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arXiv · 2406.06895

On a nonhomogeneous heat equation on the complex plane

Duong Ngoc Son·Tran Van Thuy·Pham Truong Xuan
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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.

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Duong Ngoc Son, Tran Van Thuy, Pham Truong Xuan. 2024-06-11. On a nonhomogeneous heat equation on the complex plane. https://doi.org/10.1142/s0129167x24500605

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