In this paper we consider a viscoelastic double-Kirchhoff type wave
equation of the form $$
u_{tt}-M_{1}(\|\nabla
u\|^{2})\Delta
u-M_{2}(\|\nabla
u\|_{p(x)})\Delta_{p(x)}u+(g\ast\Delta
u)(x,t)+\sigma(\|\nabla
u\|^{2})h(u_{t})=\phi(u),
$$ where the functions $M_{1},M_{2}$ and
$\sigma, \phi$ are real valued functions
and $(g\ast\nabla u)(x,t)$ is the
viscoelastic term which are introduced later. Under appropriate
conditions for the data and exponents, the general decay result and
blow-up of solutions are proved with positive initial energy. This study
extends and improves the previous results in the literature to
viscoelastic double-Kirchhoff type equation with degenerate nonlocal
damping and variable-exponent nonlinearities.
A nonlinear $r(x)$-Kirchhoff type hyperbolic equation: Stability result and blow up of solutions with positive initial energy
