Analytical investigation of a non-local mathematical model for normal and abnormal wound healing

This study considers a non-local mathematical model for normal and abnormal wound healing described by four equations: an ordinary differential equation for the dynamics of extracellular matrix, and three partial differential equations for the dynamics of growth factors, fibroblasts, and macrophages. Two of these equations include non-local (integral) terms that characterize adhesive cell-cell and cell-matrix interactions. Here, we follow up on an earlier numerical study of a similar model that showed the solutions of this class of models approaching either spatially-homogeneous steady states or spatially-heterogenous states with overgrown cell densities. The goal of this current study is to investigate (from the perspective of normal/abnormal wound healing) the linear stability of the steady states, and further prove the local in-time existence and uniqueness of solutions for this class of non-local models using the framework of $ {\rm{\rlap{--} {the}}}$ analytic semigroups of operators. We also discuss the impact of different types of kernels (smooth and non-smooth) for non-local cell-cell and cell-matrix interactions, on these analytical results. Overall, the complexity of this 2D non-local model, which incorporates various discontinuous functions, leads to new approaches for some of these analytical results.