Existence and blow-up of logarithmic-viscoelastic wave equation with fractional Kirchhoff type and variables exponents

In this paper, we investigate the existence of solutions for a Viscoelastic Wave equation with fractional kirchhof f type and logarithmic term and variables exponents. In the initial boundary value problem of this work, we focus on using the Gagliardo seminorm [.]α,2, and the fractional Laplace operator (-∆)α where 0 < α < 1. Firstly, we prove the existence of the weak solutions, concerning this issue under suitable assumptions on the variables exponents of nonlinear source term, we use the Galerkin’s approximation method. Subsequently, we establish the long-time behavior of solutions with nonpositive initial energy.