On a nonlocal quasilinear parabolic model related to a current-carrying stellarator

Abstract

An initial-boundary value problem for the nonlinear elliptic–parabolic equation (_(u))t −_u = G(u)(t, x)+J(u)(t, x) is considered. Here _(s) = min(s, 0) = −s−, G and J are nonlocal operators. This problem arises in the study of magnetic confinement of plasma in a stellarator device. An existence theorem of a weak solution defined in this paper is proved. In the course of the proof of the existence theorem with the help of the replacement of _(s) by __(s) = _s+ −s−, a family of regularized parabolic equations is constructed. It is established that the family of solutions of the regularized problems converges as _!0 to the solution of the original initial-boundary value problem. The solvability of the regularized problem with the help of Galerkin’s method is proved.