A blowup solution of a complex semi-linear heat equation with an irrational power

TL;DR

This paper constructs a finite-time blowup solution for a complex semi-linear heat equation with an arbitrary (including irrational) power, revealing new phenomena in the imaginary part's behavior near singularity.

Contribution

It extends the analysis of blowup solutions to complex heat equations with non-integer powers, introducing novel phenomena and employing a reduction to finite dimensions and topological methods.

Findings

Constructed a blowup solution with a single singularity.

Described the asymptotic profile of the solution near blowup.

Discovered a new phenomenon of sign change in the imaginary part near the singularity.

Abstract

In this paper, we consider the following semi-linear complex heat equation \begin{eqnarray*} \partial_t u = \Delta u + u^p, u \in \mathbb{C} \end{eqnarray*} in $\mathbb{R}^n,$ with an arbitrary power $p,$ $ p > 1$. In particular, $p$ can be non integer and even irrational. We construct 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 $a.$ Moreover, we also describe the asymptotics of the solution by the following final profiles: \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||} > 0 , \text{ as } x \to a. \end{eqnarray*} In addition to that, since we also have $u_1 (0,t) \to + \infty$ and $u_2(0,t) \to - \infty$ as $t \to T,$ the blowup in the imaginary part shows a new phenomenon unkown for the standard heat equation in the real case: a non constant sign near the singularity, with the existence of a vanishing surface for the imaginary part, shrinking to the origin. In our work, we have succeeded to extend for any power $p$ where the non linear term $u^p$ is not continuous if $ p$ is not integer. In particular, the solution which we have constructed has a positive real part. We study our equation as a system of the real part and the imaginary part $u_1$ and $u_2$. Our work 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.