Journal of the Korean Mathematical Society (대한수학회지)

Korean Mathematical Society (대한수학회)
권호 표지
DOI QR코드

DOI QR Code

A SUFFICIENT CONDITION FOR THE UNIQUENESS OF POSITIVE STEADY STATE TO A REACTION DIFFUSION SYSTEM

  • Published : 2002.05.01
Download PDF
⟨ Previous Next ⟩

Abstract

In this paper, we concentrate on the uniquencess of the positive solution for the general elliptic system $\Delta$u+u($g_1$(u)-$g_2$(v))=0 $\Delta$u+u($h_1$(u)-$h_2$(v))=0 in$R_{+}$ $\times$ $\Omega$, $u\mid\partial\Omega = u\mid\partial\Omega = 0$. This system is the general model for the steady state of a competitive interacting system. The techniques used in this paper are upper-lower solutions, maximum principles and spectrum estimates. The arguments also rely on some detailed properties for the solution of logistic equations.

Keywords

References

  1. Houston J. Math. v.13 On the steady-state problem for the Volterra - Lotka competition model with diffusion R. S. Cantrell;C. Cosner
  2. Houston J. Math. v.15 On the uniqueness and stability of positive solutions in the Lotka- Volterra competition model with diffusion
  3. SIAM J. Appl. Math. v.44 Stable coexistence states in the Volterra-Lotka competition model with diffusion C. Cosner;A, C. Lazer https://doi.org/10.1137/0144080
  4. Methods of mathematical physics v.Ⅱ Partial differential equations R. Courant;D. Hilbert
  5. Lecture note for applied analysis in Michigan State University D. Dunninger
  6. C. R. Acad. Sci. Paris, Ser. I Math. v.313 Existence and uniqueness for some competition models with diffusion J. L.-Gomez;R. Pardo
  7. Comm. Pure Appl. Math. v.47 no.12 Uniqueness and nonuniqueness of coexistence states in the Lotka-Volterra competition model C. Gui;Y. Lou https://doi.org/10.1002/cpa.3160471203
  8. Math. Z. v.155 On uniqueness of positive solutions of nonlinear elliptic boundary value problems P. Hess
  9. J. Math. Anal. Appl. v.73 Equilibria and stabilities for competing-species, reaction-diffusion equations with Dirichlet boundary data A. Leung https://doi.org/10.1016/0022-247X(80)90028-1
  10. Differential Integral Equations v.4 Positive solutions to general elliptic competition models L. Li;R. Logan
  11. Maximum principles in differential equations M. H. Protter;H. F. Weinberger
  12. Lecture Notes in Mathematics v.322 Nonlinear problems in nuclear reactor analysis, in nonlinear problems in the physical sciences and biology I. Stakgold;L. E. Payne https://doi.org/10.1007/BFb0060573