Smooth solutions and singularity formation for the inhomogeneous nonlinear wave equation

We study the nonlinear inhomogeneous wave equation in one space dimension: vtt−T(v,x)xx=0. By constructing some “decoupled” Riccati type equations for smooth solutions, we provide a singularity formation result without restrictions on the total variation of the data, which generalizes earlier singularity results of Lax and the first author. We apply these results to compressible Euler flows with a general pressure law and elasticity in an inhomogeneous medium.