A Study Of Scattering And Blowup Solutions For Fourth-order NLS With A Potential

In this text,we consider the radial initial-value Cauchy problem for the fourth-order NLS with focusing nonlinearity given by where 8/3<p<?,HV,?=?2-??+V,??0,V is a small range of nonnegative real-valued decaying potentials.In this paper,we focus on studying the energy scattering and blow-up in finite time of the solution for the equation with radial initial data,the main conclusions are as follows:Firstly,the corresponding solution of the radial initial-valve Cauchy problem for the fourth-order NLS with focusing nonlinearity scatters in H2(R3)in both directions.If the radial initial-valve u0 ? H2(R3)of the fourth-order NLS equation satisfy the condition:Ev,?(u0)[M(u0)]sc<E0(Q)[M(Q)]sc,?(?2+V)1/2u0?L2?u0?L2sc<??Q?L2?Q?L2sc.where ??0,V decays enough and satisfies x·?V(x)<0,Q is the ground state of the elliptic equation.Then the corresponding solution to the fourth-order NLS exists globally in time and scatters in H2(R3)in both directions,i.e.there exist u±?H2(R3),such thatSecondly,the corresponding solution of the radial initial-valve Cauchy problem for the fourth-order NLS with focusing nonlinearity blow-up in finite time.If the radial initial-valve u0 ? H2(R3)of the fourth-order NLS equation satisfy the condition:EV,?(u0)[M(u0)]sc<E0(Q)[M(Q)]sc,?(?2+V)1/2u0?L2?u0?L2sc>??Q?L2?Q?L2sc where ?>0,V decays enough,satisfies x·?V(x)<0 and |x·?V(x)+4V(x)| is enough small,Q is the ground state of the elliptic equation,then there exist 0<T<+?,such that lim t?T??u(t)?L2=+?.In the proof of this paper,we mostly used Strichartz estimates,the estimate of the Virial equation,the radial Sobolev embedding,Duhamel's lemma,the radial Morawetz estimates with a potential,the Gagliardo-Nirenberg inequality.