Sharp kato smoothing properties of weakly dissipated kdv equations with variable coefficients on a periodic domain

It is well known that the solutions of the Cauchy problem of the Korteweg-de Vries (KdV) equation on a periodic domain T, u(t) + uu(x) + u(xxx) = 0, u(x, 0) = phi(x), x is an element of T, t is an element of R, possess neither the sharp Kato smoothing property, phi E H-s(T) double right arrow partial derivative(s+1)(x) u is an element of L-x(infinity)(T, L-2(0, T)), nor the Kato smoothing property, phi is an element of H-s(T) double right arrow u is an element of L-2(0, T; Hs+1(T)). This paper shows that the solutions of the Cauchy problem of following weakly dissipated KdV equation with variable coefficients posed on a periodic domain T, u(t) + uu(x) + a(x, t)u(xxx) -(g(x, t)u(x))(x) = 0, u(x, 0) = phi(x), where a and g are given real-valued smooth functions periodic in x satisfying a(x, t) not equal 0 x is an element of T, t >= 0 and integral(T)(x, t)/vertical bar a(x, t)vertical bar dx > 0 for all t >= 0, possess the sharp Kato smoothing property, phi is an element of H-s(T) double right arrow partial derivative(s+1)(x)u is an element of L-x(infinity) (T, L-2(0, T)), for all s >= 0, and the nonlinear part of its solution u possesses the strong Kato smoothing property, phi is an element of H-s(T) double right arrow (u-v) is an element of C([0,T];Hs+1(T)), for all s > 1/2, and the sharp double Kato smoothing property, phi is an element of H-s(T) double right arrow partial derivative(s+2)(x) (u-v) is an element of L-x(infinity) (T, L-2(0, T)), for all s > 1/2, with v being the solution of the linear problem v(t) + a(x, t)v(xxx) -(g(x, t)v(x))(x) = 0, v(x, 0) = phi(x), x is an element of T, t > 0.