Modeling the isotropic growth of incompressible neo-Hookean material

The paper is devoted to the analysis of the mathematical model of the volumetric growth of incompressible neo-Hookean material. Models of this kind are used in order to describe the evolution of the human brain under the action of an external load. In the paper, we show that the space of deformation fields in the homeostatic state coincides with the Möbius group of conformal transforms in $\mathbb R^3$. We prove the well-posedness of the linear boundary value problem obtained by linearizing the governing equations on the homeostatic state. We study the behavior of solutions when the time variable tends to infinity. The main conclusion is that changes in the material, caused by a temporary increase in pressure (hydrocephalus) are irreversible.