General decay for a viscoelastic Kirchhoff equation with Balakrishnan-Taylor damping, dynamic boundary conditions and a time-varying delay term

Abstract

In this paper, we consider a viscoelastic Kirchhoff equation with Balakrishnan-Taylor damping, dynamic boundary conditions and a time-varying delay term acting on the boundary. By using the Faedo-Galerkin approximation method, we first prove the well-posedness of the solutions. By introducing suitable energy and perturbed Lyapunov functionals, we then prove the general decay results, from which the usual exponential and polynomial decay rates are only special cases. To achieve these results, we consider the following two cases according to the coefficient α of the strong damping term: for the presence of the strong damping term (α>0), we use the strong damping term to control the time-varying delay term, under a restriction of the size between the time-varying delay term and the strong damping term; for the absence of the strong damping term (α=0), we use the viscoelasticity term to control the time-varying delay term, under a restriction of the size between the time-varying delay term and the kernel function.