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 ${\bar{\mathrm{\square }}}_{\phi }=\bar{D}{\bar{D}}^{\ast }$, where $\bar{D}f=\bar{\mathrm{\partial }}f+{\phi }_{\bar{z}}f$, the function $\phi$ is smooth and subharmonic on $\mathbb{C}$, and ${\bar{D}}^{\ast }$ is the formal adjoint of $\bar{D}$. Our method combines certain estimates of heat kernel associating with the homogeneous linear equation of Raich \cite{raich06} and a fixed point argument.