Arbitrary decays in linear viscoelasticity

It is by now well known that a necessary condition for the exponential (polynomial) decay of the energy of a problem arising in viscoelasticity is that the kernel (which appears in the memory term) itself be of exponential (polynomial) type. By a kernel of exponential (polynomial) type we mean that the product of this kernel with an exponential (polynomial) function is summable. Some researchers have started from this condition to seek other (sufficient) conditions ensuring exponential or polynomial decay of the energy. In this work we generalize these works to an arbitrary decay not necessarily of an exponential or polynomial rate.