Hiroshima Mathematical Journal

Some nonlinear degenerate diffusion equations with a nonlocally convective term in ecology

Toshitaka Nagai

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Hiroshima Math. J., Volume 13, Number 1 (1983), 165-202.

First available in Project Euclid: 21 March 2008

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Zentralblatt MATH identifier

Primary: 35K55: Nonlinear parabolic equations
Secondary: 45K05: Integro-partial differential equations [See also 34K30, 35R09, 35R10, 47G20] 92A17


Nagai, Toshitaka. Some nonlinear degenerate diffusion equations with a nonlocally convective term in ecology. Hiroshima Math. J. 13 (1983), no. 1, 165--202. doi:10.32917/hmj/1206133543. https://projecteuclid.org/euclid.hmj/1206133543

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  • [1] D.G. Aronson, Regularity properties of flows through porous media, SIAM J. Appl. Math. 17 (1969), 461-467.
  • [2] D.G. Aronson, Regularity properties of flows through porous media; A counterexample, SIAM J. Apll. Math. 19 (1970), 299-307.
  • [3] D.G. Aronson, Density dependent interaction-diffusion systems, Dynamics and modelling of reactive systems, Edited by W. E. Stewart, Academic press, 1980.
  • [4] G.I. Barenblatt, On some unsteady motions of a liquid and a gas in porous medium, Prikl. Mat.Mech. 16 (1952), 67-78. (Russian).
  • [5] J. Bear, Dynamics of Fluids in Porous Media, American Elsevier, NewYork,1972.
  • [6] P. R. Beesack, Gronwall inequalities, Carleton Mathematical Lecture Notes. 11 (1975).
  • [7] A. Friedman, Partial Differential Equations of Parabolic Type, Prentice-Hall, Englewood Cliffs, N.J,1964.
  • [8] B. H. Gilding, Holder continuity of solutions of parabolic equations, J. London Math. Soc. 13 (1976), 103-106.
  • [9] B. H. Gilding, Properties of solutions of an equation in thetheory of infiltration, Arch. Rational Mech. Anal. 65 (1977), 203-225.
  • [10] B. H. Gilding and L. A. Peletier, The Cauchy problem for an equation in the theory of infiltration, Arch. Rational Mech. Anal. 61 (1976), 127-140.
  • [11] W. S. C. Gurney and R. M. Nisbet, The regulation of inhomogeneous populations, J. Theoret. Biol. 52 (1975), 441-457.
  • [12] M.E. Gurtin and R. C. MacCamy, On the diffusion of biological populations, Math. Biosci. 33 (1979), 35-49.
  • [13] A. S. Kalashnikov, On the occurrence of singularities in the solutions of the equation of non-stationary filtration, Z. Vycisl. Mat. i Mat. Fiz. 7 (1967), 440-444. (Russian).
  • [14] Y. Kuramoto, Rhythms and turbulence in populations of chemical oscillators, Physica. 106A(1981), 128-143.
  • [15] O. A. Ladyzenskaja, V. A. Solonikov and N.N. Uraceva, Linear and Quasilinear Equations of Parabolic Type, Translations of Mathematical Monographs, vol.23, American Mathematical Society, Providence , R. I. 1968.
  • [16] T. Munakata, Liquid instability and freezing-reductive perturbation approach-, J. Phys. Soc. Japan, 43 (1977), 1723-1728.
  • [17] T. Nagai and M. Mimura, Some nonlinear degenerate diffusion equation related to population dynamics, to appear in J. Math. Soc. Japan.
  • [18] T. Nagai and M. Mimura, Asymptotic behavior for a nonlinear degenerate diffusion equation in population dynamics, to appear in SIAM J. Appl. Math.
  • [19] W. I. Newman, Some exact solutions to a non-linear diffusion problem in population genetics and combustion, J. Theoret. Biol. 85 (1980), 325-334.
  • [20] O. A. Oleinik, A. S. Kalashnikov and Chzou Yui-lin, The Cauchy problem and boundary value problems for equations of the type of nonstationary filtration, Izv. Akad. Nauk, SSSR. 22 (1958), 667-704. (Russian).
  • [21] R. E. Pattle, Diffusion from an instantaneous point source with concentration-dependent coefficient, Quart J. Mech. Appl. Math. 12 (1959), 407^09.
  • [22] J. Satsuma, Exact solutions of a nonlinear diffusion equation, J. Phys. Soc. Japan. 50 (1981), 1423-1424.