Blow-up phenomena for a couple of parabolic equations with memory and source terms: Analytical and simulation

In this paper, the focus is on investigating the asymptotic behavior of the solution for a system of parabolic equations with memory terms acting in both equations. This system has many applications in various scientific fields, including heat conduction in materials with memory effects and the study of biological systems exhibiting memory phenomena. The system of parabolic equations with a memory term provides a powerful framework for understanding and predicting the behavior of such complex systems, with emphasis on the role of the memory term in capturing the system's history-dependent behavior. Firstly, we assume that the relaxation functions mu 2 (t) <= mu 1 (t), for all t >= 0, and under certain conditions regarding the function p(<middle dot>) we prove that the solution with positive initial energy blows up in finite time. Finally, we present the theoretical results as numerical findings in the form of figures that illustrate and confirm the results by studying examples in two dimensions.