Existence and asymptotic stability for generalized elasticity equation with variable exponent

Abstract

In this paper we propose a new mathematical model describing the deformations of an isotropic nonlinear elastic body with variable exponent in dynamic regime. We assume that the stress tensor $\sigma^{p(\cdot)}$ has the form $\sigma^{p(\cdot)}(u)=(2\mu +|d(u)|^{p(\cdot)-2})d(u)+\lambda Tr(d(u)) I_{3},$ where $u$ is the displacement field, $\mu$, $\lambda$ are the given coefficients $d(\cdot)$ and $I_{3}$ are the deformation tensor and the unit tensor, respectively. By using the Faedo-Galerkin techniques and a compactness result we prove the existence of the weak solutions, then we study the asymptotic behaviour stability of the solutions.