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arXiv · 1805.12367

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

Mamoru Okamoto
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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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Mamoru Okamoto. 2018-05-31. Asymptotic behavior of solutions to a higher-order KdV-type equation with critical nonlinearity. https://doi.org/10.3934/eect.2019027

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