Growth estimates for solutions to initial-boundary value problems in viscoelasticity

AbstractFor the abstract Volterra integro-differential equation utt − Nu + ∝−∞t K(t − τ) u(τ) dτ = 0 in Hilbert space, with prescribed past history u(τ) = U(τ), − ∞ < τ < 0, and associated initial data u(0) = f, ut(0) = g, we establish conditions on K(t), − ∞ < t < + ∞ which yield various growth estimates for solutions u(t), belonging to a certain uniformly bounded class, as well as lower bounds for the rate of decay of solutions. Our results are interpreted in terms of solutions to a class of initial-boundary value problems in isothermal linear viscoelasticity