Model problem of radial heating in a spherical layer with localized internal source

This study presents a mathematical model for heat distribution in a spherical layer induced by a radially symmetric distributed heat source. The model is governed by an initial-boundary value problem for a linear parabolic equation in a spherically symmetric domain with three spatial variables, subject to thermal insulation boundary conditions.
By employing the method of separation of variables and exploiting radial symmetry, the three-dimensional problem is reduced to a one-dimensional formulation, yielding an exact analytical solution expressed as a convergent Fourier series. Explicit solutions for both homogeneous and inhomogeneous cases are derived by using the eigenfunctions of the associated Sturm–Liouville problem. Furthermore, the solution’s stability is rigorously established via a priori estimates.