The global existence and blow-up of the solutions for a fractional Kirchhoff hyperbolic equations with viscoelastic term and logarithmic term

Lijun Zhou; Ning Pan; Lijun Zhou; Ning Pan

doi:10.3934/era.2025315

Electronic Research Archive

2025, Volume 33, Issue 11: 7126-7145. doi: 10.3934/era.2025315

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Research article

The global existence and blow-up of the solutions for a fractional Kirchhoff hyperbolic equations with viscoelastic term and logarithmic term

Lijun Zhou ,
Ning Pan ^,

Department of Mathematics, Northeast Forestry University, Harbin 150040, China

Received: 05 September 2025 Revised: 29 October 2025 Accepted: 14 November 2025 Published: 24 November 2025

This article investigates a type of hyperbolic equation of the fractional Kirchhoff with viscoelastic and logarithmic nonlinear terms subject to homogeneous Dirichlet-boundary:
$ \left\{ \begin{array}{ll} u_{tt} +M([u]^{2}_{s})(-\Delta)^su-\int_{0}^{t}g(t-\tau)(-\Delta)^su(\tau)d\tau+u_t = |u|^{h-2}u\ln |u|,\ &\text{in } \Omega\times (0,\infty), \\ u(\cdot,0) = u_0,\ \ \ u_t(\cdot,0) = u_1,\ \ \ & \text{in } \Omega,\\ u(\cdot,t) = 0,& \text{on } \partial \Omega \times (0,\infty), \end{array} \right. $
where $ [u]_{s} $ is the Gagliardo semi-norm of $ u, $ $ (-\Delta)^s $ is the fractional Laplacian with $ s\in(0, 1) $, $ 2 < 2\gamma < h < 2_s^* $, $ u_0 $ and $ u_1 $ are the initial functions, and $ \Omega \subset \mathbb{R}^N $ is a bounded domain with a smooth boundary. First, the global existence of solutions is established by combining the Galerkin method with the potential well theory. Subsequently, the finite-time blow-up of solutions is derived via the concavity method and a series of peculiar inequalities.
- hyperbolic,
- fractional Laplacian,
- Kirchhoff equations,
- viscoelastic,
- logarithmic nonlinear term,
- global existence,
- blow-up
Citation: Lijun Zhou, Ning Pan. The global existence and blow-up of the solutions for a fractional Kirchhoff hyperbolic equations with viscoelastic term and logarithmic term[J]. Electronic Research Archive, 2025, 33(11): 7126-7145. doi: 10.3934/era.2025315

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Abstract

This article investigates a type of hyperbolic equation of the fractional Kirchhoff with viscoelastic and logarithmic nonlinear terms subject to homogeneous Dirichlet-boundary:

$ \left\{ \begin{array}{ll} u_{tt} +M([u]^{2}_{s})(-\Delta)^su-\int_{0}^{t}g(t-\tau)(-\Delta)^su(\tau)d\tau+u_t = |u|^{h-2}u\ln |u|,\ &\text{in } \Omega\times (0,\infty), \\ u(\cdot,0) = u_0,\ \ \ u_t(\cdot,0) = u_1,\ \ \ & \text{in } \Omega,\\ u(\cdot,t) = 0,& \text{on } \partial \Omega \times (0,\infty), \end{array} \right. $

where $ [u]_{s} $ is the Gagliardo semi-norm of $ u, $ $ (-\Delta)^s $ is the fractional Laplacian with $ s\in(0, 1) $, $ 2 < 2\gamma < h < 2_s^* $, $ u_0 $ and $ u_1 $ are the initial functions, and $ \Omega \subset \mathbb{R}^N $ is a bounded domain with a smooth boundary. First, the global existence of solutions is established by combining the Galerkin method with the potential well theory. Subsequently, the finite-time blow-up of solutions is derived via the concavity method and a series of peculiar inequalities.

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