Asymptotic Behavior of the Solution to the Cauchy Problem for a Nonlinear Equation

Sufficient conditions are found for the existence and exponential decay of a global solutionto the Cauchy problem for a nonlinear partial differential equation generalizing a dampedsixth-order Boussinesq equation with double dispersion and the equation of transverse oscillationsof a viscoelastic Voigt–Kelvin beam under the action of external and internal friction whosedeformation is considered taking into account the correction for the inertia of section rotation.