Semilinear Models of Sobolev Type. Non-Uniqueness of Solution to the Showalter-Sidorov Problem

The article is of a survey nature and contains the results of a study about the morphology of the phase spaces of semilinear models of Sobolev type. The paper presents studies of the mathematical models whose phase spaces belong to smooth Banach manifolds with singularities depending on the parameters of the problem, namely, the Hoff model, the Plotnikov model, the distributed brusselator model, and the nerve impulse propagation model. In the first part of the article, we present conditions under which the phase manifolds of the considered models are simple smooth Banach manifolds, which implies the uniqueness of a solution to the Showalter-Sidorov problem. In the second part of the article, we present conditions under which the phase manifolds of the considered models contain singularities, which implies the non-uniqueness of a solution to the Showalter-Sidorov problem.

Keywords

Sobolev type equations; phase space; morphology of phase space; Banach manifold; Showalter-Sidorov problem; k-assembly Whitney.

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