Theoretical and numerical studies of the blow-up of a degenerate nonlinear reaction–diffusion problem with source terms

In this paper, we focus on the phenomenon of blow-up of solutions for semilinear and degenerate (time-derivative) parabolic equation systems with additional source terms. We show that, under suitable conditions on the initial data and on the growth of the source term, the solution exhibits finite-time blow-up. The presence of the source term significantly influences the blow-up mechanism and the localization of blow-up points. Using a MATLAB code, we compute the solution of the discretized system via a finite difference scheme adapted to the source-driven dynamics. This provides a graphical representation of the solution peaks near the blow-up points and illustrates the theoretical blow-up results.This study constitutes a first step toward determining the set of blow-up points in the presence of external forcing.