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

This study considers a non-local mathematical model for normaland 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, andmacrophages. Two of these equations include non-local (integral) terms thatcharacterize adhesive cell-cell and cell-matrix interactions. Here, we follow upon an earlier numerical study of a similar model that showed the solutions ofthis class of models approaching either spatially-homogeneous steady statesor spatially-heterogenous states with overgrown cell densities. The goal of thiscurrent study is to investigate (from the perspective of normal/abnormal woundhealing) the linear stability of the steady states, and further prove the localin-time existence and uniqueness of solutions for this class of non-local modelsusing the framework of the analytic semigroups of operators. We also discussthe impact of different types of kernels (smooth and non-smooth) for non-localcell-cell and cell-matrix interactions, on these analytical results. Overall, thecomplexity of this 2D non-local model, which incorporates various discontinuous functions, leads to new approaches for some of these analytical results.