Loss of Regularity in the $K(m,n)$ Equations

TL;DR

This paper proves that solutions to the $K(m,n)$ equations with certain initial conditions inevitably develop singularities in finite time, highlighting the link between singularity formation and the emergence of compactons.

Contribution

It establishes finite-time loss of smoothness for solutions of the $K(m,n)$ equations using a priori estimates, advancing understanding of singularity formation in nonlinear PDEs.

Findings

Solutions lose smoothness in finite time

Singularity formation is necessary for compacton emergence

Provides a priori estimates for solution behavior

Abstract

Using a priori estimates we prove that initially nonnegative, smooth and compactly supported solutions of the $K(m,n)$ equations must lose their smoothness in finite time. Formation of a singularity is a prerequisite for the emergence of compactons.