Intrinsic decay rates for the energy of a nonlinear viscoelastic equation modeling the vibrations of thin rods with variable density (Q525392)

The authors investigate the long time behaviour of a nonlinear partial differential equation with a memory term modeling the vibrations of a viscoelastic thin rode with variable density. The corresponding initial-boundary value problem has the form \[ \begin{aligned} &|u_t|^\rho u_{tt}-\Delta u_{tt}-\Delta u + \int_0^t g(t-s)\Delta u(s)\,ds=0\;\text{in}\;\Omega\times (0,\infty), \\ &u=0\;\text{on}\;\partial\Omega\times (0,\infty);\;u(0)=u^0,\;u_t(0)=u^1,\;\text{for}\;x\in \Omega, \end{aligned} \] where \(\Omega\) is bounded in \(\mathbb{R}^N,N\leq 3\). The decay is studied for the transformed equation in the abstract form \[ \frac d{dt}\rho(u_t)+Au_{tt}+\gamma A^\theta u_{t}+Au-\int_0^t g(t-s)Au(s)\,ds=0, \] where \(A\) is a self-adjoint positive definite operator in a Hilbert space, \(s\mapsto \rho(s)\) is a continuous monotone increasing function.