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.