Quarterly of Applied Mathematics

Quarterly of Applied Mathematics

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Existence and uniqueness in nonclassical diffusion


Authors: K. Kuttler and Elias C. Aifantis
Journal: Quart. Appl. Math. 45 (1987), 549-560
MSC: Primary 73B30; Secondary 80A20
DOI: https://doi.org/10.1090/qam/910461
MathSciNet review: 910461
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Abstract: We consider a class of diffusion models that arise in certain nonclassical physical situations and discuss existence and uniqueness of the resulting evolution equations.


References [Enhancements On Off] (What's this?)

  • [1] E. C. Aifantis, On the problem of diffusion in solids, Acta Mech. 37 (1980), no. 3-4, 265–296 (English, with German summary). MR 586062, https://doi.org/10.1007/BF01202949
  • [2] Tsuan Wu Ting, Parabolic and pseudo-parabolic partial differential equations, J. Math. Soc. Japan 21 (1969), 440–453. MR 0264231, https://doi.org/10.2969/jmsj/02130440
  • [3] Robert Wayne Carroll and Ralph E. Showalter, Singular and degenerate Cauchy problems, Academic Press [Harcourt Brace Jovanovich, Publishers], New York-London, 1976. Mathematics in Science and Engineering, Vol. 127. MR 0460842
  • [4] J. W. Cahn, On spinodal decomposition, Acta Metallurgica 9, 795-901 (1979)
  • [5] E. C. Aifantis, A new interpretation of diffusion in regions with high diffusivity paths--a continuum approach, Acta Metallurgica 27, 683-691 (1979)
  • [6] E. C. Aifantis and J. M. Hill, On the theory of diffusion in media with double diffusivity-I, Quart. J. Mech. Appl. Math. 33, 1-21 (1980)
  • [7] E. C. Aifantis and J. M. Hill, On the theory of diffusion in media with double diffusivity-II, Quart. J. Mech. Appl. Math. 33, 23-41 (1980)
  • [8] A. I. Lee and J. M. Hill, On the solution of boundary value problems for fourth order diffusion, Acta Mech. 46 (1983), no. 1-4, 23–35. MR 696459, https://doi.org/10.1007/BF01176762
  • [9] R. E. Showalter, Degenerate evolution equations and applications, Indiana Univ. Math. J. 23 (1973/74), 655–677. MR 0333835, https://doi.org/10.1512/iumj.1974.23.23056
  • [10] Kenneth L. Kuttler Jr., Time-dependent implicit evolution equations, Nonlinear Anal. 10 (1986), no. 5, 447–463. MR 839357, https://doi.org/10.1016/0362-546X(86)90050-7
  • [11] C. Truesdell and W. Noll, The non-linear field theories of mechanics, Handbuch der Physik, Band III/3, Springer-Verlag, Berlin, 1965, pp. 1–602. MR 0193816
  • [12] S. Lefshetz, Differential equations: Geometric theory, Dover, 1977
  • [13] Avner Friedman, Partial differential equations, Holt, Rinehart and Winston, Inc., New York-Montreal, Que.-London, 1969. MR 0445088
  • [14] Robert A. Adams, Sobolev spaces, Academic Press [A subsidiary of Harcourt Brace Jovanovich, Publishers], New York-London, 1975. Pure and Applied Mathematics, Vol. 65. MR 0450957
  • [15] R. E. Showalter, Hilbert space methods for partial differential equations, Pitman, London-San Francisco, Calif.-Melbourne, 1977. Monographs and Studies in Mathematics, Vol. 1. MR 0477394
  • [16] Walter Rudin, Functional analysis, McGraw-Hill Book Co., New York-Düsseldorf-Johannesburg, 1973. McGraw-Hill Series in Higher Mathematics. MR 0365062
  • [17] E. C. Aifantis, Maxwell and van der Waals revisited, in: Phase transformations in solids, Ed. T. Tsakalakos, MRS 21, pp. 37-49, North Holland, 1984
  • [18] E. C. Aifantis, Higher-order diffusion theory and non-classical diffusion, Lecture Notes, Univ. of Illinois Urbana, 1979
  • [19] K. L. Kuttler and E. C. Aifantis, Existence and uniqueness in non-classical diffusion, Mechanics of Microstructures (MM) Report No. 9, Department of Mechanical Engineering--Engineering Mechanics, Michigan Technological University, 1984

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DOI: https://doi.org/10.1090/qam/910461
Article copyright: © Copyright 1987 American Mathematical Society


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