In the present paper, we studied a hyperbolic $p(x)$-biharmonic problem with known no-flux values on the smooth part of the boundary in variable exponent Sobolev spaces. The global existence of a weak solution, using the Galerkin method, was proved. The blow-up of the solution with negative initial functional energy in finite time was discussed using the concavity method. The interaction between the variable exponent in the main equation and the boundary condition is crucial in determining the presence/absence of solutions.