Formation of singularities in solutions to nonlinear hyperbolic systems with general sources

TL;DR

This paper proves that solutions to certain nonlinear hyperbolic systems with general sources develop singularities in finite time, using classical methods to show the unbounded growth of derivatives.

Contribution

It extends the understanding of singularity formation in nonlinear hyperbolic systems with general sources by establishing finite-time blow-up of solutions.

Findings

Solutions blow up in finite time for smooth initial data

Derivative of the solution becomes unbounded in finite time

Builds on classical methods of John and Hörmander

Abstract

We consider nonlinear hyperbolic systems with a general source and prove that for appropriately chosen smooth initial data the lifespan of the associated $C^1$-solution $u$ cannot be infinite. We employ ideas of F. John (1974) and L. H\"ormander (1987) to show that the derivative $u_x$ of $u$ becomes unbounded in finite time.