Dispersive Smoothing Effects for KdV Type Equations

In this paper we study the smoothness properties of solutions of some nonlinear equations of Korteweg–de Vries (KdV) type, which are of the form∂tu=a(x, t) u3+f(u2, u1, u, x, t), (1)wherex R,uj=∂jxu, andkandjare nonnegative integers. Our principal condition is thata(x, t) be positive and bounded, so that the dispersion is dominant. It is shown under certain additional conditions onaandfthatC∞solutionsu(x, t) are obtained fort>0 if the initial datau(x, 0) decays faster than it does polynomially onR−and has certain initial Sobolev regularity. A quantitative relationship between the rate of decay and the amount of gain of smoothness is given. Lets0be the Sobolev index. If∫R u2(x, 0)(1+x−m) dx<∞ (2)for an integerm 0 and the solution obeys u Hs0<∞ for an existence time 0