Asymptotic behavior of solutions to a higher-order KdV-type equation with critical nonlinearity

Mamoru Okamoto

doi:10.3934/eect.2019027

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2019, Volume 8, Issue 3: 567-601. Doi: 10.3934/eect.2019027

This issue Previous Article A comparison principle for Hamilton-Jacobi equation with moving in time boundary Next Article Optimal control of evolution differential inclusions with polynomial linear differential operators

Asymptotic behavior of solutions to a higher-order KdV-type equation with critical nonlinearity

Mamoru Okamoto^,

Division of Mathematics and Physics, Faculty of Engineering, Shinshu University, 4-17-1 Wakasato, Nagano City 380-8553, Japan

^* Corresponding author: Mamoru Okamoto
^* Corresponding author: Mamoru Okamoto

Received: May 31, 2018

Revised: October 31, 2018

Early access: May 2019

Published: August 2019

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Abstract

We consider the Cauchy problem of the higher-order KdV-type equation:

$ \partial_t u + \frac{1}{ {\mathfrak{m}}} | \partial_x|^{ {\mathfrak{m}}-1} \partial_x u = \partial_x (u^{ {\mathfrak{m}}}) $

where $ {\mathfrak{m}} \ge 4 $. The nonlinearity is critical in the sense of long-time behavior. Using the method of testing by wave packets, we prove that there exists a unique global solution of the Cauchy problem satisfying the same time decay estimate as that of linear solutions. Moreover, we divide the long-time behavior of the solution into three distinct regions.

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Mathematics Subject Classification: Primary: 35Q53; Secondary: 35B40.

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