Analysis of the energy decay of a viscoelasticity type equation

Abstract

Abtract: In this paper, we study a system of thermoelasticity with a degenerated second order operator in the Heat equation. We analyze the evolution of the energy density of a family of solutions. We consider two cases: when the set of points where the ellipticity of the Heat operator fails is included in a hypersurface and when it is an open set. In the first case and under special assumptions, we prove that the evolution of the energy density is the one of a damped wave equation: propagation along the rays of geometric optic and damping according to a microlocal process. In the second case, we show that the energy density propagates along rays which are distortions of the rays of geometric optic.

Symmetry Analysis and Conservation Laws of a Generalization of the Kelvin-Voigt Viscoelasticity Equation

“…In this section, we use the optimal system of one-dimensional subalgebras of Theorem (2) to determine the symmetry reductions of Equation (1).…”

“…The pair (T, X) is called a conserved current. Two local conservation laws are considered to be locally equivalent [5,21] if they differ by a locally trivial conservation law T = D x Θ, X = −D t Θ, where T and X are evaluated on the set of solutions of Equation (1) and Θ is a function of x, t, u, and derivatives of u. Funding: This research received no external funding.…”

“…The Kelvin-Voigt model consists of a lumped parameter model similar to a spring and dashpot in parallel, while the Maxwell model describes a serially connected spring and dashpot. In addition, many papers have been published studying these models [1][2][3][4].…”

“…There is no general theory for solving non-linear PDEs. Therefore, in this work, we use Lie theory to analyse Equation (1). Lie group analysis is a powerful tool to find general solutions for PDEs.…”