Uniform decay in viscoelasticity for kernels with small non-decreasingness zones

We consider a viscoelastic problem with a relaxation function which may be strictly increasing in some sub-intervals. That is we go beyond the zero which was a bound for the derivative of the kernel in the previous works. It is proved that we have different types of decay rates (including the exponential one) provided that the rate and/or the non-decreasingness zone (including the flat zone) is small enough in a certain sense.