About nonuniqueness of solutions of the Showalter–Sidorov problem for one mathematical model of nerve impulse spread in membrane

The article is devoted to the study of the morphology of the phase space of a mathematical model of the nerve impulse spread in a membrane, based on a degenerate Fitz Hugh–Nagumo system, defined on a bounded domain with a smooth boundary. In this mathematical model, the rate of change of one of the components of the system can significantly exceed the other, which leads to a degenerate Fitz Hugh–Nagumo system. The model under inquiry belongs to a wide class of semilinear Sobolev type models. To research the problem of nonuniqueness of solutions of the Showalter–Sidorov problem, the phase space method will be used, which was developed by G.A. Sviridyuk to scrutinize the solvability of Sobolev type equations. We have shown that the phase space of the studied model contains singularity such as the Whitney fold. The conditions of existence, uniqueness or multiplicity of solutions of the Showalter–Sidorov problem depending on the parameters of the system are found.