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We demonstrate the global existence of weak solutions to a class of semilinear strongly damped wave equations possessing nonlinear hyperbolic dynamic boundary conditions. The associated linear operator is 
$(-\unicode[STIX]{x1D6E5}_{W})^{\unicode[STIX]{x1D703}}\unicode[STIX]{x2202}_{t}u$, where 
$\unicode[STIX]{x1D703}\in [\frac{1}{2},1)$ and 
$\unicode[STIX]{x1D6E5}_{W}$ is the Wentzell–Laplacian. A balance condition is assumed to hold between the nonlinearity defined on the interior of the domain and the nonlinearity on the boundary. This allows for arbitrary (supercritical) polynomial growth of each potential, as well as mixed dissipative/antidissipative behaviour.
