On a nonhomogeneous heat equation on the complex plane

TL;DR

This paper studies the existence, uniqueness, and long-term behavior of solutions to a nonhomogeneous heat equation on the complex plane involving a specific diffusion operator with subharmonic potential.

Contribution

It introduces a novel approach combining heat kernel estimates and fixed point methods to analyze a complex parabolic equation with a nonhomogeneous diffusion operator.

Findings

Established existence and uniqueness of solutions.

Analyzed asymptotic behavior of solutions.

Extended heat kernel estimates to complex nonhomogeneous operators.

Abstract

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.