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

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  • Kyoko Tomoeda

Abstract

We consider the initial value problem for the reduced fifth-order KdV-type equation: 𠜕 ð ‘¡ ð ‘¢ − 𠜕 5 ð ‘¥ ð ‘¢ − 1 0 𠜕 ð ‘¥ ( ð ‘¢ 3 ) + 1 0 𠜕 ð ‘¥ ( 𠜕 ð ‘¥ ð ‘¢ ) 2 = 0 , ð ‘¡ , ð ‘¥ ∈ â„ , ð ‘¢ ( 0 , ð ‘¥ ) = 𠜙 ( ð ‘¥ ) , ð ‘¥ ∈ â„ . This equation is obtained by removing the nonlinear term 1 0 ð ‘¢ 𠜕 3 ð ‘¥ ð ‘¢ 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 𠜙 ∈ ð » ð ‘ ( â„ ) ( ð ‘ > 1 / 8 ) satisfies the condition ∑ ∞ 𠑘 = 0 ( ð ´ ð ‘˜ 0 / 𠑘 ! ) ‖ ( ð ‘¥ 𠜕 ð ‘¥ ) 𠑘 𠜙 ‖ ð » ð ‘ < ∞ , for some constant ð ´ 0 ( 0 < ð ´ 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 𠜕 ð ‘¥ ( 𠜕 ð ‘¥ ð ‘¢ ) 2 on the Bourgain space, which is accomplished by improving Kenig et al.'s method used in (Kenig et al. 1996).

Suggested Citation

  • Kyoko Tomoeda, 2011. "Local Analyticity in the Time and Space Variables and the Smoothing Effect for the Fifth-Order KdV-Type Equation," Advances in Mathematical Physics, Hindawi, vol. 2011, pages 1-39, April.
    • Handle: RePEc:hin:jnlamp:238138
    DOI: 10.1155/2011/238138
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