Existence and asymptotic stability for a semilinear damped wave equation with dynamic boundary conditions involving variable nonlinearity

We investigate the solvability of a class of quasilinear elliptic equations characterized by a (p(x); k(x)) growth structure and nonlinear boundary conditions, speci cally in the context of Kelvin-Voigt damping with arbitrary data. Our approach involves analyzing the problem within appropriate functional spaces, utilizing Lebesgue and Sobolev spaces with variable exponents. In the rst step, we establish the existence and uniqueness of results for solutions to the model, provided the data meet certain regularity conditions. Our methodology primarily relies on xed-point theory and Faedo-Galerkin techniques, incorporating some novel strategies. In the second part, we consider scenarios with su ciently large data sets and show that the system's energy grows exponentially.