Complex-plane singularity dynamics for blow up in a nonlinear heat equation: analysis and computation

TL;DR

This paper investigates the complex-plane singularity dynamics leading to blow-up in a nonlinear heat equation, combining asymptotic analysis and numerical methods to understand the singularities' role and their impact on finite-time blow-up.

Contribution

It introduces a detailed analysis of complex singularity dynamics in a nonlinear heat equation, including numerical computation on multiple Riemann sheets and the role of a quadratic ODE in the asymptotic behavior.

Findings

Singularities in the complex plane cause finite-time blow-up.

The dynamics are distinguished between small and large nonlinear effects.

Far-field solutions are characterized by Weierstrass elliptic functions.

Abstract

Blow-up solutions to a heat equation with spatial periodicity and a quadratic nonlinearity are studied through asymptotic analyses and a variety of numerical methods. The focus is on the dynamics of the singularities in the complexified space domain. Blow up in finite time is caused by these singularities eventually reaching the real axis. The analysis provides a distinction between small and large nonlinear effects, as well as insight into the various time scales on which blow up is approached. It is shown that an ordinary differential equation with quadratic nonlinearity plays a central role in the asymptotic analysis. This equation is studied in detail, including its numerical computation on multiple Riemann sheets, and the far-field solutions are shown to be given at leading order by a Weierstrass elliptic function.