Applied and Computational Math Seminar: Rigorous numerics for nonlinear heat equations in the complex plane of time

Speaker: Jonathan Jaquette, Brandeis University
Title: Rigorous numerics for nonlinear heat equations in the complex plane of time

Abstract: In this talk we discuss a computer assisted proof of finite time blow-up in a real valued PDE defined on a compact interval. We accomplish this by developing a numerical method to rigorously compute solutions of Cauchy problems. Then by solving the equation along a contour in the complex plane of time (which transforms the PDE into a complex Ginzburg-Landau like equation) we are able to prove the existence of a branching singularity. Furthermore, using the Lyapunov-Perron method we calculate part of a codimension-0 center-stable manifold of the zero equilibrium. This allows us to prove that our same initial condition which blows up in real time will converge to zero along fixed contours, yielding the global existence of the solution.

Based on joint work with A Takayasu (University of Tsukuba), JP Lessard (McGill University), and H Okamoto (Gakushuin University).

Time: Friday, January 24, 2020, 1:30-2:30pm

Place: Exploratory Hall, Room 4106