In this paper, we consider a one-dimensional linear theory of swelling porous elastic soils damped with a single nonlinear feedback. We establish an exponential decay rate, using a multiplier method and some properties of convex functions without imposing any restrictive growth assumption near the origin on the damping term, provided that the wave speeds of the system are equal.
This paper is concerned with the study on the existence of attractors for a nonlinear porous elastic system subjected to a delay-type damping in the volume fraction equation. The study will be performed, from the point of view of quasi-stability for infinite dimensional dynamical systems and from then on we will have the result of the existence of global and exponential attractors. 1. Introduction. In recent years the study of continuous models of deformable bodies has intensified, in particular we have the elastic solids with voids. Due to its great applicability as for example in soil mechanics, petroleum industry, materials sciences and biomechanics, porous solids now play a prominent role in scientific research (cf. [27]). Among the various theories dealing with porous material, we can find a linear theory proposed by Cowin and Nunziato [14, 42] which is a generalization of the elastic theory for materials with voids, considering besides the material elasticity property, the volume fraction of the voids in the material. In this theory, the bulk density ρ = ρ(x, t) is given by the product of matrix density of the material γ = γ(x, t) and the volume fraction ν = ν(x, t) ρ(x, t) = γ(x, t)ν(x, t).
In this article we study the well-posedness and exponential stability to the one-dimensional system in the linear isothermal theory of swelling porous elastic soils subject with time-varying weights and time-varying delay. We prove existence of global solution for the problems combining semigroup theory with the Kato's variable norm technique. To prove exponential stability, we apply the energy method without the equal wave speeds assumption.
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