 
Summary: A NOTE ON POSITIVE SOLUTIONS
FOR CONSERVATION LAWS WITH SINGULAR SOURCE
D. AMADORI AND G. M. COCLITE
Abstract. We consider the Cauchy problem for the scalar conservation law
tu + xf(u) =
1
g(u)
, t > 0, x R,
with g C1
(R), g(0) = 0, g(u) > 0 for u > 0, and assume that the initial datum u0 is nonnegative.
We show the existence of entropy solutions that are positive a.e., by means of an approximation
of the equation that preserves positive solutions, and by passing to the limit using a monotonicity
argument. The difficulty lies in handling the singularity of the right hand side (the source term) as
u possibly vanishes at the initial time. The source term is shown to be locally integrable.
Moreover, we prove an uniqueness and stability result for the above equation.
1. Introduction
We consider the initial value problem for the hyperbolic conservation law with singular
source
(1.1)
