A Korteweg-de Vries type of fifth-order equations on a finite domain with point dissipation

The paper discusses initial value problem of a Korteweg-de Vries type of fifth-order equation w(t) + w(xxx) - w(xxxxx) - Sigma(n)(j=1) a(j)w(j)w(x) = 0, w(x, 0) = w(0)(x) posed on a periodic domain x is an element of [0, 2 pi] with periodic boundary conditions w(ix) (0, t) = w(ix) (2 pi, t), i = 0,2,3,4 and an L-2-stabilizing feedback control law w(x) (2 pi, t) = alpha w(x)(0,t) + (1 - alpha)w(xxx) (0, t) where n is a fixed positive integer, a(j), j = 1,2,... , n, alpha are real constants, and vertical bar alpha vertical bar < 1. It is shown that for w(0)(x) is an element of H-alpha(1) (0, 2 pi) with the boundary conditions described above, the problem is locally well-posed for w is an element of C([0,7]; H-alpha(1) (0,2 pi)) with a conserved volume of w, [w] = integral(2 pi)(0) w(x, t)dx. Moreover, the solution with small initial condition exists globally and approaches to [w(0)(x)]/(2 pi) as t -> +infinity. Published by Elsevier Inc.