Boundary Stabilization of the Korteweg-De Vries Equation

Abstract

Consider herein is the initial-boundary value problem of the KdV equation posed on the bounded interval (0, 1):

$$ (*) \left\{ {\begin{array}{*{20}{c}} {{{u}_{t}} + u{{u}_{x}} + {{u}_{{xxx, }}}u(x,0) = \emptyset (x)} \\ {u(0,t) = 0, u(1,t) = 0, {{u}_{x}}(1,t) = \alpha {{u}_{x}}(0,t)} \\ \end{array} } \right. $$

where |α| < 1. It is shown that (i) the system (*) is globally well-posed in the space

$$ H_{\alpha }^{3} = \{ \emptyset \in {{H}^{3}}(0,1), \emptyset (0,t) = 0, \emptyset (1,t) = \alpha {{\emptyset }_{x}}(0,t)\} $$

and (ii) if α ≠ 0, then the system (*) is locally well-posed in the space H ¹₀ (0, 1), but its small amplitude solutions exist globally and decay exponentially to zero as t → ∞.