Quarterly of Applied Mathematics

Quarterly of Applied Mathematics

Online ISSN 1552-4485; Print ISSN 0033-569X



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 | References | Similar Articles | Additional Information

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 Mechanica 37, 265-296 (1980) MR 586062
  • [2] T. W. Ting, Parabolic and pseudoparabolic partial differential equations, J. Math. Soc. Japan 21, 440-453 (1969) MR 0264231
  • [3] R. W. Carroll and R. E. Showalter, Singular and degenerate Cauchy problems, Academic Press, New York, 1976 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 Mechanica 46, 23-35 (1983). Some of the uniqueness results reported in this paper were first presented by E. C. Aifantis in a set of Lecture Notes (Univ. of Illinois, Urbana, 1978) MR 696459
  • [9] R. E. Showalter, Degenerate evolution equations and applications, Indiana Univ. Math. J. 23, 655-677 (1974) MR 0333835
  • [10] K. L. Kuttler, Time dependent implicit evolution equations, J. Nonlinear Analysis-Theory, Methods, and Applications 10, 447-463 (1986) MR 839357
  • [11] C. Truesdell and W. Noll, The non-linear field theories of mechanics, In Flugge's Handbuch der Physik, Band III/3, Springer-Verlag, Berlin--Heidelberg--New York (1965) MR 0193816
  • [12] S. Lefshetz, Differential equations: Geometric theory, Dover, 1977
  • [13] A. Friedman, Partial differential equations, Holt, Rinehart, and Winston, Inc., 1969 MR 0445088
  • [14] R. A. Adams, Sobolev spaces, Academic Press, Inc., 1975 MR 0450957
  • [15] R. E. Showalter, Hilbert space methods for partial differential equations, Pittman, 1977 MR 0477394
  • [16] W. Rudin, Functional analysis, McGraw Hill, 1973 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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