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.