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BV solutions for the Cauchy problem of a quasilinear hyperbolic equation with \(\sigma\)-finite Borel measure and nonlinear source. (English) Zbl 1077.35085
Summary: 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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