The effect on finite time breakdown due to modified Fourier laws
Authors:
W. Kosiński and K. Saxton
Journal:
Quart. Appl. Math. 51 (1993), 55-68
MSC:
Primary 35L05; Secondary 73B30
DOI:
https://doi.org/10.1090/qam/1205936
MathSciNet review:
MR1205936
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Abstract: This paper discusses the finite time blow-up of the amplitude of acceleration waves in the case of heat propagation in one-dimensional rigid and elastic bodies. In both cases dissipation is not strong enough to preserve the smoothness of the solutions whose initial data is far from equilibrium.
M. Chester, Second sound in solids, Phys. Rev. 131, 2013–2015 (1962)
V. A. Cimmelli and W. Kosiński, Well posedness results for a nonlinear hyperbolic heat equation, submitted
- Bernard D. Coleman and Morton E. Gurtin, Waves in materials with memory. II. On the growth and decay of one-dimensional acceleration waves, Arch. Rational Mech. Anal. 19 (1965), 239–265. MR 195336, DOI https://doi.org/10.1007/BF00250213
- 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 behaviour of continuous media, LCDS Lecture Notes, Brown University, vol. 85-1, 1985
- C. M. Dafermos and L. Hsiao, Development of singularities in solutions of the equations of nonlinear thermoelasticity, Quart. Appl. Math. 44 (1986), no. 3, 463–474. MR 860899, DOI https://doi.org/10.1090/S0033-569X-1986-0860899-8
- Morton E. Gurtin and A. C. Pipkin, A general theory of heat conduction with finite wave speeds, Arch. Rational Mech. Anal. 31 (1968), no. 2, 113–126. MR 1553521, DOI https://doi.org/10.1007/BF00281373
W. Kosiński, Elastic waves in the presence of a new temperature scale, Elastic wave propagation, M. F. McCarthy and M. A. Hayes, eds., Elsevier Science (North-Holland), Amsterdam, 1989, p. 629
- 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
- 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
M. Chester, Second sound in solids, Phys. Rev. 131, 2013–2015 (1962)
V. A. Cimmelli and W. Kosiński, Well posedness results for a nonlinear hyperbolic heat equation, submitted
B. C. Coleman and M. E. Gurtin, Waves in materials with memory, II. On the growth and decay of one-dimensional acceleration waves, Arch. Rational Mech. Anal. 19, 239–265 (1965)
B. C. Coleman and M. E. Gurtin, Waves in materials with memory, III. Thermodynamic influence on the growth and decay of acceleration waves, Arch. Rational Mech. Anal. 19, 266–298 (1965)
C. M. Dafermos, Contemporary issues in the dynamic behaviour of continuous media, LCDS Lecture Notes, Brown University, vol. 85-1, 1985
C. M. Dafermos and L. Hsiao, Development of singularities in solutions of the equations of nonlinear thermoelasticity, Quart. Appl. Math. 44, 463–474 (1986)
M. E. Gurtin and A. C. Pipkin, A general theory of heat conduction with finite wave speeds, Arch. Rational Mech. Anal. 31, 113–126 (1968)
W. Kosiński, Elastic waves in the presence of a new temperature scale, Elastic wave propagation, M. F. McCarthy and M. A. Hayes, eds., Elsevier Science (North-Holland), Amsterdam, 1989, p. 629
P. D. Lax, Development of singularities of solutions of nonlinear hyperbolic partial differential equations, J. Math. Phys. 5, 611–613 (1964)
M. Slemrod, Global existence, uniqueness and asymptotic stability of classical smooth solutions in one-dimensional, non-linear thermoelasticity, Arch.Rational Mech. Anal. 76, 97–133 (1981)
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© Copyright 1993
American Mathematical Society