@article{Zhou2025, 
author = {Lijun Zhou and Ning Pan},
title = {The global existence and blow-up of the solutions for a fractional Kirchhoff hyperbolic equations with viscoelastic term and logarithmic term},
year = {2025},
journal = {Electronic Research Archive},
volume = {33},
number = {11},
pages = {7126-7145},
keywords = {hyperbolic, fractional Laplacian, Kirchhoff equations, viscoelastic, logarithmic nonlinear term, global existence, blow-up},
url = {https://www.sciopen.com/article/10.3934/era.2025315},
doi = {10.3934/era.2025315},
abstract = {This article investigates a type of hyperbolic equation of the fractional Kirchhoff with viscoelastic and logarithmic nonlinear terms subject to homogeneous Dirichlet-boundary:         {                                        u                          t              t                                +          M          (          [          u                      ]                          s                                      2                                )          (          −          Δ                      )            s                    u          −                      ∫                          0                                      t                                g          (          t          −          τ          )          (          −          Δ                      )            s                    u          (          τ          )          d          τ          +                      u            t                    =                      |                    u                                    |                                      h              −              2                                u          ln          ⁡                      |                    u                      |                    ,                                    in          Ω          ×          (          0          ,          ∞          )          ,                                      u          (          ⋅          ,          0          )          =                      u            0                    ,                                                    u            t                    (          ⋅          ,          0          )          =                      u            1                    ,                                                        in          Ω          ,                                      u          (          ⋅          ,          t          )          =          0          ,                          on          ∂          Ω          ×          (          0          ,          ∞          )          ,                        where    [  u      ]          s       is the Gagliardo semi-norm of    u  ,    (  −  Δ      )    s   is the fractional Laplacian with    s  ∈  (  0  ,  1  ),    2  <  2  γ  <  h  <      2    s    ∗  ,        u    0   and        u    1   are the initial functions, and    Ω  ⊂            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.}
}