A new approach to abstract linear viscoelastic equation in Hilbert space

We study an abstract linear viscoelastic equation in Hilbert space. It fits some convolution type of integro-differential equations which govern dynamics in viscoelastic or thermoviscoelastic materials. A new semigroup approach is proposed which is adapted from a semigroup approach designed originally to a second-order abstract linear integro-differential equation. As compared with the existing semigroup approach, the system contraction can be obtained readily. Sufficient conditions for the system exponential stability are also established. An illustrative example is given, and numerical simulations are provided. They are in good agreement with the theoretical analysis.