Development of singularities in solutions of the equations of nonlinear thermoelasticity

Dafermos, C. M.; Hsiao, L.

doi:10.1090/qam/860899

Authors: C. M. Dafermos and L. Hsiao
Journal: Quart. Appl. Math. 44 (1986), 463-474
MSC: Primary 73U05
DOI: https://doi.org/10.1090/qam/860899
MathSciNet review: 860899
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Nicholas D. Alikakos, An application of the invariance principle to reaction-diffusion equations, J. Differential Equations 33 (1979), no. 2, 201–225. MR 542670, DOI https://doi.org/10.1016/0022-0396%2879%2990088-3
Bernard D. Coleman and Morton E. Gurtin, Waves in materials with memory. III. Thermodymanic influences on the growth and decay of acceleration waves, Arch. Rational Mech. Anal. 19 (1965), 266–298. MR 195337, DOI https://doi.org/10.1007/BF00250214

C. M. Dafermos, Contemporary Issues in the Dynamic Behavior of Continuous Media, LCDS Lecture Notes No. 85–1

C. M. Dafermos, Development of singularities in the motion of materials with fading memory, Arch. Rational Mech. Anal. 91 (1985), no. 3, 193–205. MR 806001, DOI https://doi.org/10.1007/BF00250741

C. M. Dafermos and J. A. Nohel, A nonlinear hyperbolic Volterra equation in viscoelasticity, Amer. J. Math. (suppl. dedicated to P. Hartman) 87–116 (1981)

Harumi Hattori, Breakdown of smooth solutions in dissipative nonlinear hyperbolic equations, Quart. Appl. Math. 40 (1982/83), no. 2, 113–127. MR 666668, DOI https://doi.org/10.1090/S0033-569X-1982-0666668-X
William J. Hrusa, A nonlinear functional-differential equation in Banach space with applications to materials with fading memory, Arch. Rational Mech. Anal. 84 (1984), no. 2, 99–137. MR 713121, DOI https://doi.org/10.1007/BF00252129
Sergiu Klainerman and Andrew Majda, Formation of singularities for wave equations including the nonlinear vibrating string, Comm. Pure Appl. Math. 33 (1980), no. 3, 241–263. MR 562736, DOI https://doi.org/10.1002/cpa.3160330304
O. A. Ladyženskaja, V. A. Solonnikov, and N. N. Ural′ceva, Lineĭ nye i kvazilineĭ nye uravneniya parabolicheskogo tipa, Izdat. “Nauka”, Moscow, 1967 (Russian). MR 0241821
Peter D. Lax, Development of singularities of solutions of nonlinear hyperbolic partial differential equations, J. Mathematical Phys. 5 (1964), 611–613. MR 165243, DOI https://doi.org/10.1063/1.1704154
R. C. MacCamy, A model for one-dimensional, nonlinear viscoelasticity, Quart. Appl. Math. 35 (1977/78), no. 1, 21–33. MR 478939, DOI https://doi.org/10.1090/S0033-569X-1977-0478939-6

R. S. McIntire, Ph.D. Dissertation, Brown University (1985)

T. Nishida, Global smooth solutions for the second order quasilinear equation with first order dissipation (unpublished)

M. Slemrod, Global existence, uniqueness, and asymptotic stability of classical smooth solutions in one-dimensional nonlinear thermoelasticity, Arch. Rational Mech. Anal. 76 (1981), no. 2, 97–133. MR 629700, DOI https://doi.org/10.1007/BF00251248