In this paper we study a reaction diffusion problem with anisotropic diffusion and mixed Dirichlet-Neumann boundary conditions on \(\partial \Omega \) for \(\Omega \subset \mathbb {R}^N\) , \(N\ge 1\) . First, we prove that the parabolic problem has a unique positive, bounded solution. Then, we show that this solution converges as \(t\rightarrow +\infty \) to the nonnegative solution of the elliptic associated problem. The existence of the unique positive solution to this problem depends on a principal eigenvalue of a suitable linearized problem with a sign-changing weight. Next, we study the minimization of such eigenvalue with respect to the sign-changing weight, showing that it is solved by an optimal bang-bang weight, namely a piece-wise constant weight that takes only two values. Finally, we completely solve the problem in dimension one.
