In this paper, we investigate the existence, uniqueness, and asymptotic behaviors of mild solutions of parabolic evolution equations on complex plane, in which the diffusion operator has the form ${\bar{□}}_{φ} = \bar{D} {\bar{D}}^{*}$ , where $\bar{D} f = \bar{\partial} f + φ_{\bar{z}} f$ , the function $φ$ is smooth and subharmonic on $ℂ$ , and ${\bar{D}}^{*}$ is the formal adjoint of $\bar{D}$ . Our method combines certain estimates of heat kernel associating with the homogeneous linear equation of Raich [Heat equations in $ℝ \times ℂ$ , J. Funct. Anal. 240(1) (2006) 1–35] and a fixed point argument.

Communicated by Franc Forstneric

Keywords:

AMSC: 32W30, 32W50

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