×

zbMATH — the first resource for mathematics

Diffusion vs cross-diffusion: An elliptic approach. (English) Zbl 0934.35040
Abstract: A boundary value problem for an elliptic system is used to model the segregation of two interacting species. The main purpose is to study the limiting profiles of non-constant positive solutions when one of the cross-diffusion pressures is sufficiently large.

MSC:
35J55 Systems of elliptic equations, boundary value problems (MSC2000)
92D25 Population dynamics (general)
35B05 Oscillation, zeros of solutions, mean value theorems, etc. in context of PDEs
PDF BibTeX XML Cite
Full Text: DOI
References:
[1] Gilbarg, D.; Trudinger, N., Elliptic partial differential equation of second order, (1983), Springer-Verlag Berlin/Heidelberg/New York/Tokyo
[2] Kan, Y., Stability of singularly perturbed solutions to nonlinear diffusion system arising in population dynamics, Hiroshima math. J., 23, 509-536, (1993) · Zbl 0792.35086
[3] Lin, C.S.; Ni, W.M.; Takagi, I., Large amplitude stationary solutions to a chemotais systems, J. differential equations, 72, 1-27, (1988) · Zbl 0676.35030
[4] Lou, Y.; Ni, W.M., Diffusion, self-diffusion and cross-diffusion, J. differential equations, 131, 79-131, (1996) · Zbl 0867.35032
[5] Matano, H.; Mimura, M., Pattern formation in competition-diffusion systems in non-conve domains, Publ. res. inst. math. sci. Kyoto univ., 19, 1049-1079, (1983) · Zbl 0548.35063
[6] Mimura, M., Stationary pattern of some density-dependent diffusion system with competitive dynamics, Hiroshima math. J., 11, 621-635, (1981) · Zbl 0483.35045
[7] Mimura, M.; Kawasaki, K., Spatial segregation in competitive interaction-diffusion equations, J. math. biol., 9, 49-64, (1980) · Zbl 0425.92010
[8] Mimura, M.; Nishiura, Y.; Tesei, A.; Tsujikawa, T., Coexistence problem for two competing species models with density-dependent diffusion, Hiroshima math. J., 14, 425-449, (1984) · Zbl 0562.92011
[9] Ni, Wei-Ming; Takagi, I., Point condensation generated by a reaction-diffusion system in axially symmetric domains, Japan J. indust. appl. math., 12, 327-365, (1995) · Zbl 0843.35006
[10] Ni, Wei-Ming; Takagi, I., Locating the peaks of least-energy solutions to a semilinear Neumann problem, Duke math. J., 70, 247-281, (1993) · Zbl 0796.35056
[11] Ni, Wei-Ming; Takagi, I., On the shape of least-energy solutions to a semilinear Neumann problem, Comm. pure appl. math., 44, 819-851, (1991) · Zbl 0754.35042
[12] Protter, M.H.; Weinberger, H.F., Maximum principles in differential equations, (1984), Springer-Verlag New York · Zbl 0153.13602
[13] Shigesada, N.; Kawasaki, K.; Teramoto, E., Spatial segregation of interacting species, J. theoret. biol., 79, 83-99, (1979)
[14] Stampacchia, G., Le problème de Dirichlet pour LES équations elliptiques du second ordre à coefficients discontinus, Ann. inst. Fourier, 15, 189-258, (965) · Zbl 0151.15401
[15] Y. P. Wu, Existence of stationary solutions for a class of cross-diffusion systems with small parameters, Lecture Notes on Contemporary Mathematics, Science Press, Beijing
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.