BV solutions for the Cauchy problem of a quasilinear hyperbolic equation with \(\sigma\)-finite Borel measure and nonlinear source.

*(English)*Zbl 1077.35085Summary: The aim of this paper is to discuss the Cauchy problem of the quasilinear hyperbolic equation of the form \(u_{t}+(u^{m})_{x}=u^{p}\) with nonnegative \(\sigma\)-finite Borel measures as initial conditions, where \(m > 1\), \(0<p\leqslant 1\) are some given real numbers. In particular, the existence of BV solutions for the above problem is obtained.

##### MSC:

35L60 | First-order nonlinear hyperbolic equations |

35L45 | Initial value problems for first-order hyperbolic systems |

35R05 | PDEs with low regular coefficients and/or low regular data |

35D05 | Existence of generalized solutions of PDE (MSC2000) |

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\textit{H. Yuan} et al., J. Math. Anal. Appl. 311, No. 2, 715--735 (2005; Zbl 1077.35085)

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##### References:

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