TITLE:
Existence of Weak Solutions for a Degenerate Cross-Diffusion System in Age-Structured Population Dynamics via the Faedo-Galerkin Method
AUTHORS:
Mamadou Birba
KEYWORDS:
Cross-Diffusion System, Degenerate System, Weak Solutions, Faedo-Galerkin, Degenerate Boundary Conditions
JOURNAL NAME:
Applied Mathematics,
Vol.17 No.9,
September
18,
2026
ABSTRACT: This work is devoted to the study of a class of nonlinear degenerate parabolic systems arising from the modelling of two age-structured populations interacting within a one-dimensional heterogeneous spatial environment. The system is characterized by nonlinear degenerate boundary conditions and nonlocal renewal laws. The presence of cross-diffusion operators
div(
v
∂
x
u
)
and
div(
u
∂
x
v
)
leads to coupling between the equations and, simultaneously, strong degeneracy. Consequently, we propose a method based on regularizing the system using a family of highly regular functions. We then establish the existence of global weak solutions using the Faedo-Galerkin method, with a finite element basis. Uniform estimates and compactness arguments subsequently allow us to perform a double limiting procedure. We finally show that the solution of the regularized system converges to that of the original degenerate system, while preserving in particular the nonlocal boundary conditions and the initial data.