Profile for a simultaneously blowing up solution for a complex valued semilinear heat equation

TL;DR

This paper constructs a finite-time blow-up solution for a complex semilinear heat equation, providing a detailed blow-up profile and demonstrating simultaneous blow-up of real and imaginary parts.

Contribution

It introduces a novel method to construct and analyze a finite-time blow-up solution with a precise profile for a complex nonlinear heat equation.

Findings

Solution blows up at a single point in finite time

Real and imaginary parts blow up simultaneously

Provides a sharp description of the blow-up profile

Abstract

We construct a solution to a complex nonlinear heat equation which blows up in finite time $T$ only at one blow-up point. We also give a sharp description of its blow-up profile. The proof relies on the reduction of the problem to a finite dimensional one and the use of index theory to conclude. We note that the real and imaginary parts of the constructed solution blow up simultaneously.