On a nonlocal quasilinear parabolic model related to a current-carrying stellarator
The main result is the existence of the weak solution (in the sense defined in the paper) of the nonlinear parabolic problem \[ \beta(u)_t - \Delta u = G(u)(t,x) + J(u)(t,x) \quad \text{in} \quad (0,T) \times \Omega, \tag{1} \] \[ u(t,x) = \gamma \quad \text{on} \quad (0,T) \times \partial \Omega, \tag{2} \] \[ \beta(u(0,x)) = \beta(u_0(x)) \quad \text{in} \quad \Omega. \tag{3} \] Here \(\Omega \subset \mathbb{R}^2\) is a bounded regular domain, \(T>0\), \(\beta(s) = \min (s,0)\) and \(G(u)\) is defined by means of an integral -- so it is a nonlocal operator. The formulation (1), (2), (3) is related to a problem arising in the study of the magnetic confinement of a plasma in Stellarator device, when the plasma is assumed to be a perfect conductor but with non-zero net current inside each flux magnetic surface. The reviewed paper is closely related to \textit{J. I. Diaz, J. F. Padial} and \textit{J. M. Rakotoson} [Nonlinear Anal., Theory Methods Appl. 34, 857-887 (1998; Zbl 0946.35119)], where the associated stationary problem is studied.
quasilinear nonlocal elliptic-parabolic equations
scientific article; zbMATH DE number 1880132