Stability of a continuous reaction-diffusion Cournot-Kopel Duopoly Game Model

In order to take into account the territory in which the outputs are in the market and the time-depending firms' strategies, the discrete Cournot duopoly game (with adaptive expectations, modeled by Kopel) is generalized through a non autonomous reaction-diffusion binary system of PDEs, with self and cross diffusion terms. Linear and nonlinear asymptotic $L^2$-stability, via the Lapunov Direct Method and a nonautonomous energy functional, are investigated.