Dynamics for a complex-valued heat equation with an inverse nonlinearity

We study the Cauchy problem for a parabolic system which is derived from a complex-valued heat equation with an inverse nonlinearity. First, we provide some criteria for the global existence of solutions. Then we consider the case when the initial data are asymptotically constants and obtain that, depending on the asymptotic limits, the solution quenches at space infinity or exists globally in time.