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ArticleOriginal scientific text

Title

Existence and nonexistence of solutions for a model of gravitational interaction of particles, I

Authors 1, 1

Affiliations

  1. Mathematical Institute, University of Wrocław, pl. Grunwaldzki 2/4, 50-384 WrocŁaw, Poland

Abstract

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.

Keywords

ENG
nonlinear boundary conditions, stationary solutions, global existence of solutions, parabolic-elliptic system

Bibliography

  1. R. A. Adams, Sobolev Spaces, Academic Press, New York, 1975.
  2. F. Bavaud, Equilibrium properties of the Vlasov functional: the generalized Poisson-Boltzmann-Emden equation, Rev. Modern Phys. 63 (1991), 129-149.
  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.
  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.
  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 .
  6. P. Biler, D. Hilhorst and T. Nadzieja, Existence and nonexistence of solutions for a model of gravitational interaction of particles, II, preprint, 1993.
  7. P. Biler and T. Nadzieja, A class of nonlocal parabolic problems occurring in statistical mechanics, Colloq. Math. 66 (1993), 131-145.
  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.
  9. P. Cherrier, Meilleures constantes dans des inégalités relatives aux espaces de Sobolev, Bull. Sci. Math. 108 (1984), 225-262.
  10. R. Dautray and J.-L. Lions, Mathematical Analysis and Numerical Methods for Science and Technology, Vol. 1, Springer, Berlin, 1990.
  11. D. Gilbarg and N. S. Trudinger, Elliptic Partial Differential Equations of Second Order, 2nd ed., Springer, Berlin, 1983.
  12. A. Krzywicki and T. Nadzieja, Some results concerning the Poisson-Boltzmann equation, Zastos. Mat. 21 (1991), 265-272.
  13. A. Krzywicki and T. Nadzieja, A nonstationary problem in the theory of electrolytes, Quart. Appl. Math. 50 (1992), 105-107.
  14. A. Krzywicki and T. Nadzieja, A note on the Poisson-Boltzmann equation, Zastos. Mat. 21 (1993), 591-595.
  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.
  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.
  17. S. Wang, Some nonlinear elliptic equations with subcritical growth and critical behavior, Houston J. Math. 16 (1990), 559-572.
  18. G. Wolansky, On steady distributions of self-attracting clusters under friction and fluctuations, Arch. Rational Mech. Anal. 119 (1992), 355-391.
  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.
Colloquium Mathematicum
1993/66/2
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Pages:
319-334
Main language of publication
English
Received
1993-07-06
Published
1993
Exact and natural sciences