General stability for the Kirchhoff-type equation with memory boundary and acoustic boundary conditions

Abstract

In this paper we consider the existence and general energy decay rate of global solution to the mixed problem for the Kirchhoff-type equation with memory boundary and acoustic boundary conditions. In order to prove the existence of solutions, we employ the Galerkin method and compactness arguments. Besides, we establish an explicit and general decay rate result using the perturbed modified energy method and some properties of the convex functions. Our result is obtained without imposing any restrictive assumptions on the behavior of the relaxation function at infinity. These general decay estimates extend and improve some earlier results, i.e., exponential or polynomial decay rates.