The effect of higher-order dissipation on solutions of the generalized Korteweg–de Vries equation

This paper is concerned with the generalized Korteweg–de Vries equation [Formula: see text] in the supercritical case [Formula: see text]. For such values of [Formula: see text], it is expected that large, smooth initial data can lead to locally smooth solutions that blow up in finite time. The question raised here is whether or not the addition of a suitable dissipative term could mitigate this putative blowup. The dissipative addition considered is of the form [Formula: see text] where [Formula: see text] and [Formula: see text] is a non-negative integer. It turns out that for [Formula: see text], there are values of [Formula: see text] for which global well-posedness obtains for arbitrarily large initial data for any [Formula: see text], no matter how small. In general, it is shown that for any value of [Formula: see text] and initial data [Formula: see text] whose norm is bounded by [Formula: see text], say, there is a [Formula: see text] depending on [Formula: see text] such that if [Formula: see text], then the solution emanating from [Formula: see text] is global. However, if [Formula: see text], numerical simulations show that solutions can still blow up even in the presence of dissipation if [Formula: see text] is not large enough.