A computer-assisted proof for nonlinear heat equations in the complex plane of time (Regularity and Asymptotic Analysis for Critical Cases of Partial Differential Equations)

Abstract

In the present article 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 study is to find out dynamics of blow-up phenomena with computer assistance. We numerically prove that the solution has branching singularity and globally exists on the real axis except the singular point. Such a computer-assisted proof is obtained using rigorous numerics, which consists of careful blend of functional analysis, semigroup theory, numerical analysis, fixed-point theory, the Lyapunov-Perron method and interval arithmetic. This result generalizes the previous results of the complex valued nonlinear heat equation in terms of considering the different boundary condition and without the assumption of initial data being close to a constant.