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Global attractor for a nonlocal p-Laplacian equation without uniqueness of solution
Departamento de Ecuaciones Diferenciales y Análisis Numérico, Universidad de Sevilla, Apdo. de Correos 1160,41080-Sevilla, Spain |
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.
References:
[1] |
A. Andami Ovono,
Asymptotic behaviour for a diffusion equation governed by nonlocal interactions, Electron. J. Differential Equations, 134 (2010), 1-16.
|
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[8] |
M. Chipot and B. Lovat,
On the asymptotic behaviour of some nonlocal problems, Positivity, 3 (1999), 65-81.
doi: 10.1023/A:1009706118910. |
[9] |
M. Chipot and L. Molinet,
Asymptotic behaviour of some nonlocal diffusion problems, Appl. Anal., 80 (2001), 279-315.
doi: 10.1080/00036810108840994. |
[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. |
[11] |
M. Chipot and P. Roy,
Existence results for some functional elliptic equations, Differential Integral Equations, 27 (2014), 289-300.
|
[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.
|
[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. |
[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.
|
[15] |
M. Chipot and S. Zheng,
Asymptotic behavior of solutions to nonlinear parabolic equations with nonlocal terms, Asymptot. Anal., 45 (2005), 301-312.
|
[16] |
E. A. Coddington and N. Levinson, Theory of Ordinary Differential Equations, McGraw-Hill Book Company, Inc. , New York-Toronto-London, 1955. |
[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. |
[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. |
[19] |
R. Dautray and J. L. Lions, Analyse Mathématique et Calcul Numérique pour les Sciences et les Techniques Masson, Paris, 1987. |
[20] |
J. Furter and M. Grinfeld,
Local vs. nonlocal interactions in population dynamics, J. Math. Biol., 27 (1989), 65-80.
doi: 10.1007/BF00276081. |
[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. |
[22] |
D. Hilhorst and J. F. Rodrigues,
On a nonlocal diffusion equation with discontinuous reaction, Adv. Differential Equations, 5 (2000), 657-680.
|
[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. |
[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. |
[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. |
[26] |
J. L. Lions, Quelques Méthodes de Résolution des Problémes aux Limites Non Lineaires, Dunod, Paris, 1969. |
[27] |
B. Lovat, Études de Quelques Problémes Paraboliques Non Locaux Thése, Université de Metz, 1995. |
[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. |
[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. |
[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. |
[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. |
[32] |
J. C. Robinson, Infinite-Dimensional Dynamical Systems, Cambridge University Press, Cambridge, 2001.
![]() |
[33] |
M. Růžička, Electrorheological Fluids: Modeling and Mathematical Theory, Springer-Verlag, Berlin, 2000.
![]() |
[34] |
T. Savitska, Asymptotic Behaviour of Solutions of Nonlocal Parabolic Problems Ph. D Thesis, University of Zurich, 2015. |
[35] |
R. Temam, Infinite-Dimensional Dynamical Systems in Mechanics and Physics, 2nd. ed. , Springer, New-York, 1997. |
show all references
To the memory of Pablo Cobos, Marta's grandfather, with sorrow and love
References:
[1] |
A. Andami Ovono,
Asymptotic behaviour for a diffusion equation governed by nonlocal interactions, Electron. J. Differential Equations, 134 (2010), 1-16.
|
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[8] |
M. Chipot and B. Lovat,
On the asymptotic behaviour of some nonlocal problems, Positivity, 3 (1999), 65-81.
doi: 10.1023/A:1009706118910. |
[9] |
M. Chipot and L. Molinet,
Asymptotic behaviour of some nonlocal diffusion problems, Appl. Anal., 80 (2001), 279-315.
doi: 10.1080/00036810108840994. |
[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. |
[11] |
M. Chipot and P. Roy,
Existence results for some functional elliptic equations, Differential Integral Equations, 27 (2014), 289-300.
|
[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.
|
[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. |
[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.
|
[15] |
M. Chipot and S. Zheng,
Asymptotic behavior of solutions to nonlinear parabolic equations with nonlocal terms, Asymptot. Anal., 45 (2005), 301-312.
|
[16] |
E. A. Coddington and N. Levinson, Theory of Ordinary Differential Equations, McGraw-Hill Book Company, Inc. , New York-Toronto-London, 1955. |
[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. |
[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. |
[19] |
R. Dautray and J. L. Lions, Analyse Mathématique et Calcul Numérique pour les Sciences et les Techniques Masson, Paris, 1987. |
[20] |
J. Furter and M. Grinfeld,
Local vs. nonlocal interactions in population dynamics, J. Math. Biol., 27 (1989), 65-80.
doi: 10.1007/BF00276081. |
[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. |
[22] |
D. Hilhorst and J. F. Rodrigues,
On a nonlocal diffusion equation with discontinuous reaction, Adv. Differential Equations, 5 (2000), 657-680.
|
[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. |
[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. |
[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. |
[26] |
J. L. Lions, Quelques Méthodes de Résolution des Problémes aux Limites Non Lineaires, Dunod, Paris, 1969. |
[27] |
B. Lovat, Études de Quelques Problémes Paraboliques Non Locaux Thése, Université de Metz, 1995. |
[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. |
[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. |
[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. |
[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. |
[32] |
J. C. Robinson, Infinite-Dimensional Dynamical Systems, Cambridge University Press, Cambridge, 2001.
![]() |
[33] |
M. Růžička, Electrorheological Fluids: Modeling and Mathematical Theory, Springer-Verlag, Berlin, 2000.
![]() |
[34] |
T. Savitska, Asymptotic Behaviour of Solutions of Nonlocal Parabolic Problems Ph. D Thesis, University of Zurich, 2015. |
[35] |
R. Temam, Infinite-Dimensional Dynamical Systems in Mechanics and Physics, 2nd. ed. , Springer, New-York, 1997. |
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