Local Analyticity in the Time and Space Variables and the Smoothing Effect for the Fifth-Order KdV-Type Equation

We consider the initial value problem for the reduced fifth-order KdV-type equation: partial derivative(f)u - partial derivative(5)(x)u - 10 partial derivative(x) (u(3)) + 10 partial derivative(x) (partial derivative(x)u)(2) = 0, t, x is an element of R, u(0, x) = phi(x), x is an element of R. This equation is obtained by removing the nonlinear term 10u partial derivative(3)(x)u from the fifth-order KdV equation. We show the existence of the local solution which is real analytic in both time and space variables if the initial data phi is an element of H-s (R) (s > 1/8) satisfies the condition Sigma(infinity)(k-0) (A(0)(k)/k!)parallel to(x partial derivative(x))(k) phi parallel to(Hs) < infinity for some constant A(0) (0 < A(0) < 1). Moreover, the smoothing effect for this equation is obtained. The proof of our main result is based on the contraction principle and the bootstrap argument used in the third-order KdV equation (K. Kato and Ogawa 2000). The key of the proof is to obtain the estimate of partial derivative(x) (partial derivative(x)u)(2) on the Bourgain space, which is accomplished by improving Kenig et al.'s method used in (Kenig et al. 1996).