Type: Article

Higher-order subcritical Hartree-type equations in one space dimension

Hayashi, Nakao; Naumkin, Pavel I.;

Abstract:

We study the higher-order subcritical Hartree-type equations in one space dimension i?tu??u=t??+?uJ???u??2,t>0,x?R,u(0,x)=u0(x),x?R, where ?=?(?i?x)=?12?2x+14?4x,?(?)=12?2+14?4,??(0,1),0<?<?4(1+?), the Riesz potential J? is defined by J??=??x????1??=?R?(y)|x?y|1??dy, where ? denotes the convolution in R. Our aim is to find the large time asymptotic behavior of solutions to the Cauchy problem. From the point of view of large time asymptotic behavior of solutions, ?>0 means that the nonlinearity is the subcritical one. Our method cannot be applied to the case ?=0, which is considered as the cubic nonlinearity with a time growth t?.