Instability, Nonexistence, and Uniquenessin Elasticity with Porous Dissipation

In what follows, we consider a theory for the behaviour of porous solids such that the matrix material is elastic and the interstices are void of material; it is a generalization of the classical theory of elasticity. The theory of porous elastic material has been established by Cowin andNunziato [2, 11]. In this theory, the bulk density is the product of two scalar fields, the matrix material density and the volume fraction field; it is studied in the book of Ciarletta and Ieşan [1]. Thermal effects were included in the book of Ieşan [4]. Results on linear and nonlinear problems have been obtained recently [5, 14, 15]. The aim of this paper is the study of the qualitative behaviour of the solutions of the elasticity with porous dissipation. It is worth noting that there are very few contributions to this topic. We can recall some contributions to the nonlinear problem in the book of Ciarletta and Ieşan [1], but they are in the case where porous dissipation is not present. Here we prove instability and nonexistence in the nonlinear case when some conditions on the internal energy and the dissipation function are satisfied. We also work in the linear case and we prove exponential growth of solutions whenever the initial data satisfy several conditions. A uniqueness result is also obtained in the case of the backward in time problem for the linear equations. The results of this paper are of interest from the mechanical and from the mathematical point of view: from the mechanical point of view because we obtain some qualitative results in the theory of elastic materials with voids; from the mathematical point of view because we extend some results that were known