Existence, uniqueness, and long-time behavior of materials with nonmonotone equations of state and higher-order gradients
Authors:
K. Kuttler and E. C. Aifantis
Journal:
Quart. Appl. Math. 48 (1990), 473-489
MSC:
Primary 73B30; Secondary 35B40, 35Q99, 80A22
DOI:
https://doi.org/10.1090/qam/1074962
MathSciNet review:
MR1074962
Full-text PDF Free Access
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Additional Information
V. Alexiades and E. C. Aifantis, On the thermodynamic theory of fluid interfaces, infinite intervals, equilibrium solutions and minimizers, Mechanics of Microstructures, Report No. 9, 1984
E. C. Aifantis and J. B. Serrin, The mechanical theory of fluid interfaces and Maxwellโs rule, J. Coll. Int. Sci. 96, 517โ529 (1983)
- G. Andrews and J. M. Ball, Asymptotic behaviour and changes of phase in one-dimensional nonlinear viscoelasticity, J. Differential Equations 44 (1982), no. 2, 306โ341. Special issue dedicated to J. P. LaSalle. MR 657784, DOI https://doi.org/10.1016/0022-0396%2882%2990019-5
- J. M. Ball, On the asymptotic behavior of generalized processes, with applications to nonlinear evolution equations.ations, J. Differential Equations 27 (1978), no. 2, 224โ265. MR 461576, DOI https://doi.org/10.1016/0022-0396%2878%2990032-3
J. Carr, M. E. Gurtin, and M. Slemrod, One-dimensional structured phase transformations under prescribed loads, MRC Technical Summary Report #2559, 1983
C. M. Dafermos, Contraction semigroups and trend to equilibrium in continuum mechanics, Lecture Notes in Math. 503, 295โ306 (1976)
- Jack K. Hale, Dynamical systems and stability, J. Math. Anal. Appl. 26 (1969), 39โ59. MR 244582, DOI https://doi.org/10.1016/0022-247X%2869%2990175-9
- Daniel Henry, Geometric theory of semilinear parabolic equations, Lecture Notes in Mathematics, vol. 840, Springer-Verlag, Berlin-New York, 1981. MR 610244
- Kenneth Kuttler, Regularity of weak solutions of some nonlinear conservation laws, Appl. Anal. 26 (1987), no. 1, 23โ32. MR 916896, DOI https://doi.org/10.1080/00036818708839698
- Kenneth Kuttler and Darrell Hicks, Weak solutions of initial-boundary value problems for a class of nonlinear viscoelastic equations, Appl. Anal. 26 (1987), no. 1, 33โ43. MR 916897, DOI https://doi.org/10.1080/00036818708839699
- K. Kuttler and D. Hicks, Initial-boundary value problems for the equation $u_{tt}=(\sigma (u_x))_x+(\alpha (u_x)u_{xt})_x+f$, Quart. Appl. Math. 46 (1988), no. 3, 393โ407. MR 963578, DOI https://doi.org/10.1090/S0033-569X-1988-0963578-2
- Robert L. Pego, Phase transitions in one-dimensional nonlinear viscoelasticity: admissibility and stability, Arch. Rational Mech. Anal. 97 (1987), no. 4, 353โ394. MR 865845, DOI https://doi.org/10.1007/BF00280411
- Michael Spivak, Calculus on manifolds. A modern approach to classical theorems of advanced calculus, W. A. Benjamin, Inc., New York-Amsterdam, 1965. MR 0209411
V. Alexiades and E. C. Aifantis, On the thermodynamic theory of fluid interfaces, infinite intervals, equilibrium solutions and minimizers, Mechanics of Microstructures, Report No. 9, 1984
E. C. Aifantis and J. B. Serrin, The mechanical theory of fluid interfaces and Maxwellโs rule, J. Coll. Int. Sci. 96, 517โ529 (1983)
G. Andrews and J. M. Ball, Asymptotic behavior and changes of phase in one-dimensional nonlinear viscoelasticity, J. Differential Equations 44, 306โ341 (1982)
J. M. Ball, On the asymptotic behavior of generalized processes with applications to nonlinear evolution equations, J. Differential Equations 27, 224โ265 (1978)
J. Carr, M. E. Gurtin, and M. Slemrod, One-dimensional structured phase transformations under prescribed loads, MRC Technical Summary Report #2559, 1983
C. M. Dafermos, Contraction semigroups and trend to equilibrium in continuum mechanics, Lecture Notes in Math. 503, 295โ306 (1976)
J. K. Hale, Dynamical systems and stability, J. Math. Anal. Appl., 26, 39โ59 (1969)
D. Henry, Geometric Theory of Semi linear Parabolic Equations, Lecture Notes in Math. Vol. 840, Springer, New York, 1981
K. L. Kuttler, Regularity of weak solutions of some nonlinear conservation laws, Applicable Analysis, Vol. 26, Oct. 1987
K. L. Kuttler and D. L. Hicks, Weak solutions of initial-boundary value problems for a class of nonlinear viscoelastic equations, Appl. Anal. (1) 26, 33โ43 (1987)
K. L. Kuttler and D. L. Hicks, Initial boundary value problems for the equation ${u_{tt}} = {\left ( {\alpha \left ( {{u_x}} \right ){u_{xt}} + \sigma \left ( {{u_x}} \right )} \right )_x} + f$, Quart. Appl. Math. (3) 46, 393โ407 (1988)
R. L. Pego, Phase transitions in one dimensional nonlinear viscoelasticity: admissibility and stability, Arch. Rational Mech. Anal. (4) 97, 353โ394 (1987)
M. Spivak, Calculus on Manifolds, W. A. Benjamin, 1965
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Article copyright:
© Copyright 1990
American Mathematical Society