Quarterly of Applied Mathematics

Quarterly of Applied Mathematics

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Global existence and singularity formation in solutions of a modified Fourier law


Author: K. Saxton
Journal: Quart. Appl. Math. 54 (1996), 697-707
MSC: Primary 35L60; Secondary 35Q72, 73B30, 80A20
DOI: https://doi.org/10.1090/qam/1417233
MathSciNet review: MR1417233
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Abstract | References | Similar Articles | Additional Information

Abstract: The aim of this paper is the analysis of formation of singularities from smooth initial data, for rigid conductors. The modified Fourier law used asserts that the governing second-order equation is hyperbolic with first-order dissipation. Global existence for small initial data is also examined.


References [Enhancements On Off] (What's this?)

  • [1] V. A. Cimmelli and W. Kosinski, Heat waves in continuous media at low temperatures, Quaderni del Dipartimento di Matematica, vol. 6, Universaita della Basilicata, Potenza, 1992
  • [2] V. A. Cimmelli and W. Kosinski, Well posedness for a nonlinear hyperbolic heat equation, Ricerche di Matematica 42, 49-68 (1993)
  • [3] H. Hattori, Breakdown of smooth solutions in dissipative nonlinear hyperbolic equations, Quart. Appl. Math. 40, 113-127 (1982)
  • [4] W. Kosinski, 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
  • [5] W. Kosinski, Gradient catastrophe in the solution of nonconservative hyperbolic systems, J. Math. Anal. Appl. 61, 672-688 (1977)
  • [6] W. Kosinski and K. Saxton, The effect on finite time breakdown due to modified Fourier laws, Quart Appl. Math. 51, 55-68 (1993)
  • [7] P. D. Lax, Development of singularities of solutions of nonlinear hyperbolic partial differential equations, J. Math. Phys. 5, 611-613 (1964)
  • [8] L. Longwei and Z. Yongshu, Existence and non-existence of global smooth solutions for quasilinear hyperbolic systems, Chinese Ann. of Math. 9B, 372-377 (1988)
  • [9] R. C. MacCamy, An integro-differential equation with application in heat flow, Quart. Appl. Math. 35, 1-19 (1977)
  • [10] R. Malek-Madani and J. A. Nohel, Formation of singularities for a conservation law with memory, Siam J. Math. 16, 530-540 (1985)
  • [11] A. Matsumara, Global existence and asymptotic solutions of the second order quasi-linear hyperbolic equations with first order dissipation, Publ. Res. Inst. Math. Sci. Kyoto Univ. A 13, 349-379 (1977)
  • [12] T. Nishida, Nonlinear hyperbolic equations and related topics in fluid dynamics, Publications Mathématiques D'Orsay, Université de Paris-Sud, Département de Mathématiques, vol. 78.02, 1978
  • [13] M. Slemrod, Instability of steady shearing flow in a nonlinear viscoelastic fluid, Arch. Rational Mech. Anal. 68, 211-225 (1978)

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Additional Information

DOI: https://doi.org/10.1090/qam/1417233
Article copyright: © Copyright 1996 American Mathematical Society


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