Numerical study on the non-uniqueness of solutions to the Showalter–Sidorov problem for one degenerate mathematical model of an autocatalytic reaction with diffusion

The article is devoted to a numerical study of the phase space of a mathematical model of an autocatalytic reaction with diffusion, based on a degenerate system of equations of the "distributed" brusselator. In this mathematical model, the rate of change in one of the components of the system can significantly exceed the other, which leads to a degenerate system of equations. This model belongs to a wide class of semilinear Sobolev type equations. We will identify the conditions for the existence, uniqueness or multiplicity of solutions to the Showalter–Sidorov problem, depending on the parameters of the system. The theoretical results obtained made it possible to develop an algorithm for the numerical solution of the problem based on the modified Galerkin method. The results of computational experiments are presented.