Structurally Stable Singularities for a Nonlinear Wave Equation

For the nonlinear wave equation $u_{tt}-c(u)(c(u)u_x{)}_x=0$, it is well known that solutions can develop singularities in finite time. For an open dense set of initial data, the present paper provides a detailed asymptotic description of the solution in a neighborhood of each singular point, where $\mathrm{\mid }{u}_x\mathrm{\mid }\to \mathrm{\infty }$. The different structure of conservative and dissipative solutions is analyzed.