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

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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: 

$\left\{\begin{array}{ll} u_{t t}+M\left([u]_{s}^{2}\right)(-\Delta)^{s} u-\int_{0}^{t} g(t-\tau)(-\Delta)^{s} u(\tau) d \tau+u_{t}=|u|^{h-2} u \ln |u|, & \text { in } \Omega \times(0, \infty), \\ u(\cdot, 0)=u_{0}, \quad u_{t}(\cdot, 0)=u_{1}, & \text { in } \Omega, \\ u(\cdot, t)=0, & \text { on } \partial \Omega \times(0, \infty), \end{array}\right.$

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.

Keywords

hyperbolic / fractional Laplacian / Kirchhoff equations / viscoelastic / logarithmic nonlinear term / global existence / blow-up

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