The Korteweg-de Vries equation on a periodic domain with singular-point dissipation

This paper considers the Korteweg-de Vries (KdV) equation ut + uux + uxxx = 0, 0 < x < 1, t > 0, u(0,x) = u0(x). and the periodic boundary conditions u (t,1) = u(t,0), uxx (t,0) = uxx(t,1) with an L2-stabilizing control input implemented by a feedback mechanism ux(t,1) = αux(t,0) and |α| < 1. It can be shown that the solutions conserve the volume [u] = ∫10 u(t,x)dx and the constant state [u0] possesses the smallest energy among solutions with same volume. It has been proved that the solution of the system exists and approaches [u0] as t → + ∞ when α ¬= -1/2. This paper studies the case for α = -1/2 and gives a proof of the existence and exponential decay of the solutions by deriving estimates of the corresponding Green's function and using semigroup theory. The method used here also works for the other cases with |α| < 1.