On the uniqueness of the weak solutions of a quasilinear hyperbolic system with a singular source term (Q1848444)

This paper deals with the existence and uniqueness of entropy weak solutions \((a,u)\) of the Cauchy problem for a first-order singular hyperbolic system in the plane. In their previous paper published in [Rend. Mat., 21, 245-258 (2001)], the same authors have proved the existence of an entropy weak solution and its approximation by the solutions of the Cauchy problem to an appropriate parabolic system. Under some additional conditions imposed on the Cauchy data \(({a_0\over x},{u_0 \over x}\in L_1\); \(a_0\), \(u_0\) have bounded total variation) it is proved the uniqueness of the weak entropy solution and that a.e. in \(t:\|a(\cdot,t) -a_0\|_{L_1}+ \|u(\cdot,t)-u_0 \|_{L_1} @>>t\to 0^+> 0\).