We consider a class of coupled systems with past history acting only in one equation. We show in the abstract setting that the dissipation given by the history term is not strong enough to produce exponential stability. We show that the solution decays polynomially to zero, with rates that can be improved depending on the regularity of the initial data. Some examples are given.
D.S.A. Júnior. L.P.V. Matos. M.L. Santos. "Polynomial decay to a class of abstract coupled systems with past history." Differential Integral Equations 25 (11/12) 1119 - 1134, November/December 2012. https://doi.org/10.57262/die/1356012253