Rigorous numerics for nonlinear heat equations in the complex plane of time

Abstract

In this paper, we introduce a method for computing rigorous local inclusions of solutions of Cauchy problems for nonlinear heat equations for complex time values. The proof is constructive and provides explicit bounds for the inclusion of the solution of the Cauchy problem, which is rewritten as a zero-finding problem on a certain Banach space. Using a solution map operator, we construct a simplified Newton operator and show that it has a unique fixed point. The fixed point together with its rigorous bounds provides the local inclusion of the solution of the Cauchy problem. The local inclusion technique is then applied iteratively to compute solutions over long time intervals. This technique is used to prove the existence of a branching singularity in the nonlinear heat equation. Finally, we introduce an approach based on the Lyapunov–Perron method for calculating part of a center-stable manifold and prove that an open set of solutions of the Cauchy problem converge to zero, hence yielding the global existence of the solutions in the complex plane of time.

Blow up in a periodic semilinear heat equation

“…There are five time scales that need to be considered. Setting w = w 0 + ϵw 1 + ϵ 2 w 2 + • • •, where w 0 (x, 0) = cos x, w k (x, 0) = 0, k ≥ 0, one obtains a sequence of linear PDEs: (23) and (24).…”

“…Other complex-plane studies of the NLH have been reported in [22,23]. In both papers the equation was studied numerically in the complex t-plane.…”