  In this article, we investigate the existence, uniqueness, and asymptotic
behaviors of mild solutions of a parabolic evolution equations on complex
plane, in which the diffusion operator has the form \(\overline{\Box}_{\varphi}
= \overline{D}\, \overline{D}^{\ast}\), where \(\overline{D} f =
\bar{\partial}f + \varphi_{\bar{z}} f\), the function \(\varphi\) is smooth and
subharmonic on \(\mathbb{C}\), and \(\overline{D}^{\ast}\) is the formal
adjoint of \(\overline{D}\). Our method combines certain estimates of heat
kernel associating with the homogeneous linear equation of Raich \cite{raich06}
and a fixed point argument.
