Existence and non-existence of global smooth solutions for quasilinear hyperbolic systems (Q1114862)

Consider the initial value problem (E) \(v_ t-u_ x=0\), \(u_ t+p(v)_ x=0\), (I) \(v(x,0)=v_ 0(x)\), \(u(x,0)=u_ 0(x)\), where \(A\geq 0\), \(p(v)=K^ 2v^{-\gamma}\), \(K>0\), \(0<\gamma <3\). As \(0<\gamma \leq 1\), the authors give a sufficient condition for that (E), (I) to have a unique global smooth solution. As \(1\leq \gamma <3\), a necessary condition is given for that.