Blowup For Higher-order NLS

In this text,we consider the radial initial-valve Cauchy problem for the higher-order NLS with focusing nonlinearity given by where ??R,n?2,T>0,0<?<+?for d?2n and 0<?<2n/d-2n for d?2n+ 1.In the mass-supercritical and energy-subcritical case(0<d/2-n/?<n),and energy-critical case(?= 2n/d-2n),we prove a general result on finite-time blowup for radial data.For the blowup proof,we construct a local virial a identity,and then get its estimate by use the general formula of[(-?)n,i??R],Plancherel theorem and Young inequality.The another important result in this text is that we derive a universal upper bound for the blowup rate for suitable case(0<d/2-n/?<n).For this upper bound,we first construct a higher-order local Riesz variance identity.Then,by the general formula[(-?)n,(?)k?R]and Mikhlin multiplier lemma,we derive the higher-order local Riesz variance estimate.At last,we can prove this result by this estimate and Newton theorem.