Large time behavior of solutions for a complex-valued quadratic heat equation

In this paper we study the existence and the asymptotic behavior of global solutions for a parabolic system related to the complex-valued heat equation with quadratic nonlinearity: , with initial data z (0) = u (0) + iv (0). We show that if and as with (|c| is sufficiently small), then the solution is global and converges to a self-similar solution. We also establish the existence of four different self-similar behaviors. These behaviors depend on the values of alpha(1) and . In particular, the real and the imaginary parts of the constructed solutions may have different behaviors in the L (a)-norm for large time. Also, the real part may have different behaviors from those known for the real-valued quadratic heat equation.