2023/08/07 by M. Fasondini, John R. King, Fasondini, M. +3
Computer Science · Engineering · Mathematics · #32S99 #35B44 #35C20 #41A21 #65N99 #Advanced Mathematical Modeling in Engineering #Analysis of PDEs (math.AP) #Classical Analysis and ODEs (math.CA) #Complex Variables (math.CV) #Differential Equations and Numerical Methods #FOS: Mathematics #Numerical Analysis (math.NA) #Stability and Controllability of Differential Equations
paper · pdf · doi:10.48550/arxiv.2308.03452
openalex publication_date 2023/08/07 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
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.