Inverse Coefficient Problem for a Diffusive Logistic Model With Reflecting Boundary Conditions

ABSTRACT This paper investigates the inverse problem of determining a time‐dependent coefficient that governs the spreading or growth process in a social network model represented by a nonlinear heat equation, based on user density data observed at a specific point, under the Neumann boundary conditions. This type of problem also encompasses various population dynamics described by the Fisher–KPP models. Moreover, a generalized version of the inverse problem is considered, which includes models with cubic nonlinearities such as those found in population genetics and the Allen–Cahn (or the Ginzburg–Landau) models. The existence and uniqueness of the solution are established by demonstrating the existence of a unique fixed‐point of an associated integral operator equation, using the Fourier method and applying the Banach fixed‐point theorem over small time intervals.