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.
- 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, DOI https://doi.org/10.1007/BF01202949
- Tsuan Wu Ting, Parabolic and pseudo-parabolic partial differential equations, J. Math. Soc. Japan 21 (1969), 440–453. MR 264231, DOI https://doi.org/10.2969/jmsj/02130440
- 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
J. W. Cahn, On spinodal decomposition, Acta Metallurgica 9, 795–901 (1979)
E. C. Aifantis, A new interpretation of diffusion in regions with high diffusivity paths—a continuum approach, Acta Metallurgica 27, 683–691 (1979)
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)
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)
- 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, DOI https://doi.org/10.1007/BF01176762
- R. E. Showalter, Degenerate evolution equations and applications, Indiana Univ. Math. J. 23 (1973/74), 655–677. MR 333835, DOI https://doi.org/10.1512/iumj.1974.23.23056
- Kenneth L. Kuttler Jr., Time-dependent implicit evolution equations, Nonlinear Anal. 10 (1986), no. 5, 447–463. MR 839357, DOI https://doi.org/10.1016/0362-546X%2886%2990050-7
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S. Lefshetz, Differential equations: Geometric theory, Dover, 1977
- Avner Friedman, Partial differential equations, Holt, Rinehart and Winston, Inc., New York-Montreal, Que.-London, 1969. MR 0445088
- 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
- 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
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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
E. C. Aifantis, Higher-order diffusion theory and non-classical diffusion, Lecture Notes, Univ. of Illinois Urbana, 1979
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
E. C. Aifantis, On the problem of diffusion in solids, Acta Mechanica 37, 265–296 (1980)
T. W. Ting, Parabolic and pseudoparabolic partial differential equations, J. Math. Soc. Japan 21, 440–453 (1969)
R. W. Carroll and R. E. Showalter, Singular and degenerate Cauchy problems, Academic Press, New York, 1976
J. W. Cahn, On spinodal decomposition, Acta Metallurgica 9, 795–901 (1979)
E. C. Aifantis, A new interpretation of diffusion in regions with high diffusivity paths—a continuum approach, Acta Metallurgica 27, 683–691 (1979)
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)
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)
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)
R. E. Showalter, Degenerate evolution equations and applications, Indiana Univ. Math. J. 23, 655–677 (1974)
K. L. Kuttler, Time dependent implicit evolution equations, J. Nonlinear Analysis-Theory, Methods, and Applications 10, 447–463 (1986)
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)
S. Lefshetz, Differential equations: Geometric theory, Dover, 1977
A. Friedman, Partial differential equations, Holt, Rinehart, and Winston, Inc., 1969
R. A. Adams, Sobolev spaces, Academic Press, Inc., 1975
R. E. Showalter, Hilbert space methods for partial differential equations, Pittman, 1977
W. Rudin, Functional analysis, McGraw Hill, 1973
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
E. C. Aifantis, Higher-order diffusion theory and non-classical diffusion, Lecture Notes, Univ. of Illinois Urbana, 1979
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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Article copyright:
© Copyright 1987
American Mathematical Society
