Local Analyticity in the Time and Space Variables and the Smoothing Effect for the Fifth-Order KdV-Type Equation
- Kyoko Tomoeda
DOIAbstract
We consider the initial value problem for the reduced fifth-order KdV-type equation: −5−10(3)+10()2=0, ,∈ℝ, (0,)=(), ∈ℝ. This equation is obtained by removing the nonlinear term 103 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 ∈(ℝ)(>1/8) satisfies the condition ∑∞=0(0/!)‖()‖<∞, for some constant 0(0<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 ()2 on the Bourgain space, which is accomplished by improving Ke...
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