Abstract
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.
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References
Mann, W. R. and Wolf., F., ‘Heat transfer between solids and gases under nonlinear boundary conditions,’ Quart. J. Appl. Math., 9 (1951), 163–184.
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.
] 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.
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.
Aronson, D. G., ‘A comparison method for stability analysis of nonlinear parabolic problems,’ SIAM review, 20 (1978), 245–264.
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.
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.
Okubo, A., Diffusion and Ecological Problems: Mathematical Models, Springer-Verlag, Berlin, 1980.
Levin, S., Epidemics and Population Problems, edited by S. Busenberg and K. Cooke, Academic Press, 1981.
Leung, A., ‘Nonlinear density-dependent diffusion for competing species interaction: large-time asymptotic behavior,’ Proc. Edinburg Math. Soc., 27 (1984), 131–144.
Fife, P, C., ‘Boundary and interior transition layer phenomena for pairs of second-order differential equations,’ J. Math. Anal. Appl., 54 (1976), 597–521.
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.
Howes, F. A., ‘Boundary layer behavior in perturbed second-order systems,’ J. Math. Anal. Appl., 104 (1984), 465–476.
Wasow, W., ‘The capriciousness of singular perturbations,’ Nieuw Arch. Wisk. 18 (1970), 190–210.
Leung, A., ‘Reaction-diffusion equations for competing populations, singularly perturbed by a small diffusion rate,’ Rocky Mountain J. of Math., 13 (1983), 177–190.
de Mottoni, P., Schiaffino, A. and Tesei, A., ‘On stable space-dependent stationary solutions of a competition system with diffusion,’ private communications, (1984).
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© 1989 Springer Science+Business Media Dordrecht
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Leung, A.W. (1989). Other Boundary Conditions, Nonlinear Diffusion, Asymptotics. In: Systems of Nonlinear Partial Differential Equations. Mathematics and Its Applications, vol 49. Springer, Dordrecht. https://doi.org/10.1007/978-94-015-3937-1_3
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DOI: https://doi.org/10.1007/978-94-015-3937-1_3
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