Asymptotic Properties of a Nonlinear Diffusion Process Arising in Articular Cartilage

Abstract

**Please note that the full text is embargoed** ABSTRACT: There has been considerable effort put forth to analyse degenerative Joint diseases in terms of the principles of mechanics and hydrodynamics [8]. This effort has led to mathematical models which are systems of nonlinear partial differential equations (see e.g., [6] and [7]). The mathematical models are based on assumptions concerning the nature of the physical system which are not a priori known. Therefore laboratory experiments are performed with an eye towards validating the mathematical models. This paper deals with a particular mathematical model, a nonlinear diffusion equation, which has been proposed by Mow and Monsour [4], [5] to describe the stress relaxation of articular cartilage. More precisely, the stress relaxation function f(t) is related to a solution u(x,t) of a nonlinear PDE problem (see (1.1)-(1.4) below). In this paper we analytically determine the behavior of f(t) (Theorem 1). The consistency of these results to already existing theory and experimental findings is discussed in [1].