On the time decay for an elastic problem with three porous structures

Abstract

AbstractIn this paper, we study the three-dimensional porous elastic problem in the case that three dissipative mechanisms act on the three porosity structures (one in each component). It is important to remark that we consider the case when the material is not centrosymmetric, and therefore, some coupling, not previously considered in the literature concerning the time decay of solutions in porous elasticity, can appear in the system of field equations. The new couplings provided in this situation show a strong relationship between the elastic and the porous components of the material. In this situation, we obtain an existence and uniqueness result for the solutions to the problem using the Lumer-Phillips corollary to the Hille-Yosida theorem. Later, assuming a certain condition determining a “very strong” coupling between the material components, we can use the well-known arguments for dissipative semigroups to prove the exponential stability of the solutions to the problem. It is worth emphasizing that the proposed condition allows bringing the decay of the dissipative porous structure of the problem to the macroscopic elastic structure.