Stabilization in a Finite Time to a Stationary State

Abstract

In this chapter the way of using the energy method is different from that of Chapter 1. Our aim is to study the property of finite-time stabilization to a stationary profile for solutions to nonlinear evolution problems. To be precise, let Ω ⊂ ℝ^N, N ≥ 1, be an open set (which need be neither bounded nor connected). Denote Q_∞ = Ω × ℝ₊, Σ_∞ = ∂Ω × ℝ₊. To fix ideas, let us consider the general initial and boundary-value problem

$$ \left\{ {\begin{array}{*{20}c} {u_t + A\left( u \right) = f\left( {x,t} \right) inQ\infty ,} \\ {B\left( u \right) = g\left( {x,t} \right) on\sum \infty ,} \\ {u\left( {x,0} \right) = u_0 on\Omega } \\ \end{array} } \right. $$

(1.1)

where A(u) is a differential operator on u in the space variables x, B(u) is the boundary operator, and f, g, u₀ are given functions. Our approach is applicable to the vector-valued solutions u as well.