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

Fasondini, M.; King, J. R.; Weideman, J. A.C.

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 tracking the dynamics of the singularities in the complexified space domain all the way from the initial time until the blow-up time, which occurs when the singularities reach the real axis. This widely applicable approach gives forewarning of the possibility of blow up and an understanding of the influence of singularities on the solution behaviour on the real axis, aiding the (perhaps surprisingly involved) asymptotic analysis of the real-line behaviour. The analysis provides a distinction between small and large nonlinear effects, as well as insight into the various time scales over which blow up is approached. The solution to the nonlinear heat equation in the complex spatial plane is shown to be related asymptotically to a nonlinear ordinary differential equation. This latter equation is studied in detail, including its computation on multiple Riemann sheets, providing further insight into the singularities of blow-up solutions of the nonlinear heat equation when viewed as multivalued functions in the complex space domain and illustrating the potential intricacy of singularity dynamics in such (non-integrable) nonlinear contexts.

Citation

Fasondini, M., King, J. R., & Weideman, J. A. (2024). Complex-plane singularity dynamics for blow up in a nonlinear heat equation: analysis and computation. Nonlinearity, 37(10), Article 105005. https://doi.org/10.1088/1361-6544/ad700b

• Journal Article Type: Article
• Acceptance Date: Aug 15, 2024
• Online Publication Date: Aug 28, 2024
• Publication Date: Oct 1, 2024
• Deposit Date: Nov 1, 2024
• Publicly Available Date: Nov 1, 2024
• Journal: Nonlinearity
• Print ISSN: 0951-7715
• Electronic ISSN: 1361-6544
• Publisher: IOP Publishing
• Peer Reviewed: Peer Reviewed
• Volume: 37
• Issue: 10
• Article Number: 105005
• DOI: https://doi.org/10.1088/1361-6544/ad700b
• Keywords: Padé approximation, complex singularities, 35C20, matched asymptotic expansions, 32S99, 41A21, 35B44, nonlinear blow up, 65N99, Fourier spectral methods
• Public URL: https://nottingham-repository.worktribe.com/output/39159850
• Publisher URL: https://iopscience.iop.org/article/10.1088/1361-6544/ad700b
• Additional Information: Article Title: Complex-plane singularity dynamics for blow up in a nonlinear heat equation: analysis and computation; Journal Title: Nonlinearity; Article Type: paper; Copyright Information: © 2024 The Author(s). Published by IOP Publishing Ltd and the London Mathematical Society; Date Received: 2023-08-07; Date Accepted: 2024-08-15; Online publication date: 2024-08-28