Lack of superstable trajectories in linear viscoelasticity: a numerical approach

Abstract

AbstractGiven a positive operator A on some Hilbert space, and a nonnegative decreasing summable function $$\mu $$ μ , we consider the abstract equation with memory $$\begin{aligned} \ddot{u}(t)+ A u(t)- \int _0^t \mu (s)Au(t-s) ds=0 \end{aligned}$$ u ¨ ( t ) + A u ( t ) - ∫ 0 t μ ( s ) A u ( t - s ) d s = 0 modeling the dynamics of linearly viscoelastic solids. The purpose of this work is to provide numerical evidence of the fact that the energy $$\begin{aligned} {{\textsf{E}}}(t)=\Big (1-\int _0^t\mu (s)ds\Big )\Vert u(t)\Vert ^2_1+\Vert \dot{u}(t)\Vert ^2 +\int _0^t\mu (s)\Vert u(t)-u(t-s)\Vert ^2_1ds \end{aligned}$$ E ( t ) = ( 1 - ∫ 0 t μ ( s ) d s ) ‖ u ( t ) ‖ 1 2 + ‖ u ˙ ( t ) ‖ 2 + ∫ 0 t μ ( s ) ‖ u ( t ) - u ( t - s ) ‖ 1 2 d s of any nontrivial solution cannot decay faster than exponential, no matter how fast might be the decay of the memory kernel $$\mu $$ μ . This will be accomplished by simulating the integro-differential equation for different choices of the memory kernel $$\mu $$ μ and of the initial data.