Weak Solution, Blow-up and Numerical Study of a Nonlinear Reaction Diffusion Problem with an Integral Condition

The purpose of this article is to investigate a semilinear nonlocal problem with a 2^nd-type integral condition applied in a specific category of nonlinear equations of parabolic type. The linear problem is analyzed using the Fadeo-Galarkin approach, and the primary objective of the study is to determine whether the weak solution is unique and existent. Significant results achieved for the linear problem are subjected to an iterative approach in order to extend this study to the semilinear problem. A special case of the semilinear problem and its finite-time blow-up solution are also examined in the work. Numerical examples are provided to confirm the precision and effectiveness of the suggested approaches, which employ a forward time-centered spatial scheme to solve the semilinear problem. The study is presented in a formal, technical way and contributes to the understanding of weak solutions for a particular category of nonlinear equations of parabolic type.