LARGE TIME ASYMPTOTICS OF SOLUTIONS TO THE GENERALIZED KORTEWEG-DE VRIES EQUATION

We study the asymptotic behavior for large time of solutions to the Cauchy problem for the generalized Korteweg–de Vries (gKdV) equationut+(|u|ρ−1 u)x+13uxxx=0, wherex, t∈Rwhen the initial data are small enough. If the powerρof the nonlinearity is greater than 3 then the solution of the Cauchy problem has a quasilinear asymptotic behavior for large time. More precisely, we show that the solutionu(t) satisfies the decay estimate ‖u(t)‖Lβ⩽C(1+t)−(1/3)(1−1/β)forβ∈(4, ∞], ‖uux(t)‖L∞⩽Ct−2/3(1+t)−1/3and using these estimates we prove the existence of the scattering stateu+∈L2such that ‖u(t)−U(t) u+‖L2⩽Ct−(ρ−3)/3for any small initial data belonging to the weighted Sobolev spaceH1, 1={f∈L2; ‖(1+|x|2)1/2(1−∂2x)1/2 f‖L2<∞}, whereU(t) is the Airy free evolution group.