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Asymptotic behavior of sign-changing radial solutions of a semilinear elliptic equation in $ \mathbb{R}^2 $ when exponent approaches $ +\infty $
On a discrete self-organized-criticality finite time result
| 1. | Worcester Polytechnic Institute, MA, USA |
| 2. | Department of Basic and Applied Sciences for Engineering, Sapienza University of Rome, Italy |
In this paper we deal with theoretical and numerical aspects of some nonlinear problems related to sandpile models. We introduce a purely discrete model for infinitely many particles interacting according to a toppling process on a uniform two-dimensional grid and prove the convergence of the solutions to a differential initial value problem.
References:
| [1] |
G. Aronsson, L. C. Evans and Y. Wu,
Fast/slow diffusion and growing sandpiles, J. Differential Equations, 131 (1996), 304-335.
doi: 10.1006/jdeq.1996.0166. |
| [2] |
P. Bak, C. Tang and K. Wiesenfeld,
Self-organized criticality, Phys. Rev. A (3), 38 (1988), 364-374.
doi: 10.1103/PhysRevA.38.364. |
| [3] |
P. Bántay and I. M. Jánosi,
Avalanche dynamics from anomalous diffusion, Phys. Rev. Lett., 68 (1992), 2058-2061.
|
| [4] |
V. Barbu, Nonlinear Semi-Groups and Differential Equations in Banach Spaces, Noordhoff: Leyden, 1976. |
| [5] |
V. Barbu, Nonlinear Differential Equations of Monotone Type in Banach Spaces, Springer: NewYork, 2010.
doi: 10.1007/978-1-4419-5542-5. |
| [6] |
V. Barbu,
Self-organized criticality and convergence to equilibrium of solutions to nonlinear diffusion problems, Annual Reviews in Control, JARAP, 340 (2010), 52-61.
|
| [7] |
V. Barbu,
Self-organized criticality of cellular automata model; absorbtion in finite-time of supercritical region into the critical one, Mathematical Methods in the Applied Sciences, 36 (2013), 1726-1733.
doi: 10.1002/mma.2718. |
| [8] |
V. Barbu, G. Da Prato and M. Röckner,
Stochastic porous media equations and self- organized criticality, Comm. Math. Phys., 285 (2009), 901-923.
doi: 10.1007/s00220-008-0651-x. |
| [9] |
J. W. Barrett and L. Prigozhin,
Sandpiles and superconductors: Nonconforming linear finite element approximations for mixed formulations of quasi-variational, IMA J. Numer. Anal., 35 (2015), 1-38.
doi: 10.1093/imanum/drt062. |
| [10] |
H. Brézis, Operateurs Maximaux Monotones et Semi-groupes de Contractions dans les Espaces de Hilbert, Ed. L.Nachbin North-Hollands/ Americal Elsevier, 1973. |
| [11] |
H. Brézis, Analisi funzionale, Ed. Liguori 1986. |
| [12] |
H. Brézis and A. Pazy,
Accretive sets and differential equations in Banach spaces, Israel J. Math., 8 (1970), 367-383.
doi: 10.1007/BF02798683. |
| [13] |
J. M. Carlson, J. T. Chayes, E. R. Grannan and G. H. Swindle,
Self-organized criticality in sandpiles: Nature of the critical phenomenon, Phys. Rev. A (3), 42 (1990), 2467-2470.
doi: 10.1103/PhysRevA.42.2467. |
| [14] |
J. M. Carlson and G. H. Swindle,
Self-organized criticality: Sandpiles, singularities, and scaling, Proc. Nat. Acad. Sci. USA, 92 (1995), 6712-6719.
doi: 10.1073/pnas.92.15.6712. |
| [15] |
M. G. Crandall and T. M. Liggett,
Generation of semi-groups of nonlinear transformations on general Banach spaces, Amer. J. Math., 93 (1971), 265-298.
doi: 10.2307/2373376. |
| [16] |
M. Creutz,
Abelian sandpiles, Nucl. Phys. B (Proc. Suppl.), 20 (1991), 758-761.
|
| [17] |
D. Dahr,
Self-organized critical state of sandpile automaton models, Phys. Rev. Lett., 64 (1990), 1613-1616.
doi: 10.1103/PhysRevLett.64.1613. |
| [18] |
L. C. Evans, M. Feldman and R. F. Gariepy,
Fast/slow diffusion and collapsing sandpiles, J. Differential Equations, 137 (1997), 166-209.
doi: 10.1006/jdeq.1997.3243. |
| [19] |
B. Gess,
Finite time extinction for stochastic sign fast diffusion and self-organized criticality, Comm. Math. Phys., 335 (2015), 309-344.
doi: 10.1007/s00220-014-2225-4. |
| [20] |
U. Mosco,
Convergence of convex sets and of solutions of variational inequalities, Adv. in Math., 3 (1969), 510-585.
doi: 10.1016/0001-8708(69)90009-7. |
| [21] |
U. Mosco, An Introduction to the approximate solution of variational inequalities, In: Constructive Aspects of Functional Analysis. Cime Summer Schools Erice, (ed. G. Geymonat), Springer, 57 (1971), 497–685. |
| [22] |
U. Mosco,
Finite -time self - organized - criticality on synchronized infinite grids, SIAM J. Math. Anal., 50 (2018), 2409-2440.
