On a nonhomogeneous heat equation on the complex plane

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.