Abstract
The main contributions here are to show the lack of exponential stability and to prove that the t(-1) is the sharp decay rate of problem (1.1). That is to show that for this types of materials, the dissipation produced by the weak-infinite memories are not strong enough to produce an exponential decay of the solution under usual/non usual conditions on the the relaxation functions's growth. This work extends the previous results by Bayoud et al. (Appl Sci 20:18-35, 2018), Zennir and Feng (Math Methods Appl Sci 112:6895-6906, 2018. https://doi.org/10.1002/mma.5201) to the weak-viscoelasticities in two parts. In order to fill this gaps, we use an appropriate estimates.