Stability and blow-up analysis for a Kirchhoff-type system with logarithmic dissipation mechanisms and logarithmic sources

Adel M. Al-Mahdi; Mohammad M. Al-Gharabli; Mohammad Kafini; Zaid Sawlan; Adel M. Al-Mahdi; Mohammad M. Al-Gharabli; Mohammad Kafini; Zaid Sawlan

doi:10.3934/math.2026508

AIMS Mathematics

2026, Volume 11, Issue 5: 12373-12396. doi: 10.3934/math.2026508

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Research article

Stability and blow-up analysis for a Kirchhoff-type system with logarithmic dissipation mechanisms and logarithmic sources

1.
Department of Mathematics and Statistics, King Fahd University of Petroleum and Minerals, Dhahran 31261, Saudi Arabia
2.
The Interdisciplinary Research Center in Construction and Building Materials, King Fahd University of Petroleum and Minerals, Dhahran 31261, Saudi Arabia
3.
Interdisciplinary Research Center for Refining & Advanced Chemicals, King Fahd University of Petroleum and Minerals, Dhahran 31261, Saudi Arabia

Received: 25 February 2026 Revised: 17 April 2026 Accepted: 23 April 2026 Published: 06 May 2026
MSC : 35L05, 35L20, 45K05, 93D23

This paper investigated, for the first time, a coupled Kirchhoff-type wave system incorporating logarithmic damping and logarithmic source terms (external forces). We established both energy decay and finite-time blow-up results, emphasizing the novel interaction between the two logarithmic components acting as dissipative and driving mechanisms. In the stable region, we constructed a suitable Lyapunov functional and employed refined logarithmic estimates to derive a polynomial decay rate of the total energy. Conversely, for initial data belonging to the unstable set, we proved that the corresponding solutions blow up in finite time using a concavity argument. In addition, we provided some numerical examples to illustrate the stability and blow-up theoretical results. This study presents the first comprehensive analysis of a Kirchhoff-type system with logarithmic damping, revealing how the combined effects of the nonlocal Kirchhoff tension and logarithmic nonlinearities govern the transition between global stabilization and blow-up behavior.
- Kirchhoff-type system,
- logarithmic nonlinearity,
- multiplier method,
- finite difference method
Citation: Adel M. Al-Mahdi, Mohammad M. Al-Gharabli, Mohammad Kafini, Zaid Sawlan. Stability and blow-up analysis for a Kirchhoff-type system with logarithmic dissipation mechanisms and logarithmic sources[J]. AIMS Mathematics, 2026, 11(5): 12373-12396. doi: 10.3934/math.2026508

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Abstract

This paper investigated, for the first time, a coupled Kirchhoff-type wave system incorporating logarithmic damping and logarithmic source terms (external forces). We established both energy decay and finite-time blow-up results, emphasizing the novel interaction between the two logarithmic components acting as dissipative and driving mechanisms. In the stable region, we constructed a suitable Lyapunov functional and employed refined logarithmic estimates to derive a polynomial decay rate of the total energy. Conversely, for initial data belonging to the unstable set, we proved that the corresponding solutions blow up in finite time using a concavity argument. In addition, we provided some numerical examples to illustrate the stability and blow-up theoretical results. This study presents the first comprehensive analysis of a Kirchhoff-type system with logarithmic damping, revealing how the combined effects of the nonlocal Kirchhoff tension and logarithmic nonlinearities govern the transition between global stabilization and blow-up behavior.

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