A note on positive solutions for conservation laws with singular source

We consider the Cauchy problem for the scalar conservation law ∂ t u + ∂ x f ( u ) = 1 g ( u ) , t > 0 , x ∈ R , \begin{equation*} \partial _t u+\partial _x f(u) =\displaystyle \frac {1}{g(u)},\qquad t>0,\ x\in \mathbb {R}, \end{equation*} with g ∈ C 1 ( R ) g\in C^1(\mathbb {R}) , g ( 0 ) = 0 g(0)=0 , g ( u ) > 0 g(u)>0 for u > 0 u>0 , and assume that the initial datum u 0 u_0 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 u possibly vanishes at the initial time. The source term is shown to be locally integrable. Moreover, we prove a uniqueness and stability result for the above equation