Uniform decay estimates for the semi-linear wave equation with locally distributed mixed-type damping via arbitrary local viscoelastic versus frictional dissipative effects

We are concerned with the stabilization of the wave equation with locally distributed mixed-type damping via arbitrary local viscoelastic and frictional effects. Here, one of the novelties is: the viscoelastic and frictional damping together effect only in a part of domain, not in entire domain, which is only assumed to meet the piecewise multiplier geometric condition that their summed interior and boundary measures can be arbitrarily small. Furthermore, there is no other additional restriction for the location of the viscoelastic-effect region. That is, it is dropped that the viscoelastic-effect region includes a part of the system boundary, which is the fundamental condition in almost all previous literature even if when two types of damping together cover the entire system domain. The other distinct novelty is: in this article we remove the fundamental condition that the derivative of the relaxation function is controlled by relaxation function itself, which is a necessity in the previous literature to obtain the optimal uniform decay rate. Under such weak conditions, we successfully establish a series of decay theorems, which generalize and extend essentially the previous related stability results for viscoelastic model regardless of local damping case, entire damping case and mixed-type damping case.