Lack of superstable trajectories in linear viscoelasticity: a numerical approach

Given a positive operator A on some Hilbert space, and a nonnegative decreasing summable function μ, we consider the abstract equation with memory u¨(t)+Au(t)-∫0tμ(s)Au(t-s)ds=0modeling the dynamics of linearly viscoelastic solids. The purpose of this work is to provide numerical evidence of the fact that the energy E(t)=(1-∫0tμ(s)ds)‖u(t)‖12+‖u˙(t)‖2+∫0tμ(s)‖u(t)-u(t-s)‖12dsof any nontrivial solution cannot decay faster than exponential, no matter how fast might be the decay of the memory kernel μ. This will be accomplished by simulating the integro-differential equation for different choices of the memory kernel μ and of the initial data.