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We consider the Cauchy problem of the higher-order KdV-type equation: \[\partial_t u + \frac{1}{\mathfrak{m}} |\partial_x|^{\mathfrak{m}-1} \partial_xu = \partial_x (u^{\mathfrak{m}}) \] where $\mathfrak{m} \ge 4$. Thenonlinearity is critical in the sense of long-time behavior. Using the methodof testing by wave packets, we prove that there exists a unique global solutionof the Cauchy problem satisfying the same time decay estimate as that of linearsolutions. Moreover, we divide the long-time behavior of the solution intothree distinct regions.

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