On the porous-elastic system with Kelvin–Voigt damping

In this article we are considering the one-dimensional equations of a homogeneous and isotropic porous elastic solid with Kelvin–Voigt damping. We prove that the semigroup associated with the system (1.3) with Dirichlet–Dirichlet boundary conditions or Dirichlet–Neumann boundary conditions is analytic and consequently exponentially stable. On the other hand, we prove that the system (1.3) with Dirichlet–Neumann boundary conditions has lack of exponential decay and it decays as 1t for the case γ1>0,γ2=0 or γ1=0,γ2>0. Moreover, we prove that this rate is optimal. We apply the main results for the Timoshenko model.