doi: 10.1137/17M1122955. |
| [23] |
U. Mosco and M. A. Vivaldi, On the external approximation of Sobolev spaces by M–convergence, to appear. |
show all references
References:
| [1] |
G. Aronsson, L. C. Evans and Y. Wu,
Fast/slow diffusion and growing sandpiles, J. Differential Equations, 131 (1996), 304-335.
doi: 10.1006/jdeq.1996.0166. |
| [2] |
P. Bak, C. Tang and K. Wiesenfeld,
Self-organized criticality, Phys. Rev. A (3), 38 (1988), 364-374.
doi: 10.1103/PhysRevA.38.364. |
| [3] |
P. Bántay and I. M. Jánosi,
Avalanche dynamics from anomalous diffusion, Phys. Rev. Lett., 68 (1992), 2058-2061.
|
| [4] |
V. Barbu, Nonlinear Semi-Groups and Differential Equations in Banach Spaces, Noordhoff: Leyden, 1976. |
| [5] |
V. Barbu, Nonlinear Differential Equations of Monotone Type in Banach Spaces, Springer: NewYork, 2010.
doi: 10.1007/978-1-4419-5542-5. |
| [6] |
V. Barbu,
Self-organized criticality and convergence to equilibrium of solutions to nonlinear diffusion problems, Annual Reviews in Control, JARAP, 340 (2010), 52-61.
|
| [7] |
V. Barbu,
Self-organized criticality of cellular automata model; absorbtion in finite-time of supercritical region into the critical one, Mathematical Methods in the Applied Sciences, 36 (2013), 1726-1733.
doi: 10.1002/mma.2718. |
| [8] |
V. Barbu, G. Da Prato and M. Röckner,
Stochastic porous media equations and self- organized criticality, Comm. Math. Phys., 285 (2009), 901-923.
doi: 10.1007/s00220-008-0651-x. |
| [9] |
J. W. Barrett and L. Prigozhin,
Sandpiles and superconductors: Nonconforming linear finite element approximations for mixed formulations of quasi-variational, IMA J. Numer. Anal., 35 (2015), 1-38.
doi: 10.1093/imanum/drt062. |
| [10] |
H. Brézis, Operateurs Maximaux Monotones et Semi-groupes de Contractions dans les Espaces de Hilbert, Ed. L.Nachbin North-Hollands/ Americal Elsevier, 1973. |
| [11] |
H. Brézis, Analisi funzionale, Ed. Liguori 1986. |
| [12] |
H. Brézis and A. Pazy,
Accretive sets and differential equations in Banach spaces, Israel J. Math., 8 (1970), 367-383.
doi: 10.1007/BF02798683. |
| [13] |
J. M. Carlson, J. T. Chayes, E. R. Grannan and G. H. Swindle,
Self-organized criticality in sandpiles: Nature of the critical phenomenon, Phys. Rev. A (3), 42 (1990), 2467-2470.
doi: 10.1103/PhysRevA.42.2467. |
| [14] |
J. M. Carlson and G. H. Swindle,
Self-organized criticality: Sandpiles, singularities, and scaling, Proc. Nat. Acad. Sci. USA, 92 (1995), 6712-6719.
doi: 10.1073/pnas.92.15.6712. |
| [15] |
M. G. Crandall and T. M. Liggett,
Generation of semi-groups of nonlinear transformations on general Banach spaces, Amer. J. Math., 93 (1971), 265-298.
doi: 10.2307/2373376. |
| [16] |
M. Creutz,
Abelian sandpiles, Nucl. Phys. B (Proc. Suppl.), 20 (1991), 758-761.
|
| [17] |
D. Dahr,
Self-organized critical state of sandpile automaton models, Phys. Rev. Lett., 64 (1990), 1613-1616.
doi: 10.1103/PhysRevLett.64.1613. |
| [18] |
L. C. Evans, M. Feldman and R. F. Gariepy,
Fast/slow diffusion and collapsing sandpiles, J. Differential Equations, 137 (1997), 166-209.
doi: 10.1006/jdeq.1997.3243. |
| [19] |
B. Gess,
Finite time extinction for stochastic sign fast diffusion and self-organized criticality, Comm. Math. Phys., 335 (2015), 309-344.
doi: 10.1007/s00220-014-2225-4. |
| [20] |
U. Mosco,
Convergence of convex sets and of solutions of variational inequalities, Adv. in Math., 3 (1969), 510-585.
doi: 10.1016/0001-8708(69)90009-7. |
| [21] |
U. Mosco, An Introduction to the approximate solution of variational inequalities, In: Constructive Aspects of Functional Analysis. Cime Summer Schools Erice, (ed. G. Geymonat), Springer, 57 (1971), 497–685. |
| [22] |
U. Mosco,
Finite -time self - organized - criticality on synchronized infinite grids, SIAM J. Math. Anal., 50 (2018), 2409-2440.
doi: 10.1137/17M1122955. |
| [23] |
U. Mosco and M. A. Vivaldi, On the external approximation of Sobolev spaces by M–convergence, to appear. |
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