We study the existence of stationary and evolution solutions to a parabolic-elliptic system with natural (no-flux) boundary conditions describing the gravitational interaction of particles.
@article{bwmeta1.element.bwnjournal-article-cmv66i2p319bwm,
author = {Piotr Biler and Tadeusz Nadzieja},
title = {Existence and nonexistence of solutions for a model of gravitational interaction of particles, I},
journal = {Colloquium Mathematicae},
volume = {66},
year = {1993},
pages = {319-334},
zbl = {0817.35041},
language = {en},
url = {http://dml.mathdoc.fr/item/bwmeta1.element.bwnjournal-article-cmv66i2p319bwm}
}
Biler, Piotr; Nadzieja, Tadeusz. Existence and nonexistence of solutions for a model of gravitational interaction of particles, I. Colloquium Mathematicae, Tome 66 (1993) pp. 319-334. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-cmv66i2p319bwm/
[000] [1] R. A. Adams, Sobolev Spaces, Academic Press, New York, 1975.
[001] [2] F. Bavaud, Equilibrium properties of the Vlasov functional: the generalized Poisson-Boltzmann-Emden equation, Rev. Modern Phys. 63 (1991), 129-149.
[002] [3] M.-F. Bidaut-Véron and L. Véron, Nonlinear elliptic equations on compact Riemannian manifolds and asymptotics of Emden equations, Invent. Math. 106 (1991), 489-539.
[003] [4] P. Biler, Existence and asymptotics of solutions for a parabolic-elliptic system with nonlinear no-flux boundary conditions, Nonlinear Anal. 19 (1992), 1121-1136. | Zbl 0781.35025
[004] [5] P. Biler, W. Hebisch and T. Nadzieja, The Debye system: existence and large time behavior of solutions, Mathematical Institute, University of Wrocław, Report no. 23 (1992), 1-24 . | Zbl 0814.35054
[005] [6] P. Biler, D. Hilhorst and T. Nadzieja, Existence and nonexistence of solutions for a model of gravitational interaction of particles, II, preprint, 1993. | Zbl 0832.35015
[006] [7] P. Biler and T. Nadzieja, A class of nonlocal parabolic problems occurring in statistical mechanics, Colloq. Math. 66 (1993), 131-145. | Zbl 0818.35046
[007] [8] E. Caglioti, P.-L. Lions, C. Marchioro and M. Pulvirenti, A special class of stationary flows for two-dimensional Euler equations: a statistical mechanics description, Comm. Math. Phys. 143 (1992), 501-525. | Zbl 0745.76001
[008] [9] P. Cherrier, Meilleures constantes dans des inégalités relatives aux espaces de Sobolev, Bull. Sci. Math. 108 (1984), 225-262. | Zbl 0547.58017
[009] [10] R. Dautray and J.-L. Lions, Mathematical Analysis and Numerical Methods for Science and Technology, Vol. 1, Springer, Berlin, 1990. | Zbl 0683.35001
[010] [11] D. Gilbarg and N. S. Trudinger, Elliptic Partial Differential Equations of Second Order, 2nd ed., Springer, Berlin, 1983. | Zbl 0562.35001
[011] [12] A. Krzywicki and T. Nadzieja, Some results concerning the Poisson-Boltzmann equation, Zastos. Mat. 21 (1991), 265-272. | Zbl 0756.35029
[012] [13] A. Krzywicki and T. Nadzieja, A nonstationary problem in the theory of electrolytes, Quart. Appl. Math. 50 (1992), 105-107. | Zbl 0754.35142
[013] [14] A. Krzywicki and T. Nadzieja, A note on the Poisson-Boltzmann equation, Zastos. Mat. 21 (1993), 591-595. | Zbl 0780.35033
[014] [15] O. A. Ladyženskaja, V. A. Solonnikov and N. N. Ural'ceva, Linear and Quasilinear Equations of Parabolic Type, Amer. Math. Soc., Providence, R.I., 1988.
[015] [16] T. Suzuki, Global analysis for a two-dimensional elliptic eigenvalue problem with the exponential nonlinearity, Ann. Inst. H. Poincaré Anal. Non Linéaire 9 (1992), 367-398. | Zbl 0785.35045
[016] [17] S. Wang, Some nonlinear elliptic equations with subcritical growth and critical behavior, Houston J. Math. 16 (1990), 559-572. | Zbl 0741.35016
[017] [18] G. Wolansky, On steady distributions of self-attracting clusters under friction and fluctuations, Arch. Rational Mech. Anal. 119 (1992), 355-391. | Zbl 0774.76069
[018] [19] G. Wolansky, On the evolution of self-interacting clusters and applications to semilinear equations with exponential nonlinearity, J. Analyse Math. 59 (1992), 251-272. | Zbl 0806.35134