Quarterly of Applied Mathematics

Quarterly of Applied Mathematics

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A remark on the existence of global BV solutions for a nonlinear hyperbolic wave equation


Authors: João-Paulo Dias and Mário Figueira
Journal: Quart. Appl. Math. 60 (2002), 245-250
MSC: Primary 35L70; Secondary 35D05
DOI: https://doi.org/10.1090/qam/1900492
MathSciNet review: MR1900492
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Abstract | References | Similar Articles | Additional Information

Abstract: By means of a suitable change of variables we obtain, by application of a general result by Dafermos and Hsiao, cf. [2], an existence theorem in $ {L^\infty } \cap {BV_{loc}}$ of a weak solution of the system corresponding to the quasilinear hyperbolic equation

$\displaystyle {\phi _{tt}} - p'\left( {\phi _x} \right){\phi _{xx}} + {\phi _t}... ...\qquad in \qquad \mathbb{R} \times \left[ {0, + \infty } \left[ \right. \right.$

, for small initial data in BV. This theorem is a partial extension of Dafermos's result for the case with $ F\left( \phi \right) \equiv 0$, proved in [1].

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

  • [1] C. M. Dafermos, A system of hyperbolic conservation laws with frictional damping, ZAMP 46, S294-S307 (1995) MR 1359325
  • [2] C. M. Dafermos and L. Hsiao, Hyperbolic systems of balance laws with inhomogeneity and dissipation, Indiana Univ. Math. J. 31, 471-491 (1982) MR 662914
  • [3] T.-P. Liu, Quasilinear hyperbolic systems, Commun. Math. Phys. 68, 141-172 (1979) MR 543196
  • [4] J. A. Smoller, Shock Waves and Reaction-Diffusion Equations, Springer-Verlag, New York, 1983 MR 1301779
  • [5] A. I. Vol'pert, The spaces BV and quasilinear equations, Math. USSR Sbornik 2, 225-267 (1967) MR 0216338

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DOI: https://doi.org/10.1090/qam/1900492
Article copyright: © Copyright 2002 American Mathematical Society

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