Local existence inC b 0, 1 and blow-up of the solutions of the Cauchy Problem for a quasilinear hyperbolic system with a singular source term

Abstract

In this paper we consider the Cauchy problem for the hyperbolic system $$\left\{ {\begin{array}{*{20}c} {a_t + (au)_x + \frac{{2au}}{x} = 0} \\ {x > 0,t \geqslant 0} \\ {u_t + \tfrac{1}{2}(a^2 + u^2 )_x = 0} \\ \end{array} } \right.$$ with null boundary conditions and we prove a local (in time) existence and uniqueness theorem inCb0, 1 and, for a special class of initial data, a blow-up result.