Cauchy's Problem for Degenerate Quasilinear Hyperbolic Equations with Measures as Initial Values (Journal of Partial Differential Equations, 1999)
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.
Keywords: Degenerate quasilinear hyperbolic equations; existence and uniqueness; extinction and positivity; localization