Coupled second order evolution equations with fading memory: Optimal energy decay rate

Of concern is the Cauchy problem for coupled second order semilinear evolution equations in a Hilbert space, with indirect memory-damping. We find an approach to obtain successfully an optimal rate of uniform decay for the system energy, only under basic conditions on the memory kernels. Simultaneously, the same rate is also obtained (with less difficulty) for the corresponding single memory-dissipative second order evolution equations. As can be seen, our results essentially improve the previously related ones in the literature. The abstract results are then applied to several concrete problems in the real world.