Blow up in a periodic semilinear heat equation

• Published in: Physica. D Vol. 446; p. 133660
• Main Authors: Fasondini, M., King, J.R., Weideman, J.A.C.
• Format: Journal Article
• Language: English
• Published: Elsevier B.V 01.04.2023
• Subjects: Complex singularities; Fourier spectral methods; Matched asymptotic expansions; Nonlinear blow up; Matched asymptotic expansions; Nonlinear blow up; Complex singularities; Fourier spectral methods
• ISSN: 0167-2789; 1872-8022
• DOI: 10.1016/j.physd.2023.133660

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 asymptotic approximations of the solution on different timescales that are, in combination, valid over the entire space and time interval right up to and including the blow-up time. Both the asymptotic analysis and the numerical methods benefit from a well-known reciprocal substitution that transforms the problem into one that does not blow up but remains bounded. This allows for highly accurate computations of blow-up times and the solution profile at the critical time, which are then used to confirm the asymptotics. The approach also makes it possible to continue a solution numerically beyond the singularity. The specific post-blow-up dynamics are believed to be presented here for the first time. •Asymptotics valid over the entire space and time interval right up to blow-up.•Reciprocal substitution allows highly accurate numerics which confirm asymptotics.•Continue solutions numerically beyond the blow-up time.•Specific post-blow-up dynamics are believed to be presented for the first time.