Threshold of Singularity Formation in the Semilinear Wave Equation

TL;DR

This paper numerically investigates the threshold of singularity formation in three-dimensional semilinear wave equations with different nonlinear exponents, revealing different asymptotic behaviors near criticality.

Contribution

It introduces adaptive mesh refinement for 3D simulations and analyzes singularity thresholds for p=5 and p=7, highlighting distinct self-similar and static solution approaches.

Findings

Approach to self-similarity for p=7 near singularity

Approach to scale-evolving static solution for p=5

Use of adaptive mesh refinement in 3D simulations

Abstract

Solutions of the semilinear wave equation are found numerically in three spatial dimensions with no assumed symmetry using distributed adaptive mesh refinement. The threshold of singularity formation is studied for the two cases in which the exponent of the nonlinear term is either $p=5$ or $p=7$. Near the threshold of singularity formation, numerical solutions suggest an approach to self-similarity for the $p=7$ case and an approach to a scale evolving static solution for $p=5$.