29]

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.  doi: 10.3934/dcds.2010.26.989.  Google Scholar

[30]

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

[31]

S. B. de Menezes, Remarks on weak solutions for a nonlocal parabolic problem, Int. J. Math. Math. Sci., 2006 (2006), 1-10.  doi: 10.1155/IJMMS/2006/82654.  Google Scholar

[32] J. C. Robinson, Infinite-Dimensional Dynamical Systems, Cambridge University Press, Cambridge, 2001.   Google Scholar
[33] M. Růžička, Electrorheological Fluids: Modeling and Mathematical Theory, Springer-Verlag, Berlin, 2000.   Google Scholar
[34]

T. Savitska, Asymptotic Behaviour of Solutions of Nonlocal Parabolic Problems Ph. D Thesis, University of Zurich, 2015. Google Scholar

[35]

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

A. Andami Ovono and A. Rougirel, Elliptic equations with diffusion parameterized by the range of nonlocal interactions, Ann. Mat. Pura Appl.(4), 189 (2010), 163-183.  doi: 10.1007/s10231-009-0104-y.  Google Scholar

[3]

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

[4]

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

[5]

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

[6]

Y. Chen, S. Levine and M. Rao, Variable exponent, linear growth functionals in image restoration, SIAM J. Appl. Math., 66 (2006), 1383-1406.  doi: 10.1137/050624522.  Google Scholar

[7]

M. Chipot and F. J. S. A. Corrêa, Boundary layer solutions to functional elliptic equations, Bull. Braz. Math. Soc. (N.S.), 40 (2009), 381-393.  doi: 10.1007/s00574-009-0017-9.  Google Scholar

[8]

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

[9]

M. Chipot and L. Molinet, Asymptotic behaviour of some nonlocal diffusion problems, Appl. Anal., 80 (2001), 279-315.  doi: 10.1080/00036810108840994.  Google Scholar

[10]

M. Chipot and J. F. Rodrigues, On a class of nonlocal nonlinear elliptic problems, RAIRO Modél. Math. Anal. Numér., 26 (1992), 447-467.  doi: 10.1051/m2an/1992260304471.  Google Scholar

[11]

M. Chipot and P. Roy, Existence results for some functional elliptic equations, Differential Integral Equations, 27 (2014), 289-300.   Google Scholar

[12]

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

[13]

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, pp. 431-449, Kluwer Acad. Publ. , Dordrecht, 2003. Google Scholar

[14]

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

[15]

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

[16]

E. A. Coddington and N. Levinson, Theory of Ordinary Differential Equations, McGraw-Hill Book Company, Inc. , New York-Toronto-London, 1955. Google Scholar

[17]

F. J. S. A. Corrêa, On positive solutions of nonlocal and nonvariational elliptic problems, Nonlinear Anal., 59 (2004), 1147-1155.  doi: 10.1016/S0362-546X(04)00322-0.  Google Scholar

[18]

F. J. S. A. Corrêa, S. B. de Menezes and J. Ferreira, On a class of problems involving a nonlocal operator, Appl. Math. Comput., 147 (2004), 475-489.  doi: 10.1016/S0096-3003(02)00740-3.  Google Scholar

[19]

R. Dautray and J. L. Lions, Analyse Mathématique et Calcul Numérique pour les Sciences et les Techniques Masson, Paris, 1987. Google Scholar

[20]

J. Furter and M. Grinfeld, Local vs. nonlocal interactions in population dynamics, J. Math. Biol., 27 (1989), 65-80.  doi: 10.1007/BF00276081.  Google Scholar

[21]

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

[22]

D. Hilhorst and J. F. Rodrigues, On a nonlocal diffusion equation with discontinuous reaction, Adv. Differential Equations, 5 (2000), 657-680.   Google Scholar

[23]

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

[24]

A. V. Kapustyan and J. Valero, On the connectedness and asymptotic behaviour of solutions of reaction-diffusion systems, J. Math. Anal. Appl., 323 (2006), 614-633.  doi: 10.1016/j.jmaa.2005.10.042.  Google Scholar

[25]

A. V. Kapustyan and J. Valero, Weak and strong attractors fo the 3D Navier-Stokes system, J. Differential Equations, 240 (2007), 249-278.  doi: 10.1016/j.jde.2007.06.008.  Google Scholar

[26]

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

[27]

B. Lovat, Études de Quelques Problémes Paraboliques Non Locaux Thése, Université de Metz, 1995. Google Scholar

[28]

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

[29]

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.  doi: 10.3934/dcds.2010.26.989.  Google Scholar

[30]

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

[31]

S. B. de Menezes, Remarks on weak solutions for a nonlocal parabolic problem, Int. J. Math. Math. Sci., 2006 (2006), 1-10.  doi: 10.1155/IJMMS/2006/82654.  Google Scholar

[32] J. C. Robinson, Infinite-Dimensional Dynamical Systems, Cambridge University Press, Cambridge, 2001.   Google Scholar
[33] M. Růžička, Electrorheological Fluids: Modeling and Mathematical Theory, Springer-Verlag, Berlin, 2000.   Google Scholar
[34]

T. Savitska, Asymptotic Behaviour of Solutions of Nonlocal Parabolic Problems Ph. D Thesis, University of Zurich, 2015. Google Scholar

[35]

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

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