Blow up in a periodic semilinear heat equation

TL;DR

This paper investigates the blow-up behavior in a one-dimensional periodic semilinear heat equation using combined numerical and analytical methods, providing asymptotic approximations and preliminary insights into solutions beyond singularity.

Contribution

It introduces novel asymptotic approximations valid up to blow-up and explores numerical continuation beyond the singularity in periodic semilinear heat equations.

Findings

Asymptotic approximations valid up to blow-up time

Numerical methods for continuing solutions beyond singularity

Insights into blow-up behavior in periodic settings

Abstract

Blow up in a one-dimensional semilinear heat equation is studied using a combination of numerical and analytical tools. The focus is on problems periodic in the space variable and starting out from a nearly flat, positive initial condition. Novel results include various asymptotic approximations that are, in combination, valid over the entire space and time interval right up to and including the blow-up time. Preliminary results on continuing a numerical solution beyond the singularity are also presented.