Construction of a blow-up solution for a complex nonlinear heat equation.

We construct a solution to a complex nonlinear heat equation which blows up in nite time T only at one blow-up point. We also give a sharp description of its blow-up pro le. The proof relies on the reduction of the problem to a nite 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