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.
Categories: math.AP math.CV