SUMMARYThe initial boundary value problem for linear elastodynamic system for viscoporous materials is considered. Exponential decay of solutions via the linear boundary feedback is established. Existence of solutions is obtained through the method of c 0 -semigroups. Exponential stabilization is derived via a proper collection of ideas of observability inequality, energy identity and c 0 -semigroup of contractions.

Abstract. The systems of evolution equations modelling elasticity and thermoelasticity of viscoporous bounded media are considered. The existence of c 0 -semigroups of contractions defining solutions to the systems is proved. The asymptotic vanishing of energies of solutions when t → ∞ is explained. Introduction and statement of problemsAn increasing interest is observed in recent years to determine the decay behavior of the solutions of several elasticity problems. In classical thermoelasticity theory the decay effects were studied in the book [12] [18] there was studied the decay of solutions of the one-dimensional elasticity models where besides of thermal dissipation the porosity dissipation is taken into account. The similar kind of problems (indirect internal stabilization of coupled evolution equations) has recently been the focus of interest of other authors [1], [6]. Our goal in this paper is to establish the stabilization of solutions for two-and three-dimensional elasticity and thermoelasticity systems for viscoporous materials.Let us begin from evolution equations [4], [3]where T denotes the stress tensor, u denotes the displacement vector, h denotes equilibrated stress vector, g denotes intrinsic equilibrated body force and the scalar function φ denotes the change in the volume fraction from the reference configuration, (divT ) i := n j=1 ∂ j T ij , n = 2, 3, denotes the dimension of space and u has the same dimension.In the linear theory there are considered the following constitutive relationswhere σ(u) denotes the elasticity stress tensor,E denotes the dissipation friction and is taken to be equal E := −r∂ t φ, I denotes the n × n unit matrix. The coefficients a, b, γ, µ > 0, and for simplicity of the further considerations we put ρ = 1 and J = 1. After subjecting the system (1.1) with initial and boundary conditions we obtain the following system for u and φ.(1.2)In the above D ⊂ R n , denotes a bounded domain with boundary ∂D having regularity of class C 2 , ∆ e := µ∆I + (µ + λ)∇div denotes the elliptic Lamé operator, R + := (0, +∞), Bφ = φ or Bφ = ∂ ν φ, where ν denotes the outer unit normal vector to ∂D. Physically D is the region occupied by the body in the reference configuration.To take into consideration also the thermal dissipation, the third equation is added to the system (1.1)where η denotes the entropy and q the heat flux (see [11]), T 0 > 0 is a constant. From the classical linear theory we take the following constitutive relations for q and η:and for g in (1.1) we takewhere θ denotes the temperature. The coefficients d, M, M 1 > 0, and for simplicity we put T 0 = 1. Unauthenticated Download Date | 5/9/18 9:35 PM Elasticity in viscoporous media 759After subjecting the system (1.1), (1.3) with initial and boundary conditions we obtain the following system for u, φ, θ:

We study the problem of exponential decaying for solutions of porous-thermoelasticity system, when timeIn the considerations we apply the idea of compact decoupling for the system of equations. The exponential decaying property is proved first for the corresponding decoupled system, which is simpler to handle, then the property is derived for the original system.