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

Stability for the Kirchhoff Plates Equations with Viscoelastic Boundary Conditions in Noncylindrical Domains

We study Kirchhoff plates equations with viscoelastic boundary conditions in a noncylindrical domain. This work is devoted to proving the global existence, uniqueness of solutions, and decay of the energy of solutions for Kirchhoff plates equations in a non-cylindrical domain

Stability for the Kirchhoff Plates Equations with Viscoelastic Boundary Conditions in Noncylindrical Domains

We study Kirchhoff plates equations with viscoelastic boundary conditions in a noncylindrical domain. This work is devoted to proving the global existence, uniqueness of solutions, and decay of the energy of solutions for Kirchhoff plates equations in a noncylindrical domain

General Stability for the Viscoelastic Wave Equation with Nonlinear Time-Varying Delay, Nonlinear Damping and Acoustic Boundary Conditions

This paper is focused on energy decay rates for the viscoelastic wave equation that includes nonlinear time-varying delay, nonlinear damping at the boundary, and acoustic boundary conditions. We derive general decay rate results without requiring the condition a2>0 and without imposing any restrictive growth assumption on the damping term f1, using the multiplier method and some properties of the convex functions. Here we investigate the relaxation function ψ, namely ψ′(t)≤−μ(t)G(ψ(t)), where G is a convex and increasing function near the origin, and μ is a positive nonincreasing function. Moreover, the energy decay rates depend on the functions μ and G, as well as the function F defined by f0, which characterizes the growth behavior of f1 at the origin

General decay for weak viscoelastic Kirchhoff plate equations with delay boundary conditions

Abstract We consider a weak viscoelastic Kirchhoff plate model with time-varying delay in the boundary. By using a suitable energy and Lyapunov function, we obtain a decay rate for the energy, which depends on the behavior of g and α

Controllability of the second-order differential inclusion in Banach spaces

AbstractThe purpose of this paper is to study the controllability for the second-order differential inclusion in Banach spaces. We rely on a fixed point theorem for condensing maps due to Martelli. We consider the damping term x′(·) and find a control u such that the solution satisfies x(T)=x1 and x′(T)=y1