Quarterly of Applied Mathematics

Quarterly of Applied Mathematics

Online ISSN 1552-4485; Print ISSN 0033-569X



On the influence of damping in hyperbolic equations with parabolic degeneracy

Authors: Katarzyna Saxton and Ralph Saxton
Journal: Quart. Appl. Math. 70 (2012), 171-180
MSC (2010): Primary 35L45, 35L67
DOI: https://doi.org/10.1090/S0033-569X-2011-01247-1
Published electronically: August 29, 2011
MathSciNet review: 2920622
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Abstract | References | Similar Articles | Additional Information

Abstract: This paper examines the effect of damping on a nonstrictly hyperbolic $ 2\times 2$ system. It is shown that the growth of singularities is not restricted as in the strictly hyperbolic case where dissipation can be strong enough to preserve the smoothness of small solutions globally in time. Here, irrespective of the stabilizing properties of damping, solutions are found to break down in finite time on a line where two eigenvalues coincide in state space.

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  • 1. Jack Carr, Applications of centre manifold theory, Applied Mathematical Sciences, vol. 35, Springer-Verlag, New York-Berlin, 1981. MR 635782
  • 2. C. M. Dafermos, Contemporary issues in the dynamic behaviour of continuous media, LCDS Lecture Notes, Brown University, vol. 85-1, 1985.
  • 3. John Guckenheimer and Philip Holmes, Nonlinear oscillations, dynamical systems, and bifurcations of vector fields, Applied Mathematical Sciences, vol. 42, Springer-Verlag, New York, 1983. MR 709768
  • 4. Barbara L. Keyfitz and Herbert C. Kranzer, Nonstrictly hyperbolic systems of conservation laws: formation of singularities, Nonlinear partial differential equations (Durham, N.H., 1982) Contemp. Math., vol. 17, Amer. Math. Soc., Providence, R.I., 1983, pp. 77–90. MR 706089
  • 5. Barbara Lee Keyfitz and Claudia A. Mora, Prototypes for nonstrict hyperbolicity in conservation laws, Nonlinear PDE’s, dynamics and continuum physics (South Hadley, MA, 1998) Contemp. Math., vol. 255, Amer. Math. Soc., Providence, RI, 2000, pp. 125–137. MR 1752505, https://doi.org/10.1090/conm/255/03978
  • 6. R. J. Knops, H. A. Levine, and L. E. Payne, Non-existence, instability, and growth theorems for solutions of a class of abstract nonlinear equations with applications to nonlinear elastodynamics, Arch. Rational Mech. Anal. 55 (1974), 52–72. MR 0364839, https://doi.org/10.1007/BF00282433
  • 7. Howard A. Levine, Some additional remarks on the nonexistence of global solutions to nonlinear wave equations, SIAM J. Math. Anal. 5 (1974), 138–146. MR 0399682, https://doi.org/10.1137/0505015
  • 8. Hailiang Li and Katarzyna Saxton, Asymptotic behavior of solutions to quasilinear hyperbolic equations with nonlinear damping, Quart. Appl. Math. 61 (2003), no. 2, 295–313. MR 1976371, https://doi.org/10.1090/qam/1976371
  • 9. Takaaki Nishida, Nonlinear hyperbolic equations and related topics in fluid dynamics, Département de Mathématique, Université de Paris-Sud, Orsay, 1978. Publications Mathématiques d’Orsay, No. 78-02. MR 0481578
  • 10. T. Ruggeri, A. Muracchini, and L. Seccia, Shock waves and second sound in a rigid heat conductor: A critical temperature for NaF and Bi, Phys. Rev. Lett., 64 (1990), pp. 2640-2643.
  • 11. K. Saxton and R. Saxton, Nonlinearity and memory effects in low temperature heat propagation, Arch. Mech., 52 (2000), pp. 127-142.
  • 12. Katarzyna Saxton and Ralph Saxton, Phase transitions and aspects of heat propagation in low temperature solids, EQUADIFF 2003, World Sci. Publ., Hackensack, NJ, 2005, pp. 1128–1130. MR 2185197, https://doi.org/10.1142/9789812702067_0199
  • 13. Ralph Saxton, Blow-up at the boundary of solutions to nonlinear evolution equations, Evolution equations (Baton Rouge, LA, 1992) Lecture Notes in Pure and Appl. Math., vol. 168, Dekker, New York, 1995, pp. 383–392. MR 1300444
  • 14. 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, https://doi.org/10.1007/BF00251248
  • 15. Y. Wang, Finite time blow-up results for the damped wave equations with arbitrary energy in an inhomogeneous medium, Arxiv preprint math/0702190v1 (2007).

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Additional Information

Katarzyna Saxton
Affiliation: Department of Mathematical Sciences, Loyola University, New Orleans, Louisiana 70118
Email: saxton@loyno.edu

Ralph Saxton
Affiliation: Department of Mathematics, University of New Orleans, New Orleans, Louisiana 70148
Email: rsaxton@uno.edu

DOI: https://doi.org/10.1090/S0033-569X-2011-01247-1
Received by editor(s): September 6, 2010
Published electronically: August 29, 2011
Article copyright: © Copyright 2011 Brown University
The copyright for this article reverts to public domain 28 years after publication.

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