Global Behavior of Solutions to the Focusing Generalized Hartree Equation

We study behavior of solutions to the nonlinear generalized Hartree equation, where nonlinearity is nonlocal type and expressed as a convolution iut+?u+(|x|?(N??)?|u|p)|u|p?2u=0,x?RN,t?R. Our main goal understand global this equation in various settings. In work we make an initial attempt towards H1 (finite energy) solutions. first investigate local well-posedness small data theory. then, intercritical regime (0<s<1), classify under mass-energy assumption ME[u0]<1, identifying sharp threshold for versus finite time via constant corresponding Gagliardo–Nirenberg interpolation inequality (note that uniqueness ground state not known general case). particular, depending on size mass gradient, will either exist all scatter H1, or blow up time, diverge along infinite sequence. To obtain scattering divergence infinity, paper employ well-known concentration compactness rigidity method Kenig Merle [36] with novelty studying nonlocal, nonlinearity.