Smoothness of semiflows for parabolic partial differential equations with state-dependent delay

In this paper, the smoothness properties of semiflows on C1C1-solution submanifold of a parabolic partial differential equations with state-dependent delay are investigated. The problem is formulated as an abstract ordinary retarded functional differential equation of the form du(t)/dt=Au(t)+F(ut)du(t)/dt=Au(t)+F(ut) with a continuously differentiable map G from an open subset U   of the space C1([−h,0],L2(Ω)), where A   is the infinitesimal generator of a compact C0C0-semigroup. The present study is continuation of a previous work [14] that highlights the classical solutions and C1C1-smoothness of solution manifold. Here, we further prove the continuous differentiability of the semiflow. We finally verify all hypotheses by a biological example which describes a stage structured diffusive model where the delay, which is the time taken from birth to maturity, is assumed as a function of a immature species population.