On blowup for semilinear wave equations with a focusing nonlinearity

TL;DR

This paper investigates the formation of singularities in semilinear wave equations with focusing nonlinearities, analyzing blowup patterns, critical solutions, and the influence of initial data through numerical studies.

Contribution

It provides a detailed numerical analysis of blowup phenomena, including the characterization of spatial patterns and critical solutions in three-dimensional semilinear wave equations.

Findings

Blowup patterns can be described by linearized perturbations around self-similar solutions.

Critical solutions separate blowup from dispersal for certain exponents.

Numerical evidence supports the role of initial data in determining singularity formation.

Abstract

In this paper we report on numerical studies of formation of singularities for the semilinear wave equations with a focusing power nonlinearity $u_{tt} - \Delta u = u^{p}$ in three space dimensions. We show that for generic large initial data that lead to singularities, the spatial pattern of blowup can be described in terms of linearized perturbations about the fundamental self-similar (homogeneous in space) solution. We consider also non-generic initial data which are fine-tuned to the threshold for blowup and identify critical solutions that separate blowup from dispersal for some values of the exponent $p$.