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 μ∈R,n≥2,T>0,0<σ<+∞for d≤2n and 0<σ<2n/d-2n for d≥2n+ 1.In the mass-supercritical and energy-subcritical case(0<d/2-n/σ<n),and energy-critical case(σ= 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,iΓφR],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/σ<n).For this upper bound,we first construct a higher-order local Riesz variance identity.Then,by the general formula[(-△)n,(?)kψR]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.