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,μ=Δ2-μΔ+V,μ≥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,μ(u0)[M(u0)]sc<E0(Q)[M(Q)]sc,‖(Δ2+V)1/2u0‖L2‖u0‖L2sc<‖ΔQ‖L2‖Q‖L2sc.where μ≥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,μ(u0)[M(u0)]sc<E0(Q)[M(Q)]sc,‖(Δ2+V)1/2u0‖L2‖u0‖L2sc>‖ΔQ‖L2‖Q‖L2sc where μ>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‖Δu(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.