Solvability of nonlinear boundary-value problems arising in modeling plasma diffusion across a magnetic field and its equilibrium configurations (Q2387841)

An one-dimensional model of plasma, which can be reduced to an initial boundary-value problem for a second-order parabolic equation with implicit degeneration has been studied. The simplest stationary problem in this model is: \(-\Delta g(u)=f(\lambda,u)\), where the solution is given by a (generalized) function \(u(x)\), \(x \in \Omega \subset \mathbb R^n\), with \(u | _{\partial \Omega}=0\) which is (almost) non-negative; \(\lambda\) is a real parameter linked with boundary condition by the equation \(f(\lambda,0)=0\); \(g\) is an increasing positive-valued function such that \(g(0)=0\). The existence and uniqueness of the solution is investigated by means of nonlocal (integral) operators.