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

Abstract

We consider the initial value problem for the reduced fifth-order KdV-type equation: 𝜕𝑡𝑢−𝜕5𝑥𝑢−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𝜕3𝑥𝑢 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(𝐴𝑘0/𝑘!)‖(𝑥𝜕𝑥)𝑘𝜙‖𝐻𝑠<∞, for some constant 𝐴0(0<𝐴0<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 𝜕𝑥(𝜕𝑥𝑢)2 on the Bourgain space, which is accomplished by improving Kenig et al.'s method used in (Kenig et al. 1996).