Long-time behaviour of a one-dimensional porous-elastic system with large amplitude

The long-time behaviour of a one-dimensional viscoelastic porous-elastic system with large amplitude is considered in the present paper. We prove an exponential decay result under the weak dissipation produced by the viscoelasticity in the first component (displacement of the solid elastic material) and a frictional damping acting in the second equation. The major difficulty encountered in the analysis stems from the non-applicability of the energy method at least in the usual way. This is due to the large deformation in the material which causes stiffness and increasing in the vibration frequency. This may result in wave break or blow up in finite time. This issue is to be overcome by treating strongly positive definite relaxation function.