Diffusion approximation for linear transport with multiplying boundary conditions

We consider the diffusion limit of a suitably rescaled model transport equation in a slab with multiplying boundary conditions, as the scaling parameter ε tends to zero. We show that, for sufficiently smooth data, the solution converges in the L ²-norm for each t > 0 to the solution of a diffusion equation with Robin boundary conditions corresponding to an incoming flux. The derivation of the diffusive limit is based on an asymptotic expansion, which is rigorously justified.