Stability Properties of Positive Solutions to Partial Differential Equations with Delay

Abstract

We investigate the stability of positive stationary solutions of semilinear initial-boundary value problems with delay and convex or concave nonlinearity. If the nonlinearity is monotone, then in the convex case ƒ(0) ≤ 0 implies instability and in the concave case ƒ(0) ≥ 0 implies stability. Special cases are shown where the monotonicity assumption can be weakened or omitted.