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; Dengming Liu; Fang He; Qi Chen

doi:10.3934/cam.2026003

Communications in Analysis and Mechanics

2026, Volume 18, Issue 1: 70-97. doi: 10.3934/cam.2026003

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

School of Mathematics and Statistics, Hunan University of Science and Technology, Xiangtan, Hunan 411201, P. R. China

Received: 01 July 2025 Revised: 07 October 2025 Accepted: 12 November 2025 Published: 21 January 2026
35K20, 35B44

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 $.
- local well-posedness,
- global existence,
- blow-up,
- logarithmic nonlinearity source,
- nonlinear dynamic boundary condition
Citation: Dengming Liu, Fang He, Qi Chen. Local well-posedness, global existence and blow-up for a heat equation with interior logarithmic source and nonlinear dynamic boundary condition[J]. Communications in Analysis and Mechanics, 2026, 18(1): 70-97. doi: 10.3934/cam.2026003

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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 $.

References

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