Regularity Properties of Solutions of a Model for Morphoelastic Growth in the Presence of Nutrients in One Spatial Dimension

Abstract

Regularity properties of solutions for a class of quasi-stationary models in one spatial dimension for stress-modulated growth in the presence of a nutrient field are proven. At a given point in time the configuration of a body after pure growth is determined by means of a family of ordinary differential equations in every point in space. Subsequently, an elastic deformation, which is given by the minimizer of a hyperelastic variational integral, is applied in order to restore Dirichlet boundary conditions. While the ordinary differential equations governing the growth process depend on the elastic stress and the pullback of a nutrient concentration in the current configuration, the hyperelastic variational problem is solved on the intermediate configuration after pure growth. Additionally, the coefficients of the reaction-diffusion equation determining the nutrient concentration in the current configuration depend on the elastic deformation and the deformation due to pure growth.