Gradient Catastrophe for Solutions to the Conservation Laws with Source Term

TL;DR

This paper investigates the formation of singularities in conservation laws with source terms, demonstrating finite-time blow-up under weaker conditions and linking initial data support size to solution behavior.

Contribution

It establishes new conditions for finite-time blow-up in conservation laws with source terms, extending previous results and highlighting the role of initial data support.

Findings

Finite-time blow-up occurs under weaker initial data conditions than previously known.

Small support length of initial data promotes solution blow-up.

Global solutions require sufficiently large initial support length.

Abstract

This paper studies singularity formation for conservation laws with a source term. Motivated by John (1974) and Barlin (2023), we prove finite-time blow-up under initial data conditions weaker than those in Barlin. Moreover, we show that a sufficiently small compact support length of the initial data promotes blow-up. Hence, global existence can only be achieved when the initial data have a large compact support length.