  This paper studies a longitudinal shape transformation model in which shapes
are deformed in response to an internal growth potential that evolves according
to an advection reaction diffusion process. This model extends prior works that
considered a static growth potential, i.e., the initial growth potential is
only advected by diffeomorphisms. We focus on the mathematical study of the
corresponding system of coupled PDEs describing the joint dynamics of the
diffeomorphic transformation together with the growth potential on the moving
domain. Specifically, we prove the uniqueness and long time existence of
solutions to this system with reasonable initial and boundary conditions as
well as regularization on deformation fields. In addition, we provide a few
simple simulations of this model in the case of isotropic elastic materials in
2D.
