The goal of this talk is to study a reaction diffusion problem with anisotropic diffusion and mixed Dirichlet-Neumann boundary conditions on $∂Ω$ for $Ω⊂ \mathbb{R}^N$, $N≥ 1$. First, we prove that the parabolic problem has a unique positive, bounded solution. Then, we show that this solution converges as $t→ +∞$ to the unique 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 there exists 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.