Finite extinction time property for a delayed linear problem on a manifold without boundary

Abstract

We prove that the mere presence of a delayed term is able to connect the initial state u0 on a manifold without boundary (here assumed given as the set ∂Ω where Ω is an open bounded set in RN ) with the zero state on it and in a finite time even if the dynamics is given by a linear problem. More precisely, we extend the states to the interior of Ω as harmonic functions and assume the dynamics given by a dynamic boundary condition of the type ∂u ∂t (t, x) + ∂u ∂n (t, x) + b(t)u(t − τ, x) = 0 on ∂Ω, where b : [0, ∞) → R is continuous and τ > 0. Using a suitable eigenfunction expansion, involving the Steklov BVP {∆ϕn = 0 in Ω, ∂νϕn = λnϕn on ∂Ω}, we show that if b(t)vanishes on [0, τ] ∪ [2τ, ∞) and satisfies some integral balance conditions, then the state u(t, .) corresponding to an initial datum u0(t, ·) = µ(t)ϕn(·) vanish on ∂Ω (and therefore in Ω) for t ≥ 2τ. We also analyze more general types of delayed boundary actions for which the finite extinction phenomenon holds for a much larger class of initial conditions and the associated implicit discretized problem.