Smooth solutions and singularity formation for the inhomogeneous nonlinear wave equation

We study the nonlinear inhomogeneous wave equation in one space dimension: $v_{tt}-T(v,x{)}_{xx}=0$. By constructing some “decoupled” Riccati type equations for smooth solutions, we provide a singularity formation result without restrictions on the total variation of unknown, which generalize earlier singularity results of Lax and the first author. These results are applied to several one-dimensional hyperbolic models, such as compressible Euler flows with a general pressure law, elasticity in an inhomogeneous medium, transverse MHD flow, and compressible flow in a variable area duct.