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Research Article | Open Access

Local well-posedness, global existence and blow-up for a heat equation with interior logarithmic source and nonlinear dynamic boundary condition

Dengming Liu( ), Fang He, Qi Chen
School of Mathematics and Statistics, Hunan University of Science and Technology, Xiangtan, Hunan 411201, P. R. China
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Abstract

In this article, a heat equation with an interior logarithmic source and a nonlinear dynamic boundary condition is considered. After establishing the local well-posedness, the global existence and finite time blow-up results of the solutions with different initial energy levels are given. Moreover, the lower bound of the maximal existence time of the weak solution is deduced for the special case m = 2.

Keywords

local well-posednessglobal existenceblow-uplogarithmic nonlinearity sourcenonlinear dynamic boundary condition
CLC number: 35K20, 35B44

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Communications in Analysis and Mechanics
Volume 18 Issue 1,
March 2026
Pages 70-97
DOI: 10.3934/cam.2026003

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Cite this article:
Liu D, He F, Chen Q. Local well-posedness, global existence and blow-up for a heat equation with interior logarithmic source and nonlinear dynamic boundary condition. Communications in Analysis and Mechanics, 2026, 18(1): 70-97. https://doi.org/10.3934/cam.2026003

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Received: 01 July 2025
Revised: 07 October 2025
Accepted: 12 November 2025
Published: 21 January 2026
©2026 the Author(s), licensee AIMS Press.

This is an open access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0)