Principle of linearized stability and instability for parabolic partial differential equations with state-dependent delay

Abstract

In this paper, the stability properties of a parabolic partial differential equation with state-dependent delay are investigated by the heuristic approach. The previous works [1,2] obtained a continuously differentiable semiflow with continuously differentiable solution operators defined by the classical solutions, and resolved the problem of linearization for this equation. Here, we clarify the relation between the spectral properties of the linearization of the semiflow at a stationary solution and the strong continuous semigroup defined by the solutions of the linearization of this equation, and consider the local stable and unstable invariant manifolds of the semiflow at a stationary solution. By a biological application, we finally verify all hypotheses for an age structured diffusive model with state-dependent delay and consider its stability behavior.