A blow-up problem for a nonlinear heat equation in the complex plane of time

Solutions of a nonlinear heat equation are numerically computed in the time variable t lying in the complex plane, and possible singularities are sought. It turns out that in the complex half plane \ R[t] ≥ 0 \ , where R denotes the real part of a complex number, there is no singularity other than that which exists on the real line. However, if we compute further in the Riemann surface, new singularities are found. A certain nonlinear Schrödinger equation which is associated with our problem is also computed numerically and we propose a conjecture that it is well-posed globally in time.