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: 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 \[ {\phi _{tt}} - p’\left ( {\phi _x} \right ){\phi _{xx}} + {\phi _t} + F\left ( \phi \right ) = 0 \qquad in \qquad \mathbb {R} \times \left [ {0, + \infty } \left [ \right . \right .\], 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].
- C. M. Dafermos, A system of hyperbolic conservation laws with frictional damping, Z. Angew. Math. Phys. 46 (1995), no. Special Issue, S294–S307. Theoretical, experimental, and numerical contributions to the mechanics of fluids and solids. MR 1359325
- C. M. Dafermos and L. Hsiao, Hyperbolic systems and balance laws with inhomogeneity and dissipation, Indiana Univ. Math. J. 31 (1982), no. 4, 471–491. MR 662914, DOI https://doi.org/10.1512/iumj.1982.31.31039
- Tai Ping Liu, Quasilinear hyperbolic systems, Comm. Math. Phys. 68 (1979), no. 2, 141–172. MR 543196
- Joel Smoller, Shock waves and reaction-diffusion equations, 2nd ed., Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 258, Springer-Verlag, New York, 1994. MR 1301779
- A. I. Vol′pert, Spaces ${\rm BV}$ and quasilinear equations, Mat. Sb. (N.S.) 73 (115) (1967), 255–302 (Russian). MR 0216338
C. M. Dafermos, A system of hyperbolic conservation laws with frictional damping, ZAMP 46, S294–S307 (1995)
C. M. Dafermos and L. Hsiao, Hyperbolic systems of balance laws with inhomogeneity and dissipation, Indiana Univ. Math. J. 31, 471–491 (1982)
T.-P. Liu, Quasilinear hyperbolic systems, Commun. Math. Phys. 68, 141–172 (1979)
J. A. Smoller, Shock Waves and Reaction-Diffusion Equations, Springer-Verlag, New York, 1983
A. I. Vol’pert, The spaces BV and quasilinear equations, Math. USSR Sbornik 2, 225–267 (1967)
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© Copyright 2002
American Mathematical Society