Abstract: This paper considered the global non-existence of solutions
of nonlinear $p(x)$-Kirchhoff systems with dynamic boundary
conditions, which involve nonlinear external damping terms $Q$ and
nonlinear driving forces $f$. Through the study of the natural
energy associated to the solutions $u$ of the systems, the
nonexistence of global solutions, when the initial energy is
controlled above by a critical value was proved. And the
$p$-Kirchhoff equations involving the quasilinear homogeneous
$p$-Laplace operator were extended to the $p(x)$-Kirchhoff equations
which have been used in the last decades to model various
phenomena.

Key words: $p(x)$-Kirchhoff systems, global non-existence, nonlinear source and boundary damping terms