(untitled) (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 ∂u∂t + ∂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.