Uniform Decaying Property of Solutions for Anisotropic Viscoelastic Systems

TL;DR

This paper proves the uniform decay of solutions for an anisotropic viscoelastic system with mixed boundary conditions, covering cases of polynomial and exponential decay of the relaxation tensor.

Contribution

It establishes a unified proof of uniform decay properties for the system under different decay rates of the relaxation tensor.

Findings

Proves UDP for polynomial decay of relaxation tensor.

Proves UDP for exponential decay of relaxation tensor.

Provides a unified approach applicable to both decay cases.

Abstract

The paper concerns about the uniform decaying property (abbreviated by UDP) of solutions for an anisotropic viscoelastic system in the form of integrodifferential system (abbreviated by VID system) with mixed type boundary condition. The mixed type condition consists of the homogeneous displacement boundary condition and a homogeneous traction boundary condition or with a dissipation. By using a dissipative structure of this system, we will prove the UDP in a unified way for the two cases, which are, when the time derivative of relaxation tensor decays with polynomial order and it decays with exponential order.