Parabolic smoothing effect and local well-posedness of fifth order semilinear dispersive equations on the torus

2017/07/29 by Tsugawa, Kotaro
#35A01 #35A02 #35B45 #35B65 (Secondary) #35Q53 (Primary) 35G55 #37K10 #Analysis of PDEs (math.AP) #FOS: Mathematics

paper · doi:10.48550/arxiv.1707.09550

Abstract

We consider the Cauchy problem of fifth order dispersive equations on the torus. We assume that the initial data is sufficiently smooth and the nonlinear term is a polynomial depending on ∂_x³ u, ∂_x² u, ∂_x u and u. We prove that the local well-posedness holds on [-T,T] when the nonlinear term satisfies a condition and otherwise, the local well-posedness holds with a smoothing effect only on either [0,T] or [-T,0] and nonexistence result holds on the other time interval, which means that the nonlinear term can not be treated as a perturbation of the linear part and the equation has a property of parabolic equations by an influence of the nonlinear term. As a corollary, we also have the same results for (2j+1)-st order dispersive equations.

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