TY - JOUR
T1 - Cauchy's Problem for Degenerate Quasilinear Hyperbolic Equations with Measures as Initial Values
AU - Hongjun Yuan 
JO - Journal of Partial Differential Equations
VL - 2
SP - 149
EP - 178
PY - 1999
DA - 1999/12
SN - 12
DO - http://doi.org/
UR - https://global-sci.org/intro/article_detail/jpde/5532.html
KW - Degenerate quasilinear hyperbolic equations
KW - existence and uniqueness
KW - extinction and positivity
KW - localization
AB -  The aim of this paper is to discuss the Cauchy problem for degenerate quasilinear hyperbolic equations of the form \frac{∂u}{∂t} + \frac{∂u^m}{∂x} = -u^p, m > 1, p > 0  with measures as initial conditions. The existence and uniqueness of solutions are obtained. In particular, we prove the following results: (1) 0 < p < 1 is a necessary and sufficient condition for the above equations to have extinction property; (2) 0 < p < m is a necessary and sufficient condition for the above equations to have localization property of the propagation of perturbations.