Finite Extinction Time via Delayed Feedback Actions

We study the “finite extinction phenomenon” for solutions of parabolic reaction-diffusion equations of the type ∂u ∂t −∆u+ b(t)f(u(t− τ,x)) = 0, (t,x) ∈ (0,+∞)×Ω, with a delay term τ > 0. Here Ω is an open and bounded set in R , b ≥ 0, f is a continuous function, u(t,x) satisfies a homogeneous Neumann or Dirichlet boundary condition on (0,+∞)×∂Ω and some functional initial condition u(s,x) = u0(s,x) on (−τ, 0) × Ω for a given function u0 ∈ C([−τ, 0] : L(Ω)), for some p ∈ [1,+∞] . The ”reaction term” b(t)f(u(t − τ,x)) can be understood as a delayed feedback control.