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

Abstract EN

We consider the initial value problem for the reduced fifth-order KdV-type equation: ∂tu-∂x5u-10∂x(u3)+10∂x(∂xu)2=0, t,x∈R, u(0,x)=ϕ(x), x∈R.

This equation is obtained by removing the nonlinear term 10u∂x3u 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 ϕ∈Hs(R) (s>1/8) satisfies the condition ∑k=0∞(A0k/k!)∥(x∂x)kϕ∥Hs<∞, for some constant A0 (0

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 ∂x(∂xu)2 on the Bourgain space, which is accomplished by improving Kenig et al.'s method used in (Kenig et al.

1996).