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

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.

37 pages

References in corpus (2)

• Well-posedness and scattering for fourth order nonlinear Schrödinger type equations at the scaling critical regularity
• Large time asymptotics of solutions to the short-pulse equation