Rates of decay for locally damped porous-elastic systems with history via operator semigroups

We consider a porous-elastic system with localized history damping, the memory kernel decaying exponentially. We estimate the resolvent of the generator of the associated operator semigroup along the imaginary axis and then obtain an ideal polynomial decay rate of the semigroup, regardless of wave speeds. Moreover, uniform exponential stability of the semigroup is shown if either the wave speeds are equal or an additional local frictional damping is present. These results are generalizations of the previously related ones for porous-elastic or Timoshenko systems with global history damping.