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 \(̄φ = D̄\, D̄^{}\), where \(D̄ f = ∂̄f + φz̄ f\), the function \(φ\) is smooth and subharmonic on \(C\), and \(D̄^{}\) is the formal adjoint of \(D̄\). Our method combines certain estimates of heat kernel associating with the homogeneous linear equation of Raich {raich06} and a fixed point argument.