Computer-assisted proofs for a nonlinear heat equation in the complex plane of time

Akitoshi Takayasu
Faculty of Engineering, Information and Systems, University of Tsukuba

In this talk, we consider a complex valued nonlinear heat equation. It is well-known that solutions of a real valued nonlinear heat equation blow up in finite time. Our aim of this project is to find out the dynamics of such a blow-up with computer assistance. Extending the time variable of the nonlinear heat equation into the complex plane, the blow-up point, which exists on the real line, shows a branching singularity. We give a proof of the branching singularity using a rigorous integrator based on semigroup theory. Additionally, we also show a computer-assisted proof of global existence of the solution on a straight path from the origin.
This is joint work with Jean-Philippe Lessard (McGill University), Jonathan Jaquette (Brandeis University), and Hisashi Okamoto (Gakushuin University).