Reaction–Diffusion Equations with Nonlinear Boundary Delay

We consider dissipative scalar reaction–diffusion equations that include the ones of the form ut−Δu=f(u(t)), subjected to boundary conditions that include small delays, that is, we consider boundary conditions of the form ∂u/∂na=g(u(t), u(t−r)). We show the global existence and uniqueness of solutions in a convenient fractional power space, and furthermore, we show that, for r sufficiently small, all bounded solutions are asymptotic to the set of equilibria as t tends to infinity.