Journal of the Korean Mathematical Society (대한수학회지)

Korean Mathematical Society (대한수학회)
권호 표지
DOI QR코드

DOI QR Code

A SIMPLE CHARACTERIZATION OF POSITIVITY PRESERVING SEMI-LINEAR PARABOLIC SYSTEMS

  • Received : 2016.11.01
  • Accepted : 2017.03.30
  • Published : 2017.11.01
Download PDF
⟨ Previous Next ⟩

Abstract

We give a simple and direct proof of the characterization of positivity preserving semi-flows for ordinary differential systems. The same method provides an abstract result on a class of evolution systems containing reaction-diffusion systems in a bounded domain of ${\mathbb{R}}^n$ with either Neumann or Dirichlet homogeneous boundary conditions. The conditions are exactly the same with or without diffusion. A similar approach gives the optimal result for invariant rectangles in the case of Neumann conditions.

Keywords

References

  1. N. Alikakos, An application of the invariance principle to reaction-diffusion equations, J. Differential Equations 33 (1979), no. 2, 201-225. https://doi.org/10.1016/0022-0396(79)90088-3
  2. A. Barabanova, On the global existence of solutions of a reaction-diffusion equation with exponential nonlinearity, Proc. Amer. Math. Soc. 122 (1994), no. 3, 827-831. https://doi.org/10.1090/S0002-9939-1994-1207533-6
  3. G. Bourceanu and G. Morosanu, The study of the evolution of some self-organized chemical systems, J. Chemical Phys. 82 (1985), 3685-3691. https://doi.org/10.1063/1.448904
  4. H. Brezis, On a characterization of flow-invariant sets, Comm. Pure Appl. Math. 23 (1970), 261-263. https://doi.org/10.1002/cpa.3160230211
  5. T. Cazenave and A. Haraux, An Introduction to Semilinear Evolution Equations, Oxford Lecture Series in Mathematics and its Applications, 13. The Clarendon Press, Oxford University Press, New York, 1998.
  6. A. Haraux and M. Kirane, Estimations $C^1$ pour des problemes paraboliques semi-lin'eaires, (French. English summary) [C1 estimates for semilinear parabolic problems] Ann. Fac. Sci. Toulouse Math. 5 (1983), no. 3-4, 265-280. https://doi.org/10.5802/afst.598
  7. A. Haraux and A. Youkana, On a result of K. Masuda concerning reaction-diffusion equations, Tohoku Math. J. 2 40 (1988), no. 1, 159-163. https://doi.org/10.2748/tmj/1178228084
  8. S. Hollis, R. H. Martin, Jr., and M. Pierre, Global existence and boundedness in reaction-diffusion systems, SIAM J. Math. Anal. 18 (1987), no. 3, 744-761. https://doi.org/10.1137/0518057
  9. R. H. Martin, Jr., Differential equations on closed subsets of a Banach space, Trans. Amer. Math. Soc. 179 (1973), 399-414. https://doi.org/10.1090/S0002-9947-1973-0318991-4
  10. K. Masuda, On the global existence and asymptotic behavior of solutions of reaction-diffusion equations, Hokkaido Math. J. 12 (1983), no. 3, 360-370. https://doi.org/10.14492/hokmj/1470081012
  11. N. Pavel, Invariant sets for a class of semi-linear equations of evolution, Nonlinear Anal. 1 (1976/77), no. 2, 187-196. https://doi.org/10.1016/0362-546X(77)90009-8