Nash equilibria for quasi-linear parabolic problems

This paper discusses the application of the Nash strategy to a quasi-linear parabolic equation and a quasi-linear parabolic equation with a semilinear term. First, we demonstrate the existence of a Nash quasi-equilibrium for both equations using the fixed point method. Subsequently, we establish that the functionals are convex, ensuring that the Nash quasi-equilibrium is, in fact, a Nash equilibrium. Additionally, in conjunction with the theoretical results, numerical techniques are developed for each problem. To describe the iterative algorithms, the Newton’s Method is utilized in combination with the Finite Element Method and Finite Difference Method. All algorithms are implemented using the Freefem++ software, and various results are presented through figures and comparative graphics.