| We show that if u(x, t) is a solution to the initial value problem for the modified KdV equation 6tu+63x u+6xu3 =0,ux,0 =u0x,x∈ R, with u0 ∈ H s' ∩ L2(|x| s dx), where s' ≥ ¼, and s ∈ (0, s'], then u(·, t) ∈ Hs' ∩ L 2(|x|s dx) for all t in the lifespan of u. Our method extends to the generalized KdV equations 6tu+63xu+6 xuk+1=0 for k = 4, 5, ..., and in weaker form for the remaining powers k = 1, 3. |
