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

In this paper, we consider the following complex-valued semilinear heat equation partial derivative(t)u = Delta u + u(p), u is an element of C, in the whole space R-n, where p is an element of N, p >= 2. We aim at constructing for this equation a complex solution u = u(1) + iu(2) which blows up in finite time T and only at one blowup point a, with the following estimates for the final profile u(x, T) similar to [(p - 1)(2) vertical bar x - a vertical bar(2)/8p vertical bar ln vertical bar x - a vertical bar vertical bar](-1/p-1), u(2)(x, T) similar to 2p/(p - 1)(2) [(p - 1)(2) vertical bar x - a vertical bar(2)/8p vertical bar ln vertical bar x - a vertical bar vertical bar](-1/p-1) x 1/vertical bar ln vertical bar x - a vertical bar vertical bar, as x -> a. 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. (C) 2019 Elsevier Inc. All rights reserved.