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The finite element-boundary element hybrid method developed previously for reflected systems and restricted to one or two group neutron diffusion theory has been extended to the general multigroup neutron diffusion theory by using the boundary integral equation of multigroup neutron diffusion theory. A linear or bilinear 2-D FEM formulation in the core combined with a 2-D linear BEM formulation in the reflector constitute the basic discretization procedure. Use of the boundary integral equation of multigroup diffusion theory transforms all group-to-group scattering domain integrals into surface integrals in the reflector. Hence the need for a reflector domain mesh is completely eliminated. Via comparisons with pure FEM and BEM solutions of the reflected systems within the context of three and four group diffusion theories, the present formulation is validated and assessed.

Key words Nonlinear parabolic equations, positive solutions MSC (2010) 35K55, 35H10, 35K65, 35R05We are concerned with the nonexistence of positive solutions of the nonlinear parabolic partial differential equations in a cylinder Ω × (0, T ) with initial condition u(., 0) = u0 (.) ≥ 0 and vanishing on the boundary ∂Ω × (0, T ), given bywhere Ω ∈ R N (resp. a Carnot Carathéodory metric ball in R 2 N + 1 ) with smooth boundary and the time dependent singular potential function V (x, t) ∈ L 1 lo c (Ω × (0, T )), α, β ∈ R, 1 < p < N, p − 1 + α + β > 0. We find the best lower bounds for p + β and provide proofs for the nonexistence of positive solutions.