On the stability properties of nonnegative solutions of semilinear problems with convex or concave nonlinearity

We investigate the stability of nontrivial nonnegative stationary solutions of semilinear initial-boundary value problems with convex or concave nonlinearity. In the convex case f(0)⩽0 implies instability, in the concave case f(0)⩾0 implies stability. We also discuss the necessity of the sign condition on f(0).