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<?<+? for ?>d/2 and 0<?<2?/d-2? for 1<?<d/2.The main result of this article is that we prove blowup of radial solutons for L2-supercritical case(0<sc??)and L2-critical case(sc=0),where sc=d/2-?/?.The details are as follows:(1)(L2-Supercritical Case)Let d>2,0<sc??,?<2?.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,??(1,2),if u is a radial solution of above equation,then u(t)blows up infinite time such that ?(-?)?/2u(t)?L2?Ct? for all t?t*with some constants C>0 and t*>0 that depend only on u0,?,d.For the blowup proof,we firstly construct a local virial identity M?[u(t)]:=<u(t),i??u(t)>=2Im?Rdu(t)??·?u(t)dx,where ??=-i(??·?+?·??)and ? is a suitable chosen function;and then we get its estimate by using the general for-mula of[(-?)?,i??],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.