Profile for the imaginary part of a blowup solution for a complex-valued seminar heat equation

TL;DR

This paper constructs a finite-time blowup solution for a complex semilinear heat equation in multiple dimensions, detailing the asymptotic behavior of both real and imaginary parts near the blowup point.

Contribution

It introduces a novel method to analyze the blowup profile of the imaginary part in multi-dimensional complex heat equations, combining finite-dimensional reduction and topological arguments.

Findings

Explicit blowup profile for the imaginary part near the singularity

First derivation of imaginary part blowup behavior in multiple dimensions

Methodology combining reduction and topological index theory

Abstract

In this paper, we consider the following complex-valued semilinear heat equation \begin{eqnarray*} \partial_t u = \Delta u + u^p, u \in \mathbb{C}, \end{eqnarray*} in the whole space $\mathbb{R}^n$, where $ p \in \mathbb{N}, p \geq 2$. We aim at constructing for this equation a complex solution $u = u_1 + i u_2$, which blows up in finite time $T$ and only at one blowup point $a$, with the following estimates for the final profile \begin{eqnarray*} u(x,T) &\sim & \left[ \frac{(p-1)^2 |x-a|^2}{ 8 p |\ln|x-a||}\right]^{-\frac{1}{p-1}}, u_2(x,T) &\sim & \frac{2 p}{(p-1)^2} \left[ \frac{ (p-1)^2|x-a|^2}{ 8p |\ln|x-a||}\right]^{-\frac{1}{p-1}}\frac{1}{ |\ln|x-a||} , \text{ as } x \to a. \end{eqnarray*} Note that the imaginary part is non-zero and that it blows up also at point $a$. Our method relies on two main arguments: the reduction of the problem to a finite dimensional one and a topological argument based on the index theory to get the conclusion. Up to our knowledge, this is the first time where the blowup behavior of the imaginary part is derived in multi-dimension.