A Study Of Blowup Solution For Higher-order NLS

In this text,we consider the radial initial-value Cauchy problem for the higher-order NLS with focusing nonlinearity given by (?)where 0<σ<+∞ for α>d/2 and 0<σ<2α/d-2α for 1<α<d/2.The main result of this article is that we prove blowup of radial solutons for L2-supercritical case(0<sc≤α)and L2-critical case(sc=0),where sc=d/2-α/σ.The details are as follows:(1)(L2-Supercritical Case)Let d>2,0<sc≤α,σ<2α.Assume that u is a radial solution of above equation,Furthermore,we suppose that either E[u0]<0,or,if E[u0]>0 and (?)where Q is the ground state,then u(t)blows up in finite time T with T>0,that is,u(t)satisfies(?).(2)(L2-Critical Case)Let d≥2,sc=0,α∈(1,2),if u is a radial solution of above equation,then u(t)blows up infinite time such that ‖(-Δ)α/2u(t)‖L2≥Ctα for all t≥t*with some constants C>0 and t*>0 that depend only on u0,α,d.For the blowup proof,we firstly construct a local virial identity Mφ[u(t)]:=<u(t),iΓφu(t)>=2Im∫Rdu(t)▽φ·▽u(t)dx,where Γφ=-i(▽φ·▽+▽·▽φ)and φ is a suitable chosen function;and then we get its estimate by using the general for-mula of[(-△)α,iΓφ],functional calculus.Finally,we use the estimate and the properties of the ground state solution of the high-order nonlinear Schrodinger equation to conclude the proof.