Other Boundary Conditions, Nonlinear Diffusion, Asymptotics

  • Anthony W. Leung
Part of the Mathematics and Its Applications book series (MAIA, volume 49)


In this Chapter, we consider various extensions of the theories in the last chapter in order to include more realistic and general problems. In section 3.2, we study the situation where the boundary condition is coupled, mixed and nonlinear. In prey-predator interaction for example, the predator may be under control at the boundary of the medium, while the prey cannot move across the boundary. Diffusion of predators at the boundary may be adjusted nonlinearly according to populations present, and there might also be some physical limitations to the process. In section 3.3, we analyze the problem when the diffusion rate is density dependent, and thus the Laplacian operator will be modified to become nonlinear and u-dependent. Moreover, the nonlinear nonhomogeneous terms become highly spatially dependent. In section 3.4, we consider the case when diffusion rate of some component is small. More thorough results concerning large-time behavior can be obtained by asympptotic methods. Estimates can be obtained by using an appropriate “reduced”problem.


Initial Boundary Nonlinear Diffusion Principal Eigenvalue Nonlinear Boundary Condition Open Quadrant 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. [153]
    Mann, W. R. and Wolf., F., ‘Heat transfer between solids and gases under nonlinear boundary conditions,’ Quart. J. Appl. Math., 9 (1951), 163–184.zbMATHGoogle Scholar
  2. [216]
    Thames, H. D., Jr. and Elster, A., ‘Equilibrium states and oscillations for localized two enzyme kinetics: model for circadian rhythms,’ J. Theor. Biol., 59 (1976), 415–427.MathSciNetCrossRefGoogle Scholar
  3. [15]
    ] Aronson, D. G. and Peletier, L. A., ‘Global stability of symmetric and asymmetric concentration profiles in catalyst particles,’ Arch. Rational Mech. Math., 54 (1974), 175–204.MathSciNetzbMATHGoogle Scholar
  4. [221]
    Turner, V. L. and Ames, W., F., ‘Two sided bounds for linked unknown nonlinear boundary conditions of reaction-diffusion,’ J. Math. Anal. Appl., 71 (1979), 366–378.MathSciNetzbMATHCrossRefGoogle Scholar
  5. [14]
    Aronson, D. G., ‘A comparison method for stability analysis of nonlinear parabolic problems,’ SIAM review, 20 (1978), 245–264.MathSciNetzbMATHCrossRefGoogle Scholar
  6. [138]
    Leung, A., ‘A semilinear reaction-diffusion prey-predator system with nonlinear coupled boundary conditions: equilibrium and stability,’ Indiana University Math J., 31 (1982), 223–241.zbMATHCrossRefGoogle Scholar
  7. [160]
    Mimura, M., Nishiura, Y. and Yamaguti, M.,’Some diffusive prey and predator systems and their bifurcation problems,’ Annals of the N. Y. Academy of Sciences, 316 (1979), 490–510.MathSciNetCrossRefGoogle Scholar
  8. [173]
    Okubo, A., Diffusion and Ecological Problems: Mathematical Models, Springer-Verlag, Berlin, 1980.zbMATHGoogle Scholar
  9. [150]
    Levin, S., Epidemics and Population Problems, edited by S. Busenberg and K. Cooke, Academic Press, 1981.Google Scholar
  10. [141]
    Leung, A., ‘Nonlinear density-dependent diffusion for competing species interaction: large-time asymptotic behavior,’ Proc. Edinburg Math. Soc., 27 (1984), 131–144.zbMATHGoogle Scholar
  11. [73]
    Fife, P, C., ‘Boundary and interior transition layer phenomena for pairs of second-order differential equations,’ J. Math. Anal. Appl., 54 (1976), 597–521.MathSciNetCrossRefGoogle Scholar
  12. [110]
    Howes, F. A., ‘Some old and new results on singularly perturbed boundary value problems,’ Singular Perturbations and Asymptotics, Edited by R. E. Meyer and S. V. Parter, Academic Press, N. Y., 1980.Google Scholar
  13. [111]
    Howes, F. A., ‘Boundary layer behavior in perturbed second-order systems,’ J. Math. Anal. Appl., 104 (1984), 465–476.MathSciNetCrossRefGoogle Scholar
  14. [225]
    Wasow, W., ‘The capriciousness of singular perturbations,’ Nieuw Arch. Wisk. 18 (1970), 190–210.MathSciNetzbMATHGoogle Scholar
  15. [139]
    Leung, A., ‘Reaction-diffusion equations for competing populations, singularly perturbed by a small diffusion rate,’ Rocky Mountain J. of Math., 13 (1983), 177–190.zbMATHCrossRefGoogle Scholar
  16. [64]
    de Mottoni, P., Schiaffino, A. and Tesei, A., ‘On stable space-dependent stationary solutions of a competition system with diffusion,’ private communications, (1984).Google Scholar

Copyright information

© Springer Science+Business Media Dordrecht 1989

Authors and Affiliations

  • Anthony W. Leung
    • 1
  1. 1.Department of Mathematical SciencesUniversity of CincinnatiCincinnatiUSA

Personalised recommendations