Smooth solutions and singularity formation for the inhomogeneous nonlinear wave equation

TL;DR

This paper investigates the formation of singularities in solutions to the inhomogeneous nonlinear wave equation in one dimension, extending classical results to more general models without total variation restrictions.

Contribution

It introduces Riccati-type equations for smooth solutions, enabling singularity results for a broad class of hyperbolic models without total variation constraints.

Findings

Singularity formation results for inhomogeneous wave equations

Application to various hyperbolic models like Euler flows and MHD

Generalization of classical singularity results

Abstract

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.